Alexandre A. P. Rodrigues

Switching near a network of rotating nodes

We study the dynamics of a Z 2 ⊕ Z 2-equivariant vector field in the neighbourhood of a heteroclinic network with a periodic trajectory and symmetric equilibria. We assume that around each equilibrium the linearization of the vector field has non-real eigenvalues. Trajectories starting near each node of the network turn around in space either following the periodic trajectory or due to the complex eigenvalues near the equilibria. Thus, in a network with rotating nodes, the rotations combine with transverse intersections of two-dimensional invariant manifolds to create switching near the network; close to the network, there are trajectories that visit neighbourhoods of the saddles following all the heteroclinic connections of the network in any given order. Our results are motivated by an example where switching was observed numerically by forced symmetry breaking of an asymptotically stable network with O(2) symmetry.

Chaotic double cycling

We study the dynamics of a generic vector field in the neighbourhood of a heteroclinic cycle of non-trivial periodic solutions whose invariant manifolds meet transversely. The main result is the existence of chaotic double cycling: there are trajectories that follow the cycle making any prescribed number of turns near the periodic solutions, for any given bi-infinite sequence of turns. Using symbolic dynamics, arbitrarily close to the cycle, we find a robust and transitive set of initial conditions whose trajectories follow the cycle for all time and that is conjugate to a Markov shift over a finite alphabet. This conjugacy allows us to prove the existence of infinitely many heteroclinic and homoclinic subsidiary connections, which give rise to a heteroclinic network with infinitely many cycles and chaotic dynamics near them, exhibiting themselves switching and cycling. We construct an example of a vector field with Z 3 symmetry in a five-dimensional sphere with a heteroclinic cycle having this property.

Spiralling dynamics near heteroclinic networks

Abstract There are few explicit examples in the literature of vector fields exhibiting complex dynamics that may be proved analytically. We construct explicitly a two parameter family of vector fields on the three-dimensional sphere S 3 , whose flow has a spiralling attractor containing the following: two hyperbolic equilibria, heteroclinic trajectories connecting them transversely and a non-trivial hyperbolic, invariant and transitive set. The spiralling set unfolds a heteroclinic network between two symmetric saddle-foci and contains a sequence of topological horseshoes semiconjugate to full shifts over an alphabet with more and more symbols, coexisting with Newhouse phenomena. The vector field is the restriction to S 3 of a polynomial vector field in R 4 . In this article, we also identify global bifurcations that induce chaotic dynamics of different types.

Dense heteroclinic tangencies near a Bykov cycle

Abstract This article presents a mechanism for the coexistence of hyperbolic and non-hyperbolic dynamics arising in a neighbourhood of a Bykov cycle where trajectories turn in opposite directions near the two nodes — we say that the nodes have different chirality. We show that in the set of vector fields defined on a three-dimensional manifold, there is a class where tangencies of the invariant manifolds of two hyperbolic saddle-foci occur densely. The class is defined by the presence of the Bykov cycle, and by a condition on the parameters that determine the linear part of the vector field at the equilibria. This has important consequences: the global dynamics is persistently dominated by heteroclinic tangencies and by Newhouse phenomena, coexisting with hyperbolic dynamics arising from transversality. The coexistence gives rise to linked suspensions of Cantor sets, with hyperbolic and non-hyperbolic dynamics, in contrast with the case where the nodes have the same chirality. We illustrate our theory with an explicit example where tangencies arise in the unfolding of a symmetric vector field on the three-dimensional sphere.

Partial Symmetry Breaking and Heteroclinic Tangencies

We study some global aspects of the bifurcation of an equivariant family of volume-contracting vector fields on the three-dimensional sphere. When part of the symmetry is broken, the vector fields exhibit Bykov cycles. Close to the symmetry, we investigate the mechanism of the emergence of heteroclinic tangencies coexisting with transverse connections. We find persistent suspended horseshoes accompanied by attracting periodic trajectories with long periods.

On the Dynamics Near a Homoclinic Network to a Bifocus: Switching and Horseshoes

We study a homoclinic network associated to a nonresonant hyperbolic bifocus. It is proved that on combining rotation with a nondegeneracy condition concerning the intersection of the two-dimensional invariant manifolds of the equilibrium, switching behavior is created: close to the network, there are trajectories that visit the neighborhood of the bifocus following connections in any prescribed order. We discuss the existence of suspended horseshoes which accumulate on the network and the relation between these horseshoes and the switching behavior.

Dynamics near the product of planar heteroclinic attractors

Motivated by problems in equivariant dynamics and connection selection in heteroclinic networks, Ashwin and Field investigated the product of planar dynamics where at least one of the factors was a planar homoclinic attractor. However, they were only able to obtain partial results in the case of a product of two planar homoclinic attractors. We give general results for the product of planar homoclinic and heteroclinic attractors. We show that the likely limit set of the basin of attraction of the product of two planar heteroclinic attractors is always the unique one-dimensional heteroclinic network which covers the heteroclinic attractors in the factors. The method we use is general and likely to apply to products of higher dimensional heteroclinic attractors as well as to situations where the product structure is broken but the cycles are preserved.

Complete Set of Invariants for a Bykov Attractor

In this paper we consider an attracting heteroclinic cycle made by a 1-dimensional and a 2-dimensional separatrices between two hyperbolic saddles having complex eigenvalues. The basin of the global attractor exhibits historic behavior and, from the asymptotic properties of these nonconverging time averages, we obtain a complete set of invariants under topological conjugacy in a neighborhood of the cycle. These invariants are determined by the quotient of the real parts of the eigenvalues of the equilibria, a linear combination of their imaginary components and also the transition maps between two cross sections on the separatrices.

Generic area-preserving reversible diffeomorphisms

Let M be a surface and R an involution in M whose set of fixed points is a submanifold with dimension 1 and such that R is an isometry. We will show that there is a residual subset of C1 area-preserving R-reversible diffeomorphisms which are either Anosov or have zero Lyapunov exponents at almost every point.

Boundary crisis for degenerate singular cycles

The term boundary crisis refers to the destruction or creation of a chaotic attractor when parameters vary. The locus of a boundary crisis may contain regions of positive Lebesgue measure marking the transition from regular dynamics to the chaotic regime. This article investigates the dynamics occurring near a heteroclinic cycle involving a hyperbolic equilibrium point E and a hyperbolic periodic solution P, such that the connection from E to P is of codimension one and the connection from P to E occurs at a quadratic tangency (also of codimension one). We study these cycles as organizing centers of two-parameter bifurcation scenarios and, depending on properties of the transition maps, we find different types of shift dynamics that appear near the cycle. Breaking one or both of the connections we further explore the bifurcation diagrams previously begun by other authors. In particular, we identify the region of crisis near the cycle, by giving information on multipulse homoclinic solutions to E and P as well as multipulse heteroclinic tangencies from P to E, and bifurcating periodic solutions, giving partial answers to the problems (Q1)-(Q3) of E. Knobloch (2008), Spatially localised structures in dissipative systems: open problems, Nonlinearity, 21, 45-60. Throughout our analysis, we focus on the case where E has real eigenvalues and P has positive Floquet multipliers.