Toral CW complexes and bifurcation control in Eulerian flows with multiple Hopf singularities
TToral CW complexes and bifurcation control in Eulerianflows with multiple Hopf singularities
Majid Gazor † and Ahmad Shoghi Department of Mathematical Sciences, Isfahan University of TechnologyIsfahan 84156-83111, Iran
February 26, 2020
Abstract
We are concerned with bifurcation analysis and control of nonlinear Eulerian flows withnon-resonant n -tuple Hopf singularity. The analysis is involved with CW complex bifurcations of flow-invariant
Clifford hypertori , where we refer to these toral manifolds by toral CW com-plexes . We observe from primary to tertiary flow-invariant toral CW complex bifurcationsfor one-parametric systems associated with two most generic cases. In a particular case, atertiary toral CW complex bifurcates from and resides outside a secondary toral CW com-plex. When the parameter varies, the secondary internal toral CW complex collapses with theorigin. However, the tertiary external toral CW complex continues to live even after the sec-ondary internal toral manifold disappears. Our analysis starts with a flow-invariant primarycell-decomposition of the state space. Each open cell admits a secondary cell-decompositionvia a smooth flow-invariant foliation. Each leaf of the foliations is a minimal flow-invariantrealization of the state space configuration for all Eulerian flows with n -tuple Hopf singulari-ties. Permissible leaf-vector field preserving transformations are introduced via a Lie algebrastructure for nonlinear vector fields on the leaf-manifold. Complete parametric leaf-normalform classification is provided for singular leaf-flows. Leaf-bifurcation analysis of leaf-normalforms are performed for three most leaf-generic cases associated with one to three unfold-ing bifurcation-parameters. Leaf-bifurcation varieties are derived. Leaf-bifurcations providesa venue for cell-bifurcation control of invariant toral CW complexes. The results are imple-mented and verified using
Maple for practical bifurcation control of such parametric nonlinearoscillators.
Keywords:
Bifurcation control; Toral CW complexes; Flow-invariant foliation; Lie algebras onmanifolds; Leaf-systems and Leaf-normal forms; Leaf and cell-bifurcation. : 34H20; 55U10; 57N15; 34C20.
Some parts of the proofs are omitted here for briefness. Full version of this paper is available uponrequest from authors. † Corresponding author. Phone: (98-31) 33913634; Fax: (98-31) 33912602; Email: [email protected]; Email:[email protected]. a r X i v : . [ m a t h . D S ] F e b . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity real-ization within the state space configuration is controlled by permissible families of flow-invariantleaf-manifolds. Permissible leaves represent the desired realizations of the state space configurationand are determined by integral foliations. Hence, our approach requires the study of integral folia-tions and the bifurcation control of the governing dynamics on the individual leaves of the foliations;also see [3–6, 18–20, 24, 25, 27–29, 31, 36, 37]. Bifurcation refers to a qualitative type change in the dynamics of a system when some parametersof the system slightly change around their critical values. Hence, a change in the number and/or thestability types of the equilibria, periodic and/or quasi-periodic orbits and flow-invariant manifoldsis called a bifurcation . We are concerned with non-resonant n -tuple Hopf singularity with nonlinearEulerian (and rotational) type coupling throughout this paper. Hopf bifurcation is an importantvenue to generate limit cycles from an equilibrium while a coupled n -tuple Hopf bifurcation forEulerian flows generates a flow-invariant toral-manifold bifurcated from a steady-state solution. Aperiodic orbit as an α or ω -limit set is a limit cycle while an invariant torus in non-resonant casesconsists of quasi-periodic solutions. A flow-invariant toral manifold here refers to a family of flow-invariant Clifford hypertori that are smoothly parameterized by open CW-cells of a CW complex.Thus, these toral manifolds are called by toral CW complexes . A toral CW complex refers to aflow-invariant connected topological space with a partition into hypertorus bundles over open CW-cells of a regular CW complex where they are topologically glued together; see Definition 6.5. Byconsidering a regular CW complex and its hypertorus bundle, we construct a toral CW complex . Thetoral CW complex is the quotient space of the hypertorus bundle over an appropriated constructedequivalence relation; see Lemma 6.7. This provides an actual description for what singular Eulerianflows in this paper experience. As the parameters slightly change, a parametric Eulerian flow with amultiple Hopf singularity may experience bifurcations of invariant toral CW complexes and, thus, wehave a complex oscillating dynamics for the nearby orbits. The cell-bifurcation analysis of invarianttoral CW complexes provide the information for our proposed parametric state-feedback controllerdesign and suitable leaf-choices in its control applications. Yet, leaf-bifurcations are chosen andcontrolled by initial data and bifurcation controller parameters. This approach lays the ground for . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity
3a desired controlled realization of the oscillating dynamics in the state space; see [19, 20].We are concerned with hypernormalization, bifurcation analysis and control of flows of nonlinear n -tuple Hopf singularities given byΘ + v ( x ) , where v ( x ) := (cid:80) ni =1 Θ if i + E g , Θ := (cid:80) ni =1 ω i Θ i , (cid:81) ni =1 ω i (cid:54) = 0 , f i (0) = g (0) = 0 , (1.1) E := (cid:80) ni =1 x i ∂∂x i + y i ∂∂y i , E g := gE , Θ i := − y i ∂∂x i + x i ∂∂y i , Θ if i := f i Θ i ,f i , g ∈ R [[ x ]] , ω i ω j / ∈ Q for 1 ≤ i < j ≤ n , and x := ( x , y , . . . , x n , y n ). We call E g and Θ if i , an Eulerian and a rotational vector field, respectively. Any such vector field is associated witha formal autonomous differential system and vice versa. Thus, the terminology and notations ofvectors, vector fields, formal flows, and differential systems are interchangeably used throughout thispaper. We simply refer to the formal flows associated with (1.1) by Eulerian flows. A bifurcationcontrol of systems of type (1.1) does not simply follow the classical normal form theory. This isbecause classical normal forms usually destroy the Eulerian and rotating structure of these vectorfields. As a result, the bifurcation analysis of the truncated classical normal forms does not reflectwhat occurs in the actual dynamics of the system. Our goal in this paper is to do the analysis bytaking into account the structural symmetry of such systems. This paper is the second draft in ourproject on bifurcation control of singular Eulerian flows with applications in parametric oscillatorsystems , robotic team control , computer harmonic music design and analysis ; also see [18–20].We first provide a flow-invariant primary cell-decomposition of the state space into open-cells invariant under all Eulerian flows . Each cell is a 2 k -manifold for some k ≤ n and is diffeomorphicto the product of k -copies of the cylinder S × R + ; see Lemma 2.2. A reduction of a given Eulerian(plus rotational) vector field over a 2 k - closed cell (the closure of an open k -cell ) gives rise to anEulerian (plus rotational) 2 k -cell vector field . Each open cell admits an irreducible flow-invariant k + 1-dimensional foliation. Each leaf of the foliations is an integral manifold whose tangent bundle is spanned by all Eulerian vector fields. In other words, leaves are minimal realizations of thestate space configuration for all Eulerian flows with multiple Hopf singularity. Each of the leavesis a manifold homeomorphic to T k × R + , where T k is a Clifford hypertorus of k -dimension for1 ≤ k ≤ n . These leaves are parameterized by positive vectors C from the k − n -permutation σ , say M Ck,σ ; see Theorem 2.3. We refer to a vector field reduced to aninvariant leaf by a leaf-vector field . Using permissible changes of state variables, the associated leaf-dependent simplified system is called a leaf-normal form system. We further allow time rescalingand dependence on bifurcation parameters to obtain (formal) parametric leaf-normal forms . Finitedeterminacy analysis and bifurcation analysis of (formal) parametric (leaf) normal forms gives riseto a comprehensive understanding about the local dynamics of the original singular system. Wedistinguish between a leaf-bifurcation , a 2 k -cell bifurcation , and a toral CW complex bifurcation . A leaf-bifurcation or a leaf-transition variety is associated with a leaf-vector field . A 2 k -cell bifurcation is concerned with the dynamics of an Eulerian 2 k -cell system. When k = n, a cell bifurcation is simply called a bifurcation . However, a toral CW complex bifurcation refers to appearance or . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity disappearance of a flow-invariant toral CW complex through a bifurcation or a cell bifurcation . Thisterminology is similar in a sense to the classical limit cycle bifurcations . The vector field (1.1) mayexperience the leaf-bifurcation of multiple invariant k -hypertori for 1 ≤ k ≤ n and cell-bifurcationsof multiple toral CW complexes.There are topologically equivalent systems associated different parameters of a parametric Eu-lerian flow in 2 n -dimension whose M Ck,σ -leaf dynamics are different; see Theorems 5.1-5.2, andcompare them with Theorems 6.11, 6.9, 6.12, and 6.15. Therefore, a leaf-bifurcation variety is notnecessarily a bifurcation variety for the 2 k -dimensional cell systems for k ≤ n . This is what enforcesour distinction between leaf-bifurcations and cell-bifurcations . Cell-bifurcations are here involvedwith flow-invariant toral manifolds. Due to the complexity of these bifurcated flow-invariant mani-folds, we introduce toral CW complexes as a technical means for their comprehensive description.Our definition is specific to the actual flow-invariant manifold cell-bifurcations in Section 6. Weobserve a 2 k -cell-bifurcation of an isolated secondary flow-invariant toral CW complex whose leaf-sections within the state space are l -hypertori for 1 ≤ l ≤ k . Secondary flow-invariant toral CWcomplexes refer to the flow-invariant manifolds bifurcated from an equilibrium (the origin) of a 2 k -cell system. The secondary invariant toral CW complex may further undergo a cell-bifurcation. Anexternal tertiary toral CW complex is born from the secondary toral CW complex in a specific case, i.e., the secondary toral manifold lives inside the tertiary manifold. Tertiary cell-bifurcations referto toral CW complexes bifurcating from a secondary toral CW complex (but not from the origin).When the origin is stable, the external toral CW complex is also stable and solutions approach toeither the origin or to the external invariant manifold.The basic tools for analysis is the derivation of normal forms. The idea in normal form theory isto use near-identity changes of state variables to transform a singular vector field into a qualitativelyequivalent but simple vector field, that is called normal form vector field . This facilitates thebifurcation analysis of a given nonlinear singular vector field. It is known that there is a one-to-onecorrespondence between near-identity transformations and the time-one mappings associated withthe flow of nonlinear vector fields without constant and linear parts ; see equations (3.1)-(3.2). In thelatter case, nonlinear vector fields without constant and linear parts are called by transformationgenerators . Due to the Lie bracket formulation (exp ad by equation (3.2)) in updating vectorfields using time-one mappings, it is an advantage to use the transformation generators in normalform theory. Then, Lie algebraic structures and Lie subalgebras are important tools for structuralsymmetry-preserving of a given singular system in its normalization process; see [18]. Our proposedapproach here is to make an invariant leaf-reduction and then, obtain leaf-normal forms. Hence, wedesign a Lie algebra structure for leaf-vector fields on M Ck,σ through a linear-epimorphism betweenEulerian (plus rotational) vector fields on C k × R + . This Lie algebra structure is required forintroducing permissible leaf-preserving transformations for the leaf-normal form derivation. Next,a complete parametric leaf-normal form classification for vector field types (1.1) are provided.This paper is organized as follows. Flow-invariant cell-decomposition of the state space isprovided in Section 2. We show that each cell admits an irreducible foliation that is flow-invariant . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity k -cell systems are described in Section 6. A comprehensive descriptionof the 2 k -cell bifurcations is achieved by an introduction of toral CW complexes associated withCW complexes. Our normal form approach provides an algorithmically computable method inbifurcation controller design for singular oscillator systems. This section first presents a primary decomposition of the state space into 2 k -dimensional cellsinvariant under all Eulerian flows. Each cell is homeomorphic to the product of k -copies of thecylinder R + × S . Next, we show that each 2 k -cell admits a k + 1-dimensional foliation. This furthersplits the 2 k -cells into the minimal flow-invariant manifolds. More precisely, each leaf of the foliationis a minimal manifold that is invariant under all singular Eulerian flows and is homeomorphic toa T k × R + , where T k is a k -dimensional Clifford hypertorus. More precisely, minimal invariantmanifolds are integral leaves of the foliations associated with all Eulerian flows. Since the celldecomposition and leaves of the foliations are flow-invariant under all Eulerian flows, we employthese in sequence as a reduction technique for their normal from, bifurcation analysis and controlin sections 4 and 5. Notation 2.1.
1. We use (cid:116) i A i for the union of disjoint sets A i . The set R + stands for nonzeropositive real numbers and T k for a k -dimensional Clifford torus. Notation S n − stands for the n − R n . Denote the n -vector ( c i ) ni =1 by ( c , c , . . . , c n ) and S n − > := (cid:8) ( c , . . . , c n ) ∈ S n − | c i > ≤ i ≤ n (cid:9) . (2.1)2. Denote S kn := { σ ∈ S n | σ ( i ) < σ ( j ) for i < j ≤ k and for k < i < j ≤ n } (2.2)where S n is the group of permutations over { , , . . . , n } . We denote the identity map in S kn by I. Thus, S kn has (cid:0) nk (cid:1) -number of elements and S nn = S n = { I } . For σ ∈ S kn , denote S k − ,σ> := (cid:8) ( c , . . . , c n ) ∈ R n | ( c σ (1) , . . . , c σ ( k ) ) ∈ S k − and c σ ( i ) > ≤ i ≤ k (cid:9) . (2.3)Here, σ ( i ) for i = 1 , . . . , k represents nonzero elements c σ ( i ) (cid:54) = 0 from c ∈ S k − ,σ> while c σ ( i ) = 0for i = k + 1 , . . . , n. For an instance, S ,σ> = { e nσ (1) } and S k − ,σ> = S k − > . Here, e ni ∈ R n standsfor the i -th element from the standard basis of R n . Lemma 2.2 (Flow-invariant cell-decomposition of the state space) . There exists a disjoint Eulerianflow-invariant decomposition of the state space into open k -manifolds M k,σ for ≤ k ≤ n, and σ ∈ S kn , i.e., R n = (cid:70) nk =0 (cid:70) σ ∈ S kn M k,σ and M ,I = { } . For each k = 1 , . . . , n , there are (cid:0) nk (cid:1) . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity number of k -dimensional cells corresponding to permutations σ ∈ S kn , while cells of odd dimensionare empty. Each M k,σ is diffeomorphic to the product of k number of cylinders S × R + , i.e.,( S × R + ) k . Proof.
For any (cid:54) = x = ( x, y ) = ( x , x , . . . , x n , y , y , . . . , y n ) ∈ R n , denote ( x n j , y n j ) (say 1 ≤ j ≤ k ) for the nonzero pairs of x and let m j with 1 ≤ j ≤ n − k stand for the remaining indices, i.e., ( x m j , y m j ) = (0 , . These spaces are pairwise disjoint andtheir union is the whole state space R n . Let ρ i ( t ) = (cid:107) ( x i ( t ) , y i ( t )) (cid:107) and v := E f + (cid:80) ni =1 Θ ig i .Consider the initial value problem associated with v and initial condition x ( t ◦ ) ∈ M k,σ . Hence,the manifold { x ∈ R n |(cid:107) ( x i , y i ) (cid:107) = 0 } , for an i ≤ k, and its complement are both invariant underthe flow associated with v. Thus, ρ σ ( i ) ( t ) (cid:54) = 0 for all t when x ( t ◦ ) ∈ M k,σ and 1 ≤ i ≤ k. When1 ≤ j ≤ n − k and t ≥ t ◦ , ρ m j ( t ) = 0 for all trajectories starting from a point x ( t ◦ ) on the manifold M k,σ . Hence, M k,σ is a 2 k -dimensional flow-invariant subspace and the union of M k,σ over k and σ ∈ S kn gives rise to R n \ { } . Define the map φ M k,σ : M k,σ → ( S × R + ) k by φ M k,σ ( x ) := (cid:16) ( x σ ( i ) ( t ) ,y σ ( i ) ( t )) (cid:107) ( x σ ( i ) ( t ) ,y σ ( i ) ( t )) (cid:107) , (cid:107) ( x σ ( i ) ( t ) , y σ ( i ) ( t )) (cid:107) (cid:17) ki =1 . (2.4)This is well-defined and smooth. Here, the cylinder S × R + is parameterized by (cos θ, sin θ, r ) for r ∈ R + and θ ∈ R π Z . Therefore, the inverse function φ − M k,σ : ( S × R + ) k → M k,σ is a diffeomorphism.Now we show that each 2 k -manifold M k,σ admits a secondary decomposition (via a smoothfoliation) as a union of disjoint flow-invariant connected k + 1-submanifolds denoted by M Ck,σ .Each submanifold M Ck,σ is called a leaf. Here, every point has an open neighborhood U and alocal coordinate-system, say ( y , · · · , y k ) : U → R k , so that each leaf within U can be describedby y k +2 = constant, . . . , y k = constant. The leaves M Ck,σ of the foliations are parameterized by1 ≤ k ≤ n and C ∈ S k − > . Each leaf M Ck,σ is a minimal manifold that is invariant under everyEulerian flow with multiple Hopf singularity. Then, each vector field type in equation (1.1) canbe reduced on these individual flow-invariant minimal leaves. Next, leaf parametric normal formclassifications provide further reduction of the vector field (1.1). Hence, the analysis of the infinitelevel leaf parametric normal form v ( ∞ ) k,C for 1 ≤ k ≤ n and C ∈ S k − > provides all bifurcation scenariosof the vector field v. Theorem 2.3 (Irreducible flow-invariant foliations) . There is a smooth k + 1 -dimensional foliationfor further refinements of each M k,σ into the disjoint leaves M Ck,σ of the foliations parameterizedby C ∈ S k − ,σ> , indeed, M k,σ = (cid:116) C ∈ S k − > M Ck,σ . Each leaf M Ck,σ is a flow-invariant manifold homeo-morphic to T k × R + and M k,σ is homeomorphic to T k × R + × S k − > . The set M Ck,σ is a minimalmanifold that is invariant under all flows associated with Eulerian and rotational vector fields.Proof.
Consider a point (cid:54) = x ◦ ∈ M k,σ . We parameterize the leaves of the foliation by C ∈ S k − ,σ> . Let c ◦ σ ( j ) := || ( x ◦ σ ( j ) ,y ◦ σ ( j ) ) |||| x ◦ || (cid:54) = 0 for j ≤ k, and c ◦ σ ( j ) = 0 for k < j. Thus, C x ◦ := ( c ◦ , . . . , c ◦ n ) ∈ S k − ,σ> . . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity θ σ ( j ) , ρ σ ( j ) ) for ( x σ ( j ) , y σ ( j ) ) in the polar coordinates and N x ◦ ⊂ M k,σ for a small openneighborhood around x ◦ . Note that for all x ∈ N x ◦ , ( x σ ( i ) , y σ ( i ) ) (cid:54) = 0 when i ≤ k and ( x σ ( i ) , y σ ( i ) ) = 0for i > k. Then, we introduce the map ϕ x ◦ : N x ◦ ⊂ M k,σ → R k by ϕ x ◦ ( x ) := (cid:32) θ σ (1) , θ σ (2) , . . . , θ σ ( k ) , || ( x σ (1) , y σ (1) ) || , c ◦ σ (1) ρ σ (2) c ◦ σ (2) − ρ σ (1) , . . . , c ◦ σ ( k − ρ σ ( k ) c ◦ σ ( k ) − ρ σ ( k − (cid:33) . The neighborhood N x ◦ can be chosen small enough so that the family ϕ x ◦ would construct a smooth system of local coordinates within the invariant M k,σ . Let x (cid:54) = y . Thus, ∅ (cid:54) = ϕ − x ( R k +1 × k − ) ∩ ϕ − y ( R k +1 × k − ) if and only if C x = aC y for some 0 (cid:54) = a ∈ R + . Since C x , C y ∈ S k − ,σ> , we have a = 1 . Hence, the family ∪ { x | C x = C } ϕ − x ( R k +1 × k − ) for C ∈ S k − ,σ> partitions M k,σ into disjoint connected sub-manifolds. Thereby, these sub-manifolds can be parameterized by C ∈ S k − ,σ> . The leaf M Ck,σ is a k + 1-manifold invariant under all Eulerian flows. This provides adiffeomorphism between M Ck,σ and the toral cylinder T k × R + . This section is devoted to leaf-reduction of vector fields and the study of leaf-preserving transfor-mation generators using a Lie algebra structure for leaf-vector fields. We first recall how a nonlinear(formal) vector field Y ( x ) generates a near-identity transformation when Y (0) = D x Y (0) = 0 , and D x stands for derivatives with respect x ; e.g., see [38, format 2b]. Consider the initial value problem ddt x ( t, y ) = Y ( x ( t, y )) , x (0 , y ) = y . (3.1)Then, the time-one mapping x := φ Y ( y ) = x (1; y ) is a near-identity coordinate transformationgenerated by Y . Assume that this transforms the new variable y to the old variable x . Then, avector field v ( x ) is transformed to w ( y ) := [( D y φ Y )( y )] − v ( φ Y ( y )) = exp ad Y v = v + [ Y, v ] + (cid:2) Y, [ Y, v ] (cid:3) + · · · , (3.2)where ad Y v = [ Y, v ] := Wronskian( v, Y ) = ( D x Y ) v − ( D x v ) Y ; e.g., see [34, 35, 38]. Then, system(3.1) is transformed into ˙ y = w ( y ) . Therefore, a Lie subalgebra structure for transformation gen-erators is sufficient to preserve a structural symmetry using these types of transformations. Hence,the time-one flows associated with nonlinear vector fields of type Y := p ( u ) E + (cid:80) ni =1 h i ( u )Θ i , for u i = ¯ v i , p ( ) = h ( ) = 0 , (3.3) . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity k -truncated simplestparametric normal form with least number of parameters to fully unfold a system with respect to the k -equivalence relation; e.g., see [16, 17]. However, the truncated simplest parametric normal formsystems are not yet sufficiently simple for bifurcation analysis and control. Thus, we alternativelyuse the flow-invariant leaf reduction of the vector fields to obtain leaf-vector fields in this section.Given transformation generators described above, we further study the leaf-vector field preservingtransformation generators through a Lie algebra structure for leaf-vector fields. Then, in section4, we do the parametric normal form of the leaf-vector fields on the manifold T k × R + . Next, theestimated transition varieties associated with parametric leaf-normal forms establish a bifurcationcontrol criteria in a parametric state-feedback controlled system. These are indeed necessary forany meaningful bifurcation analysis and bifurcation control of an Eulerian flow with multiple Hopfsingularity; see sections 4 and 5. Now we add some extra notations to those described in Notation2.1. Notation 3.1.
Given σ ∈ S kn , n -vectors a = ( a , a , . . . , a n ) and b := ( b , . . . , b n ) , we denoteˆ a := (ˆ a , ˆ a , . . . , ˆ a k ) := ( a σ (1) , a σ (2) , . . . a σ ( k ) ) , a b := Π ni =1 a ib i , cos a := (cos a , . . . , cos a n ) , sin ˆ b ˆ a :=Π kj =1 sin b nj a n j , and | a | = (cid:80) ni =1 | a i | . Lemma 3.2 ( M Ck,σ -leaf reduction) . Consider v given in equation (1.1) , g ( x ) := (cid:80) | α | + | β |≥ a α,β x α y β ,f i ( x ) := (cid:80) | α | + | β |≥ b iα,β x α y β , σ ∈ S kn , C ∈ S k − ,σ> , k ≤ n, and ( x, y ) ∈ M k,σ . Then, the M Ck,σ -leafreduction of
Θ + v is given by Θ k + v σ,C where Θ k := (cid:80) kj =1 ω σ ( j ) ∂∂θ σ ( j ) and v σ,C := (cid:80) ∞| ˆ α | + | ˆ β | =1 ρ σ ( k ) | ˆ α | + | ˆ β | cos ˆ α ˆ θ sin ˆ β ˆ θ (cid:16) ˜ a k ˆ α, ˆ β ( C ) (cid:80) kj =1 c σ ( j ) c σ ( k ) ρ σ ( k ) ∂∂ρ σ ( j ) + (cid:80) kj =1 ˜ b k,j ˆ α, ˆ β ( C ) ∂∂θ σ ( j ) (cid:17) . (3.4) Here, ρ σ ( k ) ∈ R + and ˆ θ := ( θ σ (1) , θ σ (2) , . . . , θ σ ( k ) ) ∈ T k . When | ˆ α || ˆ β | < | α || β | , we have ˜ a k ˆ α, ˆ β ( C ) =˜ b k,j ˆ α, ˆ β ( C ) = 0 . Otherwise, ˜ a k ˆ α, ˆ β ( C ) := (cid:81) kj =1 c σ ( j ) ασ ( j )+ βσ ( j ) c σ ( k ) | α | + | β | a α,β , ˜ b k,j ˆ α, ˆ β ( C ) := (cid:81) kj =1 c σ ( j ) ασ ( j )+ βσ ( j ) c σ ( k ) | α | + | β | b jα,β for ≤ j ≤ k. (3.5) Proof.
Proof is omitted.Since M Ck,σ is homeomorphic to T k × R + , we may identify the leaf-vector field (3.4) with anEulerian type vector field plus a rotational vector field on T k × R + . This is described as follows. Let( (cid:37) j , ϑ j ) stands for ( ρ σ ( j ) , θ σ ( j ) ) . Thus, let ( X i , Y i ) := (cos ϑ i , sin ϑ i ) , ( X , Y ) ∈ T k = S × S · · · × S and denote L T k × R + := span ≤ i ≤ k (cid:110) q i ∂∂ϑ i , (cid:80) kj =1 ˆ c j (cid:37) k h∂ ˆ c k ∂(cid:37) j (cid:12)(cid:12)(cid:12) h, q i ∈ R [[ X , Y , (cid:37) k ]] , q i ( ) = h ( ) = 0 (cid:111) , (3.6) R := span { g ∈ C [[ υ , . . . , υ k , r ]] | υ i ∈ { z i , w i } for i ≤ k } , . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity z := ( z, w ) = ( z , . . . , z k , w , . . . , w k ) , w i = z i . We remark that the monomials appearing in R are in terms of either z i or w i but, a monomial cannot include both z i and w i for the same index i , e.g., z z w ∈ R and z w / ∈ R . The main goal in the next lemma is to construct a leaf-dependentLie algebra structure over the class of leaf-vector fields. The idea is to use the pushforward mapsassociated with projection of coordinate changes from a complex coordinate system on C k × R + onto T k × R + . The Lie algebra structure introduces permissible transformation generators for theirleaf-normal form classification. Theorem 3.3 (Lie algebra structure on invariant leaves) . There exists a linear-isomorphism
Ψ : L T k × R + → J := span ≤ i ≤ k (cid:110) rg∂∂r , f i w i ∂ i ∂w i − f i z i ∂ i ∂z i (cid:12)(cid:12)(cid:12) g, f i ∈ R (cid:111) , (3.7) where f i ( z, w, r ) = f i ( z, w, r ) , g ( z, w, r ) = g ( z, w, r ) , f i ( ) = g ( ) = 0 , and (cid:37), r ∈ R + . Further,there are Lie algebra structures on L T k × R + and J so that Ψ is a Lie isomorphism.Proof. Proof is omitted for briefness.
Remark 3.4.
An alternative dynamics reduction can be made using projective space of the statespace. However, this is fruitless due to the fact that the dynamics on the projective space is trivial.
This section is devoted to derive the formal leaf-normal forms of singular systems with multiple Hopfsingularity. We use near-identity changes of the state variables. For the parametric vector fields, wealso use the rescaling of time and it is also important to allow the state- and time-transformationsto depend on the bifurcation parameters; see [4, 14, 26, 29–37, 41, 43, 43–47, 50] for a recent literatureon normal forms, convergence and their optimal truncations. Given the proof of Theorem 3.3 andthe convenience of notations, we identify leaf-vector fields with those on T k × R + . Then, our leaf-normal form derivation uses the map ˆ ψ − ∗ to transform a vector field on T k × R + into a vector fieldon C k × R + . Next, normal forms are derived and then, the pushforward map ψ ∗ is applied to projectthe normal form vector field back to a normal form vector field on T k × R + . Theorem 4.1 (The first level M Ck,σ -leaf normal form) . For any C ∈ S k − > , there exists a near-identity changes of the state variables transforming the M Ck,σ -leaf vector field
Θ + v σ,C given by (3.4) into a first level M Ck,σ -leaf normal form Θ k + v (1) σ,C , where v (1) σ,C := (cid:80) ∞ p =0 ( x σ ( k )2 + y σ ( k )2 ) p A σp ( x σ (1) , y σ (1) , . . . , x σ ( k ) , y σ ( k ) ) t , (4.1) A σp := diag( A σ ,p , A σ ,p , . . . , A σk,p ) , A σi,p := ( a p + b ip ) R θ ip ,R θ ip is the standard counterclockwise rotation matrix, θ ip := tan − (cid:16) b ip a p (cid:17) is the rotation angle, and a = 0 , b i = ω σ ( i ) for ≤ i ≤ k. The coefficients a p and b jp are C -dependent polynomials in terms of ˜ a k ˆ α, ˆ β and ˜ b k,j ˆ α, ˆ β given in equation (3.5) for | ˆ α + ˆ β | ≤ p . . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity Proof.
The proof follows direct calculations.The family of the first level leaf-normal form vector fields of type ( ?? ) is a Lie subalgebrain L T k × R + . Hence, we denote them by L := span ≤ i ≤ k (cid:110) q i ∂∂ϑ i , (cid:80) kj =1 h ˆ c j (cid:37) k ∂ ˆ c k ∂(cid:37) j (cid:12)(cid:12) q i , h ∈ R [[ (cid:37) k ]] , m ≥ (cid:111) . Apermissible direction-preserving time-rescaling is given by τ := (1 + T ) t, where T ( x σ ( k )2 + y σ ( k )2 ) is aformal power series without constant terms. This time rescaling transforms a vector field v ∈ L into v + T v ∈ L . Hence, we consider R as the formal power series generated by Z i := (cid:0) x σ ( k )2 + y σ ( k )2 (cid:1) i and denote ˆ E := (cid:80) kj =1 ˆ c j (cid:37) k ∂ ˆ c k ∂(cid:37) j . Lemma 4.2.
The vector space L constitutes a Lie subalgebra in L T k × R + so that for every α, β ∈ R , m, n ∈ Z ≥ and ≤ i ≤ j ≤ k , the structure constants are given by (cid:104) ρ σ ( k )2 m ˆ E , α ρ σ ( k )2 n ˆ E + β ρ σ ( k )2 l Θ σ ( i ) (cid:105) = ( m − n ) αρ σ ( k )2( m + n ) ˆ E − lβρ σ ( k )2( m + l ) Θ σ ( i ) , (cid:104) ρ σ ( k )2 m Θ σ ( i ) , ρ σ ( k )2 n Θ σ ( j ) (cid:105) = 0 . (4.2) The Lie algebra L is also an R -module that is consistent with time rescaling of vector fields, wherefor every β i ∈ R and ≤ i ≤ k,Z m ρ σ ( k )2 n ˆ E = ρ σ ( k )2( m + n ) ˆ E , Z m (cid:80) ki =1 β i ρ σ ( k )2 σ ( i ) Θ σ ( i ) = (cid:80) ki =1 β i ρ σ ( k )2( m + σ ( i )) Θ σ ( i ) . (4.3) Proof.
The argument for relations (4.2) and (4.3) follows direct calculations.The leaf-normal form (4.1) in polar coordinates readsΘ k + (cid:80) ki =1 (cid:80) p ≥ ρ σ ( k )2 p a p c σ ( i ) ρ σ ( k ) ∂c σ ( k ) ∂ρ σ ( i ) + (cid:80) kj =1 (cid:80) p ≥ b jp ρ σ ( k )2 p ∂∂ Θ σ ( j ) . (4.4)The first level parametric normal form of every vector field (1.1) is similar to equation (4.4), exceptthat the coefficients a m and b im depend on the parameter vector µ := ( µ , µ , . . . , µ r ) and we denotethem by a m ( µ, C ) and b im ( µ, C ) where a (0 , C ) = 0 , b i (0 , C ) = 0 for 1 ≤ i ≤ n. Hence, parametric version of (4.4) with respect to Eulerian and rotational vector fields in L isexpressed byΘ k + v (1) σ,C := (cid:80) ki =1 ω σ ( i ) Θ σ ( i ) + (cid:80) j ≥ a j ( µ, C ) ρ σ ( k )2 j ˆ E + (cid:80) ki =1 (cid:80) j ≥ b ij ( µ, C ) ρ σ ( k )2 j Θ σ ( i ) , (4.5)where we take the notation m := ( m , . . . , m r ), µ m := µ m · · · µ rm r ,a j ( µ, C ) = (cid:80) | m |≥ a j, m ( C ) µ m , and b ij ( µ, C ) = (cid:80) | m |≥ b ij, m ( C ) µ m for 1 ≤ i ≤ k. In order to do the hyper-normalisation of vector fields, we define s := min { j | a j ( , C ) (cid:54) = 0 } (4.6) . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity δ ( µ m ρ σ ( k )2 j ˆ E ) := | m | ( s + 1) + j, δ ( µ m ρ σ ( k )2 j Θ σ ( i ) ) := | m | ( s + 1) + s + j for 1 ≤ i ≤ k. The grading δ decomposes the Lie algebra L = (cid:80) L i into δ -homogeneous spaces as a graded Liealgebra, [ L i , L j ] ⊆ L i + j . Further, it will be a R -graded module. Let Θ k + v (1) σ,C := (cid:80) ∞ i =0 v i and v i ∈ L i . The map d i, ( T i , S i ) := T i v + [ S i , v ] is defined for ( T i , S i ) ∈ R i × L i . Using a slight abuseof notation, we inductively define the map d i,r : ker d i − ,r − × R i × L i → L i by d i,r ( T r − i − r +1 , · · · , T r − i − , T i , S r − i − r +1 , · · · , S r − i − , S i ) := (cid:80) r − i =1 (cid:0) T r − i − i v i + [ S r − i − i , v i ] (cid:1) + T i v + [ S i , v ] , where ( T r − i − r +1 , · · · , T r − i − , S r − i − r +1 , · · · , S r − i − ) ∈ ker d i − ,r − , i ≥ r, and r ≥
2. The map d i,r computesall possible spectral data available as transformation generators to simplify terms in grade i byusing not only the linear part of the vector field but also all terms in the normalising vector fieldup-to grade r − . Thus, im d i,r represents the space that can be simplified from the vector fieldin grade r while a complement space C ri to im d i,r , for any i, stands for all possible terms thatmay not be simplified in the r -level normal form step; e.g., see [21, Theorem 4.3 and Lemma 4.2].Hence, using near-identity changes of state variable and time rescaling (direction preserving), thevector field Θ k + v (1) σ,C can be transformed into a r -th level parametric normal form v ( r ) := (cid:80) v ( r ) i ,where v ri ∈ C ri . Deriving d i,i and C ii for any i ≥ e.g., see [21, 35]. Theorem 4.3 (Infinite-level parametric leaf-normal forms) . Consider C ∈ S k − > and a M Ck,σ -leaf.Then, there are a natural number s, near-identity changes of state variables and time-rescalingsuch that they transform the parametric leaf-normal form (4.5) into the infinite-level parametricleaf-normal form Θ k + w ( ∞ ) σ,C where w ( ∞ ) σ,C is given by (cid:80) k,si =1 ,j =0 ( x σ ( k )2 + y σ ( k )2 ) j (cid:16) (ˆ a j ( µ,C ) x σ ( i ) − ˆ b ij ( µ,C ) y σ ( i ) ) ∂∂x σ ( i ) +(ˆ a j ( µ,C ) y σ ( i ) +ˆ b ij ( µ,C ) x σ ( i ) ) ∂∂y σ ( i ) (cid:17) , (4.7) where ˆ a j ( ,C ) = ˆ b i ( ,C ) = ˆ b j ( µ,C ) = 0 for ≤ j ≤ s − , ≤ i ≤ k, and (cid:54) = a s := a s ( ,C ) = a s ( µ,C ) . Proof.
The index for zero vectors indicate their dimension. For ρ := ρ σ ( k ) , we have d | m | ( s +1)+ s + j,s +1 (cid:16) γ j, m µ m Z j , s , α j, m µ m ρ j E + (cid:80) ki =1 β ij − s, m µ m ρ j − s ) Θ σ ( i ) , s (cid:17) = a s ( γ j, m + 2 α j, m ( j − s )) µ m ρ s + j ) ˆ E + (cid:80) ki =1 (cid:0) ω σ ( i ) γ j, m + 2 β ij − s, m ( j − s ) a s (cid:1) µ m ρ j Θ σ ( i ) . This implies that all terms µ m ρ s + j ) ˆ E for j ≥ µ m ρ j Θ σ (1) for j ≥ s + 1-th level orbital leaf-normal form system. Consider the case j = s . Hence, we take α s, m = β i , m = 0 for i = 1 , . . . , k and γ s, m := − a s a s, m ( C ). Therefore, in s + 1-th level parametricleaf-normal form, the Eulerian terms µ m ρ s + j ) ˆ E for j ≥ , | m | ≥ µ m ρ s ˆ E for | m | > µ m ρ s + j ) Θ σ ( i ) for j ≥ , | m | ≥ , ≤ i ≤ k . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity µ m ρ j Θ σ (1) for all s (cid:54) = j ∈ { } ∪ N are simplified. This is indeed the equation (4.7) in polarcoordinates. Time rescaling generators µ m Z j and state transformation generators µ m ρ j ˆ E for j > s do not have any influence in enlarging the space im d s + l,s + j +1 in normalization levels higher than s + 1 . On the other hand, terms µ m Z j for j ≤ s and µ m ρ j ˆ E for j < s have already contributed inim d s + l,s +1 . Further, im d s + l,s +1 ⊆ im d s + l,s + j +1 . Therefore, im d s + l,s + j +1 = im d s + l,s +1 for all l > j. This completes the proof.
When specific initial values are chosen, the state space configuration is realized within an individualflow-invariant leaf. Then, leaf-transition varieties make a partition for the parameter space intoconnected regions. All parameters from an open connected region corresponds to qualitativelythe same dynamics for the parametric leaf-vector field. Hence, leaf-varieties classify the persistent qualitative dynamics of the leaf vector field subjected to small parameter-perturbation . Thus, theindividual leaf-choices and leaf-bifurcations contribute into the state-feedback controller designs inpractical bifurcation control applications. This section studies the leaf-bifurcations associated witha leaf-parametric normal form (4.7) for three most generic cases s = 1 , , s = 1 and s = 2 Theorem 5.1 (Leaf case s = 1) . Consider the parametric leaf-normal form (4.7) when s = 1 inequation (4.6) for some ≤ k ≤ n and C ∈ S k − ,σ> . Then, there is a leaf-dependent bifurcation ofan invariant T k -torus from the origin. This leaf-bifurcation is three-determined and its associatedbifurcation variety is given by T P ch := { ν | ν = 0 } . When ν > and a < , the invariant torus isstable while the origin is unstable. For a > and ν < , the origin is stable and the T k -torus isrepelling.Proof. The assumption is equivalent with a := a (0 , C ) (cid:54) = 0. For i = k and only looking forthe steady-state solutions, this represents a normal form for subcritical and supercritical pitchforkbifurcation at ( ρ σ ( k ) , ν ) = (0 ,
0) when a > a < , respectively. Thus, this is a three-determined differential system. The none-zero equilibrium of this system for a ν < ρ ∗ = ( ρ , . . . , ρ n ) = (cid:113) − ν a Cc σ ( k ) . Since ddρ σ ( k ) f k,σ ( k ) (cid:16)(cid:113) − ν a , ν , C (cid:17) = 2 a (cid:113) − ν a < ν > a < , an asymptotically stable T k -torus bifurcates from the origin and the origin is unstable. For ν < a > , an unstable T k -torus bifurcates from the origin while the origin is asymptoticallystable. Theorem 5.2 (Leaf case s = 2) . Assume that the leaf parametric normal form (4.7) is associatedwith s = 2 for some ≤ k ≤ n and C ∈ S k − ,σ> . A secondary stable flow-invariant k -hypertorusbifurcates from the origin on M Ck,σ -leaf variety given by T SupP := { ( ν , ν ) | ν = 0 , ν < } . (5.1) . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity (a) Transition sets when a = − . (b) Curve Γ and bifurcation varietiesfor a = 1. . . . • r • r γ := tan − ν ν +0 . γ (c) Root locus correspond-ing with Γ from Figure 1b. Figure 1: Bifurcation varieties and root loci for the vector field (4.7), leaf case s = 2. Here,( ν , ν ) = ( − , ) in Figure 1b corresponds with roots r = 0 .
085 and r = 0 .
245 in Figure 1c.
A secondary unstable invariant k -hypertori bifurcates from the origin on the variety T SubP := { ( ν , ν ) | ν = 0 , ν > } . (5.2) There is a secondary double saddle-node type bifurcation of flow-invariant k -hypertori at the leaf-transition set T SD := (cid:110) ( ν , ν ) | (cid:16) ν a (cid:17) − ν a = 0 , a ν < (cid:111) ; (5.3) see Figures 1. Here, two invariant hypertori (bifurcated at T SupP and T SubP ) collide when theirradiuses converge, and then, they both disappear as similar to a saddle-node type bifurcation. Thesebifurcations are five-determined. One of the invariant hypertori always live inside the other oneuntil the radiuses of the inner hypertorus and the outer hypertorus converge. When ν > a ν and ν > , the origin and the outer hypertorus are unstable while the inner invariant hypertorus isstable. For ν > a ν and ν < , the origin and the outer invariant hypertorus are stable whilethe inner invariant T k -torus is repelling.Proof. Let a := a ( , C ) (cid:54) = 0 and ν (cid:54) = 0 . Consider the Z -equivariant differential equation˙ ρ σ ( k ) = f k,σ ( k ) ( ρ σ ( k ) , ν ) := ρ σ ( k ) ( ν + ν ρ σ ( k )2 + a ρ σ ( k )4 ) . Then, f k,σ ( k ) (0 ,
0) = ∂∂ρ σ ( k ) f k,σ ( k ) (0 ,
0) = 0 , ∂∂ν f k,σ ( k ) (0 ,
0) = ∂ ∂ρ σ ( k )2 f k,σ ( k ) (0 ,
0) = 0 , ∂ ∂ρ σ ( k ) ∂ν f k,σ ( k ) (0 ,
0) = 1 (cid:54) = 0 , and ∂ ∂ρ σ ( k )3 f k,σ ( k ) (0 ,
0) = ν (cid:54) = 0 . This is a five-determined Z -equivariant type bifurcation and corresponds with the σ ( k )-th am-plitude dynamics for trajectories on the M Ck,σ -leaf manifold. When ν < , we have a super-critical pitchfork bifurcation at the origin while ν > ν a < , take µ := ν a − ν a such that f k,σ ( k ) ( ρ σ ( k ) , ν ) is read by g ( ρ σ ( k ) , µ ) := . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity (a) ν = − .
1. (b) ν = 0. (c) ν = 0 . Figure 2: Transition sets for leaf case s = 3 ρ σ ( k ) (cid:0) a ( ρ σ ( k )2 + ν a ) + µ (cid:1) . This function takes its minima at ρ ± σ ( k ) = ± (cid:16) − ν a (cid:17) . Since g ( ρ ± σ ( k ) ,
0) = ∂∂ρ σ ( k ) g ( ρ ± σ ( k ) ,
0) = 0 , ∂∂µ g ( ρ ± σ ( k ) ,
0) = ρ ± σ ( k ) (cid:54) = 0 , ∂ ∂ρ σ ( k )2 g ( ρ ± σ ( k ) ,
0) = 8 a ρ ± σ ( k )3 (cid:54) = 0 , the points ( ρ, µ ) = ( ρ ± σ ( k ) ,
0) are bifurcation points and each of them represents a saddle-node typebifurcation. s = 3 Theorem 5.3 (Leaf case s = 3) . Consider the M Ck,σ -leaf parametric normal form (4.7) , where s = 3 for ≤ k ≤ n and C ∈ S k − ,σ> . A k -hypertorus bifurcates from the origin at M Ck,σ -leaf bifurcationvariety T P sup := { ( ν , ν , ν ) | ν = 0 when either ν < ν = 0 , ν < } ,T P sub := { ( ν , ν , ν ) | ν = 0 when either ν > ν = 0 , ν > } . (5.4) This hypertorus is unstable when ν > , and stable for ν < . There is a double saddle-nodebifurcation variety of hypertori at T SN := (cid:26) ( ν , ν , ν ) (cid:12)(cid:12)(cid:12) D = 0 , and either ( ν a < , ν a > or ( ν a > , ν a ≤ (cid:27) , (5.5) where D := 4 (cid:0) ν a − ν a (cid:1) + 27 (cid:0) ν a − ν ν a + ν a (cid:1) . These bifurcations are seven-determined.Proof.
Consider the Z -equivariant equation ˙ ρ σ ( k ) = f k,σ ( k ) ( ρ σ ( k ) , ν , ν , ν ) := ν ρ σ ( k ) + ν ρ σ ( k )3 + ν ρ σ ( k )5 + a ρ σ ( k )7 , where a (cid:54) = 0 . It is easy to prove that this system is a seven-determined differentialequation and so is the M Ck,σ -leaf parametric normal form (4.7). We first prove the following claims:Claim 1. The function f k,σ ( k ) has 3 distinct positive roots if and only if ( ν , ν , ν ) ∈ A . Claim 2. The parameters ( ν , ν , ν ) ∈ A ∪ B if and only if f k,σ ( k ) has 2 distinct positiveroots. . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity (a) Corresponding with Γ fromFigure 2a for ν = − .
1. (b) Associated with Γ on Figure2b for ν = 0. (c) Radius loci for the ellipse Γ from Figure2c when ν = 0 . Figure 3: Radius loci of the invariant tori on M Ck,σ along the ellipses Γ i , i = 1 , , γ := ν . ν − . , γ := ν . ν +0 . , and γ := ν . ν +0 . , associated with equation (4.7) where s = 3 . Claim 3. The map f k,σ ( k ) has one positive root if and only if ( ν , ν , ν ) ∈ A ∪ B . Claim 4. The function f k,σ ( k ) has no positive root if and only if ( ν , ν , ν ) ∈ A ∪ B . We take R := ρ σ ( k )2 and consider ν a + ν a R + ν a R + R = 0 . (5.6)By substitution r := R + ν a , we obtain r + pr + q = 0 , where p := − ν a + ν a and q := − ν ν a + ν a + ν a . (5.7)The number of real roots of the equations (5.6) and (5.7) are equal. Let D = (cid:0) p (cid:1) + (cid:0) q (cid:1) . Wediscuss the number of real roots by sign ( D ) while the number of roots with positive real part areaddressed by Routh-Hurwitz Theorem. The number 0 ≤ n ≤ n = var(1 , ∆ , ∆ ∆ , ∆ ∆ ) where the functionvar represents the number of sign variations in its arguments while Hurwitz determinants follow∆ := ν a , ∆ := ν ν a − ν a , ∆ := ν a ν ν − a ν a . (5.8)We only discuss the singular cases of either ∆ = 0 or ∆ = 0 . The case ∆ = 0 leads to either∆ = 0 or ν = 0. The latter gives rise to ∆ := ν a , ∆ := ν ν a , and the claim is straightforward.For ∆ = 0 , we have ν = 0 . Thus, we may use the modified Hurwitz determinants ∆ ∗ and ∆ ∗ aslong as they retain the same signs as of ∆ and ∆ , respectively. Hence, we introduce ν = (cid:15) where (cid:15) is a positive small number and the modified Hurwitz determinants by ∆ ∗ := (cid:15), ∆ ∗ := (cid:15)ν a − ν a and∆ ∗ := ν ( (cid:15)ν − ν ) a . Therefore, we havesign(∆ ∗ ) > , sign( ∆ ∗ ∆ ∗ ) = sign( − ν (cid:15)a ) , and sign( ∆ ∗ ∆ ∗ ) = sign( ν a ) . (5.9)Hence, we can alternatively study the sign variation function var(1 , ∆ ∗ , ∆ ∗ ∆ ∗ , ∆ ∗ ∆ ∗ ) . . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity = 0 , we have ν ν = a ν . Thus, equation (5.6) can be factored and isread as ( ρ σ ( k )2 + ν a )( ρ σ ( k )4 + ν a ) = 0. Hence, we discuss the roots of this reduced equation.Claim 1. The equations (5.6) has 3 distinct positive roots iff D < n = 3 . Sincevar(1 , ∆ , ∆ ∆ , ∆ ∆ ) = 3 , ∆ < , ∆ ∆ > , ∆ ∆ < . By Hurwitz determinants (5.8), we introduce S := { ( ν , ν , ν ) | ν a < , ν ν − a ν a ν > , ν a < , D < } . Therefore, ν a − ν ν > ν a > ν ν > . This implies that S ⊆ A . Let ( ν , ν , ν ) ∈ A . Hence, ν a < , ν a < , ν a > , and D < . We claim that ν ν − a ν a ν = ∆ ∆ >
0. Our argument is bycontradiction. Suppose that ∆ ∆ = ν ν − a ν a ν < . Since a ν < , a ν < ν ν < . Hence, we have D = − ν ν a − ν ν ν a + ν a + ν ν a + ν a ≥ ν a + ν ν a + ν a > . (5.10)This, however, is a contradiction. Therefore, ∆ ∆ > A = S .Claim 2. Let ν (cid:54) = 0. The equation (5.7) has two distinct positive roots iff D < n = 2 . We define S := { ( ν , ν , ν ) | var(1 , ∆ , ∆ ∆ , ∆ ∆ ) = 2 , D < } and show that S = A ∪ B . The condition n = 2 for none-zero Hurwitz determinants gives rise tothree different cases:(i) ∆ > , ∆ ∆ < , ∆ ∆ > , (ii) ∆ < , ∆ ∆ > , ∆ ∆ > , (iii) ∆ < , ∆ ∆ < , ∆ ∆ > . The singular cases ∆ ∗ = 0 and ∆ ∗ = 0 adds the following two more cases:(iv) ∆ ∗ = (cid:15) > , ∆ ∗ ∆ ∗ = − ν (cid:15)a < , ∆ ∗ ∆ ∗ = ν a > , and (v) ν a < , ν a < . Let ( ν , ν , ν ) ∈ S satisfy case (i). So, ν a > , ν a > , and ν ν − a ν a ν < . The latter inequalityimplies that ν ν < a ν . These conditions along with D < ν a < ν a ≥ ν a < ν , ν , ν ) ∈ B . If ( ν , ν , ν ) ∈ S satisfies case (ii), we have ν a < , ν a > , and ν ν − a ν a ν > . Similar to the case(i), the condition
D < ν a > ν , ν , ν ) ∈ A . Now assume that ( ν , ν , ν ) ∈ S satisfies the case (iii). Then, ν a < , ν a > , and ν a < ν ν <
0. This implies ( ν , ν , ν ) ∈ B . Either ofthe singular cases (iv) and (v) along with
D < ν , ν , ν ) ∈ B . For ( ν , ν , ν ) ∈ A , we have ν a > > ν ν . So, ( ν , ν , ν ) satisfies case (iii) and as a result ( ν , ν , ν ) ∈ S . When( ν , ν , ν ) ∈ B and ν a >
0, this results in ∆ ∆ = ν ν − a ν a ν < . Thereby, the parameters satisfythe case (i). When ( ν , ν , ν ) ∈ B , ν a <
0, and ∆ ∆ = ν ν − a ν a ν is either positive or negative, theparameters satisfy one of cases (ii) and (iii). Hence, ( ν , ν , ν ) ∈ S . For ( ν , ν , ν ) ∈ B , ν a < ∆ ∆ = ν ν − a ν a ν = 0 , singular case (v) is satisfied. Finally when ( ν , ν , ν ) ∈ B and ν = 0 , thesingular case (iv) results.Claim 3. Let S := { ( ν , ν , ν ) | Either var(1 , ∆ , ∆ ∆ , ∆ ∆ ) = 1 or (var(1 , ∆ , ∆ ∆ , ∆ ∆ ) = 3 when D > } . . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity ν , ν , ν ) ∈ S . Hence, it sufficesto prove S = A ∪ B . Let ( ν , ν , ν ) ∈ S . Since var(1 , ∆ , ∆ ∆ , ∆ ∆ ) = 1 or 3 , we have 4 possibledifferent cases: (a) ∆ < , ∆ ∆ > , ∆ ∆ < > , ∆ ∆ > , ∆ ∆ < , (c) ∆ > , ∆ ∆ < , ∆ ∆ < , (d) ∆ < , ∆ ∆ < , ∆ ∆ < . (5.11)For the singular case ∆ = ν ν a − ν a = 0 , the equation ( ρ σ ( k )2 + ν a )( ρ σ ( k )4 + ν a ) = 0 has only onepositive root iff ν ν < . Since D = ν a ( ν a + ν a ) , D > ν a > , ν a < , and ν a < . This results in ( ν , ν , ν ) ∈ A . Similarly, for
D < ν , ν , ν ) ∈ B . Given equations(5.9), for the singular case ∆ = ν a = 0 and var (1 , ∆ ∗ , ∆ ∗ ∆ ∗ , ∆ ∗ ∆ ∗ ) = 1 , the only possible case is ν a < . When ν >
0, ( ν , ν , ν ) ∈ A while ν ≤ ν , ν , ν ) ∈ B . When
D > ν a < , ν a < , and ν a > ν ν > . So,( ν , ν , ν ) ∈ A . For the case (b), we have ν a > , ν a < ν a > ν ν . Hence, for ν a > ν , ν , ν ) ∈ A and when ν a ≤ , ( ν , ν , ν ) ∈ B . The case (c) leads to ν a > , ν a < ν a < ν ν < . This concludes that ( ν , ν , ν ) ∈ B . Finally, the conditions (d)implies that ν a < , ν a < ν a < ν ν . When ν a ≤
0, ( ν , ν , ν ) ∈ B . By inequalities in (5.10),we have D > ν a >
0. Therefore, ( ν , ν , ν ) ∈ A . Hence, S ⊆ A ∪ B . Now assume that ( ν , ν , ν ) ∈ A ∪ B . If ( ν , ν , ν ) ∈ A , we have the following conditions:( ν a < , ν a > , ν a < , D > , ( ν a < , ν a > , ν a > , or ( ν a < , ν a > , ν = 0) . From the first group of inequalities, for ν a < ν ν , we have ∆ ∆ = ν ν − a ν a ν < ν a > ν ν , ∆ ∆ > ν a = ν ν , the condition ∆ = 0is met. Further, recall that ν ν <
0. Second group gives rise to ∆ ∆ = ν ν − a ν a ν > = 0. By thelatter, we have var(1 , ∆ ∗ , ∆ ∗ ∆ ∗ , ∆ ∗ ∆ ∗ ) = 1 and ν a >
0. Hence, ( ν , ν , ν ) ∈ S . Let ( ν , ν , ν ) ∈ B and ν a >
0. We decompose the conditions in B into the following cases:( ν a < , ν ν < a ν , ν a < , ( ν a < , ν ν = a ν , ν a < , ( ν a ≤ , ν ν > a ν , ν a < . Each group of the above inequalities for nonsingular cases gives rise to var(1 , ∆ , ∆ ∆ , ∆ ∆ ) = 1 . Forthe singular case ν ν = a ν , we have ν ν < . Thus, the equation ( ρ σ ( k )2 + ν a )( ρ σ ( k )4 + ν a ) = 0have one root for both cases D ≤ D >
0. Hence, ( ν , ν , ν ) ∈ S . Since ν a ≤ ν a < , the condition ν a < ∆ ∆ = ν ν − a ν a ν <
0. Thereby, var(1 , ∆ , ∆ ∆ , ∆ ∆ ) = 1 . So, for both cases D ≤ D >
0, ( ν , ν , ν ) ∈ S . Since ν a < ν = 0, the conditionvar(1 , ∆ ∗ , ∆ ∗ ∆ ∗ , ∆ ∗ ∆ ∗ ) = 1 holds. Similar to the above, ( ν , ν , ν ) ∈ S . Then, A ∪ B = S . . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity ν , ν , ν ) ∈ A ∪ B . The boundaries of the sets A , A ∪ B , A ∪ B and A ∪ B is expressed by the varieties T P introduced in (5.4) and T SN given by (5.5). Further, R \ ( ∪ i =0 A i (cid:116) ∪ i =0 B i ) = T P ∪ T SN . There is always pitchfork bifurcationtype on the variety T P and double saddle node bifurcation on the variety T SD . The saddle-nodebifurcation corresponds with equation ddt ρ σ ( k ) = f k,σ ( k ) ( ρ σ ( k ) , ν , ν , ν ) and the equilibria are givenby ( γ ± , D ) = (cid:16)(cid:0) − ν a ± (cid:113) − p (cid:1) , (cid:17) and ( γ ± , D ) = (cid:16) − (cid:0) − ν a ± (cid:113) − p (cid:1) , (cid:17) . Let g ( ρ σ ( k ) , D ) := a ρ σ ( k ) (cid:16)(cid:0) ρ σ ( k )2 + ν a ± (cid:113) − p (cid:1)(cid:0) ρ σ ( k )2 + ν a ∓ (cid:113) − p (cid:1) + Dq ± − p ) (cid:17) . Here, for i = 1 , , we have g ( γ ± i ,
0) = ∂∂ρ σ ( k ) g ( γ ± i ,
0) = 0 , ∂∂D g ( γ ± i ,
0) = a γ ± i q ± ( − p ) (cid:54) = 0 , and ∂ ∂ρ σ ( k )2 g ( γ ± i ,
0) = ± (cid:113) − p γ ± i (cid:54) = 0 . These correspond with a saddle-node bifurcation. Let ν (cid:54) = 0 . Then, f k,σ ( k ) (0 ,
0) = ∂∂ρ σ ( k ) f k,σ ( k ) (0 ,
0) = 0 , ∂∂ν f k,σ ( k ) (0 ,
0) = ∂ ∂ρ σ ( k )2 f k,σ ( k ) (0 ,
0) = 0 , ∂ ∂ρ σ ( k ) ∂ν f k,σ ( k ) (0 ,
0) = 1 (cid:54) = 0 , and ∂ ∂ρ σ ( k )3 f k,σ ( k ) (0 ,
0) = ν (cid:54) = 0The invariant hypertorus is stable for ν > ν < ν (cid:54) = 0 , ν = 0 and D ( ν , , ν ) (cid:54) = 0 , D does not change its sign as ν slightly varies. Therefore, there is no qualitative type change in the vicinity of ν = 0. The case( ν , D ( ν , , ν )) = (0 ,
0) implies that ν ν a + ν a = 0 and thereby, ( ν , , ν ) ∈ T SN . Consider the equation ddt x = (cid:80) i =1 ω i Θ i + f ( µ, x ) E (5.12)where ω i = √ i for 1 ≤ i ≤ , x := ( x , y , x , y , x , y ) ∈ R , f ( µ, x ) := α + α x + α y + α x + α y + α x + α y + α x x + α x y (5.13)and α i = a i + µ i for 0 ≤ i ≤ a = 0 . Thus, n = 3 . . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity (a) Leaf case s = 1 in example 5.4. Leaf-bifurcation varieties for C and C . (b) The transition sets for theleaf case s = 2 in example 5.5. (c) The leaf-transition sets forleaf case s = 3 in example 5.5. Figure 4: Leaf-bifurcation varieties as ( c , c , c ) varies in examples 5.4 and 5.5. Example 5.4.
We take k = 2 and σ ∈ S for σ := I. Hence, we have M ,σ = { x ∈ R | ( x , y ) (cid:54) = 0 , ( x , y ) (cid:54) = 0 , ( x , y ) = 0 } . For notation brevity, C = ( c , c , ∈ S ≥ is denoted by ( c , c ) . Then, the leaf M C ,σ follows M ( c ,c )2 ,σ = { x ∈ M ,σ | c (cid:107) ( x , y ) (cid:107) = c (cid:107) ( x , y ) (cid:107)} . The M ( c ,c )2 ,σ -leaf reduction of the differential system (5.12) in polar coordinates is associated with (cid:80) i =1 (cid:16) ω i ∂∂θ i + (cid:0) α + ρ ( α cos θ + α sin θ )+ ρ ( α cos θ + α sin θ + α cos θ + α sin θ c c − ) (cid:1) c i ρ ∂c ∂ρ i (cid:17) . The associated vector field in the Lie algebra J via the homeomorphism Ψ is given by (cid:80) i =1 (cid:16) √ iw i ∂ i ∂w i − √ iz i ∂ i ∂z i (cid:17) + (cid:0) µ + (cid:0) α − i α (cid:1) rz + (cid:0) α + i α (cid:1) rw + (cid:0) α − α (cid:1) r z + (cid:0) α − α (cid:1) r w (cid:1) r∂∂r + (cid:16)(cid:16) c ( α + α )+ c ( α + α )2 c (cid:17) r + c c (cid:0) α − α (cid:1) r z + c c (cid:0) α − α (cid:1) r w (cid:17) r∂∂r . We omit the parameters µ i for i = 1 , , , , Maple implemen-tation of Theorem 4.1 and its proof, a truncated first level parametric leaf-normal form in L T k × R + (via the map Ψ − ) up to grade-seven is (cid:80) i =1 √ i∂∂ϑ i + (cid:80) i =1 ( µ + b (cid:37) + b (cid:37) ) c i (cid:37) ∂c ∂(cid:37) i where b ( µ, C ) := c ( a + a )+ c ( a + a )2 c + µ + µ and b ( µ, C ) := a ( a +3 a )+ a (3 a + a )8 ω + ( a + a )( a + a )4 ω . When b (0 , C ) (cid:54) = 0 , we have the generic leaf case s = 1 . By Theorem 4.3, the third level (infinite-level) parametric leaf-normal forms is given by (cid:80) i =1 √ i∂∂θ i + (cid:80) i =1 (cid:16) µ + c ( a + a )+ c ( a + a )2 c ρ (cid:17) c i ρ ∂c ∂ρ i . . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity (a) Leaf case s = 1 . Two numerical phase portrait trajecto-ries ( x ( t ) versus y ( t )) and ( x ( t ) versus y ( t )) convergingto an invariant torus when ( c , c ) = ( √ , √ ) . (b) Leaf case s = 2 . Three numerical trajectories depictedin ( x ( t ) , y ( t ))-plane phase portrait. There are two dif-ferent invariant limit 4-tori for ( c , c ) = ( √ , √ ) . Figure 5: The controlled numerical phase portraits in ( x , y )- and ( x , y )-planes for the system(5.12), constants given by (5.14) in Example 5.4 and µ = 0 . T -torus on the leaf- M ( √ , √ )2 ,σ . Figure 5b illustrate two tori living on the leaf- M (0 . √ , . √ ,σ . Let a := 1 , a = 0 , a := − , a := 6 , a := − , and a = − . (5.14)Hence, b (0 , C ) = c − c c . By Theorem 5.1, T P ch := { ( µ , µ ) | µ = 0 } and for µ c c − c < µ , µ ), an invariant T -torus bifurcates from origin; see Figures 4a and 5a.For numerical bifurcation control of the system (5.12), we take the leaf corresponding with ( c , c ) =( √ , √ ) and µ = 0 . . Thus, the initial condition ( x , y , x , y , x , y ) = (0 . , , . , , ,
0) frominside the invariant torus and ( x , y , x , y , x , y ) = (0 . , , . , , ,
0) from outside the stable in-variant torus give rise to the numerical phase portraits in ( x , y )-plane and ( x , y )-plane depictedby Figures 5a, respectively.Now take c = c = √ . Then, b (0 , C ) = 0 and b (0 , C ) = c − c c = (cid:54) = 0. Hence, we havethe leaf case s = 2. Then, the amplitude equation of fourteenth-grade truncation of parametricleaf-normal form is (cid:80) i =1 √ i∂∂θ i + (cid:80) i =1 (cid:0) µ + µ + µ µ + (cid:0) µ − µ (cid:1) ρ + ρ (cid:1) c i ρ ∂c ∂ρ i . Then, the estimated transition varieties are given by T SupP = { ( µ , µ ) | µ = 0 , µ < } , T SubP = { ( µ , µ ) | µ = 0 , µ > } and T SD = { ( µ , µ ) | (cid:0) µ − µ (cid:1) − µ − µ − µ µ , µ < } . Note that T SD is not a transition set for the leaf-system associated with ( c , c ) = ( √ , √ ). Thetransition variety T P ch changes into two intrinsically different transition varieties T SupP and T SubP for the case ( c , c ) = ( √ , √ ). Figure 5b depicts the bifurcation of two invariant T -tori from originliving on the leaf M (0 . √ , . √ ,σ . There are three trajectories in Figure 5b when µ = 0 . , µ = − .
35: 1) the blue trajectory starts from the initial condition ( − . , . , − . , . , ,
0) out-side the external unstable torus. 2) the green trajectory is associated with the initial condi-tion ( − . , . , − . , . , ,
0) and converges to the stable internal invariant torus. Figure 5b . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity (a) Forward-time series for x ( t ) and y ( t ); also see Figure 6d. (b) Two trajectories for x ( t ) and y ( t ) . They converge to two invarianttori in backward and forward time. (c) Trajectories for x ( t ) and y ( t ) ininverse time converge to the internalinvariant torus. (d) Time series for x ( t ) and y ( t ).These along with Figure 6a convergeto the external T -torus. (e) Two trajectories for x ( t ) and y ( t ) whose α - and ω -limit sets arethe internal and external 4-tori. (f) Backward-time trajectories of x ( t ) and y ( t ). Figure 6: The controlled numerical trajectories depicting two invariant tori for system (5.12),( c , c , c ) = ( √ , √ , ∈ S ≥ , µ := − . , µ := 0 . , µ := 0 , and constants in example 5.5.depicts green trajectory in both forward and backward time. 3) the red trajectory starts at(0 . , , . , , ,
0) from inside the internal stable torus. In order to illustrate the invariant tori, thetrajectories associated with blue and red are plotted with inverse time (backward-time trajectory).
Example 5.5.
Let k = 2 and σ ∈ S where ( σ (1) , σ (2) , σ (3)) = (1 , , . As a result, M ,σ = { x ∈ R | ( x , y ) (cid:54) = 0 , ( x , y ) (cid:54) = 0 , ( x , y ) = 0 } and M ( c ,c )2 ,σ = { x ∈ M ,σ | c (cid:107) ( x , y ) (cid:107) = c (cid:107) ( x , y ) (cid:107)} where ( c , c ) is denoted on behalf of ( c , , c ) ∈ S . By transforming the M ( c ,c )2 ,σ -leaf vector fieldassociated with (5.12) into the Lie algebra J via the homeomorphism Ψ, we obtain (cid:80) i =1 , (cid:16) √ iw i ∂ i ∂w i − √ iz i ∂ i ∂z i (cid:17) + (cid:0) µ + (cid:0) α − i α (cid:1) rz + (cid:0) α + i α (cid:1) rw + (cid:0) α − α (cid:1) r z + (cid:0) α − α (cid:1) r w (cid:1) r∂∂r + (cid:16)(cid:0) α + α (cid:1) r + c c (cid:0) z + w (cid:1) (cid:0)(cid:0) α − α (cid:1) r z + (cid:0) α + α (cid:1) r + (cid:0) α − α (cid:1) r w (cid:1)(cid:17) r∂∂r . . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity a = 2 , a = 1 , a = − , a = 1 , a = a = 3 andset µ i = 0 for i (cid:54) = 0 , , . Then, the parametric leaf normal form up to degree seven is given by (cid:80) i =1 , √ i∂∂θ i + (cid:80) i =1 , (cid:16) µ + µ + µ ρ + (cid:0) µ +13 µ + c +3 c ) µ c + c − c )4 c (cid:1) ρ + c − c )4 c ρ (cid:17) c i ρ ∂c ∂ρ i . Since b (0 , C ) = 0, we choose C = ( c , , c ) ∈ S such that b (0 , C ) = c − c ) c (cid:54) = 0. Then, theleaf case s = 2 is satisfied. Hence, we take c := c := √ and have b (0 , C ) = − (cid:54) = 0 . The infinitelevel parametric leaf-normal form up to twelfth-grade is as follows: (cid:80) i =1 , √ i∂∂θ i + (cid:80) i =1 , (cid:16) µ + µ − µ µ + ( − µ + µ ) ρ − ρ (cid:17) c i ρ ∂c ∂ρ i . For numerical simulation in Figures 6, let µ := − . , µ := 0 . , and µ := 0 . Solutions start frominitial solutions ( − . , . , , , − . , . , ( − . , . , , , − . , . ,
0) and ( − . , , , , − . , M ( √ , √ , ,σ while trajectories in both backward and forward time are depicted in Figures 6d and 6e. These inforward time/backward time converge to the external/internal invariant torus. Figures 6c and 6edepict a solution converging to the internal 4-torus in backward time.Alternatively, we take c = 2 c = √ . Thereby, b (0 , C ) = 0 and b (0 , C ) = − . This leads tothe leaf case s = 3 . Next, the fourteenth-grade truncation of the infinite level parametric leaf-normalform is (cid:80) i =1 , √ i∂∂θ i + (cid:80) i =1 , (cid:0) µ + ( − µ + µ + µ ) ρ + (cid:0) µ − µ − µ (cid:1) ρ − r (cid:1) c i ρ ∂c ∂ρ i . We let µ = − . µ . The associated transition sets are depicted in Figures 4b and 4c for the cases s = 2 and s = 3. The estimated transition varieties corresponding with equations (5.1), (5.2) and(5.3) are given by T SN = { ( µ , µ ) | − ν + ν ) + 27( µ + ν + ν ν ) = 0 , µ > } T P sup = { ( µ , µ ) | µ = 0 , µ > − . } , T P sub = { ( µ , µ ) | µ = 0 , µ < − . } where ν := µ + µ and ν := µ + µ ; see Figure 4c. For a numerical bifurca-tion control simulation, we take µ = 5 e − , µ = − .
4. Figure 7 illustrates the existing threetori on the leaf M ( √ , , √ )2 ,σ . The forward-time trajectories of ρ ( t ) and ρ ( t ) with the initial val-ues ( − . , . , , , − . , .
4) in Figure 7a demonstrates a stable (external flow-invariant) T -torus.Forward and backward-time trajectories of ρ ( t ) and ρ ( t ) corresponding with the initial values( − . , . , , , − . , . ρ ( t ) and ρ ( t )with the initial values ( − . , . , , , − . , . − . , . , , , − . , . . . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity (a) Forward-time trajectories of ρ ( t ) and ρ ( t ) . (b) Forward and backward-time trajectories of ρ ( t )and ρ ( t ) . ; (c) Trajectories of ρ ( t ) and ρ ( t ). (d) Inverse-time trajectories of ρ ( t ) and ρ ( t ) . Figure 7: Three flow-invariant T -tori on the leaf- M ( √ , , √ )2 ,σ associated with example 5.5.The local bifurcation of Eulerian flows are not only associated with the corresponding bifur-cations of the reduced leaf-systems but also they are associated with the changes of the invariantleaf-manifolds. However, the analysis of the 2 n -dimensional system is not a straightforward corol-lary of those on individual leaves. Section 6 deals with the bifurcation analysis of 2 k -dimensionalcell-systems for k ≤ n. Bifurcation varieties for a 2 n -dimensional vector field are not necessarily the same as leaf-transitionsets. Leaf-transition sets provide a partition to the parameter space according to the topologicalqualitative changes in parametric leaf-vector fields. However, cell-bifurcation transition varietieshere refer to the partition of the parameter space according to the dynamics of the Eulerian systemon a closed cell M k,σ , that is, the closure of an open k -cell M k,σ for k ≤ n and σ ∈ S kn . Cell-bifurcations are involved with flow-invariant toral CW complexes. Hence, we first describe toral CWcomplexes and then, deal with their cell-bifurcations for two most generic truncated one-parametric2 k -cell normal form systems; also see [18]. . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity Notation 6.1. • Using the notation S kn in equation (2.2), we introduce S l,σn := (cid:8) γ ∈ S ln | { σ ( i ) | i > k } ⊆ { γ ( i ) | i > l } (cid:9) for σ ∈ S kn and l ≤ k. (6.1)Hence, S l,σn has (cid:0) kk − l (cid:1) -number of elements and S k,σn = { σ } for σ ∈ S kn . For instance, let n = 4 , k = 2 , σ (1) := 2 , σ (2) := 3 , σ (3) := 1 , and σ (4) := 4 . Then, σ ∈ S n and S ,σn = { σ , σ } , where σ (1) = 2 , σ (2) = 1 , σ ( j ) = j for j = 3 , , and σ (1) = 3 , σ (2) = 1 , σ (3) = 2 , and σ (4) = 4 . • Denote B k ⊂ R k for the k -open ball when k > , and B := { } . Notation B k is used for the k -closed ball in R k while T xn +1 stands for an x -dependent n + 1-dimensional Clifford torus.For γ ∈ S l,σn , denote B l,γ := { ( c i ) ni =1 ∈ R n | (cid:80) lj =1 c γ ( j ) = 1 , c γ ( j ) = 0 for j > l } , and S l − ,γ> by equation (2.3).Our main goal in the next lemma (and the illustrations in examples 6.3-6.4) is to providea regular CW complex decomposition for S k − ,σ> . This decomposition is the actual decompositionimposed by the closed cell-dynamics associated with Eulerian flows latter in this section.
Lemma 6.2.
The space S k − ,σ> is a regular CW complex.Proof. Recall that S k − ,σ> := { C = ( c , · · · , c n ) | (cid:80) ki =1 c σ ( i )2 = 1 , c σ ( i ) > i ≤ k and c σ ( i ) =0 for j > k } . Let ∂ i S l − ,γ> := (cid:116) ¯ γ ∈ S l − i,γn S l − i − , ¯ γ> (6.2)be a union of disjoint l − i − S l − ,γ> , where γ ∈ S l,σn ,l ≤ k, and i ≥ . Then, ∂ S l − ,γ> = S l − ,γ> , ∂ S l − ,γ> = ∂ S l − ,γ> (cid:116) ∂∂ S l − ,γ> , ∂∂ S l − ,γ> = ∂ S l − ,γ> (cid:116) ∂∂ S l − ,γ> ,∂∂ i S l − ,γ> = ∂ i +1 S l − ,γ> (cid:116) ∂∂ i +1 S l − ,γ> and ∂ S l − ,γ> = (cid:116) l − i =1 ∂ i S l − ,γ> , for l ≤ k and γ ∈ S l,σn . (6.3)Further, ∂ S l − ,σ> = (cid:116) l − i =1 (cid:116) ¯ γ ∈ S l − i,γn S l − i − , ¯ γ> and S l − ,γ> = (cid:116) li =1 (cid:116) ¯ γ ∈ S i,γn S i − , ¯ γ> . We claim that each S l − ,γ> represents an l − B l − , and this cell decomposition constitutes a CWcomplex structure for S l − ,σ> . For each l and γ, we need to introduce an attaching map˜Φ l,γ : B l − → S l − ,γ> = (cid:116) lj =1 (cid:116) ¯ γ ∈ S j,γn S j − , ¯ γ> , (6.4)that is a homeomorphism. Since S l − > and S l − ,γ> are homeomorphic, let q l = (cid:80) li =1 1 √ l e li ∈ S l − > ⊂ R l . For a construction of ˜Φ l,γ , consider the l -disc B √ l ( q l ) centered at q l and radius √ l , the intersection . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity B √ l ( q l ) with the l − i.e., (cid:0) q l + √ l B l (cid:1) ∩ S l − , and the family of all l − H passing through the origin and q l . The spaces ( q l + √ l B l ) ∩ S l − and S l − > are homeomorphic.A homeomorphism can be constructed via a uniform rescaling of the arcs obtained from the inter-section of H with S l − > and B √ l ( q l ) ∩ S l − , respectively. Since there is a homeomorphism between( q l + √ l B l ) ∩ S l − and B l − , the combination of these homeomorphisms constructs the expectedhomeomorphism ˜Φ l,γ . Hence, the space S k − ,σ> is a regular CW complex. Example 6.3.
Let k = n = 2 and B = [ − , . The space S ,I> = { C = ( c , c ) | c + c =1 , c i ≥ i = 1 , } is a regular CW complex, where S ,I> = { C = ( c , c ) | c + c = 1 , c i > i = 1 , } . We have S ,In = { γ , γ } for ( γ (1) , γ (2)) := (1 ,
2) and ( γ (1) , γ (2)) := (2 , . Hence, ∂ S ,I> = S ,γ > (cid:116) S ,γ > = { (1 , , (0 , } . A continuous attaching map associated with S ,I> follows ˜Φ ,I : B → S ,I> = S ,I> (cid:116) ( S ,γ > (cid:116) S ,γ > ) , by ˜Φ ,γ ( s ) = (cid:0) cos( π s + π ) , sin( π s + π ) (cid:1) . (6.5)Here, ˜Φ ,I is a homeomorphism. Example 6.4 (A CW-decomposition for S ,I> ) . Let k = n = 3 and consider S ,I> = { C ∈ S | c i > i = 1 , , } . Let S ,I = { γ , γ , γ } , where( γ (1) , γ (2) , γ (3)) := (2 , , , ( γ (1) , γ (2) , γ (3)) := (1 , , , and ( γ (1) , γ (2) , γ (3)) := (1 , , . Hence, ∂ S ,I> = ( S ,γ > (cid:116) S ,γ > ) (cid:116) S ,γ > , where S ,γ i > = { C ∈ S | c j > j (cid:54) = i, c i = 0 } . Further, S ,In = { ¯ γ , ¯ γ , ¯ γ } for (¯ γ (1) , ¯ γ (2) , ¯ γ (3)) := (1 , , , (¯ γ (1) , ¯ γ (2) , ¯ γ (3)) := (2 , , , and(¯ γ (1) , ¯ γ (2) , ¯ γ (3)) := (3 , , . Hence, ∂ S ,I> = { (1 , , , (0 , , , (0 , , } , where S , ¯ γ > = { (1 , , } , S , ¯ γ > = { (0 , , } , and S , ¯ γ > = { (0 , , } . The space S ,I> is homeomorphic with the sector in the ( x, y )-plane obtainedby the projection map (see Figure 8a)Π : D := { ( c , c ) | c + c ≤ , c , c ≥ } → S ,I> , Π ( c , c ) = (cid:18) c , c , (cid:113) − c − c (cid:19) . The projected (green) sector is circumscribed by a full circle as illustrated in Figure 8b. Then, auniform rescaling of segments of the circle’s radius inside the (green) sector makes a homeomorphismbetween the sector and B ; see Figure 8b. This is given by Π : √ B + ( , ) → D, Π ( s , s ) =( c , c ) , where( c , c ) := ( , ) − (cid:16) (cos θ +sin θ ) −√ sin 2 θ +3 √ (cid:17) ( s − , s − ) , θ ∈ [ − π , π ] , r √ {| s − | , | s − |} (2 s − , s −
1) + ( , ) , π ≤ θ ≤ π , . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity c c c Π S ,I> Π ( S ,I> ) ∂ S ,I> Π ( ∂ S ,I> ) ∂ S ,I> (a) S > and its Π -projection on ( c , c )-plane • •• ( s ,s ) Π θ • • Υ c c − ◦ Π ( ∂ S ,I> )Π − ◦ Π ( ∂ S ,I> )Υ ⊂ Π ( S ,I> )Π − (Υ) ⊂ B √ ( , )1 (b) Π ( S > ) circumscribed by a circle in ( c , c )-plane Figure 8: A CW complex decomposition for S > . θ := tan − s − s − . Further, ψ (0 ,
0) = (0 , , Π − ( c , c ) = ( , ) + (cid:18) − c + c + √ c + c )+2 c c − c + c )+2 √ (cid:19) (cos θ, sin θ ) θ ∈ [ − π , π ] , max {| c − | , | c − |}√ √ (2 c − +(2 c − (2 c − , c −
1) + ( , ) π ≤ θ ≤ π . We can transform B into √ B + ( , ) using a shift and a rescaling. Then, the combination of thiswith Π and Π provides an attaching homeomorphism between B and S ,I> . A toral CW complex X can be constructed by attaching toral cells associated with a celldecomposition of a regular CW complex ˜ X . A toral cell is a smooth toral bundle and it here refersto a space homeomorphic to the Tychonoff product of an open CW-cell in a CW decomposition witha Clifford hypertorus. Thereby, a toral cell is a torus bundle over an open CW-cell whose fiber is ahypertorus. In this paper we encounter toral CW complexes as flow-invariant manifolds bifurcatedfrom singular Eulerian cell-flows. Hence, we merely describe the toral CW complexes specifically towhat appears in those cases. A toral cell here is homeomorphic to a Tychonoff product of an openCW k -cell with either a k + 1-hypertorus or a k + 2-hypertorus. Therefore, an even dimensionaltoral cell is always homeomorphic to a Tychonoff product of T k +2 with a CW k -cell while odddimensional toral cells are homeomorphic to the product of T k +1 with an open CW k -cell. Hence,we denote a regular CW complex ˜ X as˜ X := (cid:116) m,i m ∈ I m ˜ C i m (cid:98) m − (cid:99) when ˜ C i m (cid:98) m − (cid:99) is an open (cid:98) m − (cid:99) − dimensional CW cell in ˜ X (6.6)and the corresponding toral cell associated with ˜ C i m (cid:98) m − (cid:99) is homeomorphic to the hypertorus bundle T m −(cid:98) m − (cid:99) × ˜ C i m (cid:98) m − (cid:99) . In other words, m stands for the dimension of the toral cell associated with theCW cell ˜ C i m (cid:98) m − (cid:99) . This representation for the CW cell decomposition splits CW (cid:98) m − (cid:99) -cells into twocategories based on their association with odd and even dimensional toral cells. . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity n,i n : B n → ˜ C i n (cid:98) n − (cid:99) ⊆ (cid:116) m ≤ n,i m ∈ I m ˜ C i m (cid:98) m − (cid:99) ⊆ ˜ X (6.7)be the attaching homeomorphism associated with ˜ C i n (cid:98) n − (cid:99) . We shall correspond ˜Φ n,i n to an attachingmap Φ i n n associated with C i n (cid:98) n − (cid:99) in the introduction of a toral CW complex X := (cid:116) i n ∈ I n ,n C i n n . Nowwe look for a regular CW decomposition on the closed disc B n corresponding with toral cells ofdifferent odd and even dimensions. Denote ∂ o,i n B n := B n and ∂ e,i n B n := ∅ while for 1 ≤ k < n,∂ o,i n n − k B n := ˜Φ − n,i n ( (cid:116) i k +2 ∈ I k +2 ˜ C i k +2 k ) ⊆ ∂ B n and ∂ e,i n n − k B n := ˜Φ − n,i n ( (cid:116) i k +1 ∈ I k +1 ˜ C i k +1 k ) ⊆ ∂ B n . (6.8)Here, super-indices o and e stand for their associations with odd and even dimensional toral cellcases. For instance, the space ∂ o,i n n − k B n will be associated with a 2 k + 1-dimensional toral cell in X. Further, B n = ∪ nk =0 ,δ ∈{ o,e } ∂ δ,i n k B n . We assume that the space (cid:116) nk =0 (cid:116) x ∈ ∂ o,ink B n (cid:0) T xn − k +1 × { x } (cid:1) (cid:116) (cid:116) nk =1 (cid:116) x ∈ ∂ e,ink B n (cid:0) T xn − k +2 × { x } (cid:1) carries a metrizable topology so that • (cid:116) x ∈ B n T xn +1 × { x } is homeomorphic to T n +1 × B n . • (cid:116) x ∈ B n T xn +1 × { x } is dense and a relatively compact open subset, i.e., (cid:116) x ∈ B n T xn +1 × { x } := (cid:116) nk =0 ,x ∈ ∂ o,ink B n ( T xn − k +1 × { x } ) (cid:116) (cid:116) nk =1 ,x ∈ ∂ e,ink B n ( T xn − k +2 × { x } ) . (6.9)For brevity of notations, we shall denote the space (cid:116) x ∈ B n T xn +1 × { x } by its homeomorphic space T n +1 × B n . We now describe the assumed metrizable topology in more details as follows. Euclidiantopology demonstrates the convergent sequences within individual open cells. More precisely, open2 k + 2-toral cell (cid:116) x ∈ ˜Φ − n,in (˜ C i k +2 k ) ( T xk +2 × { x } ) ⊂ T n +1 × B n is homeomorphic to T k +2 × ˜Φ − n,i n (˜ C i k +2 k ) (6.10)and open 2 k + 1-toral cell (cid:116) x ∈ ˜Φ − n,in (˜ C i k +1 k ) ( T xk +1 × { x } ) is homeomorphic to T k +1 × ˜Φ − n,i n (˜ C i k +1 k ) for k ≤ n . No sequence from an open cell in T n +1 × B n converges to a point on another open cell withan equal or higher dimension, e.g., ∂ (cid:0) (cid:116) x ∈ ∂ e,inl B n T xn − l +2 × { x } (cid:1) ⊆ (cid:116) nk = l +1 ,x ∈ ∂ o,ink B n T xn − k +1 × { x } (cid:116) (cid:116) nk = l +1 ,x ∈ ∂ e,ink B n T xn − k +2 × { x } . Convergent sequences from a higher dimensional cell to a lower dimensional cell are as follows. For x j ∈ ∂ δ ,i n k B n and 1 ≤ j ≤ ∞ , a sequence( s x j , . . . , s x j n − k +1 , x j ) ∈ T x j n − k +1 × { x j } = ( (cid:81) n − k +1 j =1 S ,x j ) × { x j } , . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity S ,x j is an x j -dependent circle, approaches( s y , . . . , s yn − l +1 , y ) ∈ T yn − l +1 × { y } , for k < l ≤ n, δ , δ ∈ { o, e } , when x j ∈ ∂ δ ,i n k B n ⊂ B n converges to y ∈ ∂ δ ,i n l B n ⊂ B n and n − l +1-number of s x j i -components from( s x j , . . . , s x j n − k +1 ) correspond with and converge to ( s y , . . . , s yn − l +1 ) as j approaches infinity. However,the sequence of s x j i -components from ( s x j , . . . , s x j n − k +1 ) corresponding with the l − k -remaining indicescollapse to a point as j converges to infinity. Roughly speaking, some S -components of the toralfiber collapse to a point as points from a toral cell in T n +1 × B n approaches a point on a neighboringlower dimensional toral cell. This naturally reduces the dimension of the corresponding torus. Definition 6.5 (Toral CW complexes) . We refer to a Hausdorff space X with a finite partition { C i m m } , i.e., X = (cid:116) i m ∈ I m ,m C i m m for finite number of finite index sets I m , as a toral CW complex andcall each C i k +1 k +1 and C i k k by an open 2 k + 1-toral cell and an open 2 k -toral cell in X when thefollowing conditions hold. • There exist a regular CW complex ˜ X given by (6.6) and homeomorphisms g i m m so that C i k +2 k +2 in X is g i k +2 k +2 -homeomorphic to T k +2 × ˜ C i k +2 k and each open 2 k + 1-toral cell C i k +1 k +1 is g i k +1 k +1 -homeomorphic to T k +1 × ˜ C i k +1 k . • Consider the attaching map ˜Φ n,i n in (6.7) and the space T n +1 × B n described by (6.8)-(6.9).Then, there exists an attaching homeomorphismΦ i n n : T n +1 × B n (cid:16) C i n n ⊆ X := (cid:116) m,i m ∈ I m C i m m for each toral cell C i n n . Furthermore,Φ i n n (cid:0) ∂ ( T n +1 × B n ) (cid:1) ⊆ (cid:116) k There exists a toral CW complex associated with the space S k − ,σ> and its CW-celldecomposition described by equations (6.2) and (6.3) for l = k and σ = γ . Elements of the partitionassociated with this toral CW complex are homomorphic to those in (cid:110) T l +1 × S l,γ> | γ ∈ S l +1 ,σn , ≤ l ≤ k − (cid:111) . (6.12) Here, there is no even dimensional toral cell. . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity Proof. The main idea of the proof is to construct a toral CW complex via a quotient space Y / ∼ of the space Y = T k × S k − ,σ> over an equivalence relation ∼ , where ∼ is generated by identifyingthe extra dimensions of the hypertori corresponding with the lower dimensional open cells in theboundary set ∂ S k − ,σ> . By Lemma 6.2, for l ≤ k and γ ∈ S l,σn , we have S k − ,σ> = S k − ,σ> (cid:116) (cid:116) k − i =1 ∂ i S k − ,σ> , where ∂ i S k − ,σ> = (cid:116) γ ∈ S k − i,σn S k − i − ,γ> and ∂ S k − ,γ> = (cid:116) k − i =1 ∂ i S k − ,γ> . We consider the Tychonoff product space Y := (cid:16)(cid:81) kj =1 S σ ( j ) (cid:17) × S k − ,σ> for S σ ( j ) = S . Given the S -components collapsing criteria for the converging sequences to lower dimensional cells,the equivalence relation ∼ is generated by identifying elements of (cid:98) Y ( s γ ( j ) ) k − ij =1 x,i,γ := (cid:8)(cid:0) ( s σ ( j ) ) kj =1 , x (cid:1) ∈ Y | s γ ( l ) ∈ S γ ( l ) for l > k − i (cid:9) as an equivalent class for x ∈ S k − i − ,γ> , ≤ i < k, and γ ∈ S k − i,σn . This gives rise to the quotientspace Y / ∼ . Thus, the open cell S k − i − ,γ> in ∂ S k − ,γ> corresponds with a space in the quotientspace Y / ∼ that is homeomorphic to T k − i × S k − i − ,γ> . This introduces the homeomorphism g i k − i − k − i − for i k − i − := γ ∈ S k − i,σn . The quotient space Y / ∼ is the desired toral CW complex associatedwith CW-cell decomposition described by equations (6.2) and (6.3). The homeomorphism ˜Φ k,γ given in equation (6.4) induces the corresponding topology from Y / ∼ onto T k × B k − and thehomeomorphism Φ γ k − i − . The equivalence relation ∼ is designed such that the homeomorphisms g γ k − i − and Φ γ k − i − follow equations (6.11). Theorem 6.8 (A toral CW complex bifurcation associated with S k − ,σ> ) . Consider k > , σ ∈ S kn ,s = 1 , sign( a e σ ( j ) a e σ ( i ) ) = 1 for all ≤ i < j ≤ k, and the closure of an open k -cell M k,σ . Then,there is a cell-bifurcation variety at T P ch := { ν | ν = 0 } (6.13) for the one-parametric Eulerian flow associated with (See the normal form in [18, Theorem 4.7]) Θ + v := Θ + (cid:80) ni =1 (cid:16) ν + (cid:80) nj =1 a e j ( x j + y j ) (cid:17) (cid:0) x i ∂∂x i + y i ∂∂y i (cid:1) . (6.14) Here, a flow-invariant toral CW complex associated with the CW complex S k − ,σ> = (cid:116) k − l =0 (cid:116) γ ∈ S l +1 ,σn S l,γ> bifurcates from the origin corresponding with the dynamics on M k,σ . This toral CW com-plex and its partition are homeomorphic to the one given in Lemma 6.7 and the partition (6.12) .This flow-invariant toral manifold exists when a e σ ( k ) ν < . This is asymptotically stable when . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity a e σ ( k ) < and otherwise, it is unstable. A trajectory associated with ρ σ ( i ) , for ≤ i ≤ k, con-verges to/diverges from the stable/unstable toral CW complex at frequency ω σ ( i ) π [hz] and radial ve-locity c i c k ρ σ ( k ) (cid:16) ν + ρ σ ( k ) (cid:80) nj =1 a e j c j c k (cid:17) [m/s], where C is determined by the initial conditions and ρ σ ( k )2 = x j + y j .Proof. The dynamics of (6.14) on M k,σ follows the governing dynamics on open 2 l -cells M l,γ for γ ∈ S l,σn and l ≤ k. The hypertorus exists when a σ ( l )1 ( C ) ν < a σ ( l )1 ( C ) < X, inside the invariant space M k,σ (the closure ofa 2 k -cell) bifurcates from the origin when the parameter ν crosses the variety T P ch given by (6.13).Now we introduce a toral CW decomposition associated with S k − ,σ> for the flow-invariantmanifold X. For γ ∈ S l,σn and by equation (6.4), the attaching map associated with a CW complex S l − ,γ> in S k − ,σ> is the homeomorphism˜Φ l,γ : B l − (cid:16) S l − ,γ> = (cid:116) l − j =0 (cid:116) ¯ γ ∈ S j +1 ,γn S j, ¯ γ> ⊆ (cid:116) k − j =0 (cid:116) γ ∈ S j +1 ,σn S j,γ> for 1 ≤ l ≤ k − . Since the index set I l − j +1 := S l − j +1 ,γn , we may replace the index i l − j +1 ∈ I l − j +1 with ¯ γ ∈ S l − j +1 ,γn . The flow-invariant toral manifold X is determined by its sectional hypertorus in each M Ck,σ -leaf. For each open toral cell C γ l − := (cid:116) C ∈ S l − ,γ> T C,γl × { C } for γ ∈ I l − = S l,σn , the hypertorus T C,γl is determined by the radius vector ρ C = ( ρ , . . . ρ n ) , where ρ γ ( q ) = (cid:115) − ν a σ ( q )1 ( C ) for q = 1 , . . . l, and ρ γ ( q ) = 0 for q > l. Hence, for any γ ∈ S l,σn we have lim c γ ( q ) → ρ γ ( q ) = 0 andlim c γ ( q ) → ,q = l +1 ,...,k ρ σ ( j )2 = − ν c σ ( j )2 (cid:80) ki =1 ,i/ ∈{ γ ( q ) } kq = l +1 a e σ ( i ) c σ ( i )2 (cid:54) = 0 for σ ( j ) ∈ { γ ( q ) } lq =1 . (6.15)Thus, the squared radiuses corresponding with bifurcated k -hypertori within M Ck,σ converge to − ν c σ ( j )2 (cid:80) ki =1 ,i (cid:54) = { γ ( q ) } kq = l +1 a e σ ( i ) c σ ( i )2 when C approaches S l − ,γ> ⊂ ∂ k − l S k − ,σ> . On the other hand, ∂ o,γj − B l − := (cid:116) ¯ γ ∈ S l − j +1 ,γn ˜Φ − l,γ (cid:0) S l − j, ¯ γ> (cid:1) and ∂ e,i l j B l = ∅ for 1 ≤ j ≤ l and 1 ≤ l ≤ k. The j -th nonzero squared radius of the bifurcated hypertorus for the M Ck,σ -leaf normal form is ρ σ ( j )2 := − ν a σ ( j )1 ( C ) . Hence, we introduce (cid:116) x ∈ ˜Φ − l,γ (cid:0) S l − j, ¯ γ> (cid:1) T x,γ, ¯ γl − j +1 × { x } and for notation simplicity denote . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity T γ, ¯ γl − j +1 × ˜Φ − l,γ (cid:0) S l − j, ¯ γ> (cid:1) . Similarly, T γl − j +1 × ∂ o,γj − B l − stands for (cid:116) x ∈ ∂ o,γj − B l − T x,γl − j +1 × { x } and T x,γl − j +1 is a l − j + 1-dimensional torus. Here, T x,γ, ¯ γl − j +1 × { x } := (cid:110) ( ρ x , θ, x ) (cid:12)(cid:12) < θ γ ( i ) < π, ρ ¯ γ ( i )2 := − ν a ¯ γ ( i )1 ( ˜Φ l, ¯ γ ( x )) for 0 ≤ i ≤ l − j + 1 (cid:111) , (6.16)where x ∈ ∂ o,γj − B l − , ( ρ x , θ ) := ( ρ , ρ , . . . , ρ n , θ , θ , . . . , θ n ) represents an action-angle coordinatesystem for the l − j + 1-hypertorus T x,γ, ¯ γl − j +1 , and ρ ¯ γ ( i ) = 0 for i > l − j + 1 . Thus, T γ, ¯ γl − j +1 × ˜Φ − l,γ (cid:0) S l − j, ¯ γ> (cid:1) is homeomorphic to T l − j +1 × B l − j for any γ ∈ S l,σn , ¯ γ ∈ S l − j +1 ,γn and l ≤ k. Now we claim that T γl − j +1 × ∂ o,γj − B l − := (cid:116) ¯ γ ∈ S l − j +1 ,γn T γ, ¯ γl − j +1 × ˜Φ − l,γ (cid:0) S l − j, ¯ γ> (cid:1) . (6.17)Therefore, T γl × B l − ,γ = (cid:116) lj =1 (cid:0) T γl − j +1 × ∂ o,γj − B l − (cid:1) = (cid:116) lj =1 (cid:116) ¯ γ ∈ S l − j +1 ,γn T γ, ¯ γl − j +1 × ˜Φ − l,γ (cid:0) S l − j, ¯ γ> (cid:1) . (6.18)Furthermore, the space T γl +1 × B l,γ is homeomorphic to the toral CW complex constructed in Lemma6.7. Following Definition 6.5, we defineΦ γl +1 : T γl × B l − ,γ (cid:16) C γ l − = (cid:116) ¯ γ ∈ S m,γn ,m ≤ l C γ m +1 by Φ γl +1 ( r, θ, x ) := ( ρ ˜Φ l +1 ,γ ( x ) , θ, ˜Φ l +1 ,γ ( x )) . Here, Φ γl +1 is a homeomorphism. This completes the proof. Theorem 6.9. Let the hypotheses of Theorem 6.8 hold. Then, the parametric k -cell vector fields v k ( ν ) and v k ( ν ) for either ν , ν ∈ { ν | ν > } or ν , ν ∈ { ν | ν < } are orbitally equivalent.Hence, the variety given by (6.13) is the only k -cell bifurcation variety for the k -cell truncatednormal form system.Proof. Consider the following two differential equations ddt r = r (cid:16) ν + (cid:80) ki =1 a e σ ( i ) r σ ( i )2 (cid:17) and ddt R = R (cid:16) ν + (cid:80) ki =1 a e σ ( i ) R σ ( i )2 (cid:17) , (6.19)where ν ν > , r = ( r σ (1) , · · · , r σ ( k ) ) , and R = ( R σ (1) , · · · , R σ ( k ) ) . We show that these equationsare orbitally equivalent. Consider the homeomorphism h : R k → R k and the map τ : R k × R → R defined by h ( r ) = ( h ( r ) , · · · , h k ( r )) := (cid:113) ν ν r and τ ( r , t ) = ν ν t. The flow associated with the first equation in (6.19) follows r ( t, r ) = (cid:18) − ν r σ ( k )2 exp(2 ν ( t − t )) (cid:80) ki =1 a e σ ( i ) r σ ( i )2 ( exp(2 ν ( t − t )) − ) − ν (cid:19) r r σ ( k ) for r ( t , r ) = r . . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity R ( t , h ( r )) = h ( r ) and h ( r ( τ ( r , t ) , r )) = (cid:16) ν ν (cid:17) r ( τ ( r , t ) , r ) = (cid:16) − ν r σ ( k )2 exp(2 ν ( t − t ) (cid:80) ki =1 a e σ ( i ) r σ ( i )2 (cid:0) exp(2 ν ( t − t )) − (cid:1) − ν (cid:17) r r σ ( k ) = (cid:16) − ν h k ( r ) exp(2 ν ( t − t )) (cid:80) ki =1 a e σ ( i ) h i ( r ) (exp(2 ν ( t − t )) − − ν (cid:17) h ( r ) h k ( r ) = R ( t, h ( r )) . This completes the proof.Now we consider a one-parameter 5-degree truncated 2 k -cell normal form (see [18]) given byΘ k + v σk for v σk := (cid:80) ki =1 (cid:16) ν (cid:0) x σ ( i ) ∂∂x σ ( i ) + y σ ( i ) ∂∂y σ ( i ) (cid:1) + (cid:80) | m | =1 (cid:81) kj =1 ( x σ ( j )2 + y σ ( j )2 ) m σ ( j ) (cid:0) a m x σ ( i ) ∂∂x σ ( i ) + a m y σ ( i ) ∂∂y σ ( i ) (cid:1)(cid:17) , (6.20)where a = ν , m = ( m i ) ni =1 ∈ R n , m σ ( i ) = 0 for i > k. Using a leaf-invariant M Ck,σ , we haveΘ k + v σk,C := Θ k + (cid:80) j =0 (cid:80) kl =1 (cid:0) a σ ( l ) j ( C ) ( x σ ( l )2 + y σ ( l )2 ) j x σ ( l ) ∂∂x σ ( l ) + a σ ( l ) j ( C ) ( x σ ( l )2 + y σ ( l )2 ) j y σ ( l ) ∂∂y σ ( l ) (cid:1) . (6.21)Here, c σ ( j ) (cid:54) = 0 for j ≤ k and c σ ( j ) = 0 for j > k. Then, for any l ≤ k we have a σ ( l )0 ( C ) = ν ,a σ ( l )1 ( C ) = c σ ( l )2 (cid:80) ki =1 c σ ( i )2 a e σ ( i ) , a σ ( l )2 ( C ) = c σ ( l )4 (cid:80) ≤ i ≤ j ≤ k c σ ( i )2 c σ ( j )2 a e σ ( i ) + e σ ( j ) . (6.22)Denote diag( a e σ ( i ) ) for a n × n diagonal matrix where a e σ ( i ) (1 ≤ i ≤ k ) is the σ ( i )-th diagonalentry and the rest of entries are zero. Further, denote L ( a e σ ( i ) ) ki =1 := (cid:8) C ∈ R n | (cid:10) diag( a e σ ( i ) ) C, C (cid:11) = 0 (cid:9) for a quadric k -hypersurface passing through the origin. For a γ ∈ S l,σn , letΓ l,γa := L ( a e σ ( i ) ) ki =1 ∩ S l − ,γ> and Γ l,γ, ± a := (cid:110) C ∈ S l − ,γ> | sign (cid:16) a e σ (1) (cid:10) diag( a e σ ( i ) ) C, C (cid:11)(cid:17) = ± (cid:111) . (6.23)Hence, Γ l,γ, − a := S l − ,γ> \ (Γ l,γa (cid:116) Γ l,γ, + a ) . Lemma 6.10 (CW complex structures for Γ k,σ, ± a ) . Assume that k ≥ , σ ∈ S kn , and for at least apair of indices ( i, j ) , ≤ i < j ≤ k, sign( a e σ ( j ) a e σ ( i ) ) = − , while sign( a e σ ( i + e σ ( i )sign( a e σ ( j + e σ ( j ) = 1 (6.24) for all ≤ i ≤ i ≤ k and ≤ j ≤ j ≤ k. Then, a σ ( i )1 ( C ∗ ) = 0 for any C ∗ ∈ Γ k,σa = L ( a e σ ( i ) ) ki =1 ∩ S k − > and i ≤ k. Besides, the C -parameter space S k − > is partitioned into a union ofdisjoint three topological subspaces Γ k,σ, − a , Γ k,σ, + a , and Γ k,σa given by (6.23) . Each of the topologicalclosures of Γ k,σ, + a and Γ k,σ, − a constitutes a CW complex. . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity Proof. There exists a unique natural number l < k and a γ + ∈ S l,σn so that a e σ ( γ +( i )) a e σ ( i > ≤ i ≤ l and a e σ ( j ) a e σ ( i < j / ∈ { γ + ( i ) | ≤ i ≤ l } . Similarly, there is a unique γ − ∈ S k − l,σn so that { , , . . . , k } \ { γ + ( i ) | ≤ i ≤ l } = { γ − ( i ) | ≤ i ≤ k − l } . Hence, the CW decomposition ofΓ k,σ, + a is given by the disjoint sets appearing in the equationΓ k,σ, + a = ( (cid:116) lj =1 , ¯ γ ∈ S j,γ + n S j − , ¯ γ> ) (cid:116) ( (cid:116) kj =1 , ¯ γ ∈ S j,σn \ ( S j,γ + n ∪ S j,γ − n ) (Γ j, ¯ γ, + a (cid:116) Γ j, ¯ γa )) . (6.25)Note that S j,γ + n = ∅ for j > l while S j,γ − n = ∅ for j > k − l. Similarly, the CW-decompositions forΓ k,σ, − a and Γ k,σa are derived by the disjoint subsets appearing inΓ k,σ, − a = (cid:116) lj =1 , ¯ γ ∈ S j,γ − n S j − , ¯ γ> (cid:116) (cid:116) kj =1 , ¯ γ ∈ S j,σn \ ( S j,γ + n ∪ S j,γ − n ) (Γ j, ¯ γ, − a (cid:116) Γ j, ¯ γa )and Γ k,σa = (cid:116) kj =1 ,γ ∈ S j,σn Γ j,γa . (6.26)Remark that the spaces Γ k,σ, + a and Γ k,σ, − a are relatively compact connected open subsets of S k − > . Further, the spaces Γ k,σ, + a and Γ k,σ, − a are homeomorphic to B k − while Γ k,σa is homeomorphic to B k − . We need to introduce the attaching maps to complete the proof. The attaching map ˜Φ j, ¯ γ given by (6.4) works fine in the cases of j − S j − , ¯ γ> for ¯ γ ∈ S j,γ ± n and 1 ≤ j ≤ l. Thus,we only refine the attaching maps to work for Γ j, ¯ γ, − a -cells. The other cases are similar. We firstintroduce a homeomorphism h j,γ, − a : S j − ,γ> → Γ j,γ, − a . The idea is to choose a point, say P , fromthe interior of Γ j,γ, − a ⊂ S j − ,γ> . Consider all two dimensional planes passing through the origin and P. The intersections of each of these planes with S j − ,γ> and Γ j,γ, − a give rise to two open arcs (anarc here refers to a one-manifold). The point P divides each of these two arcs into two connectedarc-pieces and the homeomorphism h j,γ, − a is defined as identity on one piece while it compresses theother piece in S j − ,γ> to homeomorphically match it with the corresponding arc-piece in Γ j,γ, − a . Thehomeomorphism h j,γ, − a is readily defined as a uniformly continuous map on S j − ,γ> . Thus, it can alsobe uniquely extended to h j,γ, − a : S j − ,γ> → Γ j,γ, − a . Using this map and the attaching map ˜Φ j, ¯ γ from(6.4), we introduce an attaching homeomorphism for the space decomposition (6.26) by (cid:103) Φ j,γ : B j − → Γ j,γ, − a where (cid:103) Φ j,γ := h j,γ, − a ◦ ˜Φ j, ¯ γ . (6.27)The CW-decomposition (6.26) and the attaching map (6.27) provide a CW complex structure forΓ j,γ, − a . Hence, the proof is complete. Theorem 6.11 (Cell-bifurcation of a toral CW complex associated with the CW complex Γ k,σ, + a ) . Assume that the hypotheses described by (6.24) hold and σ ∈ S kn . Let M k,σ be an open cell and M k,σ as its closure. Consider a one-parametric (normal form) vector field Θ + v ( r , θ, ν ) given by Θ+ (cid:80) ni =1 (cid:16) ν + (cid:80) nj =1 a e j ( x j + y j ) + (cid:80) ≤ j ≤ l ≤ n a e j + e l ( x j + y j )( x l + y l ) (cid:17) ( x i ∂∂x i + y i ∂∂y i ) . (6.28) Then, there is a primary cell-bifurcation variety given by T P ch := { ν | ν = 0 } . (6.29) . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity 1. When ν a e σ ( k ) < . A secondary flow-invariant toral CW complex associated with the CWcomplex Γ k,σ, + a bifurcates from the origin exactly when ν a e σ ( k ) < . There exist a naturalnumber l < k, γ + ∈ S l,σn and a γ − ∈ S k − l,σn so that its toral CW decomposition is homeomorphicto { T j × S j − , ¯ γ> | ¯ γ ∈ S j,γ + n } lj =1 (cid:116) { ( T j × Γ j, ¯ γ, + a ) (cid:116) ( T j × Γ j, ¯ γa ) | ¯ γ ∈ S j,σn \ ( S j,γ + n ∪ S j,γ − n ) } kj =1 . (6.30) This manifold is unbounded when | ν | approaches to infinity and ν a e σ ( k ) < . This toralmanifold, when it exists, is asymptotically stable for a e σ ( k ) < and is unstable otherwise.2. For ν a e σ ( k ) ≥ , there is no flow-invariant hypertorus for the system ( ?? ) when C ∈ Γ k,σ, + a . Proof. The possible radiuses of flow-invariant tori are given by r σ ( i )2 = − a σ ( i )1 ( C ) ± (cid:113) a σ ( i )1 ( C ) − ν a σ ( i )2 ( C )2 a σ ( i )2 ( C ) , and a σ ( i )1 ( C ) − ν a σ ( i )2 ( C ) > ν a σ ( k )2 ( C ) < . (6.31) The assumption (6.24) implies that either a σ ( i )2 ( C ) > C and i ≤ k, or a σ ( i )2 ( C ) < C and i ≤ k. Further, a σ ( i )1 ( C ) a σ ( i )2 ( C ) = a σ ( j )1 ( C ) a σ ( j )2 ( C ) for all i ≤ j ≤ k. Thereby, there is always precisely onehypertorus on each leaf M Ck,σ for C ∈ Γ k + a ∪ Γ ka as long as ν a σ ( k )2 ( C ) < . A secondary toral CWcomplex parameterized by C ∈ Γ k,σ + a (cid:116) Γ k,σa bifurcates from the origin at T P ch . This secondarymanifold exists when ν a σ ( k )2 ( C ) < ν a σ ( k )2 ( C ) > . Hence, the flow-invarianttoral manifold bifurcates from the origin via a simultaneous M Ck,σ -leaf bifurcation of hypertori atthe variety T P ch for all C ∈ Γ k,σ, + a . The attaching maps for toral cells indexed with ¯ γ ∈ S j,γ + n issimilar to the cases in the proof of Theorem 6.8 and we skip them here. We instead assume that ν a σ ( k )2 ( C ) < γ ∈ S j,σn \ ( S j,γ + n ∪ S j,γ − n ) . Then, we introduce T σ, ¯ γj × (cid:101) Φ − k,σ (Γ j, ¯ γ, + a (cid:116) Γ j, ¯ γa ) := (cid:8) ( r x , θ, x ) | < θ γ ( i ) < π, ¯ r γ ( i ) := 0 for i > j (cid:9) , where x ∈ ∂ o,γj B l , r x := (¯ r , ¯ r , . . . , ¯ r n ) and ¯ r γ ( i ) follows (6.31) for i ≤ j and C = (cid:101) Φ k,σ ( x ).A sequence (( r x p , θ p ) , . . . , ( r x p l , θ pl ) , x p ) ∞ p =1 ⊂ T σ,γl × (cid:101) Φ − k,σ (Γ l,γ, + a (cid:116) Γ l,γa ) , approaches (( R y , ϑ ) , . . . , ( R yj , ϑ j ) , y ) ∈ T σ, ¯ γj × (cid:101) Φ − k,σ (Γ j, ¯ γ, + a (cid:116) Γ j, ¯ γa ) , for j < l, when x p converges to y and j -number of angles from the sequence ( θ p , . . . , θ pl ) correspond with andconverge to ( ϑ , . . . , ϑ j ) . Further, the radiuses corresponding with the same j -number of indicesfrom the sequence of ( r x p , . . . , r x p l ) converges to those in ( R y , . . . , R yj ) . However, the sequence ofradiuses corresponding with the l − j -remaining indices either converges to zero. Hence, we have T σk × Γ k,σ, + a := (cid:116) kj =1 , ¯ γ ∈ S j,σn \ ( S j,γ + n ∪ S j,γ − n ) T σ, ¯ γj × (cid:101) Φ − k,σ (Γ j, ¯ γ, + a (cid:116) Γ j, ¯ γa ) (cid:116)(cid:116) lj =1 , ¯ γ ∈ S j,γ + n T σ, ¯ γj × (cid:101) Φ − k,σ ( S j − , ¯ γ> ) . (6.32) . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity C ∈ Γ k + a ∪ Γ ka , the vector radius of the hypertorus approaches the origin and then, the invarianthypertorus vanishes as ν converges to and crosses the transit variety T P ch . In other words, thereis no invariant hypertorus on M Ck,σ -leaves when ν a σ ( k )2 ( C ) > a σ ( k )1 ( C ) a σ ( k )2 ( C ) ≥ 0. Theradiuses of the hypertorus diverges to infinity, as ν a σ ( k )2 ( C ) diverges to the negative infinity.For any γ ∈ S l,σn define D γν := (cid:26) C ∈ Γ l,γ, − a | < ν < a γ ( l )1 ( C ) a γ ( l )2 ( C ) when a e γ ( l ) > , while a γ ( l )1 ( C ) a γ ( l )2 ( C ) < ν < a e γ ( l ) < (cid:27) ,D γ,∂ν := (cid:26) C ∈ Γ l,γ, − a | ν = a γ ( l )1 ( C ) a γ ( l )2 ( C ) (cid:27) , and N γν := Γ l,γ, − a \ ( D γν (cid:116) D γ,∂ν ) . (6.33)For an instance, we illustrate D σν by assuming that ν a σ ( k )2 ( e σ (1) ) > a e σ ( k ) > . (Thecase for the conditions ν a σ ( k )2 ( e σ (1) ) > a e σ ( k ) < ≤ l ≤ k and a (cid:101) γ ∈ S l,σn such that4 ν a (cid:101) γ ( l )2 ( e (cid:101) γ ( i ) ) < a (cid:101) γ ( l )1 ( e (cid:101) γ ( i ) ) for all i ≤ l, and 4 ν a (cid:101) γ ( l )2 ( e (cid:101) γ ( i ) ) ≥ a (cid:101) γ ( l )1 ( e (cid:101) γ ( i ) ) for i > l. (6.34)In this case, e (cid:101) σ ( i ) ∈ D σν for all i ≤ l, and e (cid:101) γ ( i ) / ∈ D σν for all i > l. When l = 0 , D σν = ∅ . For l = k,D σν = Γ k,σ, − a . Let 0 < l < k. Since Γ k,σ, − a , D σν and N σν are three connected k − D σ,∂ν is a connected k − l,σ, − a , D σν and N σν are all homeomorphic to B k − , while D σ,∂ν is homeomorphic to B k − . The closure of D σν is a regularCW complex whose CW decomposition is given by the disjoint sets D σν := ( (cid:116) lj =1 ,γ ∈ S j, (cid:101) γn S j − ,γ> ) (cid:116) ( (cid:116) kj =1 ,γ ∈ S j,σn \ S j, (cid:101) γn ( D γν (cid:116) D γ,∂ν )) . (6.35)Here, ˜ γ follows the conditions (6.34). The associated attaching maps is defined similar to what isgiven in the proof of Lemma 6.10.Consider the symmetric matrix M γ := [ M γ , . . . , M lγ ] , M jγ := a e γ ( j ) e j + (cid:80) li =1 ,i (cid:54) = j a e γ ( i ) + e γ ( l ) e i , and a γ := (cid:0) a γ (1) , . . . , a γ ( l ) (cid:1) t (6.36) for γ ∈ S l,σn . Then by equation (6.22), a γ ( l )2 ( C ) = c γ ( l )4 (cid:104)C γ , M γ C γ (cid:105) for C γ := (cid:0) c γ (1)2 , . . . , c γ ( l )2 (cid:1) t . Theorem 6.12 (Toral CW complex bifurcations associated with CW complex subspaces in Γ k,σ, − a ) . Consider the closed cell M k,σ and the vector field (6.28) along with the assumptions in Lemma 6.11.Further, assume that M σ given by (6.36) is either a positive definite or a negative definite matrixand ν min := min (cid:110) , (cid:104) a γ ,M γ − a γ (cid:105) | γ ∈ S l,σn , l ≤ k (cid:111) , ν max := max (cid:110) , (cid:104) a γ ,M γ − a γ (cid:105) | γ ∈ S l,σn , l ≤ k (cid:111) . (6.37) 1. When ν a e σ ( k ) > , there is a bifurcation variety given by T SN := { ν | ν = ν min0 when a e k < , and ν = ν max0 for a e k > } . (6.38) . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity Two flow-invariant toral CW complex manifolds ( T intν and T extν ) associated with the topolog-ical closure of D σν ⊆ Γ k,σ, − a simultaneously exists when ν min0 < ν < ν max0 and ν a e σ ( k ) > . The hypertoral manifold T intν lives inside T extν . There is no flow-invariant hypertori corre-sponding with C ∈ N σν for positive values of ν a e σ ( k ) . The external toral CW complex T intν isasymptotically stable when a e k < while it is unstable for a e k > . The internal toral CWcomplex T intν is asymptotically unstable/stable when T extν is asymptotically stable/unstable. As ν approaches T SN when ν a e σ ( k ) > , the space N σν enlarges and converges to Γ k,σ, − a . 2. The toral manifolds T intν and T extν collide (intersect) on D σ,∂ν and construct a flow-invariantbi-stable toral CW complex associated with the CW complex D σ,∂ν .3. When ν a e σ ( k ) > and ν is outside of the interval [ ν min0 , ν max0 ] , the vector field ( ?? ) does notadmit any flow-invariant hypertorus.4. For ν a e σ ( k ) < , there is precisely one flow-invariant toral CW complex associated with Γ k,σ, − a . When sign of ν changes, i.e., ν a e σ ( k ) > , this toral CW complex turns to be T extν . In otherwords, the CW complex associated with this toral CW complex shrinks to the CW complex D σν (cid:40) Γ k,σ, − a . This toral CW complex coalesces with the secondary toral CW complex T intν on D σν and disappear when the parameter crosses the transition variety T SN defined by (6.38) .Proof. Let ν a e σ ( k ) > 0. The squared radiuses r ± σ ( i )2 are positive when a σ ( i )1 ( C ) a σ ( i )2 ( C ) < ν for a σ ( i )2 ( C ) < . Hence, for C ∈ D σν ⊆ Γ k,σ, − a , two Clifford hypertori bifurcate from the origin through a secondarysaddle-node type leaf-bifurcation at T CSN := (cid:110) ν (cid:12)(cid:12) ν = a σ ( k )1 ( C ) a σ ( k )2 ( C ) (cid:111) . (6.39)Here, one of the hypertori live inside the other one. On the other hand, a σ ( i )1 ( C ) a σ ( i )2 ( C ) = (cid:16)(cid:80) kl =1 a e σ ( l ) c σ ( l )2 (cid:17) (cid:80) ≤ l ≤ j ≤ k a e σ ( l )+ e σ ( j ) c σ ( l )2 c σ ( j )2 and (cid:80) ki =1 c σ ( i )2 = 1 . In order to find possible critical values of the parameter ν , we consider the Lagrange function L ( C, λ ) := (cid:16)(cid:80) kl =1 a e σ ( l ) c σ ( l )2 (cid:17) (cid:80) ≤ j ≤ l ≤ k a e σ ( l )+ e σ ( j ) c σ ( l )2 c σ ( j )2 + λ (cid:16)(cid:80) ki =1 c σ ( i )2 − (cid:17) , (6.40)where λ is the Lagrange multiplier. Let a := (cid:80) kl =1 a e σ ( l ) c σ ( l )2 and a := (cid:80) ≤ j ≤ l ≤ k a e σ ( l ) + e σ ( j ) c σ ( l )2 c σ ( j )2 . Then, we have ∇ C,λ L = a a ( ∇ C a − a a ∇ C a ) + 2 λC + (cid:0) (cid:80) ki =1 c σ ( i )2 − (cid:1) e k +1 . We show that ∇ C,λ L = 0 has no roots on the manifold S k − ,σ> . Since (cid:104) C, ∇ C a (cid:105) = 2 a , (cid:104) C, ∇ C a (cid:105) =4 a , and (cid:104) C, C (cid:105) = (cid:80) ki =1 c σ ( i )2 = 1 , we compute (cid:104) C, ∇ C,λ L (cid:105) = a a (cid:16) (cid:104) C, ∇ C a (cid:105) − a a (cid:104) C, ∇ C a (cid:105) (cid:17) + 2 λ (cid:104) C, C (cid:105) = 2 λ (cid:104) C, C (cid:105) = 0 . . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity λ = 0 . Now assume that C ∈ S k − ,σ> is a solution of ∇ C,λ L = 0 = a a ( ∇ C a − a a ∇ C a ) . Thus, a e σ ( i ) c σ ( i ) = c σ ( i ) a (cid:80) ≤ l ≤ k a e σ ( i )+ e σ ( l ) c σ ( l )2 a + a e σ ( i ) a c σ ( i )3 a and a e σ ( i ) = a a ( (cid:80) ≤ l ≤ k a e σ ( i ) + e σ ( l ) c σ ( l )2 + a e σ ( i ) c σ ( i )2 ) if c σ ( i ) (cid:54) = 0 . Hence, for c σ ( i ) c σ ( j ) (cid:54) = 0 we have sign( a e σ ( i ) a e σ ( j ) ) > . Otherwise, c σ ( i ) = 0 or c σ ( j ) = 0 . Therefore,the critical values do not occur for C -values on S k − ,σ> .Let γ ∈ S l,σn , c γ ( i ) c γ ( j ) (cid:54) = 0 for all i (cid:54) = j ≤ l < k and c γ ( i ) = 0 for i > l. Since a a = a γ ( l )1 ( C ) a γ ( l )2 ( C ) , ∇ C a = a a ∇ C a , a = (cid:104)C γ , M γ C γ (cid:105) , and a as given by (6.36) , we have a γ = a γ ( l )1 ( C ) a γ ( l )2 ( C ) M γ C γ . Since M γ is invertible, M γ − a γ = a γ ( l )1 ( C ) a γ ( l )2 ( C ) C γ and (cid:104) a γ , M γ − a γ (cid:105) = a γ ( l )1 ( C ) a γ ( l )2 ( C ) (cid:104) a γ , C γ (cid:105) = a γ ( l )1 ( C ) a γ ( l )2 ( C ) . Therefore, the local extremum values of a γ ( l )1 ( C ) a γ ( l )2 ( C ) is given by (cid:104) a γ , M γ − a γ (cid:105) . Thereby, the criticalvalues of parameters are ν min and ν max given by equations (6.37) and we always have ν min ≤ a σ ( i )1 ( C ) a σ ( i )2 ( C ) ≤ ν max . When ν a e σ ( k ) > , a secondary flow-invariant internal hypertoral CW complex manifold T intν associated with D σ, ◦ ν ⊂ Γ k,σ, − a defined in (6.33) bifurcates from the origin through an instant bifurca-tion at T P ch given by (6.29). It shrinks through a continuous leaf-dependent family of saddle-nodetype bifurcation of hypertori. In other words, the external hypertoral manifold T extν bifurcates fromthis leaf-dependent continuous hypertoral saddle-node type bifurcation when ν a σ ( k )2 ( C ) = a σ ( k )1 ( C ) . Therefore, we call the internal manifold by a secondary hypertoral manifold while the external man-ifold is referred by a tertiary hypertoral manifold . Part of these flow-invariant manifolds associatedwith S k − ,σ> is homeomorphic to B k − × T k and is relatively compact. Their topological closure,however, represent the actual flow-invariant toral CW complex bifurcated compact manifolds T intν and T extν . The proof for the toral CW complex structure of T intν and T extν is similar to Theorem6.11 and is thus omitted for briefness. For ν a ( C ) < , there is always a hypertorus correspondingwith the CW complex Γ k,σ, − a in the 2 k -cell.The radiuses of the tori converge to − a σ ( i )1 ( C ) a σ ( i )2 ( C ) for a σ ( k )1 ( C ) a σ ( k )2 ( C ) < i ≤ k, when ν approaches zero. Hence, the existing hypertorus for ν a ( C ) < ν changes its sign and remains sufficiently small associated with C ∈ D σν ⊂ Γ k,σ, − a . More precisely, when ν a ( C ) > ν lie in the interval ( ν min , ν max ), there are always twohypertori (one inside the other one) corresponding to the M Ck,σ -leaf for C ∈ D σν ⊂ Γ k,σ, − a ⊂ S k − > , while there is no hypertorus associated with C ∈ Γ k,σ, + a ⊂ S k − > . When ν a ( C ) > ν is outsidethe interval ( ν min , ν max ) , there is no flow-invariant hypertorus throughout the 2 k -cell. . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity Example 6.13 (Case n = k = 2) . Let n = k = 2 , a e = − a e = 1 , and a e = a e = a e + e = 1 . Figure 9: The space S ,I> and its partition givenby the spaces Γ ,Ia , Γ ,I, + a , Γ ,I, − a , N Iν and D I . When ν > , there exist two invariant toralCW complex over D Iν and there is no hyper-torus corresponding with C ∈ N Iν ∪ Γ ,I, + a . There is a toral CW complex over the entireCW complex S ,I> for ν < . A bistable toralCW complex exists over the CW complex D I,∂ν when 0 < ν ≤ ν max . 12 1 √ √ √ ν := c c ,I, − a Γ ,I, + a D Iν N Iν Γ ,Ia D I,∂ν Then, for C = ( c , c ) ∈ S ,I> , c (cid:54) = 0 , we have( a ( C ) , a ( C )) = ( c − c c , c + c c + c c ) and ( a ( C ) , a ( C )) = ( c − c c , c + c c + c c ) . √ . . . . . . c ρ ν . − . − . − . − . (a) Red curves represents the ρ -radiuses of two torifor ν > . √ . . . . . . . c ρ ν . − . − . − . − . (b) Blue curves represents the ρ -radiuses of a torusfor ν < . Figure 10: The radius curves of the invariant toral CW complexes versus c , where ( c , c ) ∈ S ,I> . Thus, Γ ,Ia = { ( √ , √ ) } and Γ ,I, ± a = (cid:110) C ∈ S ,I> | sign( c − c ) = ± (cid:111) . In particular, Γ ,I, + a = (cid:110) C ∈ S ,I> | √ < c ≤ (cid:111) . Further, D I := (cid:110) ( c , c ) ∈ S ,I> | ≤ c < (cid:111) , D I,∂ := (cid:110) ( , √ ) (cid:111) , and N I := (cid:110) C ∈ Γ ,I, − a | < c < √ (cid:111) . These are depicted in Figure 11. The ρ - and ρ -radiuses of tori corresponding with positive andnegative values of ν > . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity D Iν for ν = 0 . , , . The blue curvesdemonstrates an invariant toral CW complex on S ,I> for ν = − . , − . , − . , − . . Part of theblue curves corresponding with Γ ,I, + a coalesces to the origin and disappear when ν approaches tozero, i.e., the green curve corresponds with ν = 0. The family of tori collapse to an invariant limitcycle when c converges to either zero or 1 . Example 6.14. Let n = k = 3 , a e = a e = − a e = 1 , a e i + e j = 1 for all 1 ≤ i ≤ j ≤ , and σ = I as the identity permutation.Figure 11: Case k = n = 3 , a e = a e = − a e = 1 , and a e i + e j = 1for all 1 ≤ i ≤ j ≤ . The space S ,I> and Γ ,Ia , Γ ,I, + a , Γ ,I, − a , N Iν and D Iν . The space D I,∂ν is indexed with ν = 0 . . , while Γ ,Ia = { ( c , c , c ) | c = √ , c + c = 0 . } . There are twotoral CW complexes T intν and T extν over D I,∂ν and no other toral object else-where when ν > . For negative valuesof ν , the toral CW complex is associ-ated with the whole space S ,I> . . 510 0 . . 51 Γ ,I, + a Γ ,Ia Γ ,I, − a D I,∂ . D I . D I,∂ . Γ ,I − a N Iν c c c Thus, Γ ,Ia = L ( a e i ) i =1 ∩ S ,I> = { ( c , c , c ) ∈ S ,I> | c = √ , c + c = 0 . } andΓ ,I, ± a = (cid:110) C ∈ S ,I> | sign(1 − c ) = ± (cid:111) . Further, D Iν := (cid:110) C ∈ Γ ,I, − a | ν < a ( C ) a ( C ) ≤ (cid:111) ,D I,∂ν := (cid:110) C ∈ Γ ,I, − a | ν = a ( C ) a ( C ) (cid:111) , and N Iν := Γ ,I, − a \ ( D Iν (cid:116) D I,∂ν ) . where ν min = 0 and ν max = ; see Figure 11. Two toral CW complex T intν and T extν exists when ν > . These are associated with D Iν , that is depicted in Figure 11 with yellow bullets on partof the blue region; the blue region stands for Γ ,I, − a . For positive values of ν , there is a toralCW complex associated with S ,I> . Part of this toral manifold associated with Γ ,I, + a simultaneouslycollapses with the origin ( i.e., all radiuses of the tori converge to zero) as soon as the parameter ν converges to zero. In this case, we only have a toral CW complex associated with Γ ,I, − a . When ν further increases from zero, this toral manifold shrinks to be only associated with D Iν ; this turnsout to be T extν . More precisely, both the internal and external toral CW complexes T intν and T extν exist over D Iν . The intersection of the manifolds T intν and T extν is a bistable toral CW complex on . Gazor and A. Shoghi Bifurcation control with multiple Hopf singularity D I,∂ν for all 0 < ν ≤ ν max = 0 . 25. When ν > . , D I,∂ν shrinks tothe point D I,∂ . = { (0 , , } ; this is depicted by red bullet in Figure 11. Hence, the toral manifolds T intν and T extν shrink and collapse to a bistable limit cycle. Then, the limit cycle disappears when ν > T SN = { ν = 0 . } given by (6.38). Theorem 6.15. Consider the parametric vector field (6.28) , and assume that the condition (6.24) and hypotheses in Theorem 6.12 hold. Then, the varieties (6.29) and (6.38) are the only k -cellbifurcation varieties for the differential system corresponding with (6.28) . More precisely, the para-metric vector fields v σ ( r , θ, ν ) and v σ ( r , θ, ν ) are topologically equivalent when one of the followingholds.1. For δ = 1 , , ν min0 < ν δ < ν max0 and ν δ a e σ ( k ) > . ν δ a e σ ( k ) < for both δ = 1 , . 3. When ν δ a e σ ( k ) > and ν δ for δ = 1 , is outside of the interval [ ν min0 , ν max0 ] . Proof. The idea is to use a homeomorphism on the CW complex subspaces of S k − ,σ> to transformflow-invariant leaves associated with v σ ( r , θ, ν ) to those associated with v σ ( r , θ, ν ).Let ν min0 < ν δ < ν max0 and ν δ a e σ ( k ) > δ = 1 , . The CW complexes D σν and D σν are homeomorphic. Further, the complement of these spaces are also homeomorphic; see theproof of Lemma 6.10 and argument above Theorem 6.12. Assume that these are given by thehomeomorphism h : S k − ,σ> → S k − ,σ> , where h (cid:0) D σν (cid:1) = D σν and h (cid:0) D σν c (cid:1) = D σν c . Given the radiuses r ± σ ( i )2 ( ν , C ) in equation (6.31), the map ˜ h : T intν → T intν defined by ˜ h ( r ( ν , C ) , θ ) :=( r ( ν , h ( C )) , θ ) is a homeomorphism between the toral CW complex manifolds T intν and T intν . Sim-ilarly, we may assume that T extν and T extν are ˜ h -homeomorphic.We shall extend the homeomorphism ˜ h to a flow-invariant homeomorphism ˜ h : M k,σ → M k,σ . Thus, we merely need to consider the extension to the space M l,γ for any γ ∈ S l,σn and l ≤ k. Let r ( t, r δ,γ , ν δ ) stand for the trajectory of r in action-angle ( r , θ )-coordinates correspondingwith v σ ( r , θ, ν δ ) with the initial condition r (0 , r δ,γ , ν δ ) = r δ,γ , δ = 1 , 2. Denote r γ ( l ) for γ ( l )-thcomponent of r . Assume that ( r ∗ , θ ∗ ) ∈ M Cl,γ ⊂ M l,γ , C ∈ D γν , and r ∗ γ ( l ) = r ∗ . 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