Heteroclinic Cycles in ODEs with the Symmetry of the Quaternionic \mathbf{Q}_8 Group
aa r X i v : . [ m a t h . D S ] J u l HETEROCLINIC CYCLES IN ODES WITH THE SYMMETRY OF THEQUATERNION Q GROUP
ADRIAN C. MURZA
Abstract.
In this paper we analyze the heteroclinic cycle and the Hopf bifurcation of a genericdynamical system with the symmetry of the group Q , constructed via a Cayley graph. While theHopf bifurcation is similar to that of a D –equivariant system, our main result comes from analyzingthe system under weak coupling. We identify the conditions for heteroclinic cycle between threeequilibria in the three–dimensional fixed point subspace of a certain isotropy subgroup of Q × S . We also analyze the stability of the heteroclinic cycle. Introduction
Heteroclinic cycles with in systems with symmetry have been widely studied over the large decades[1, 7, 11, 12, 13, 14]. During the last couple of years a special interest has received the existenceof heteroclinic cycles in systems related with quaternionic symmetry, see for example the worksof X. Zhang [15] and O. Podvigina [12, 13]. This is basically due to two facts. On the one handquaternions are involved in the study of heteroclinic cycles in ODEs with symmetry in a naturalway, owing to the easy representation of the dynamics in R in terms of quaternions. Many of thedynamical systems giving rise to heteroclinic cycles studied so far are D n –equivariant; the action of D n in R is absolutely irreducible, so R is D n –simple. Therefore, quaternionic representations in R turned out to be very useful. On the other hand there is the intrinsic interest in the differentialequations where the variables are the quaternions. We relate the study of heteroclinic cycles withthe dynamics of networks of n coupled oscillators with symmetry. Ashwin and Swift [2] showed thatthe symmetry group of the network can be considered a subgroup of S n , as long as the oscillatorstaken individually have no internal symmetries. Besides these two main reasons, there are also otherones that stimulates the analysis of dynamical systems with the quaternionic symmetry, and theseare related to applications to other sciences. For example we can cite the heteroclinic phenomenaobserved in systems with quaternionic symmetry such as nematic liquid crystals [3], particle physics[5] and improving computational efficiency [6]; however, these heteroclinic behaviors in such systemshave never been encountered a theoretical explanation. This is one of our major motivation, togetherwith the intrinsic value of the mathematical theory developed around this subject.An important step in designing oscillatory networks with the symmetry of a specific group hasbeen developed by Stork [14]. The authors have shown how to construct an oscillatory networkwith certain designed symmetry, by the Cayley graph of the symmetry group.In this paper we analyze the heteroclinic cycles and Hopf bifurcation in ODEs with the symmetryof the quaternionic group Q of order 16 . We use the methodology developed by Ashwin and Stork[1] to construct a network of differential systems with Q symmetry. We investigate the dynamicalbehavior of the system under the weak coupling. In this case we reduce the asymptotic dynamics toa flow on an sixteen-dimensional torus T . We prove the existence of heteroclinic cycles between thethree steady–states existing within a three–dimensional fixed–point subspace of one of the isotropysubgroups of Q × S , namely Z . We also classify the stability of heteroclinic cycles.The paper is organized as follows. In Section 2 we construct the most general oscillatory systemwith the Q symmetry by using the Cayley graph of this group. In Section 3 we analyze the Hopfbifurcation of the constructed array. In Section 4 we prove the existence of heteroclinic cycles in Mathematics Subject Classification.
Key words and phrases. equivariant dynamical system, Cayley graph, Q quaternions, heteroclinic cycle. some of the subspaces which are invariant under the action of certain isotropy subgroups of Q . Wealso analyze their stability. 2.
The Cayley graph of the Q group In this section we construct an oscillatory system with the Q symmetry and describe the elementsof this group, as the relationships between them. For more details about the use of the Cayley graphin constructing the network with the prescribed symmetry see [14] or [11]. The Cayley graphs for Q is shown in Figure (1). Figure 1.
A Cayley graph of the Q group. Solid arrows represent left-multiplication with a, dot-and-dashed arrows left multiplication with b, the twogenerators of this group. The action of the group Q on the cells can be written as(1) Ida = (1 2 3 4 9 10 11 12)(5 16 15 14 13 8 7 6) b = (1 5 9 13)(2 6 10 14)(3 7 11 15)(4 8 12 16) ab = (1 16 9 8)(2 5 10 13)(3 6 11 14)(4 7 12 15) b = (1 9)(2 10)(2 11)(4 12)(5 13)(6 14)(7 15)(8 16) a = (1 3 9 11)(2 4 10 12)(5 15 13 7)(6 16 14 8) a = (1 4 11 2 9 12 3 10)(5 14 7 16 13 6 15 8) ab = (1 10 3 12 9 2 11 4)(5 8 15 6 13 16 7 4) a b = (1 11 9 3)(2 12 10 4)(5 7 13 15)(6 8 14 16) a b = (1 12 11 10 9 4 3 2)(5 6 7 8 13 14 15 16) ba = (1 6 9 14)(2 7 10 15)(3 8 11 16)(4 13 12 15) ba = (1 7 9 15)(2 8 10 16)(3 13 11 5)(4 14 12 16) b = (1 13 9 5)(2 14)(3 15 11 7)(4 16 12 8)(6 10) ab = (1 8 9 16)(2 13 10 5)(3 4 15 14)(6 11 12 7) a b = (1 14 9 6)(2 15 10 7)(3 16 11 8)(4 5 12 13) a b = (1 15 9 7)(2 16 10 8)(3 5 11 13)(4 6 12 14)with the relationship between them(2) a = Id, a = b = abab, aba = b If we assign coupling g between cells related by a and coupling h between cells related by b , fromthe permutations in (1), we can build the following pairwise system in with the Q symmetry. ETEROCLINIC CYCLES IN ODES WITH THE SYMMETRY OF THE QUATERNIONIC Q GROUP 3 (3) ˙ x = f ( x ) + g ( x , x ) + h ( x , x ) , ˙ x = f ( x ) + g ( x , x ) + h ( x , x ) , ˙ x = f ( x ) + g ( x , x ) + h ( x , x ) , ˙ x = f ( x ) + g ( x , x ) + h ( x , x ) , ˙ x = f ( x ) + g ( x , x ) + h ( x , x ) , ˙ x = f ( x ) + g ( x , x ) + h ( x , x ) , ˙ x = f ( x ) + g ( x , x ) + h ( x , x ) , ˙ x = f ( x ) + g ( x , x ) + h ( x , x ) , ˙ x = f ( x ) + g ( x , x ) + h ( x , x ) , ˙ x = f ( x ) + g ( x , x ) + h ( x , x ) , ˙ x = f ( x ) + g ( x , x ) + h ( x , x ) , ˙ x = f ( x ) + g ( x , x ) + h ( x , x ) , ˙ x = f ( x ) + g ( x , x ) + h ( x , x ) , ˙ x = f ( x ) + g ( x , x ) + h ( x , x ) , ˙ x = f ( x ) + g ( x , x ) + h ( x , x ) , ˙ x = f ( x ) + g ( x , x ) + h ( x , x ) , where f : R → R and g, h : R → R . As shown by Ashwin and Stork [1] we can think of f, g, h asbeing generic functions that assure that the isotropy of this vector field under the action of O isgenerically Q . 3. Hopf bifurcation
In order to consider generic one-parameter Hopf bifurcation in systems with Q , we need toanalyze the complex irreducible representations of Q . Based on the work of Golubitsky and Stewart[8], these representations are of one or two dimensions. From their theory, we have that the linearrepresentation of a group Γ α Γ : Γ × W → W on the complex vector space W is irreducible if and only if Γ − invariant subspaces are trivial; itis to say, { } or W itself. It is important to notice, that (a) there need be no faithful irreduciblerepresentations, and (b) this is typical.In addition, the amount by which the representation fails tobe faithful is the kernel of the action α Γ . The group Q has five irreducible representations; four of them are one-dimensional and theremaining one is two-dimensional. The one-dimensional representations can be interpreted as Hopfbifurcation with trivial or Z symmetry, which correspond to a quotient group of Q . From [9] the two generators of Q are(4) a = (cid:18) ω
00 ¯ ω (cid:19) , b = (cid:18) −
11 0 (cid:19) , where ω = exp (cid:18) πi (cid:19) . Therefore the standard irreducible action of Q on C is given by(5) a ( z + , z − ) = √
22 (1 + i ) z + , √
22 (1 − i ) z − ! b ( z + , z − ) = ( − z − , z + )and there is a phase shift action of S given by R φ ( z + , z − ) = ( e iφ z + , e iφ z − ) , for φ ∈ S . The action of Q × S is similar to the action of D × S . This action is generated by(6) κ ( z + , z − ) = ( z − , z + ) , ρ ( z + , z − ) = ( iz + , − iz − ) , where ρ = κ = 1 and ρκ = κρ . In our case of the group Q we have a = b = 1 , a = b and aba = b. The kernel of this action in the 2 − cycle in D × S is generated by ( ρ , π ), while the kernelof the action of Q × S is generated by ( a = b = 1 , π ). Therefore, it is possible to check that Q × S /kerα Q × S ≡ Q × S /kerα D × S . ADRIAN C. MURZA
Table 1.
Table that relates the isotropies of points in C for identical actions of Q × S and D × S .Isotropy in Q × S Isotropy in D × S Fix dim C F ix
Name ( D ) Q × S D × S (0 ,
0) 0 Trivial solution˜ Z a ˜ Z ( ρ ) ( z,
0) 1 Rotating Wave˜ Z b ˜ Z ( ρ ) × ˜ Z ( κ ) ( z, z ) 1 Edge Solution˜ Z c ˜ Z ( ρ ) × ˜ Z ( ρκ ) ( z, iz ) 1 Vertex Oscillation˜ Z ˜ Z ( ρ ) ( w, z ) 2 Submaximal Table 2.
Branching Equations for D Hopf Bifurcation.Orbit type Branching Equations Signs of Eigenvalues(0 ,
0) - Re A (0, λ )( a, A + Ba = 0 Re( A N + B ) + O ( a ) − Re( B ) [twice]( a, a ) A + Ba + Ca + Da = 0 Re(2 A N + B ) + O ( a ) (cid:26) trace = Re( B ) + O ( a )det = − Re( B ¯ C ) + O ( a )( a, e πi/ a ) A + Ba − Ca − Da = 0 Re(2 A N + B ) + O ( a ) (cid:26) trace = Re( B ) + O ( a )det = Re( B ¯ C ) + O ( a )This means that we use the results obtained in [8] for Hopf bifurcation in systems with D symmetry,with a re-interpretation of the branches. Proposition 1.
There are exactly three branches of periodic solutions that bifurcate from (0 , , corresponding to the isotropy subgroups D × S with two-dimensional fixed-point subspaces.Proof. The proof is a direct application to the group D of Theorem 4 . D symmetry. (cid:3) From Proposition 2 . , page 372 in [8] we have that every smooth D × S − equivariant map germ g : C → C has the form(7) g ( z , z ) = A (cid:20) z z (cid:21) + B (cid:20) z ¯ z z ¯ z (cid:21) + C (cid:20) ¯ z z z ¯ z (cid:21) + D (cid:20) z ¯ z ¯ z z (cid:21) , where A, B, C, D are complex-valued D × S − invariant functions. The branching equations for D − equivariant Hopf bifurcation may be rewritten g ( z , z ) = 0 . These branching equations areshown in Table (2).3.1.
Bifurcating branches.
We now use the information in Table (2) to derive the bifurcationdiagrams describing the generic D − equivariant Hopf bifurcation. Assume (8) ( a ) Re( A N + B ) = 0 , ( b ) Re( B ) = 0 , ( c ) Re(2 A N + B ) = 0 , ( d ) Re( B ¯ C ) = 0 , ( c ) Re( A λ ) = 0 , where each term is evaluated at the origin.Assuming nondegeneracy conditions (8) and the trivial branch is stable subcritically and losesstability as bifurcation parameter λ passes through 0 . We summarize these facts into the nexttheorem.
ETEROCLINIC CYCLES IN ODES WITH THE SYMMETRY OF THE QUATERNIONIC Q GROUP 5
Table 3.
Isotropy subgroups and fixed point subspaces for the Q × S action on T . Σ Fix(Σ) Generators dim Fix(Σ) Q (0 , , , , , , , , , , , , , , ,
0) ( a, , ( b,
0) 0˜ Q a (0 , , , , , , , , π, π, π, π, π, π, π, π ) ( b, π ) , ( ab, π ) 0˜ Q b (0 , π, , π, , π, , π, , π, , π, , π, , π ) ( a, π ) , ( ab, π ) 0˜ Q ab (0 , π, , π, , π, , π, π, , π, , π, , π,
0) ( a, π ) , ( b, π ) 0 Z a (0 , , , , , , , , φ, φ, φ, φ, φ, φ, φ, φ ) ( a,
0) 1 Z b (0 , φ, , φ, , φ, , φ, , φ, , φ, , φ, , φ ) ( b,
0) 1 Z ab (0 , φ, , φ, , φ, , φ, φ, , φ, , φ, , φ,
0) ( ab,
0) 1˜ Z a (0 , π, , π, , π, , π, φ, φ + π, φ, φ + π, φ, φ + π, φ, φ + π ) ( a, π ) 1˜ Z b (0 , φ, , φ, , φ, , φ, φ + π, φ, φ + π, φ, φ + π, φ, φ + π, π ) ( b, π ) 1˜ Z ab (0 , φ, , φ, , φ, , φ + π, φ, φ + π, φ, φ + π, φ, φ + π, π, φ + π ) ( ab, π ) 1˜ Z a/ (0 , π , π , π , π, π , π , π , φ, φ + π , φ + π , φ + π , φ + π, φ + π , φ + π , φ + π ) ( a, π ) 1˜ Z b/ (0 , φ, π, φ + π, φ + π , π , φ + π , π , φ + π , φ, φ + π , φ, φ + π , φ, φ + π , π ) ( b, π ) 1˜ Z b/ (0 , φ, π, φ + π, π , φ + π , π , φ + π , π , φ + π , π , φ + π , π , φ + π , π , φ + π ) ( ab, π ) 1 Z (0 , φ , , φ , , φ , , φ , φ , φ , φ , φ , φ , φ , φ , φ ) ( b ,
0) 3˜ Z (0 , φ , π, φ + π, , φ , π, φ + π, φ , φ , φ + π, φ + π, φ , φ , φ + π, φ + π ) ( b , π ) 3 Theorem 1.
The following statements hold. (a)
The ˜ Z branch is super- or subcritical according to whether Re( A N (0) + B (0)) is positive ornegative. It is stable if Re( A N (0) + B (0)) > , Re( B (0)) < . (b) The Z ( κ )[ ⊕ Z c ] is super- or subcritical according to wether Re(2 A N (0) + B (0)) is positiveor negative. It is stable if Re(2 A N (0) + B (0)) > , Re( B (0)) > and Re(2 B (0) ¯ C (0)) < . (c) The Z ( κ, π )[ ⊕ Z c ] or Z ( κ, ξ ) ⊕ Z c branch is super- or subcritical according to wether Re(2 A N (0)+ B (0)) is positive or negative. It is stable if Re(2 A N (0)+ B (0)) > , Re( B (0)) > and Re(2 B (0) ¯ C (0)) < . Proof.
The proof is a direct application to the case D of the Theorem 3 . (cid:3) Weak Coupling
The idea of studying ODEs in the weak coupling limit was introduced by Ashwin and Swift[2]. This situation can be uderstood as follows. In the no coupling case there is an attracting n –dimensional torus with one angle for every oscillator. The situation is completely different to theHopf bifurcation. Instead of examining small amplitude oscillations near a Hopf bifurcation point,we make a weak coupling approximation. There is a slow evolution of the phase differences in theweak coupling. Another improvement with respect to the Hopf bifurcation is that while the Hopfbifurcation theory gives local information, the weak coupling case the yields global results on then–dimensional torus.System (3) can be rewritten under weak coupling case as an ODE of the form:(9) ˙ x i = f ( x i ) + ǫg i ( x , . . . , x )for i = 1 , . . . , , x i ∈ V and commuting with the permutation action of Q on V , both f and g i being of the class C ∞ . The constant ǫ represents the coupling strength and we have ǫ ≪
1. As in[2], or [1] we may assume ˙ x = f ( x ) has an hyperbolic stable limit cycle.It follows that if the coupling is weak, we should not just take into account the irreduciblerepresentations of Q . Since there are 16 stable hyperbolic limit cycles in the limit of ǫ = 0 , itmeans that the asymptotic dynamics of the system factors into the asymptotic dynamics of 16 limitcycles. We assume that each limit cycle taken individually is hyperbolic for small enough valuesof the coupling parameter. This justifies expressing the dynamics of the system only in terms ofphases, i.e. an ODE on T which is Q − equivariant.When considering the weakly coupled system we can average it over the phases [14]. This is thesame as introducing and phase shift symmetry by translation along the diagonal; R θ ( φ , . . . , φ ) := ( φ + θ, . . . , φ + θ ) , ADRIAN C. MURZA for θ ∈ S . We obtained an ODE on that is equivariant under the action of Q × S , and we have to classifythe isotropy types of points under this action. This is done in Table (3). Since now on, ourinterest focuses in the three-dimensional space Fix( Z ); it does not contain two-dimensional fixed-point subspaces. In turn, it contains several one- and zero-dimensional subspaces fixed by theisotropy subgroups ˜ Q i , ˜ Z i and Z i , respectively, where i = { a, b, ab, a/ , b/ } as in Table (3). Thesesymmetries are not in Z ; however, they are in the normalizer of Z .4.1. Dynamics of the θ , θ and θ angles in Fix( Z ) . We can define coordinates in Fix( Z ) bytaking a basis(10) e = − (1 , , , , , , , , − , − , − , − , − , − , − , − e = − (1 , − , , − , , − , , − , , − , , − , , − , , − e = − (1 , − , , − , , − , , − , − , , − , , − , , − , { e , e , e } parametrized by { θ , θ , θ } : P n =1 θ n e n . By using these coordinates, we construct the following family of three-dimensional differentialsystems which satisfies the symmetry of Fix( Z ).(11) ˙ θ = u sin θ cos θ + ǫ sin 2 θ cos 2 θ ˙ θ = u sin θ cos θ + ǫ sin 2 θ cos 2 θ ˙ θ = u sin θ cos θ + ǫ sin 2 θ cos 2 θ + q (1 − cos θ ) sin 2 θ , where u, ǫ, q ∈ R . We will show that this vector field contains structurally stable, attracting heteroclinic cycles whichmay be asymptotically stable, essentially asymptotically stable or completely unstable, depending onthe values of u, ǫ and q. We can assume, without loss of genericity that the space Fix( Z ) is normallyattracting for the dynamics and therefore the dynamics within the fixed-point space determines thestability of the full system. In the following we will show that the planes θ i = 0 (mod π ) , i = 1 , , X be the vector field of system (11). Definition 1.
We call a trigonometric invariant algebraic surface h ( θ , θ , θ ) = 0 , if it is invariantby the flow of (11) , i.e. there exists a function K ( θ , θ , θ ) such that (12) X h = ∂h∂θ ˙ θ + ∂h∂θ ˙ θ + ∂h∂θ ˙ θ = Kh.
Lemma 1.
Functions sin θ , sin θ and sin θ are trigonometric invariant algebraic surfaces forsystem (11) .Proof. We can write the system (11) in the form(13) ˙ θ = sin θ ( u cos θ + 2 ǫ cos θ cos 2 θ )˙ θ = sin θ ( u cos θ + 2 ǫ cos θ cos 2 θ )˙ θ = sin θ ( u cos θ + 2 ǫ cos 2 θ cos θ + 2 q (1 − cos θ ) cos θ )Now if we choose h = sin θ , then X h = cos θ sin θ ( u cos θ + 2 ǫ cos θ cos 2 θ ) so K =cos θ ( u cos θ + 2 ǫ cos θ cos 2 θ ) . The remaining cases follow similarly. (cid:3)
Since the planes θ i = 0(mod π ) are invariant under the flow of (11), it is clear that (0 , , , ( π, , , (0 , π, , , π ) are equilibria for (11). To check the possibility of heteroclinic cyclesin system (11), we linearize about the equilibria (i.e. the zero-dimensional fixed points). The ideais proving that there are three-dimensional fixed-point spaces Fix( Z ) and Fix( ˜ Z ) which connect ETEROCLINIC CYCLES IN ODES WITH THE SYMMETRY OF THE QUATERNIONIC Q GROUP 7
Table 4.
Eigenvalues of the flow of equation (11), at the four non-conjugate zero-dimensional fixed points.Σ Fix(Σ) with coordinates ( φ , φ , φ ) λ λ λ Q (0 , , u + 2 ǫ u + 2 ǫ u + 2 ǫ ˜ Q a ( π, , − u + 2 ǫ u + 2 ǫ − u + 2 ǫ + 4 q ˜ Q b (0 , π, − u + 2 ǫ − u + 2 ǫ u + 2 ǫ ˜ Q ab (0 , , π ) u + 2 ǫ − u + 2 ǫ − u + 2 ǫ these fixed points, allowing the existence of such a heteroclinic network between the equilibria.Let’s assume(14) | ǫ | < u and | ǫ + 2 q | < u . We use the criteria of Krupa and Melbourne [10] to study the stability of the heteroclinic cycle.
Theorem 2.
In the following we will prove that there exists the possibility of a heteroclinic cyclein the following way: (15) · · ·
Fix(˜ Z b ) −−−−−→ Fix( ˜ Q a ) Fix(˜ Z ab ) −−−−−→ Fix( ˜ Q b ) Fix(˜ Z a ) −−−−−→ Fix( ˜ Q ab ) Fix(˜ Z b ) −−−−−→ · · · The stability of the heteroclinic cycle is: (a) asymptotically stable if (16) u < q < u − ǫ , (b) unstable but essentially asymptotically stable if (17) u < u − ǫ < q < u − ( u + 2 ǫ ) ( − u + 2 ǫ ) . (c) completely unstable if u > . Proof.
The stability is expressed by(18) ρ = Y i =1 ρ i , where ρ i = min { c i /e i , − t i /e i } . In equation (18), e i is the expanding eigenvalue at the i th point of the cycle, − c i is the contractingeigenvalue and t i is the transverse eigenvalue of the linearization. For the heteroclinic cycle we have(19) ρ = u − qu + 2 ǫ if q < u − ǫ , − u + 2 ǫu + 2 ǫ if q > u − ǫ , ρ = ρ = − u + 2 ǫu + 2 ǫ , so from equations (18) and (19) we obtain(20) ρ = ( − u + 2 ǫ ) (2 u − q )( u + 2 ǫ ) if u < q < u − ǫ , ( − u + 2 ǫ ) ( u + 2 ǫ ) if u < q > u − ǫ . Then the proof follows by applying Theorem 2 . (cid:3) ADRIAN C. MURZA
For any u < u − ǫ < q < u − ( u + 2 ǫ ) ( − u + 2 ǫ ) and therefore there exist values of q forwhich there exist essentially asymptotic stable heteroclinic connections. In consequence, thereexists an attracting heteroclinic cycle even though the linear stability of Fix( ˜ Q a ) has an expandingtransverse eigenvalue. 5. Conclusions
We prove the existence of stable heteroclinic cycles in the most general coupled ordinary differ-ential equations with quaternionic symmetry Q . Our approach is generic and offers for the firsttime as far as we know, evidence of these phenomena in systems with this symmetry. While theresults stands on its own from a mathematical point of view, it might also contribute to a betterunderstanding of these intermittent behaviors experimentally observed in nematic liquid crystals [3]and particle physics [5]. Acknowledgements
In first place I would like to address many thanks to the Referee, whose helpful indicationsand comments greatly improved the presentation of the paper. I acknowledge a BITDEFENDERpostdoctoral fellowship from the Institute of Mathematics Simion Stoilow of the Romanian Academy,Contract of Sponsorship No. 262/2016 as well as economical support from a grant of the RomanianNational Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-II-RU-TE-2014-4-0657.
References [1]
P. Ashwin, P. Stork , Permissible Symmetries of Coupled Cell Networks , Math. Proc. Camb. Phil. Soc., ,(1994), 27–36.[2]
P. Ashwin, J.W. Swift , The Dynamics of n Identical Oscillators with Symmetric Coupling , J. Nonlin. Sci., ,(1992), 69–108.[3] S. ˘Copar, S. ˘Zumer , Quaternions and Hybrid Nematic Disclinations , Proc. Royal Soc. A, , (2013), 1–10.[4]
A.P.S. Dias, R.C. Paiva , A Note on Hopf Bifurcation with Dihedral Group Symmetry , Glasgow Math. J., ,(2006), 41–51.[5] S. Dev, S. Verma , Leptogenesis in a Hybrid Texture Neutrino Mass Model , Mod. Phys. Lett., , (2010),2837–2848.[6] J. Funda, R.H. Taylor, R.P.Paul , On Homogeneous Transforms, Quaternions and Computational Efficiency ,IEEE Trans., , (1990), 382–387.[7] M. Golubitsky, M. Pivato, I. Stewart , Interior Symmetry and Local Bifurcation in Coupled Cell Networks ,Dyn. Syst, , (2004), 389–407.[8] M. Golubitsky, I. Stewart, D.G. Schaeffer , Singularities and Groups in Bifurcation Theory
II, AppliedMathematical Sciences , Springer–Verlag, (1988).[9] D.L. Johnson , Topics in the Theory of Group Presentations , Lecture Notes Series , Cambridge UniversityPress, (1980).[10] M. Krupa, I. Melbourne , Asymptotic Stability of Heteroclinic Cycles in Systems with Symmetry , ErgodicTheory Dyn. Syst., , (1995), 121–147.[11] A.C. Murza , Hopf Bifurcation and Heteroclinic Cycles in a Class of D –Equivariant Systems , Math. Rep., ,(2015), 369–383.[12] O. Podvigina , Stability and Bifurcations of Heteroclinic Cycles of Type Z , Nonlinearity, , (2012), 1887–1917.[13] O. Podvigina, P. Chossat , Simple Heteroclinic Cycles in R , Nonlinearity, , (2015), 901–926.[14] P. Stork , Statische Verzweigung in Gradientelfeldern mit Symetrien vom Komplexen oder QuaternionischenTyp mit Numerischer Behandlung , in Wissenschaftliche Beitr¨age aus europ¨aichen Hochschulen, Vol. 11, (1993),Verlag an der Lottbeck.[15]
X. Zhang , Global Structure of Quaternion Ppolynomial Differential Equations , Comm. Math. Phys., , (2011),301–316.
Adrian C. Murza, Institute of Mathematics “Simion Stoilow” of the Romanian Academy, CaleaGrivit¸ei 21, 010702 Bucharest, Romania
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