Robust homoclinic cycles in  ⁴

Abstract

The conditions of existence of robust homoclinic cycles for G-equivariant vector fields in R 4 with G a finite group are investigated. Depending on the action of G, such cycles are either of type A, B or C. We first introduce a notion of minimal admissible group. The existence of robust homoclinic cycles for vector fields which are equivariant by such a group is generic. Then we show that for type A cycles, the number n of equilibria is either even or equal to three. In the case of type B cycles, n can only be equal to two, three or six. Finally, for those of type C, n is either four or eight. Moreover, we provide expressions for the generators of the minimal admissible groups for each of the above mentioned cycles and we show the explicit form of vector fields generating three type A homoclinic cycles with six, eight and 24 equilibria, respectively.

Homoclinic and Heteroclinic Bifurcations in Vector Fields

“…Simple homoclinic cycles of type B and C in R 4 were investigated in [239], while Sottocornola [380][381][382] studied type A simple homoclinic cycles in R 4 . To state the classification result, we need the definition of structure angles that was introduced in [380][381][382].…”

“…To state the classification result, we need the definition of structure angles that was introduced in [380][381][382]. Recall that the twist is the group element γ ∈ Γ for which p j+1 = γp j .…”

Simple homoclinic cycles in low-dimensional spaces

“…The case R 3 is quite simple and it was already known in the 1980s when people started to look at these objects. The problem in dimension four is first discussed in [9] and completed in [16][17][18]. In this paper we resume these results and we give the classification in R 5 in the case of pure rotations symmetry group.…”

“…For a more detailed analysis of such cycles, see also [3,11,18,19]. We can introduce a direct orthonormal basis B = {e 1 , e 2 , e 3 } where the action of the twist has the form P 1 = e 1 , e 2 − → P 2 = e 2 , e 3 − → P 3 = cos(t) e 2 + sin(t) e 3 , e 1 .…”

Construction of heteroclinic networks in ${{\mathbb{R}}^{4}}$

“…At this point, we must distinguish between homoclinic and heteroclinic cycles: the former exhibit connections of a node to itself (or to another node in the same group orbit). Homoclinic cycles were systematically addressed by Sottocornola [31] and Podvigina [24] (see also Homburg et al [14]), while networks involving homoclinic cycles appear in the results of Driesse and Homburg [10] and Podvigina and Chossat [26]. We focus on a particular type of heteroclinic cycle, in the context of symmetry, made of what we call elementary building blocks, see section 2.…”