Sofia B. S. D. Castro

Dynamics near a heteroclinic network

We study the dynamical behaviour of a smooth vector field on a three-manifold near a heteroclinic network. Under some generic assumptions on the network, we prove that every path on the network is followed by a neighbouring trajectory of the vector field—there is switching on the network. We also show that near the network there is an infinite number of hyperbolic suspended horseshoes. This leads to the existence of a horseshoe of suspended horseshoes with the shape of the network.Our results are motivated by an example constructed by Field (1996 Lectures on Bifurcations, Dynamics, and Symmetry (Pitman Research Notes in Mathematics Series vol 356) (Harlow: Longman)), where we have observed, numerically, the existence of such a network.

SIMPLE VECTOR FIELDS WITH COMPLEX BEHAVIOR

We construct examples of vector fields on a three-sphere, amenable to analytic proof of properties that guarantee the existence of complex behavior. The examples are restrictions of symmetric polynomial vector fields in R4 and possess heteroclinic networks producing switching and nearby suspended horseshoes. The heteroclinic networks in our examples are persistent under symmetry preserving perturbations. We prove that some of the connections in the networks are the transverse intersection of invariant manifolds. The remaining connections are symmetry-induced. The networks lie in an invariant three-sphere and may involve connections exclusively between equilibria or between equilibria and periodic trajectories. The same construction technique may be applied to obtain other examples with similar features.

Stability in simple heteroclinic networks in

We describe all heteroclinic networks in made of simple heteroclinic cycles of types B or C, with at least one common connecting trajectory. For networks made of cycles of type B, we study the stability of the cycles that make up the network as well as the stability of the network. We show that even when none of the cycles has strong stability properties, the network as a whole may be quite stable. We prove, and provide illustrative examples of, the fact that the stability of the network does not depend a priori uniquely on the stability of the individual cycles.

A heteroclinic network in mode interaction with symmetry

We study a robust heteroclinic network existing in generic mode interactions of symmetric dynamical systems. Each mode lies in C 3 and is equivariant under the action of D 6 ⋉ T 2 × Z 2. With this symmetry there are eight different types of non-trivial steady states. This work is motivated by Boussinesq convection on a plane layer with periodic boundary conditions on a hexagonal lattice. The mode interaction takes place in a centre eigenspace isomorphic to C 6 when the trivial steady state becomes unstable to two modes of the form of rolls with spatial periods in the ratio. Due to relations between the normal form coefficients, only four types of steady states can be involved in the network. We examine the normal form restricted to R 6, a flow-invariant subspace, then we describe the dynamics near the network and discuss subnetworks and switching near them.

Switching in Heteroclinic Networks

We study the dynamics near heteroclinic networks for which all eigenvalues of the linearization at the equilibria are real. A common connection and an assumption on the geometry of its incoming and outgoing directions exclude even the weakest forms of switching (i.e. along this connection). The form of the global transition maps, and thus the type of the heteroclinic cycle, plays a crucial role in this. We look at two examples in

Construction of heteroclinic networks in

\mathbb{R}^5

Existence of a Markov perfect equilibrium in a third market model

, the House and Bowtie networks, to illustrate complex dynamics that may occur when either of these conditions is broken. For the House network, there is switching along the common connection, while for the Bowtie network we find switching along a cycle.

Bifurcation, symmetry and patterns

We study heteroclinic networks in

MODE INTERACTIONS WITH SPHERICAL SYMMETRY

\mathbb{R}^4

Thom-Boardman Stratification of Aggregate Excess Demand and Finiteness of Equilibria

, made of a certain type of simple robust heteroclinic cycle. In simple cycles all the connections are of saddle-sink type in two-dimensional fixed-point spaces. We show that there exist only very few ways to join such cycles together in a network and provide the list of all possible such networks in