arXiv subjects

Sofia Castro

Publications and source records attributed to Sofia Castro.

3 recordsLinked to original sources

arXiv2019

Almost complete and equable heteroclinic networks

Heteroclinic connections are trajectories that link invariant sets for an autonomous dynamical flow: these connections can robustly form networks between equilibria, for systems with flow-invariant spaces. In this paper we examine the relation between the heteroclinic network as a flow-invariant set and directed graphs of possible connections between nodes. We consider realizations of a large class of transitive digraphs as robust heteroclinic networks and show that although robust realizations are typically not complete (i.e. not all unstable manifolds of nodes are part of the network), they can be almost complete (i.e. complete up to a set of zero measure within the unstable manifold) and equable (i.e. all sets of connections from a node have the same dimension). We show there are almost complete and equable realizations that can be closed by adding a number of extra nodes and connections. We discuss some examples and describe a sense in which an equable almost complete network embedding is an optimal description of stochastically perturbed motion on the network.

math.DS↗

arXiv2018

A hybrid heteroclinic cycle

Using a vector field in $\mathbb{R}^4$, we provide an example of a robust heteroclinic cycle between two equilibria that displays a mix of features exhibited by well-known types of low-dimensional heteroclinic structures, including simple, quasi-simple and pseudo-simple cycles. Our cycle consists of two equilibria on one coordinate axis and two connections. One of the connections is one-dimensional while the other is two-dimensional. We compare our heteroclinic cycle to others in the literature that are similar in architecture, and illustrate how the standard methods used to analyse those cycles fail to provide sufficient information on the attraction properties of our example. The instability of two subcycles contained in invariant three-dimensional subspaces seems to indicate that our cycle is generically completely unstable. Although this cycle is one of the simplest possible and exists in low-dimension, the complete study of the stability of our cycle by using the standard techniques for return map reduction is not possible given the hybrid nature of the return map.

math.DS↗

arXiv2016

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 $\mathbb{R}^5$, 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.

math.DS↗