Pseudo-simple heteroclinic cycles in $R^4$
Pascal Chossat, Alexander Lohse, Olga Podvigina
TL;DR
This paper classifies pseudo-simple heteroclinic cycles in four-dimensional space with symmetry, introduces a method to construct systems with such cycles, and explores their stability and nearby dynamics.
Contribution
It provides a classification of symmetry groups admitting pseudo-simple cycles, introduces a constructive method for such systems, and analyzes their stability and periodic orbits.
Findings
Identified all finite subgroups of O(4) with pseudo-simple cycles.
Developed a constructive method to build equivariant systems with heteroclinic cycles.
Found subgroups with fragmentarily asymptotically stable pseudo-simple heteroclinic cycles.
Abstract
We study pseudo-simple heteroclinic cycles for a -equivariant system in with finite , and their nearby dynamics. In particular, in a first step towards a full classification - analogous to that which exists already for the class of simple cycles - we identify all finite subgroups of admitting pseudo-simple cycles. To this end we introduce a constructive method to build equivariant dynamical systems possessing a robust heteroclinic cycle. Extending a previous study we also investigate the existence of periodic orbits close to a pseudo-simple cycle, which depends on the symmetry groups of equilibria in the cycle. Moreover, we identify subgroups , , admitting fragmentarily asymptotically stable pseudo-simple heteroclinic cycles. (It has been previously shown that for pseudo-simple…
| # | group | order | # | group | order | # | group | order | ||
| 1 | 15 | 29 | 2880 | |||||||
| 2 | 16 | 30 | 7200 | |||||||
| 3 | 17 | 31 | 120 | |||||||
| 4 | 18 | 32 | 120 | |||||||
| 5 | 19 | 33 | nkr | |||||||
| 6 | 20 | 288 | ||||||||
| 7 | 21 | 24 | 34 | 2nkr | ||||||
| 8 | 22 | 96 | (mod 2) | |||||||
| 9 | 23 | 576 | 35 | 12 | ||||||
| 10 | 24 | 1440 | 36 | 24 | ||||||
| 11 | 25 | 1152 | 37 | 24 | ||||||
| 12 | 26 | 48 | 38 | 60 | ||||||
| 13 | 27 | 192 | 39 | 60 | ||||||
| 14 | 28 | 576 |
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Pseudo-simple heteroclinic cycles in
Pascal Chossat
Université Côte d’Azur - CNRS, Parc Valrose, 06108 Nice cedex, France
Alexander Lohse
University of Hamburg, Bundesstraße 55, 20146 Hamburg, Germany
Olga Podvigina
Institute of Earthquake Prediction Theory and Mathematical Geophysics, 84/32 Profsoyuznaya St, 117997 Moscow, Russian Federation
Abstract
We study pseudo-simple heteroclinic cycles for a -equivariant system in with finite , and their nearby dynamics. In particular, in a first step towards a full classification – analogous to that which exists already for the class of simple cycles – we identify all finite subgroups of admitting pseudo-simple cycles. To this end we introduce a constructive method to build equivariant dynamical systems possessing a robust heteroclinic cycle. Extending a previous study we also investigate the existence of periodic orbits close to a pseudo-simple cycle, which depends on the symmetry groups of equilibria in the cycle. Moreover, we identify subgroups , , admitting fragmentarily asymptotically stable pseudo-simple heteroclinic cycles. (It has been previously shown that for pseudo-simple cycles generically are completely unstable.) Finally, we study a generalized heteroclinic cycle, which involves a pseudo-simple cycle as a subset.
Keywords: equivariant dynamics, quaternions, heteroclinic cycle, periodic orbit, stability
Mathematics Subject Classification: 34C14, 34C37, 37C29, 37C75, 37C80, 37G15, 37G40
1 Introduction
A heteroclinic cycle is an invariant set of a dynamical system comprised of equilibria and heteroclinic orbits from to , with the convention . For several decades these objects have been of keen interest to the nonlinear science community. A heteroclinic cycle is associated with intermittent dynamics, where the system alternates between states of almost stationary behaviour and phases of quick change. It is well-known that a heteroclinic cycle can exist robustly in equivariant dynamical systems, i.e. persist under generic equivariant perturbations, namely when all heteroclinic orbits are saddle-sink connections in (flow-invariant) fixed-point subspaces. Robust heteroclinic cycles, their nearby dynamics and attraction properties have been thorougly studied, especially in low dimensions. See [4, 8] for a general overview. In , there are comparatively few possibilities for heteroclinic dynamics and these are rather well-understood. In , the situation is significantly more involved. We therefore consider systems
[TABLE]
where is a smooth map that is equivariant with respect to the action of a finite group , i.e.
[TABLE]
In this setting, much attention has been paid to so-called simple cycles, see e.g. [1, 2, 9, 10], for which (i) all connections lie in two-dimensional fixed-point spaces with , and (ii) the cycle intersects each connected component of at most once. This definition was introduced by [10], who also suggested several examples of subgroups of that admit such a cycle (in the sense that there is an open set of -equivariant vector fields possessing such an invariant set). The classification of simple cycles was completed in [17, 18] (for homoclinic cycles) and finally in [15] by finding all groups admitting such a cycle. In [15] it was also discovered that the original definition of simple cycles from [10] implicitly assumed a condition on the isotypic decomposition of with respect to the isotropy subgroup of an equilibrium, see subsection 2.1 for details. This prompted them to define pseudo-simple heteroclinic cycles as those satisfying (i) and (ii) above, but not this implicit condition.
It is the primary aim of the present paper to carry out a systematic study of pseudo-simple cycles in , by establishing a complete list of all groups that admit such a cycle. This is done in a similar fashion to the classification of simple cycles in [15], by using a quaternionic approach to describe finite subgroups of . First examples for pseudo-simple cycles were investigated in [15, 16]. The latter of those also addressed stability issues: it was shown that a pseudo-simple cycle with is generically completely unstable, while for the case a cycle displaying a weak form of stability, called fragmentary asymptotic stability, was found. A fragmentarily asymptotically stable (f.a.s.) cycle has a positive measure basin of attraction that does not necessarily include a full neighbourhood of the cycle. We extend this stability study by showing an example of group which admits an asymptotically stable generalized heteroclinic cycle and pseudo-simple subcycles that are f.a.s.. Moreover, we look at the dynamics near a pseudo-simple cycle and discover that asymptotically stable periodic orbits may bifurcate from it. Whether or not this happens depends on the isotropy subgroup , of equilibria comprising the cycle. The case was already considered in [16]. We illustrate our more general results by numerical simulations for an example with in the case .
This paper is organized as follows. Section 2 recalls background information on (pseudo-simple) heteroclinic cycles and useful properties of quaternions as a means to describe finite subgroups of . Then, in section 3 we give conditions that allow us to decide whether or not such a group admits pseudo-simple heteroclinic cycles. Section 4 contains the statement and proofs of theorems 1 and 2, which use the previous results to list all subgroups of admitting pseudo-simple heteroclinic cycles. The proof of theorem 1 relies on properties of finite subgroups of that are given in appendices A-C. In section 5 we investigate the existence of asymptotically stable periodic orbits close to a pseudo-simple cycle, depending on the symmetry groups , of equilibria. The cases and are covered by theorems 3 and 4, respectively. In section 6 we employ the ideas of the previous sections to provide a numerical example of a pseudo-simple cycle with a nearby attracting periodic orbit. Finally, in section 7 for a family of subgroups we construct a generalized heteroclinic cycle (i.e., a cycle with multidimensional connection(s)) and prove conditions for its asymptotic stability in theorem 5. This cycle involves as a subset a pseudo-simple heteroclinic cycle, that can be fragmentarily asymptotically stable. Section 8 concludes and identifies possible continuations of this study. The appendices contain additional information on subgroups of that is relevant for the proof of theorem 1.
2 Background
Here we briefly review basic concepts and terminology for pseudo-simple heteroclinic cycles and the quaternionic approach to describing subgroups of as needed in this paper.
2.1 Pseudo-simple heteroclinic cycles
In this subsection we give the precise framework in which we investigate robust heteroclinic cycles and the associated dynamics. Given an equivariant system (1) with finite recall that for the isotropy subgroup of is the subgroup of all elements in that fix . On the other hand, given a subgroup we denote by its fixed point space, i.e. the space of points in that are fixed by all elements of .
Let be hyperbolic equilibria of a system (1) with stable and unstable manifolds and , respectively. Also, let for be connections between them, where we set . Then the union of equilibria and connecting trajectories is called a heteroclinic cycle. Following [9] we say it is structurally stable or robust if for all there are subgroups such that is a sink in and is contained in . We also employ the established notation , with a subgroup . As usual we divide the eigenvalues of the Jacobian into radial (eigenspace belonging to ), contracting (belonging to ), expanding (belonging to ) and transverse (all others), where we write for a complementary subspace of in . In accordance with [9] our interest lies in cycles where
- (H1)
for all ,
- (H2)
the heteroclinic cycle intersects each connected component of at most once.
Then there is one eigenvalue of each type and we denote the corresponding contracting, expanding and transverse eigenspaces of by , and , respectively. In [15] it is shown that under these conditions there are three possibilities for the unique -isotypic decomposition of :
- (1)
2. (2)
, where is two-dimensional 3. (3)
, where is two-dimensional
Here denotes the orthogonal direct sum. This prompts the following definition.
Definition 1** ([15])**
We call a heteroclinic cycle satisfying conditions (H1) and (H2) above simple if case 1 holds true for all , and pseudo-simple otherwise.
Remark 1
In case 1 the group acts as on each one-dimensional component other than and (which is always the case if ) or . In cases 2 and 3 the group acts on the two-dimensional isotypic component as a dihedral group in for some and (always for ) or . For in case 2 the element acts as a -fold rotation on and trivially on , while acts as on and trivially on . In case 3 the element acts as a -fold rotation on and trivially on , while acts as on and trivially on .
Remark 2
The existence of a two-dimensional isotypic component implies that in case 2 the contracting and transverse eigenvalues are equal () and the associated eigenspace is two-dimensional, while in case 3 the expanding and transverse eigenvalues are equal () and the associated eigenspace is two-dimensional. Hence, we say that has a multiple contracting or expanding eigenvalue in cases 2 or 3, respectively.
We are interested in identifying all subgroups of that admit pseudo-simple heteroclinic cycles in the following sense. For simple cycles this task has been achieved step by step in [10, 15, 17, 18].
Definition 2** ([15])**
We say that a subgroup of O() admits (pseudo-)simple heteroclinic cycles if there exists an open subset of the set of smooth -equivariant vector fields in , such that all vector fields in this subset possess a (pseudo-)simple heteroclinic cycle.
In order to establish the existence of a heteroclinic cycle it is sufficient to find a sequence of connections with some finite order , that is minimal in the sense that no satisfy for any .
Definition 3** ([16])**
Such a sequence together with the element is called a building block of the heteroclinic cycle.
Remark 3
Heteroclinic cycle in an equivariant system can be decomposed as a union of building blocks. Usually, it is tacitly assumed that all blocks in such a decomposition can be obtained from just one building block by applying the associated symmetry . We also make this assumption.
2.2 Asymptotic stability
Given a heteroclinic cycle and writing the flow of (1) as , the -basin of attraction of is the set
[TABLE]
Definition 4
A heteroclinic cycle is asymptotically stable if for any there exists such that , where denotes -neighbourhood of .
Definition 5
A heteroclinic cycle is completely unstable if there exists such that , where denotes Lebesgue measure on .
Definition 6
A heteroclinic cycle is fragmentarily asymptotically stable if for any .
2.3 Quaternions and subgroups of
We briefly recall some information on quaternions and their relation to subgroups of , mainly following the notation and exposition in [6, Chapter 3]. For a more detailed background on this in general, and in the context of heteroclinic cycles, we also refer the reader to [5] and [15], respectively.
A quaternion may be described by four real numbers as . With the convention , , and any can be written as . We denote the conjugate of as . Multiplication is defined in the standard way through the rules , , , , such that for we have
[TABLE]
By we denote the multiplicative group of unit quaternions, with identity element . There is a 2-to-1 homomorphism from to , relating to the map , which is a rotation in the three-dimensional space of points . Any finite subgroup of then falls into one of the following cases, which are pre-images of the respective subgroups of under this homomorphism:
[TABLE]
where . Double parenthesis denote all even permutations of quantities within the parenthesis.
The four numbers can be regarded as Euclidean coordinates of a point in . For any pair of unit quaternions , the transformation is a rotation in , i.e. an element of the group . The mapping that relates the pair with the rotation is a homomorphism onto, whose kernel consists of two elements, and ; thus the homomorphism is two to one. Therefore, a finite subgroup of is a subgroup of a product of two finite subgroups of . Following [6] we write for the group . The isomorphism between and may not be unique and different isomorphisms give rise to different subgroups of . The complete list of finite subgroups of is given in table 1, where the subscript distinguishes subgroups obtained by different isomorphisms for and prime to . This is explained in more detail in [6, Chapter 3] and in section 2.2 of [15].
The superscript is employed to denote subgroups of where the isomorphism between the quotient groups and is not the identity. The group , isomorphic to , involves the elements , where . The groups 1-32 contain the central rotation , and the groups 33-39 do not.
A reflection in can be expressed in the quaternionic presentation as , where and is a pair of unit quaternions. We write this reflection as . The transformations and are respectively the axial reflection in the -axis (leaving unchanged all vectors parallel to the axis and reversing all those perpendicular to it) and reflection through the hyperplane orthogonal to the vector .
2.4 Lemmas
In this subsection we recall lemmas 1-5 from [6, 13, 15] and prove lemmas 6 and 7. They provide basic geometric information that is used to prove theorem 1 in section 4.
Lemma 1** (see proof in [13])**
Let and be two planes in and , , be the elements of which act on as identity, and on as , and , where is the homomorphism defined in the previous subsection. Denote by and the first components of the respective quaternion products. The planes and intersect if and only if and is the angle between the planes.
Lemma 2** (see proof in [15])**
Let and be two planes in , , O() is a plane reflection about and O() maps into . Suppose that and are elements of a finite subgroup O(). Then , where .
Lemma 3** (see proof in [15])**
*Consider , .
Then if and only if .*
Lemma 4** (see proof in [15])**
Consider , where and . Then .
Lemma 5** (see proof in [6])**
If and , then the transformation is a rotation of angles in a pair of absolutely perpendicular planes.
Lemma 6
If , , admits pseudo-simple heteroclinic cycles then and , where .
**Proof: **Let and , where is the group discussed in remark 1. Existence of at least one such follows from definitions 1 and 2. Lemma 5 implies that the order of the elements and is , while the order of and is 2. Since is a homomorphism, and . QED
Lemma 7
If admits pseudo-simple heteroclinic cycles, then it has a symmetry axis , where is a maximal isotropy subgroup such that with .
The proof follows from definitions 1 and 2 and remark 1.
3 Construction of a -equivariant system, possessing
a heteroclinic cycle
In this section we prove the following lemma:
Lemma 8
- (i)
If for a given finite subgroup there exist two sequences of isotropy subgroups , , , and an element satisfying the following conditions:
- C1.
Denote and . Then and for all .
- C2.
For , and are not conjugate.
- C3.
For , , and . We set .
- C4.
For all , the subspaces , and belong to different isotypic components in the isotypic decomposition of in .
- C5.
* with for at least one .*
Consider , the dihedral group of order , where we write for a trivial or for . Let be the number of isotropy types of axes that are not fixed by an element of and , .
- C6.
Depending on one of the following takes place:
(a) if then either is even and the groups , are not conjugate, or is odd;
(b) if then the groups and are not conjugate and one of or is conjugate to ;
(c) if then and are conjugate to and .
then admits pseudo-simple heteroclinic cycles.
- (ii)
If admits pseudo-simple heteroclinic cycles, then there are two sequences of isotropy subgroups , , , where , and an element , satisfying conditions C1,C3,C4, C5 and C6∗.
- C6∗.
If , then and are not conjugate for any , .
In [15] we proved a similar lemma, stating necessary and sufficient conditions for a group ) to admit simple heteroclinic cycles. As noted in [16], with minor modifications the proof can be used to prove sufficient conditions for a group to admit pseudo-simple heteroclinic cycles. Here, our proof of lemma 8 employs a different idea. We explicitly build a -equivariant dynamical system possessing a pseudo-simple heteroclinic cycle and prove that the cycle persists under small -equivariant perturbations.
**Proof: ** Starting with the proof of (i), we show that for any group satisfying conditions C1-C6 there is a vector field f such that the associated dynamics possess a heteroclinic cycle between equilibria in with connections in the fixed-point planes .
As a first step, for each plane , , that contains the axes and (in agreement with C3, ), we define a two-dimensional vector field , which in the polar coordinates is:
[TABLE]
where are the angles of all fixed-point axes in other than , and the angle of is . For the flow of each of these axes is invariant and has an equilibrium which is attracting along the direction of . Moreover, there are heteroclinic connections between equilibria on neighbouring axes, since the sign of changes when an axis is crossed.
We extend the vector fields to as follows: Denote by and the projections onto the plane and its orthogonal complement in , respectively. We set
[TABLE]
with a positive constant (to be chosen sufficiently large later). The vector field is then defined as
[TABLE]
where and is the normalizer of in .
As the second step, we show that the system
[TABLE]
possesses steady states , , and heteroclinic connections , where and . Note, that by construction the system (7) is -equivariant, which implies invariance of the axes and planes .
The system (7) restricted to is
[TABLE]
where is the coordinate along and is the angle between and . We split the sum in (8) into two, for and , and write
[TABLE]
where is the number of planes that contain . Hence, for sufficiently large positive there exists an equilibrium with , attracting in .
To prove existence of a heteroclinic connection , we consider a sector in between and , where with such that there are no invariant axes between and . Existence of such a sector follows from C6. (In case (a) the axes and are invariant axes of , they are the only invariant axes in . In case (b) invariant axes of alternate with (symmetric copies of) . In case (c) there is one and one between any two neighbouring invariant axes of .) We choose a small number and divide this sector into three subsets, as sketched in Figure 1:
- •
: a strip of width near
- •
: a strip of width near
- •
: the rest of the sector
We now consider the dynamics of system (7) in each of these regions.
- •
For we distinguish three cases: (a) is a simple equilibrium, i.e. the isotypic decomposition of w.r.t. has only 1D components, (b) is a pseudo-simple equilibrium, i.e. the isotypic decomposition of w.r.t. has a 2D component, and the component is the contracting eigenspace, (c) is a pseudo-simple equilibrium with 2D expanding eigenspace.
In we employ the coordinates . Choosing sufficiently small and sufficiently large, to approximate the dynamics near , we take into account only leading terms in and and in (7) we omit the terms corresponding to the planes that do not contain . The condition C4 implies that in case (a) the axis belongs to planes and only; in case (b) it also belongs to several symmetric copies of ; in case (c) to , and several symmetric copies of . In case (a) we have
[TABLE]
and since . In case (b), is orthogonal to and its symmetric copies. Hence, near we have
[TABLE]
In case (c), assuming that there are symmetric copies of containing , hence the angles between neighbouring planes are , we approximate the dynamics in as
[TABLE]
Thus, in all cases (a)-(c) the equilibrium is attracting in and we have in , so trajectories leave and enter . At the point of entrance, .
- •
The region is bounded away from the axes and . Then, for any given there exists such that for all , the dynamics away from the fixed-point axes are essentially those of . Namely, the trajectories through are attracted by and at the entrance point .
- •
In , for sufficiently small and sufficiently large all trajectories with are attracted by by arguments that are analogous to those for .
So we have shown that for the dynamics of (7) in each there is a connection . Taking into account their symmetric copies we obtain a sequence of connections forming a building block of a heteroclinic cycle. The cycle is pseudo-simple because of C5, and robust since by construction all connections lie in fixed-point subspaces that persist under equivariant perturbations.
For (ii) we note that necessity of conditions C1, C3, C4, C5 and C6∗ follows directly from the definition of pseudo-simple heteroclinic cycles. The two sequences of subgroups are found by choosing as the isotropy subgroups of the planes and as the isotropy subgroups of the equilibria . QED
In the following lemma we state sufficient conditions for a group to admit pseudo-simple cycles, that are slightly different from the ones proven in lemma 8(i). The difference is that in lemma 9 the subgroups can be conjugate in . Since the proof of lemma 9 is similar to the one of lemma 8(i), it is omitted.
Lemma 9
If for a given finite subgroup , with , there exist two sequences of isotropy subgroups , , , and an element satisfying conditions C1, C2’, C3, C4, C5 and C6’, where
- C2’.
* and are not conjugate for any .*
- C6’.
For any , there exists a sector in , bounded by and , that does not contain any other isotropy axes of .
then admits pseudo-simple heteroclinic cycles.
Remark 4
*Note that lemma 9 can be generalised to as follows:
If for a given finite subgroup , with , there exist two sequences of isotropy subgroups , , , and an element satisfying conditions C1, C2’, C3, C4 and C6’, then admits heteroclinic cycles.*
4 List of groups
4.1 The groups in
In this subsection we prove theorem 1 that exhibits all finite subgroups of , admitting robust pseudo-simple heteroclinic cycles. The proof employs lemmas 8(i) and 9, that give sufficient conditions for to admit pseudo-simple cycles, and lemma 8(ii) that gives necessary conditions. The lemmas allow us to split subgroups of into two classes, those admitting and those not admitting pseudo-simple heteroclinic cycles. Similarly to [15], we use the quaternionic presentation for subgroups of , see subsection 2.3. Appendices A-C contain detailed information on the geometry of various subgroups of which are used for proving the theorem.
Theorem 1
A group admits pseudo-simple heteroclinic cycles, if and only if it is one of those listed in table 2.
To prove the theorem, we proceed in four steps:
In step [i], using lemmas 6 and 7 we identify subgroups of , that do not satisfy necessary conditions for existence of pseudo-simple heteroclinic cycles stated in the lemmas. The groups 1-9 and 33 (see table 1) do not satisfy conditions of lemma 6. The groups 14, 20-32 and 35-39 do not satisfy conditions of lemma 7. The groups 10-13, 15-19 and 34 should satisfy extra conditions on , , , and .
In step [ii], using lemmas 1-5 and the correspondence between and (see section 2.3 ), we identify all subgroups such that , which are elements of groups found in step [i]. The results are listed in appendix A.
In step [iii], using the results obtained at step [ii], we determine the (maximal) conjugacy classes of subgroups of , isomorphic to , which have two-dimensional fixed-point subspaces and (maximal) conjugacy classes of such that . The results are listed in appendix B.
Finally, in step [iv], using the list in appendix B, we identify all groups that possess sequences of subgroups and satisfying conditions C1-C6 of lemmas 8(i) or 9 (they are presented in appendix C). All the other groups do not have sequences satisfying conditions C1,C3, C4,C5 and C6∗. In fact, the only groups satisfying the conditions of lemma 9, but not those of lemma 8(i), are and with odd .
**Proof of the theorem
**Step [i]
The groups that satisfy conditions of lemmas 6 and 7 are:
[TABLE]
Below we show that the groups and admit heteroclinic cycles, while the groups , where at least one of or is odd, do not. For other groups the proofs are similar and we omit them.
The group .
- [ii
] The group (see (3) ) is comprised of the elements
[TABLE]
The pairs satisfy , , where all possible combinations are elements of the group. If both and are odd, then the elements satisfying are
[TABLE]
where , , , , , and . The elements are plane reflections, while is a rotation by in the plane orthogonal to . For even the group possesses an additional set of plane reflections
[TABLE]
- [iii
] In the group the elements split into two conjugacy classes, corresponding to odd and even . Since , in the case when both and are odd the group has three maximal isotropy types of subgroups satisfying . The subgroups are
[TABLE]
The subgroups and are conjugate, e.g. by
.
For any plane the only symmetry axes are the intersections with . The axes are conjugate by . Therefore, the group has two maximal isotropy types of subgroups satisfying :
[TABLE]
Since planes do not satisfy the condition C4∗ and the remaining planes and do not intersect, the group with odd does not admit heteroclinic cycles.
In the case when is odd and is even, a plane fixed by the reflection does not intersect with any of and (see lemma 1). Moreover, does dot intersect with for any . Similar arguments apply when is even and is odd. Therefore, the group does not admit heteroclinic cycles when at least one of or is odd.
The group .
- [ii
] The group can be decomposed as , see (3). Therefore, the group is comprised of the following elements:
[TABLE]
where and . For odd the elements satisfying are
[TABLE]
where . Here are plane reflections and are rotations by in the planes orthogonal to . For even the group possesses an additional set of plane reflections
[TABLE]
- [iii
] Since and in the elements and are conjugate, for odd all are conjugate in . The elements
split into two conjugacy classes, depending on whether is even or odd. Hence, for odd the group has three maximal isotropy types of subgroups satisfying :
[TABLE]
Each contains isotropy axes, each of them are intersections with three , where . Hence, the isotropy groups of symmetry axes can be written as
[TABLE]
They split into two isotropy types, depending on whether is even or odd. Any plane contains four isotropy axes which are intersections with . Since (this can be checked directly using the list (11) ), all four isotropy axes are of different types. Therefore, the group has four types of isotropy subgroups (14) safisfying , corresponding to odd and even and .
In the case when is even there exist five isotropy types of subgroups satisfying :
[TABLE]
A plane orthogonally intersects with the ones (and also with ), hence for odd and even the isotropy axes are different. Namely, for odd they are
[TABLE]
while for even the second set of isotropy axes is
[TABLE]
- [iv
] According to (iii), for odd the group does not have isotropy subgroups satisfying conditions C1-C6 of lemma 8(i). Let us show that we can find subgroups satisfying conditions of lemma 9. Set
[TABLE]
By construction and due to (14) and (13), the subgroups satisfy conditions C1,C2’,C3,C4 and C5.
To show that satisfies condition C6’, we recall (see (iii)) that the plane involves four isotropy axes, all non-conjugate, that are intersections with , , and . To determine the angles between axes, we use lemmas 4 and 5. By lemma 4,
, where is a plane reflection about a plane that intersects with orthogonally. Therefore, is -invariant. Composition of two reflections about axes, intersecting with the angle , is a rotation by . Since , by lemma 5 the angle in between the lines of intersections with and is , while the lines of intersections with and are orthogonal. Hence, in no other isotropy axes belong to the smaller sector bounded by and . Similarly, it can be shown that the condition C6’ holds true for as well.
For even we apply lemma 8. We choose
[TABLE]
[TABLE]
[TABLE]
which together with satisfy conditions C1-C6, as follows from (15), (16) and (17).
The group .
- [ii
] The group is comprised of the pairs , where and . Since for odd all elements are conjugate, the group has the following the elements satisfying :
[TABLE]
where , , and . By and we denote elements that are conjugate in . Here and are plane reflections, and are rotations by , is a rotation by and is a rotation by . For even the group possesses additional set of plane reflections:
[TABLE]
- [iii
] For odd all plane reflections are conjugate in . The rotations by are conjugate, the rotations by and are conjugate as well. Hence, the group has three maximal isotropy types of subgroups satisfying :
[TABLE]
Each contains isotropy axes, each of them is an (orthogonal) intersection with three . Each contains isotropy axes, each of them in an (orthogonal) intersection with five . Hence, the isotropy groups of symmetry axes can be written as
[TABLE]
In the case when is even there exist five isotropy types of subgroups satisfying :
[TABLE]
Each of the planes has twelve isotropy axes. Four of them (of two isotropy types) are orthogonal intersections with , therefore . The other eight axes (of two isotropy types) are intersections with and . The respective isotropy subgroups are different for odd or even , as stated in appendix B. A plane or involves two isotropy types (with odd or even ) of symmetry axes, which are intersections with .
- [iv
] For odd we show that the groups
[TABLE]
and satisfy conditions of lemma 9. By construction and due to (20) and (21), the subgroups satisfy conditions C1,C2’,C3,C4 and C5.
Consider . Denote by , and the angles between the intersection with and the following three axes: intersections with , and , respectively. By lemmas 4 and 5,
[TABLE]
which implies that .
Since for old and due to (21), in the angle between the intersection with and any other isotropy axes is not smaller that . Therefore, satisfies the condition C6’. Similar arguments imply that this condition is satisfied for as well.
Since for even the elements and are not conjugate, the set (23) satisfies conditions C1-C6 of lemma 8.
QED
4.2 The groups in but not in
In this subsection we prove theorem 2 that completes the list of finite subgroups of , admitting pseudo-simple heteroclinic cycles.
A reflection in can be expressed in the quaternionic presentation as , where and is a pair of unit quaternions (see [6, 15]). We write this reflection as . The transformations and are respectively the reflections about the axis and through the hyperplane orthogonal to the vector .
A group , , can be decomposed as , where and . If is finite, then in the quaternionic form of
[TABLE]
Theorem 2
A group ,
[TABLE]
admits pseudo-simple heteroclinic cycles, if and only if and are listed in table 3.
**Proof: **Lemma 8 in [15] states that if a group admits simple heteroclinic cycles, then so does . By similar arguments the same holds true for pseudo-simple heteroclinic cycles. Therefore (see lemma 8), the group has two sequences of isotropy subgroups , , , satisfying conditions C1,C3,C4,C5 and C6∗. Let be the subgroup satisfying C5, i.e. with . An element , , maps either to itself, or to another with .
First, we assume the existence of , such that . Hence, there exists which is a reflection through a hyperplane that contains . Let the hyperplane be spanned by , and and . The hyperplane is mapped by elements of to
[TABLE]
Any isotropy plane of , that intersects with , is . An isotropy plane , such that with , is orthogonal to all hyperplanes (26). Therefore (if such an isotropy plane exists), it is . Any other isotropy plane of (different from , and ) either intersects all hyperplanes (26) orthogonally, or the line of intersection belongs to or . Since there is no isotropy plane that satisfies these conditions, we conclude that the only isotropy planes of are , and . The groups listed in table 2 satisfying these conditions and (24) are , . The element acting as a reflection through is .
Second, we assume that there is no , , such that . Therefore, satisfies , where with , and the subgroups and are not conjugate in . The only groups in table 2 that contain such and are , . Moreover, (see appendix C). The element maps a symmetry axis in to a symmetry axis in . For definiteness, we assume that maps to , where according to the appendices
, ,
, , , and . Such is . QED
Remark 5
A heteroclinic cycle in a -equivariant system, where in the decomposition (25) and , in general is completely unstable. The proof follows the same arguments as the proof of theorem 1 in [16]. Similarly, the conditions for existence of a nearby periodic orbit are the ones given in theorems 3 and 4 in section 5 below.
Remark 6
A heteroclinic cycle in a -equivariant system, where in the decomposition (25) and , can be fragmentarily asymptotically stable. The conditions for stability can be obtained by algebra, which is standard (see, e.g., theorem 3 in [16]) but tedious; we do not present it here.
5 Existence of nearby periodic orbits when
As shown in [16], despite complete instability of a pseudo-simple heteroclinic cycle in a -equivariant system for , trajectories staying in a small neighbourhood of a pseudo-simple cycle for all can possibly exist. Namely, it was proven ibid that in a one-parameter dynamical system an asymptotically stable periodic orbit can bifurcate from a cycle. More specifically, in their example such an asymptotically stable periodic orbit exists as long as a double positive eigenvalue is sufficiently small. Building blocks of the considered cycles were comprised of two equilibria, whose isotropy groups were isomorphic to . One of these equilibria had a multiple expanding eigenvalue, while the other equilibrium had a multiple contracting one. In this section we prove that similar periodic orbits can bifurcate in a more general setup – we do not restrict the number of equilibria in a building block (note that building block of a pseudo-simple cycle in is comprised of at least two equilibria) and assume that their isotropy groups are isomorphic to with . However, we assume that building block of a heteroclinic cycle involves only one equilibrium with a multiple expanding eigenvalue. In the case of several such equilibria, the bifurcation of a periodic orbit has codimension two or higher, which is beyond the scope of this paper. By contrast, no such periodic orbits bifurcate in a codimension one bifurcation if a building block involves an equilibrium with the isotropy group with .
5.1 The case and
Consider the -equivariant system
[TABLE]
and is a smooth map. We assume that the system possesses a pseudo-simple heteroclinic cycle with a building block . By , and we denote the non-radial eigenvalues of , . Let be an equilibrium with a two-dimensional expanding eigenspace (hence, ) and a symmetry group , or 4, acting naturally on the expanding eigenspace, and all other equilibria have one-dimensional expanding eigenspaces. A general -equivariant dynamical system in in the leading order is (for ) and (). A necessary condition for existence of a heteroclinic trajectory along the direction of real is that and or (for or , respectively). Suppose that there exists such that
- (i)
for and for ;
- (ii)
for any there exist heteroclinic connections , for all , where .
Denote by the group orbit of heteroclinic connections :
[TABLE]
is the product , where we set if , and (for ) or (for ).
Theorem 3
If then there exist and , such that for any almost all trajectories escape from as .
If then generically there exists a periodic orbit bifurcating from at . To be more precise, for any we can find such that for all the system (27) possesses an asymptotically stable periodic orbit that belongs to .
We give the proof only for , for it can be obtained by a simple modification combined with results of [16]. Since it follows closely the proof of theorem 2 ibid, some details are omitted and the reader is referred to that paper. We first formulate lemma 10 below, describing properties of trajectories of a generic -equivariant systems in , which in the leading order is
[TABLE]
In polar coordinates, , it takes the form
[TABLE]
We assume that
[TABLE]
The system has four invariant axes with , . The two axes with even are symmetric images of one another, as are the two axes with odd . In case there are four equilibria that are not at the origin with and , . We consider the system in the sector , the complement part of is related to this sector by symmetries of the group .
Trajectories of the system satisfy
[TABLE]
where we have denoted . Re-writing this equation as
[TABLE]
multiplying it by and integrating, we obtain that
[TABLE]
which implies that
[TABLE]
for the trajectory through the point .
Lemma 10
Let denote the time it takes the trajectory of the system (29),(30) starting at to reach and denote the value of at . Then
- (i)
* satisfies*
[TABLE]
- (ii)
* satisfies*
[TABLE]
- (iii)
* satisfies*
[TABLE]
- (iv)
Given , and , for sufficiently small and
[TABLE]
The proof is similar to the proof of lemma 3(i-iv) in [16] and is omitted.
**Proof of the theorem
**As usual, we approximate trajectories in the vicinity of the cycle by superposition of local and global maps, and , respectively, where and are cross sections transversal to the incoming and outgoing connections at an equilibrium . We consider , where the is the symmetry in the definition of a building block. Since the expanding eigenspace of is two-dimensional, the contracting eigenspace of is two-dimensional as well. By the assumption of the theorem, other equilibria in the cycle have one-dimensional expanding and contracting eigenspaces. We employ the coordinates in and in , similarly to [16]. We also employ the coordinates and , in and , respectively, such that , , and . In the leading order the maps is
[TABLE]
which in polar coordinates takes the form
[TABLE]
The maps , , are
[TABLE]
(Here superscripts indicate coordinates in or . Below, where it does not create ambiguity, we do not use superscripts.) In the leading order the map is
[TABLE]
where generically for . The maps , , are
[TABLE]
Because of (i), for small the expanding eigenvalue of depends linearly on , therefore without restriction of generality we can assume that . Generically, all other eigenvalues and coefficients in the expressions for local and global maps do not vanish for sufficiently small and are of the order of one. We assume them to be constants independent of . From (ii), the eigenvalues satisfy , and .
For small enough , in the scaled neighbourhoods the restriction of the system to the unstable manifold of in the leading order is , where we have denoted . We assume that the local bases near and are chosen in such a way that the heteroclinic connections and go along the directions for both . In the complement subspace the system is approximated by the contractions and . In terms of the functions and introduced in lemma 10, the map is
[TABLE]
According to lemma 10(iv), for small and
[TABLE]
which implies that the superposition can be approximated as , where and the constants and depend on , , and eigenvalues of , . For small we have . Taking into account (36), (38) and lemma 10(iii), we obtain that
[TABLE]
where we have denoted , and .
(a) From (40), the -component of satisfies
[TABLE]
hence if then for any the iterates with initial satisfy for sufficiently large .
(b) Assume that . Existence and stability of a fixed point of the map (40) for small can be proven by the same arguments as employed to prove theorem 2(b) in [16]. We omit the proof. The fixed point can be approximated by . This fixed point is an intersection of a periodic orbit with . The distance from to depends on as , therefore the trajectory approaches as . QED
5.2 The case ,
In this subsection we prove that a bifurcation of a periodic orbit, that was discussed in the previous subsection, does not take place for :
Theorem 4
Suppose that for the system (27) possesses a pseudo-simple heteroclinic cycle , where has a two-dimensional expanding eigenspace with the associated eigenvalue and the symmetry group , , acting naturally on the expanding eigenspace. There exist and , such that for any almost all trajectories of the system (27), such that , satisfy for some . By we denoted the distance between a point and a set.
**Proof: **Similarly to the proof of theorem 1 in [16], we consider the map and prove existence of such that
[TABLE]
where
[TABLE]
Equation (41) shows that all points in are mapped outside , which implies the statement of the theorem.
The maps and are the same as for the system, they are given by (36) and (38), respectively. In (38) generically for . Moreover, there exist and , such that for all sufficiently small and (recall that is the distance from and to ).
For small enough , in a -neighbourhoods of the restriction of the system to the unstable manifold of in the leading order is , where . In polar coordinates the system takes the form , , which implies that the map satisfies and
[TABLE]
We choose and set
[TABLE]
Any satisfies , therefore (36) and (43) imply that . Hence, due to (38),(42) and (43), for any . The steady state has symmetric copies (under the action of symmetries ) of the heteroclinic connection which belong to the hyperplanes with some integer ’s. Due to (42) and (43), the distance of to any of these hyperplanes is larger than , which implies (41). QED
6 Example: periodic orbit near a heteroclinic cycle in
a -equivariant system
We proved (see section 5) that an attracting periodic orbit can exist near a pseudo-simple heteroclinic cycle if the isotropy subgroup of one of its equilibria is or . For the case when the isotropy subgroup is , examples of -equivariant systems possessing periodic orbit in a neighbourhood of a heteroclinic cycle were given in [16] for and . The vector fields considered ibid were third order normal forms commuting with the considered actions of . Here we present a numerical example of a heteroclinic cycle with a nearby attracting periodic orbit, where the isotropy subgroup of an equilibrium is and a -equivariant vector field is constructed using ideas employed in the proofs of lemma 8 and theorem 3.
We consider a -equivariant dynamical system where (recall that the quaternionic group is usually denoted by ). The elements of are:
[TABLE]
The group has five isotropy types of subgroups satisfying (see appendix A). In agreement with appendix C, we take and . For convenience, we use different notation for generating elements. Namely, we write and , where
[TABLE]
[TABLE]
The action on of (some) elements of is
[TABLE]
The isotropy planes can be labelled as follows:
[TABLE]
hence there exist two different planes with and eight different planes corresponding to . A plane contain four symmetry axes of two isotropy types with isotropy groups of the axes isomorphic to . An axis is an intersection of with two planes (and also with two other planes fixed by , that is irrelevant), namely intersects with and . The axes split into two isotropy classes, with odd or even . A plane contains two isotropy axes which are intersections with and .
We choose , and, in agreement with (4), set
[TABLE]
where and in and and in . Hence, is unstable in and stable in ; is stable in and unstable in . Following the proof of lemma 8, we construct the system (5)-(7) that possesses a heteroclinic cycle with a building block , where . In agreement with theorem 1 in [16], the cycle is not asymptotically stable, hence trajectories starting near the cycle escape from it (see fig. 2(a)).
Theorem 3 states that a periodic orbit exists near a heteroclinic cycle with if the multiple expanding eigenvalue is sufficiently small and (recall that , and are the coefficients of the system (28) ). To be more precise, in the proof we use the fact that the ratio is small. Therefore, we introduce a modified system
[TABLE]
[TABLE]
and .
In a small neighbourhood of the projection of the local field (46) into the plane is
[TABLE]
Comparing the above expression with (28), we obtain that
[TABLE]
If the coefficient in (47) is sufficiently large, then by the same arguments as applied in the proof in lemma 8, the system (46) possesses the heteroclinic cycle . Theorem 3 indicates that for sufficiently large there exists a stable periodic orbit close to the cycle. Therefore, we set
[TABLE]
In agreement with our arguments, the system (46)-(48) has an attracting periodic orbit near the heteroclinic cycle, as shown on fig. 2(b).
7 An example of stability when
In this section we show that a family of subgroups , , admits heteroclinic cycles involving multidimensional heteroclinic orbits. Following [3], we call such heteroclinic cycles generalized. We derive conditions for asymptotic stability of such generalized cycle and show that it involves as a subset a pseudo-simple heteroclinic cycle, that can be fragmentarily asymptotically stable. Numerical studies indicate that addition of small perturbation that breaks an symmetry can result on emergence of asymptotically stable periodic orbit or on chaotic dynamics in the vicinity of a pseudo-simple heteroclinic cycle.
We shall in fact consider a class of subgroups of defined as follows.
Let and . Fix an integer and let be the group generated by the transformations
[TABLE]
(Choosing coordinates and , we obtain that in quaternionic presentention the subgroup of is , in agreement with theorem 2.) This group action decomposes into the direct sum of three irreducible representations of the dihedral group :
(i) the trivial representation acting on the component ,
(ii) the one-dimensional representation acting on by ,
(iii) the two-dimensional natural representation of acting on .
There are four types of fixed-point subspaces for this action:
- •
,
- •
,
- •
,
- •
,
- •
.
When is even there are two more types of invariant subspaces:
- •
,
- •
.
Note that is fixed by . When is odd and have symmetric copies , , . When is even each of , , , has symmetric copies.
It can be shown that for an open set of -equivariant vector fields, there exists an equilibrium on the negative semi-axis in , an equilibrium on the positive semi-axis, and heteroclinic orbits lying in the planes and and realizing a cycle between and . Moreover this cycle is pseudo-simple due to the action of the rotation on the plane , which forces the eigenvalues along the direction in to be double. To fix ideas we assume the double eigenvalue is stable at and unstable at . In order to study the stability of this pseudo-simple cycle we shall exploit a property that was observed in the case in [16] and appears to also occur when . First, the two dimensional unstable manifold at lies entirely in the invariant subspace , which also contains the axis . Second, for an open set of vector fields any orbit on this unstable manifold lies in the stable manifold of , hence realizing a two dimensional manifold of saddle-sink connections in . Therefore the pseudo-simple heteroclinic cycle is part of a cycle involving multidimensional heteroclinic orbits, which was called a generalized heteroclinic cycle in [3]. Let us prove this claim.
Proposition 1
There exists an open set of -equivariant smooth vector fields which possess a generalized heteroclinic cycle. This cycle, which we denote by , connects two equilibria and which lie on the negative, resp. positive semi axis in . It is composed of a single heteroclinic orbit in and a two dimensional manifold of heteroclinic orbits in the space . This manifold in contains heteroclinic orbits in and in (when is even), which realize two isotropy types of pseudo-simple heteroclinic cycles.
**Proof: **Let us consider the group defined by relations (49) where we replace the transformation by , . This group has the same invariant subspaces as , but in addition any copy of the plane by is also invariant, and moreover is spanned by letting rotate with any . Therefore if a saddle-sink connection between equilibria , lying on exists in , then a two dimensional manifold of connections exists in . The fact that such equilibria and connections exist for an open set of smooth vector fields follows from a slight adaptation of lemma 8, which shows that the group admits robust heteroclinic cycles with connections in and . Since any equivariant perturbation of this vector field leaves , and invariant, we conclude by structural stability that generalized heteroclinic cycles persist for an open set of equivariant smooth vector fields. The same argument applies if replacing by when is even. QED
We denote by and the non-radial eigenvalues at , , and further assume that and are the double eigenvalues. Hence is a source while is a sink in , along the eigendirections .
Theorem 5
The generalized heteroclinic cycle defined in Proposition 1 is asymptotically stable if and is completely unstable if . Moreover there exists an open subset of such that for any vector field in this subset, a pseudo-simple heteroclinic subcycle of is fragmentarily asymptotically stable.
**Proof: **As usual we want to define a first return map in the vicinity of the heteroclinic cycle, and to do so we decompose the dynamics close to into local maps and global transition maps between suitably chosen cross-sections to the heteroclinic orbits near the equilibria. Possibly after a smooth -equivariant change of coordinates we can always assume that in a neighborhood of the equilibria their stable and unstable manifolds are linear. Let , resp. denote the local coordinates near along , resp. . The ”radial” direction (along the axis , coordinate ) can be neglected. We define the cross-sections along the (single) heteroclinic orbit from to (Fig. 3) by
[TABLE]
where is a small constant value.
Similarly we define the cross-sections along the two-dimensional manifold of connections from to by (see Fig. 4):
[TABLE]
The boundaries of the cross-sections at the limit values () lie in the space while at they lie in the space (when is odd) or (when is even). Since these spaces are flow-invariant, the sections defined above are mapped to each other by the flow in the order . We can therefore define the local first hit maps and global maps , .
By choosing small enough and if non resonance conditions are satisfied between the eigenvalues at each equilibrium, we can approximate the local vector fields by their linear parts. Therefore near the flow is defined by the equations
[TABLE]
which gives
[TABLE]
and near the flow is defined by
[TABLE]
which gives
[TABLE]
The far map is a -equivariant near identity diffeomorphism which can be linearized under generic conditions. We therefore have
[TABLE]
where is a positive constant.
The far map is also -equivariant, however it is not near identity and it can’t be expressed as simply as . Let us set
[TABLE]
The component vanishes when , hence there exists a smooth function such that . Moreover because is a diffeomorphism , which allows by a smooth change of variables to set where is a bounded function. The map is differentiably defined in the interval and has fixed-points at [math] and .
Now we can define the first return map in by and we write .
Applying the above expressions for and one obtains
[TABLE]
Since is a bounded function the iterates of the first component of tend to 0 if and only if . This proves the first part of the theorem.
The second component of has the form
[TABLE]
Assume , then by iteration the first argument of the function tends to 0. Therefore the dynamics of converges to the dynamics of the map . By an argument similar to Prop. 4.9 of [9], has generically hyperbolic fixed points at [math] and . Moreover there exists an open subset of such that for vector fields in this subset, has no fixed point inside . In this case we can conclude that the iterates of converge to a pseudo-simple heteroclinic cycle. QED
In order to illustrate this result we built a equivariant polynomial system with satisfying the hypotheses of the theorem and performed numerical simulations. We use bifurcation method to find the equilibria and corresponding heteroclinic orbits. Applying classical methods in computing equivariant bifurcation systems [7] we construct
[TABLE]
where , and are small parameters. Suitable coefficient values for the system to possess generalized heteroclinic cycles can be found as was done in [16] in the case.
We additionally assume that is close to 0 in order to ensure supercritical bifurcation of two equilibria on the axis. There is no loss of generality to take this sum equal to 0, so that the bifurcation is a pitchfork. Moreover in this bifurcation context it is suitable to take negative cubic coefficients in both equations, in order to keep the dynamics bounded. We normalize these coefficients to . Then the bifurcated equilibria are and . The non radial eigenvalues at and are
[TABLE]
The heteroclinic cycles exist for a range of coefficient values which includes the following:
[TABLE]
The eigenvalues are
[TABLE]
so that the generalized heteroclinic cycle is an attractor. The numerical simulations (with Matlab) were done with and . The two pictures in Figure 5 show the dynamics of the variable in polar coordinates: . The horizontal axis is the radial variable while the vertical axis is the angle (in degrees). Observe that in both cases, taking an initial condition close to even with a small angle (hence close to the plane ) the trajectory comes back to the vertical axis sequentially (as expected since it corresponds to going close to ), but with an increasing value of the angle. In the case the angle converges to while in the case it converges to . In both cases this corresponds to convergence to a pseudo-simple cycle with a connection in .
It is clear from this figure that when the convergence to the pseudo-simple cycle is faster and in particular the trajectory near the equilibria (near the vertical coordinate axis in the figure) is oblique while it is nearly horizontal in the case. This is consistent with the results of [16] where the case was studied using a different approach in which the double unstable eigenvalue is small enough for nonlinear effects to be felt by the flow near . This argument doesn’t work however when because one essential property of the case is that on the center manifold which exists at when is small enough, an unstable equilibrium point always exists near in , which obliges the flow to ”bend” back to or to in the vicinity of . A similar idea holds when . The advantage of the method of [16] is that it does not require the existence of a generalized heteroclinic cycle, however only fragmentarily asymptotic stability can be proved in such case.
Let us assume now that a perturbation is added to the vector field, which breaks the symmetry . The symmetry group is therefore reduced to the action of generated by the transformations and . The invariant planes , (and its copies by ) and are preserved, but not the invariant space . If the perturbation is not too large the equilibria in and their heteroclinic connections in the invariant planes persist, hence a pseudosimple heteroclinic cycle exists, however we know it is completely unstable. The question is what happens to the dynamics when this perturbation is switched on. Some preliminary numerical experiments have been performed on the system (58), where and the perturbation consists in replacing the terms by and by , where and are small but non zero. Other coefficients are the same as in (60) except . It has been observed that the dynamics remains in a neighborhood of the cycle and converges in certain cases to a periodic orbit (Fig. 6) while in other cases it exhibits a clear a aperiodic, possibly chaotic behavior (Fig. 7). The mathematical analysis of this behavior will be a subject for future study.
8 Conclusion
In this paper we completed the study of pseudo-simple heteroclinic cycles in , which have been discovered and distinguished from simple cycles only recently [15, 16]. Our primary contribution is a complete list of finite subgroups of admitting pseudo-simple heteroclinic cycles. Similar to the completion of the classification of simple cycles in [15], and as projected ibid, this was achieved using the quaternionic presentation of such groups.
Up to now stability of pseudo-simple cycles had only been addressed in [16], where generic complete instability for the case was shown, and an example of a fragmentarily asymptotically stable cycle, an intermediate weak form of stability, with was given. We extended the stability analysis for pseudo-simple cycles in subsection 4.2 by identifying all subgroups of admitting f.a.s. pseudo-simple heteroclinic cycles. A more comprehensive study, e.g. derivation of conditions for fragmentary asymptotic stability or calculation of stability indices along the heteroclinic connections as defined in [14], is beyond the scope of this work.
We have also studied the behaviour of trajectories close to pseudo-simple cycles. Namely, we proved that asymptotically stable periodic orbits can bifurcate from the cycle in a codimension one bifurcation at a point where a multiple expanding eigenvalue vanishes. Necessary and sufficient conditions for such a bifurcation are given in theorems 3 and 4. In section 6 we illustrated this through a numerical example of a heteroclinic cycle with a nearby attracting periodic orbit with symmetry group .
In contrast with [15], the proof of lemma 8 to characterize conditions for a group to be admissible relies upon an explicit construction of corresponding equivariant systems. This allows us to build examples of pseudo-simple heteroclinic cycles for any admissible group. As we noted (see remark 4), lemma 9 can be generalized to with to provide sufficient conditions for a subgroup of to admit heteroclinic cycles. Moreover, the explicit construction of an equivariant system in is applicable for this subgroup.
In addition to simple and pseudo-simple heteroclinic cycles other types of structurally stable heteroclinic cycles can exist in . One example is the generalized heteroclinic cycle that we studied in section 7. Another example is the cycle considered in [11]. To describe all robust heteroclinic cycles existing in is an open problem which is beyond the scope of this paper.
Other possible continuations of our work include the full classification of pseudo-simple cycles in , similar to the full classification of homoclinic cycles in [13], as well as the study of networks, which are connected unions of more than one cycle. In principle we think this can be achieved by the same means as we used here, even though a complete classification of networks has not even been done for simple cycles yet, partial results to this end can be found in [2].
Appendix A Elements of for the groups listed in Table 1,
satisfying
We denote the quaternions , , , , , , , where , and the permutation .
Annex A continued.
Annex A continued.
Annex A continued.
Appendix B Conjugacy classes of isotropy subgroups of finite groups satisfying
and
We list subgroups and of that satisfy and .
[FIGURE:]
[FIGURE:]
[FIGURE:]
[FIGURE:]
Appendix C Isotropy subgroups and and the element
satisfying conditions C1-C6 of lemma 8, where with . For the groups and with odd we list subgroups satisfying conditions of lemma 9.
[FIGURE:]
[FIGURE:]
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] S. Castro, A. Lohse. Stability in simple heteroclinic networks in ℝ 4 superscript ℝ 4 {\mathbb{R}}^{4} . Dynamical Systems 29 , 451–481 (2014).
- 2[2] S. Castro, A. Lohse. Construction of heteroclinic networks in ℝ 4 superscript ℝ 4 {\mathbb{R}}^{4} . Nonlinearity 29 , 3677–3695 (2016).
- 3[3] P. Chossat, F. Guyard and R. Lauterbach. Generalized Heteroclinic Cycles in Spherically Invariant Systems and Their Perturbations. J. Nonlinear Sci. 9 , 479 524 (1999).
- 4[4] P. Chossat and R. Lauterbach. Methods in Equivariant Bifurcations and Dynamical Systems. World Scientific Publishing Company , 2000.
- 5[5] J. H. Conway, D. Smith. On Quaternions and Octonions. A K Peters: Natick, Massachusets, 2003.
- 6[6] P. Du Val. Homographies, Quaternions and Rotations. OUP: Oxford, 1964.
- 7[7] M. Golubitsky, I. Stewart and D. Schaeffer. Singularities and Groups in Bifurcation Theory . Springer, 1988.
- 8[8] M. Krupa. Robust heteroclinic cycles. J. Nonlinear Science , 7 , 129 – 176 (1997).