Unfolding of nilpotent equilibria of degree 4 in Hamiltonian systems with 2 degrees of freedom
UUnfolding of nilpotent equilibria of degree 4 inHamiltonian systems with 2 degrees of freedom
Giannis MoutsinasMathematics Institute, University of Warwick
E-mail: [email protected]
September 28, 2018
Abstract
We consider Hamiltonian systems of two degrees of freedome havinga nilpotent equilibrium point with only one eigenvector. We provide theuniversal unfolding of such equilibrium, provided a non-degeneracy con-dition holds. We show that the only co-dimension 1 bifurcations thathappen in the unfolding are of two types: the normally hyperbolic or el-liptic centre-saddle bifurcations and the supercritical Hamiltonian-Hopfbifurcation.
Contents R . . . . . . . . . . . . . . . . . . . . . 21.2 Linear normal form . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Versal deformations of linear systems . . . . . . . . . . . . . . . . 31.4 Non-linear normal form . . . . . . . . . . . . . . . . . . . . . . . 61.4.1 Normal forms near equilibria . . . . . . . . . . . . . . . . 71.5 Bifurcations of equilibria . . . . . . . . . . . . . . . . . . . . . . . 81.5.1 Centre-saddle bifurcation . . . . . . . . . . . . . . . . . . 81.5.2 Hamiltonian-Hopf bifurcation . . . . . . . . . . . . . . . . 9 a r X i v : . [ m a t h . D S ] J un Parameter reduction 21
One of the few possible methods to study the dynamics of Hamiltonian sys-tems is to focus on neighbourhoods of their equilibria. In the case where theeigenvalues of the linearized system have non-vanishing real parts, the motionis completely determined by the linearization. In the case of imaginary eigen-values, the presence of resonances affects dramatically the dynamics.In the present paper we will focus on equilibria with vanishing eigenvalues.The case with degree 2 nilpotent matrix arises when modeling optics in an χ -medium and was studied in [Wag02]. A degree 3 nilpotent matrix cannot appearin two degrees of freedom and a degree 1 nilpotent matrix is the zero matrix,which has co-dimension 10, see [Gal82].We will consider a system of two degrees of freedom with an equilibrium,such that the linearized system has a degree 4 nilpotent matrix. A fist steptowards the study of the unfolding was performed in [Han07]. In the presentpaper we complete the study. Structure of the paper
Section 1 reiterates known results that are used in the study. In section 2the non-linear unfolding of the system is studied. We will see that even thoughthe linear unfolding is of co-dimension 2, the non-linear unfolding requires 3parameters. In section 3 we will see that under a non-degeneracy condition,the system can be simplified by truncating certain 3rd order terms. Finally insection 4 we will see how the parameters of the truncated system can be reducedfrom 3 to 2. R Here we review briefly some basic facts from the theory of Hamiltonian systems.Due to the scope of the paper we concentrate on R . For a thorough treatmentof the theory see [Arn90].We can view R as a symplectic manifold by defining the symplectic form ω = dq ∧ dp + dq ∧ dp . Here we use the canonical coordinates, i.e. a point on R is represented as( q , q , p , p ). The symplectic form can equivalently be defined as ω ( u, v ) = u (cid:124) Ω v, where Ω is the 2 m × m matrix (cid:0) E m − E m (cid:1) , with E m the m × m identity matrix.Then the Hamiltonian vector field can be written as X H = Ω ∇ H, with ∇ H the divergence vector of H . 2 change of coordinates is called symplectic if it preserves the symplecticform. In particular, on ( R , ω ) a linear transformation is symplectic if and onlyif its matrix P satisfies P (cid:124) Ω P = Ω . A linear Hamiltonian system can be defined by a Hamiltonian function whichis a polynomial of degree 2. Naturally we can assume that the equilibrium islocated at the origin, so we can restrict our class of Hamiltonian functions tohomogeneous polynomials of degree 2. Such Hamiltonian, H , defines a linearsystem with the matrix A = Ω H H ,where H H is the Hessian matrix of H .Using symplectic transformations we can define equivalence classes of Hamil-tonian systems. Then we can choose one representative for each class. We callthis representative normal form of the class. The following proposition givesone such choice for nilpotent equilibria. Proposition 1.1 ([Wil36]) . If A is a m × m real symmetric matrix whoseJordan form is a Jordan block of dimension m and eigenvalue zero, then thereexists a symplectic matrix P such that A P = P A and A corresponds to theHamiltonian function H ( x ) = 12 x (cid:124) A x = ± (cid:32) m − (cid:88) i =1 p i p m − i − m (cid:88) i =1 q i q m +1 − i (cid:33) − m − (cid:88) i =1 p i q i +1 . (1)In the case of 2 degrees of freedom the Hamiltonian function (1) becomes H ( x ) = 12 p − q q − p q . (2)An equivalent Hamiltonian function, used in [CS87] , is H ( x ) = p − p q . (3)From now on, the Hamiltonian (2) will be refered to as Williamson normal form and the Hamiltonian (3) will be refered to as standard normal form .Let H ( x ) = x (cid:124) A x and H ( x ) = x (cid:124) A x . Then a symplectic matrix P that changes H ( x ) to H ( x ) has to satisfy P Ω A = Ω A P and P (cid:124) Ω P = Ω.One such matrix is P = −
10 0 1 00 0 0 1 − . (4)From here on, if the linear part of a Hamiltonian system in 2 degrees of freedomis nilpotent of degree 4, then it will be assumed to be in the standard normalform. Given a linear system, the Jordan form of its matrix provides all the informationneeded to solve it. However, if the system depends on parameters, the transform3o Jordan form may be discontinuous with respect to the parameters. In orderto define an appropriate normal form we need some additional notions, [Arn71].
Definition 1.2.
Let µ ∈ R n and A ( µ ) ∈ R m be a matrix whose elements areformal power series in µ . If A (0) = A , then we say that A ( µ ) is a deformationof the matrix A . Definition 1.3.
A deformation A ( µ ) of A is called versal , if for any otherdeformation B ( ν ) of A , there exists a map φ : ν → µ smooth in a neighbour-hood of the point ν = 0 and a symplectic matrix S ( ν ) depending smoothly on ν , such that φ (0) = 0 , S (0) = E,B ( ν ) = S ( ν ) A ( φ ( ν )) S − ( ν ) . Moreover, a versa deformation is called universal if the change of parameters φ is uniquely determined by the matrix B ( ν ).A versal deformation is the most general deformation there can be for a givenmatrix A in the sense that it can be transformed into any other deformation.Here we are interested in the deformations of Hamiltonian matrices. A versaldeformation for every Hamiltonian matrix in Williamson normal form is givenin [Gal82]. Versal deformation of nilpotent systems in 2 degrees of freedom
It is proven in [Gal82] that a versal deformation of a linear nilpotent systemin 2 degrees of freedom in Williamson normal form (2) is the Hamiltonian H w ( x ) = 12 p − q q − p q + µ p p + ν p . (5)This shows that the co-dimension of this linear system is 2. Proposition 1.4.
The Hamiltonian function H ( x ) = p − p q + µ q ν (cid:18) q p q (cid:19) (6) is a versal deformation of a linear nilpotent Hamiltonian system in the standardnormal form (3) .Proof. With a symplectic transformation by the matrix P in (4) the Hamiltonian(5) becomes H ( x ) = p − p q + ν q µ p q and therefore a versal deformation of a nilpotent system at the standard normalform.Let J ( µ , ν ) and J ( µ , ν ) be the Hamiltonian matrices of the Hamiltonianfunctions H ( x ) and H ( x ), respectively. Since H ( x ) is a versal deformation, itis sufficient to show that there exists a map φ : ( µ , ν ) → ( µ , ν ), smooth in a The name versal is obtained by the word universal discarding the prefix uni indicatinguniqueness. , S ( µ , ν ) dependingsmoothly on ( µ , ν ), such that φ (0 ,
0) = (0 , , S (0 ,
0) =
E,J ( µ , ν ) = S (( µ , ν )) J ( φ (( µ , ν ))) S − (( µ , ν )) . The eigenvalues of J ( µ , ν ) are ± (cid:113) − ν ± (cid:112) µ + ν and the eigenvaluesof J ( µ , ν ) are ± (cid:112) − µ ± √ ν . From this we find( µ , ν ) = φ ( µ , ν ) = ( ν − µ , − µ ) . Then we search for a matrix that satisfies both J ( µ , ν ) S (( µ , ν )) = S (( µ , ν )) J ( φ (( µ , ν )))and S (cid:124) (( µ , ν )) Ω S (( µ , ν )) = Ω . Recall that Ω is the 4 × (cid:0) E − E (cid:1) and E the 2 × S ( µ , ν ) = − µ − µ . The eigenvalue configurations of the Hamiltonian (6) is given in Figure 1.Zero eigenvalues occur when µ = ν and two pairs of double eigenvaluesoccur when µ = − ν . Similarly to the linear case, we can define equivalence classes in the set of Hamil-tonian systems using canonical transformations and we can choose one repre-sentative of each class. This representative is called the normal form . We canconstruct the required canonical transformations by using the flow of chosenHamiltonian systems.Let M be a symplectic manifold. A Hamiltonian system on M defines an one-parameter flow on M and by fixing the parameter it can be viewed as map from M to M . The key observation is that this defines a canonical transformation.We will describe briefly the procedure in this section. For a more detaileddescription see [AKN06]. Definition 1.5.
Let (
M, ω ) be a symplectic 2 m -dimensional manifold, f, g ∈ C ∞ ( M ) and X f , X g the Hamiltonian vector fields of f and g respectively. Thebilinear map { , } : C ∞ ( M ) × C ∞ ( M ) → C ∞ ( M ) defined by { f, g } = ω ( X f , X g )is called the Poisson bracket of f and g . In local canonical coordinates thePoisson bracket takes the form { f, g } = m (cid:88) i =1 ∂ q i f ∂ p i g − ∂ p i f ∂ q i g. efinition 1.6. By fixing f we get the linear map { f, } : C ∞ ( M ) → C ∞ ( M ).This is called the adjoint map of f ad f ( ) := { f, } . Let x = ( q , . . . , q m , p , . . . , p m ) (cid:124) . Abusing notation, let us denote the func-tion x (cid:55)→ q i by q i . Similarly for p i . Then Hamilton’s equations can be writtenas ˙ x = − ad H ( x ) . The formal solution of this equation is x ( t ) = φ tH ( x ) := exp( − t ad H )( x ) . The exponential defined byexp( − t ad f )( ) := ∞ (cid:88) n =0 ( − n n ! t n ad nf ( )and has the property exp(ad f )( { g, h } ) = { exp(ad f )( g ) , exp(ad f )( h ) } , and bythis it can be shown that the transformation x (cid:55)→ exp(ad f )( x ) defines a (formal)canonical transformation, see [BV10]. In order to study the dynamics in a neighbourhood of an equilibrium we recallthat any neighbourhood on M is diffeomorphic to a neighbourhood in R m . Sowithout loss of generality we can study the corresponding Hamiltonian systemon a neighbourhood of the origin in R m .Let P n be the vector space of homogeneous polynomials of degree n on R m .Then for f ∈ P n the adjoint map ad f : P m → P m + n − maps homogeneouspolynomials to homogeneous polynomials. In particular when f ∈ P , it holdsthat ad f : P m → P m .Let H be a Hamiltonian function without constant term. Since the origin isan equilibrium, it has no first order terms either, so we have H = m (cid:88) k =2 H k + R m , with H k ∈ P k and R m satisfying R m (0) = DR m (0) = · · · = D m R m (0) = 0.We define the canonical transformation φ f := φ f = exp( − ad f ), with f ∈ C ∞ ( R m ), and we have H ◦ φ f = ν (cid:88) k =2 H k + { H , f } + ν (cid:88) k =3 { H k , f } + . . . , see [Han07].This shows that at the term H k any element of the image of ad H can beadded. The key observation is that if f ∈ P n , then any { H k , f } with k > n . Then f can be chosen tobe such that { H , f } gives the desired terms and H n can be normalized without7aving any effect on the lower order terms. This means that f can be choseninitially to be in P to normalize H without producing terms in P , then it canbe chosen to be in P to normalize H without producing terms in P and P .One may continue and inductively normalize all H n up to any power.We see that since we can freely add elements of im ad H to H k by choosing f , we can choose to transform it to ˆ H k such that span( ˆ H k ) ∩ im ad H = { } ,with span( ˆ H k ) being the linear subspace of P k spanned by the monomials in H k . In this way H determines the normal form of H .A standard result from representation theory states that if H = S + N is the Jordan decomposition of H , then ad H = ad S + ad N is the Jordandecomposition of ad H , see [Hum78]. So in order to check whether ad H is semi-simple it is sufficient to check whether the linearized system at the equilibriumhas a semi-simple matrix.If ad H : P k → P k is semi-simple, its eigenvectors span the whole space and itcan be diagonalized over C . We immediately see that P k = im ad H ⊕ ker ad H ,which holds also over R .On the other hand, if the linear part of H is H = S + N , with S, N ∈ P , N (cid:54) = 0, such that ad H = ad S + ad N is the Jordan decomposition of ad H andad N is nilpotent. Then the system can be normalized further.It was shown in [CS87] that N can be embedded in a subalgebra of ker ad S that is isomorphic to sl (2 , R ), i.e. there are elements M, T ∈ ker ad S ∩ P suchthat { N, M } = T, { N, T } = 2 N, { M, T } = − M. By this the splitting(ker ad S ∩ ker ad M ) ⊕ (ker ad S ∩ im ad N ) = ker ad S is derived, which ensures that the normalization can be done in two steps.Initially, the system is normalized with respect to S and then with respect to N without undoing the achievements of the first step, see [vdM82]. We will briefly discuss the two bifurcations of equilibria that will appear in thepresent analysis.
Intuitively, the centre-saddle bifurcation happens when two equilibria collideand disappear. In order for this to happen, the eigenvalues of the two equilibriahave to converge and at least one eigenvalue must vanish.In Hamiltonian systems, since if λ is an eigenvalue then − λ , λ and − λ arealso eigenvalues, the eigenvalues vanish always in pairs. The simplest case ofthis bifurcation is in a one degree of freedom system, as shown in Example 1.7.This bifurcation is called centre-saddle bifurcation because at this simple caseit involves a center and a saddle. For reasons that will be apparent later, thisbifurcation is also called fold bifurcation .8 xample 1.7. Consider the Hamiltonian H = p − q + µq . Its equationsof motion are ˙ q = p, ˙ p = q − µ. If µ >
0, the equilibria are ( q , p ) = ( ±√ µ, √ µ,
0) is asaddle and the equilibrium ( −√ µ,
0) is a centre. When µ = 0, there is only oneequilibrium (0 ,
0) with a 2 × µ < centre manifold , on which the systemtakes the form of the Example 1.7. If the non-vanishing eigenvalues are imagi-nary, then it was shown in [BCKV93] that the same can be done in integrableand near integrable systems.However, the linear part of the system can always be transformed to the formof two separable linear systems, one nilpotent and another either hyperbolic orelliptic. Then a centre-saddle bifurcation happens as long as the coefficient ofthe term q does not vanish. The simplest generic case of the Hamiltonian-Hopf bifurcation happens in 2degrees of freedom. The system has to have a double eigenvalue at the bifur-cation. There are two possibilities of a system having double non semi-simpleeigenvalues: • A system with four imaginary eigenvalues that meet on the imaginary axisand split into four complex eigenvalues. • A system with four real eigenvalues that meet on the real axis and splitagain into four complex.The second case is not viewed as a bifurcation because the equilibrium is hy-perbolic throughout the process. A system of the first form is given in Example1.8.
Example 1.8.
Consider the Hamiltonian system on R given by the Hamil-tonian H = q p − q p + ( q + q ) + µ ( p + p ). This gives the followingequations of motion ˙ q = − q + µp , ˙ q = q + µp , ˙ p = − q − p , ˙ p = − q + p . The equilibrium is the point (0 ,
0) with eigenvalues ± (cid:112) − − µ ± √ µ . If µ < µ goes to zero the eigenvalues tendto ± i and then they split to pairs ± λ i and ± λ i as µ becomes positive.9 Hamiltonian system with linear part as in Example 1.8 generically under-goes the Hamiltonian-Hopf bifurcation. In general the linear part of the normalform of such systems is H = ωS + αN + µM, where S = q p − q p ,N = 12 (cid:0) q + q (cid:1) ,M = 12 (cid:0) p + p (cid:1) . Since S corresponds to a semi-simple Hamiltonian matrix, it is called the semi-simple part of the Hamiltonian and N is called the nilpotent part.The normal form of such a system generically is H = ωS + αN + µM + bM + cSM + dS + . . . . Under the condition that α and b are non-zero, the terms in the second line donot influence qualitatively the system.Depending on the sign of αb , there are two types of the bifurcation. If αb is positive the bifurcation is called supercritical and almost all orbits of thesystem are quasi-periodic. This is called soft loss of stability , since the orbits arebounded before and after the bifurcation even if the stability of the equilibriumchanges. If αb is negative the bifurcation is called subcritical and the majorityof the orbits leave any neighbourhood of the origin. This is called hard loss ofstability . See [vdM82, vdM85, vdM86]. Let H be a Hamiltonian function on R and let the linear part be nilpotent ofdegree 4. Then there is a linear symplectic transformation that will transformthe linear part into H = p − p q . It is shown in [CS87] that the monomials in the kernel of ad H are generatedby q , q p q , p q p q q q p q + 32 p p q q − p q + 23 p q − p q . Thus a Hamiltonian with H as linear part in normal form will be of the form H κ,µ,ν = p − p q + κq + µ q ν (cid:18) q p q (cid:19) + a q a q (cid:18) q p q (cid:19) + a (cid:18) p q p q q q (cid:19) + . . . (7)10otice that κ , µ and ν can change the nilpotence of the equilibrium, but a , a and a cannot. For this reason a , a and a will be considered to beparameters and a , a and a will be considered to be constants. Comparing theabove Hamiltonian with the versal deformation of the linear system, we noticethat we have an extra parameter κ . For this reason one can expect that theparameters can be reduced to two. However there is no obvious way in whichthis can be done at the present stage. Later in the article we will reconcile thelinear with the non-linear theory. Theorem 2.1. If a (cid:54) = 0 then for a small enough neighbourhood of the originin ( q , q , p , p ) and a small enough neighbourhood of the origin in ( κ, µ, ν ) , theversal unfolding of the nilpotent equilibrium is given by surfaces diffeomorphicto the surfaces depicted in Figure 2. On the red surface the Hamiltonian-Hopfbifurcation happens and it is always of the supercritical type. On the blue surfacethe fold bifurcation happens. On the black line where the two surfaces meet, theequilibrium is nilpotent. Notice that even though the Hamiltonian system has 3 parameters onlybifurcations of co-dimension 1 appear in the unfolding.At the origin the two surfaces have a 3rd order tangency. This impliesthat as the neighbourhood in the space of parameters becomes small, the spacebetween them shrinks fast. The Hamiltonian-Hopf bifurcation happens in asystem at 1:-1 resonance and the elliptic fold bifurcation happens in a systemat 1:0 resonance. This implies that in the space between the two surfaces everyresonance appears.This section and the next one are dedicated to the proof of the theorem.From now on we will assume that the coefficient a does not vanish.Since we are interested only in the local behaviour, we will truncate theHamiltonian to order 3. Then the equations of motion become˙ q = 3 a q − q , ˙ q = p + 3 νq a q a q q , ˙ p = − κ − νp − µq − a p q − a p q − a q − a p q − a q , ˙ p = p − a p q − νq − a q q − a q . If ( q ∗ , q ∗ , p ∗ , p ∗ ) is an equilibrium of the above equations, it holds that q ∗ = q ,q ∗ = 32 a q ,p ∗ = 38 ( a νq + a a q + 3 a q ) ,p ∗ = −
34 ( νq + a q + 3 a q ) , (8)with q satisfying2716 a q + 4516 a a q + 94 a νq + 98 a q + 2716 a νq − a q + 916 ν q − µq − κ = 0 . (9)11o equilibriaFigure 2: The versal unfolding of the Hamiltonian (7).12 .1 Reparametrization In order to obtain the equilibria of the Hamiltonian system, one has to solvethe 5th order polynomial (9). However, there exists no algebraic formula for itsroots. Thus, we proceed by changing the parameters of the system.First we observe that equation (9) defines a smooth hyper-surface, which willbe called surface of equilibria , in the 4-dimensional space spanned by ( κ, µ, ν, q ).Since there exists a global coordinate chart ( µ, ν, q ), we view the above equationas the definition of a function κ ( µ, ν, q ) = 2716 a q + 4516 a a q + 94 a νq + 98 a q + 2716 a νq − a q + 916 ν q − µq . (10)By replacing κ with the right hand side of equation (10), equations (8) stilldefine an equilibrium. However now q is not viewed as a value that needs to becomputed, but rather as a parameter on its own. So we shift the parametriza-tion of the system from ( κ, µ, ν ) to ( µ, ν, q ) and we restrict our analysis in aneighbourhood of the origin in the space spanned by ( µ, ν, q ).One possible problem with this method is that q is not guaranteed to besmall when κ , µ and ν are small. Setting κ = µ = ν = 0 gives q (cid:32) a q + 4516 a a q + 98 a q − a (cid:33) = 0 . We see that q = 0 is a double root. This implies that in any neighbourhood ofthe origin in ( µ, ν, κ ) there are always at least 2 possible values for q . So weare indeed able to restrict our analysis in a region where q is small.Notice that each triplet ( µ, ν, q ) defines exactly one equilibrium and eachequilibrium can be described by one such triplet. Also each such triplet definesuniquely the Hamiltonian H κ,µ,ν . So there is a bijection between the set of theHamiltonian systems in this family paired with one of its equilibrium points andthe points on the surface, { H κ,µ,ν , q } ↔ ( κ, µ, ν, q ). The eigenvalues of thisequilibrium are ± √ (cid:113) Q ( µ, ν, q ) ± (cid:112) P ( µ, ν, q ) , with Q ( µ, ν, q ) =3 a q − a q − νP ( µ, ν, q ) = − a q − a a q − a νq + 40 a q + 104 a νq + 64 ν + 64 a q + 64 µ. Using the eigenvalues, one may search for Hamiltonian-Hopf and centre-saddlebifurcations. If Q ( µ, ν, q ) is negative and P ( µ, ν, q ) changes sign, then the eigenvalues changein the same way as the eigenvalues of a system undergoing a Hamiltonian-Hopf13ifurcation. So the next step is to seek whether the bifurcation actually takesplace.Let Q ( µ, ν, q ) = 3 a q − a q − ν = − ω , with ω >
0. By taking P ( µ, ν, q ) = 0 and solving it with respect to µ , it yields µ h = − a q − ν − a νq − a q + 12316 a νq + 514 a a q + 53164 a q . (11)By substituting µ by the above, one gets equations of motion with a non-semi-simple linear part and eigenvalues ± i ω (2 √ − .Let J (cid:48) be the Jacobian matrix at the equilibrium and let J = ω √ − ω √ − ω √ − − ω √ . Then the system
P J = J (cid:48) P and P (cid:124) Ω P = Ω can be solved . One solution isthe matrix P (cid:124) = ω − ( a q − a q ω ) ω − a q ω √ ω √ a q ω − a q ( a q − a q − ω ) √ ω − ( a q a q ω ) √ ω ω a q ω − a q ( a q − a q ω ) ω − a q a q − ω ω √ − a q a q ω √ − a q √ . Shifting the axes so that the equilibrium is always at zero and using the abovetransformation, the linear part of the Hamiltonian can be transformed to ω √ p q − p q ) + 12 (cid:0) q + q (cid:1) . Now instead of using equation (11), we use µ = β + µ h , where the unfolding parameter β is added. With the same transformation asabove the linear part of Hamiltonian becomes ω √ p q − p q ) + 12 (cid:0) q + q (cid:1) + 2 βω (2 q + 2 √ ωp q + ω p ) . Then the Hamiltonian is normalized with the algorithm described in [vdM82].The parameter β is counted for the degree of the monomial. After the normal-ization, the linear part of the Hamiltonian becomes H = (cid:32) ω √ − √ βω − √ β ω (cid:33) ( p q − p q )+ (cid:18)
12 + 2 βω + 96 β ω (cid:19) (cid:0) q + q (cid:1) + (cid:18) βω + 16 β ω (cid:19) (cid:0) p + p (cid:1) . Recall that Ω is the 2 m × m matrix (cid:0) E − E (cid:1) . C m be the coefficient of M in the normal form. It satisfies ω C m = 3203 a − a a q + 9725 a q − a a a q + 189545 a a q − a a q + 58733120 a a q + 5307125 a a q + 7620485 a q + 163 a a ω − a q ω − a a q ω + 1190750 a a q ω + 18051350 a a q ω + 60909350 a q ω − a ω − a a q ω + 25959100 a q ω + 132875 a ω . (12)In order to determine the type of the Hamiltonian-Hopf bifurcation the sign of C m is needed. The polynomial (12) has the same sign as C m . However, we areinterested only in what is happening around 0 in parameters. Since ω is of thesame order as q and ν , only the term a determines the sign of C m . Thisimplies that a Hamiltonian-Hopf bifurcation actually happens in the system andit is of the supercritical type as long as a does not vanish. There are two ways we can search for the centre-saddle bifurcation. One is tolook at the surfaces Q ( µ, ν, q ) = P ( µ, ν, q ) and Q ( µ, ν, q ) = − P ( µ, ν, q ).Another is to search where both ∂ q κ µ,ν,q = 0 and ∂ q κ µ,ν,q (cid:54) = 0 hold, with κ µ,ν,q given by equation (10). The latter gives the relation µ = 916 ν − a q + 278 a νq + 278 a q + 274 a νq + 454 a a q + 13516 a q , (13)while ∂ q κ µ,ν,q = − a + 278 a ν + 274 a q + 272 a νq + 1354 a a q + 1354 a q . This shows that there exists a neighbourhood around zero in parameters, inwhich the above second derivative does not vanish as long as a does not vanish.Substituting µ by the relation (13), one finds that the equilibrium haseigenvalues 0, corresponding to a Jordan block of degree 2, and ±√ λ with λ = − ν − a q + 6 a q . So there are two distinct possibilities for thenon-zero eigenvalues. We will see that in both cases a centre-saddle bifurcationactually happens. If λ >
0, the non-zero eigenvalues of the equilibrium are ± ω , where ω ∈ R is such that ω = λ . Then the system can be transformed to one having anequilibrium with its Jacobian matrix being J = ω ω . J (cid:48) be the Jacobian matrix at the equilibrium. Then one may search for amatrix P that satisfies P J = J (cid:48) P and P (cid:124) Ω P = Ω. One such matrix is P (cid:124) = − ω − a q ω a q ( a q − a q ω ) ω ( a q a q − ω ) ω − √ ω − ( − a q a q ω ) √ ω a q √ ω ω − a q − a q ω ω − a q ω ω / a q ω / a q ( − a q a q ω ) ω / − a q − a q − ω ω / . With a shift in the axes so that the equilibrium is always at zero and usingthe above matrix as transformation, the linear part of the Hamiltonian can betransformed to p + ω (cid:0) p − q (cid:1) . Then for C + q , the coefficient of q in theHamiltonian, it holds ω C + q = − a a q + 351160 a a q + 6310 a q − a ω − a q ω . Since ω is of the same order as q and ν , C + q does not vanish as long as a does not vanish. If λ <
0, the non-zero eigenvalues of the equilibrium are ± i ω , where ω ∈ R is such that ω = − λ . Then the system can be transformed to one havingequilibrium with Jacobian matrix being J = − ω − ω . Let J (cid:48) be the Jacobian matrix at the equilibrium, then one may search for amatrix P that satisfies P J = J (cid:48) P and P (cid:124) Ω P = Ω. One such matrix is P (cid:124) = ω a q ω − a a q ω + a q ω + a q ω − a q ω − a q ω − ω √ ω − a q √ ω + a q √ ω − ω / − a q √ ω ω − a q ω + a q ω + ω − a q ω ω / a q ω / − a a q ω / + a q ω / − a q √ ω − a q ω / − a q ω / + √ ω . With a shift in the axes so the equilibrium is always at zero and using the abovematrix as transformation, the linear part of the Hamiltonian can be transformedto - p + ω (cid:0) p + q (cid:1) . Then if C − q is the coefficient of q in the Hamiltonian,it holds ω C − q = − a a q + 351160 a a q + 6310 a q − a ω − a q ω . Exactly as above, C − q does not vanish as long as a does not vanish.16 .4 Nilpotent equilibrium We have found so far that the equilibria of the system are on the 3-dimensionalsurface of equilibria living in the 4-dimensional parameter space ( κ, µ, ν, q ),defined by equation (9). Moreover there is a 2-dimensional surface living on thesurface of equilibria, defined by equation (13), on which the system undergoesa centre-saddle or fold bifurcation, when a (cid:54) = 0. This surface will be called foldsurface Because of the fold bifurcation, the Hamiltonian has different valueson the two equilibria close to the origin. This implies that there can be noheteroclinic connections between them.There is also another 2-dimensional surface living on the surface of equilibria,defined by P ( κ, µ, ν, q ) = 0, on which the system undergoes a Hamiltonian-Hopfbifurcation if the eigenvalues are not real and a (cid:54) = 0, this will be called Hopfsurface . It should be stressed here that the Hamiltonian-Hopf bifurcation doesnot happen on the whole Hopf surface. As we will see in the next sections thereare 2 possible transitions for a system passing through the Hopf surface. Oneis the actual Hamilton-Hopf bifurcation, where all the eigenvalues are initiallyimaginary and they become complex. In the other case, all the eigenvalues arereal and they become complex. In the second case the equilibrium stays unstablethroughout the bifurcation, so this transition is not of particular interest to us.It is clear by the eigenvalue configurations on the two aforementioned sur-faces, that the eigenvalues of the equilibrium will vanish when the two surfacesmeet. The two surfaces are tangent along the line defined by ν = − a q + a q and they do not meet anywhere else in a neighbourhood of the origin.On that line it can be checked that the equilibrium has a nilpotent Jacobian.So we see that the nilpotent equilibrium happens on a line. This is of coursedue to the fact that there are three parameters in our system, instead of thetwo that the linear unfolding requires. Let κ a ,a ,a , f a ,a ,a and h a ,a ,a denote the surface of equilibria, the foldsurface and the Hopf surface, respectively. Let f κa ,a ,a , h κa ,a ,a denote theprojection to ( κ, µ, ν )-hyperplane and f q a ,a ,a , h q a ,a ,a denote the projectionto the ( µ, ν, q )-hyperplane. Lastly, the absence of an index implies that it iszero, for example κ a ,a ≡ κ a ,a , and κ a ≡ κ a , , . Lemma 2.2. If a (cid:54) = 0 , the surfaces f κa ,a ,a and h κa ,a ,a are diffeomorphic tothe surfaces f κa and h κa , respectively. Since the surfaces κ a ,a ,a , f a ,a ,a and h a ,a ,a are graphs of functionsthey are trivially diffeomorphic to κ a , f a and h a respectively. Then if a (cid:54) = 0,there exists a neighbourhood of the origin in which the projection onto ( κ, µ, ν )is smooth. The above lemma implies that setting a = a = 0 does not restrictthe genericity of the results.In a system where a vanishes Hamiltonian-Hopf bifurcation of both typescan appear. Also the fold bifurcation and other bifurcations of co-dimension 2may happen generically. 17 Truncated Hamiltonian
Using the above lemma we can set a = a = 0 and the Hamiltonian becomes H = p − p q + κq + µ q ν (cid:18) q p q (cid:19) + a q . (14)Then the equilibria satisfy q = q , q = 0 , p = 0 , p = − νq . So the equation defining the surface of equilibria takes the much simpler form − a q + 916 ν q − µq − κ = 0 . The equations can be simplified further, since if a is positive a time rescalingcan transform a to 1. If a is negative it can be transformed to -1 and a changeof sign on q transforms it to 1.Notice that there can only be two equilibria instead of five. We already knowthat qualitative changes of the eigenvalues happen only on the two surfaces, thefold and the Hopf. Thus for a description of the eigenvalue configurations thetwo surfaces need to be drawn in the parameter space.In the forthcoming sections we assume that a = 1. The Hopf surface
The surface of equilibria is a 3-dimensional hypersurface so it cannot bedrawn without a projection. The Hopf surface is a 2-dimensional surface livingon the surface of equilibria, which can be written in parametric form as (cid:18) ν q − q , − ν − q , ν, q (cid:19) . Its projection on the ( µ, ν, q )-hyperplane is shown in Figure 4a and its projec-tion on the ( κ, µ, ν )-hyperplane is shown in Figure 4b. The fold surface
Th fold surface can be written in parametric form as (cid:18) q , ν − q , ν, q (cid:19) . Its projection on the ( µ, ν, q )-hyperplane is shown in Figure 3a and its projec-tion on the ( κ, µ, ν )-hyperplane is shown in Figure 3b. The eigenvalue configuration
Together the two surfaces reveal the eigenvalue configurations of the sys-tem. In Figure 5 their projection to the ( µ, ν, q )-hyperplane is shown. In thisprojection each point corresponds to exactly one equilibrium point.The fold surface has different meanings in different projections. Recall thateach point on the ( µ, ν, q )-hyperplane gets mapped to exactly one equilibrium.On the other hand, in the ( κ, µ, ν ) projection the fold surface separates the spaceinto two regions, one where 2 equilibria exist and another where no equilibriumexists. The eigenvalue configurations of this projection is shown in Figure 2.18 a) The Hopf surface projected on the( µ, ν, q )-hyperplane. (b) The Hopf surface projected on the( κ, µ, ν )-hyperplane. Figure 3: Projections of the Hopf surface. (a) The fold surface projected on the( µ, ν, q )-hyperplane. (b) The fold surface projected on the( κ, µ, ν )-hyperplane. Figure 4: Projections of the fold surface.19igure 5: The eigenvalue configurations on the µ, ν, q -hyperplane.20 Parameter reduction
Since q is the only third order term in the Hamiltonian (14), we can use it tocancel the term q . This will reduce the parameters to two, as expected fromthe linear theory. Using the translation q (cid:55)→ q − µ/a , p (cid:55)→ p + 3 µν/ a , time reparametri-zation and possibly a sign change of q , the Hamiltonian (14) gets transformedto H t = p − p q + αq + β (cid:18) q p q (cid:19) + q , with α = κ − µ / µν /
16 and β = ν being considered as the new parameters.The equations of motion take the form˙ q = − q , ˙ q = p + 34 βq , ˙ p = − α − βp − q , ˙ p = p − βq . An equilibrium of these equations has the form ( q , , , − βq ) with q satis-fying 12 q − β q + α = 0 , (15)which defines the surface of equilibria, a saddle surface in this case.As usually, q can be viewed as a new parameter and equation (15) definesa smooth 2-dimensional surface in the 3-dimensional parameter space ( q , α, β ).We define a surjection from the ( q , β )-plane to the set of equilibria by mappingthe pair ( q , β ) to the equilibrium ( q , , , − βq ). The eigenvalues of thisequilibrium are ± (cid:113) − β ± (cid:112) β + q .We know by the previous analysis that the Hamiltonian-Hopf bifurcationand the centre-saddle bifurcation always happen in this system. Using the sametechniques as before we find that the Hamiltonian-Hopf bifurcation happenson a line on the surface of equilibria, defined by q = − β , which is the Hopf“surface”. The fold “surface” can be found to be the line defined by q = β .Since in this case the surface of equilibria is two dimensional, it can be drawnwithout the need of projections. This is done in Figures 6. The different colourson the surface correspond to different eigenvalue configurations of the system.The solid line is the fold “surface” and the dashed line is the Hopf “surface”.Figure 6c corresponds to the projection of the surface on the ( β, q ) plane.The eigenvalue configurations of the system in this parametrization on this planeis shown is Figure 7. Notice that this is basically identical to the eigenvalueconfigurations of the unfolding of the linear system, shown in Figure 1.In order to find the eigenvalue configurations of the system in the originalparameters, we need to project the surface of equilibria on the plane ( α, β ).Figure 6d corresponds to this projection. The eigenvalue configurations of the21 a) (b)(c) (d) Figure 6: The surface of equilibria viewed from different angles.22igure 7: The eigenvalue configurations on the plane ( β, q ).23o equilibriaFigure 8: The eigenvalue configurations on the plane ( α, β ).24 β (a) r < αβ (b) r = 0 αβ (c) r > Figure 9: Eigenvalue configurations for different values of r .system is shown in Figure 8. On this projection the fold “surface” is given by theequation 512 α = 81 β and the Hopf “surface” by the equation 16 α = − β .It should be noted that in [Han07] the two “surfaces” are wrongly depicted tohave a 1st order tangency instead of 3rd. The fact that the two “surfaces” are defined by fourth degree equations is non-generic. This non-genericity comes from the fact that the reduction that wasperformed in section 4.1 was also non-generic.Let us consider again the Hamiltonian H = p − p q + κq + µ q ν (cid:18) q p q (cid:19) + a q . Fix an r ∈ R small enough. Then the translation q (cid:55)→ q + r − µa , p (cid:55)→ p + 3 ν ( µ − r )4 a transforms the above Hamiltonian to H r = p − p q + αq + r q β (cid:18) q p q (cid:19) + a q , with β = ν , α = r a + κ − µ a − ν r a + µν a . As above we can scale a to 1.An equilibrium of the system is of the form ( q , , , − βq ), but now q satisfies 12 q − β q + rq + α = 0and its eigenvalues are ± (cid:113) − β ± (cid:112) r + β + q . Now it is clear that r can be set to any (small) number and all these seeminglydifferent systems are equivalent. The reduction in section 4.1 is the obvious one,hence the name natural, but it is only one from a continuum of possibilities.In this system the fold “surface” is given by the equation q = − r + β andthe Hopf “surface” by the equation q = − r − β . Since α = β − q − rq ,25e find α f ( β ) = 12 r − rβ − β and α h ( β ) = 12 r − rβ + 81512 β . Since the functions α f ( β ) and α h ( β ) are even and share the same constantand second order terms, they are tangent to degree 3 for every r . The reduc-tion in section 4.1 is special in the sense that it makes this second order termdisappear. The eigenvalue configurations of the system for various values of r is shown in Figures 9. Notice that by choosing different values for r one takesdifferent cuts of constant µ of the surfaces in Figure 2. Acknowledgement
The author thanks Heinz Hanßmann for introducing the problem and for manyfruitful discussions.
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