594 research outputs found
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 , 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
Heteroclinic switching between chimeras
Functional oscillator networks, such as neuronal networks in the brain,
exhibit switching between metastable states involving many oscillators. We give
exact results how such global dynamics can arise in paradigmatic phase
oscillator networks: higher-order network interaction gives rise to metastable
chimeras - localized frequency synchrony patterns - which are joined by
heteroclinic connections. Moreover, we illuminate the mechanisms that underly
the switching dynamics in these experimentally accessible networks
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
Designing heteroclinic and excitable networks in phase space using two populations of coupled cells
We give a constructive method for realizing an arbitrary directed graph (with
no one-cycles) as a heteroclinic or an excitable dynamic network in the phase
space of a system of coupled cells of two types. In each case, the system is
expressed as a system of first order differential equations. One of the cell
types (the -cells) interacts by mutual inhibition and classifies which
vertex (state) we are currently close to, while the other cell type (the
-cells) excites the -cells selectively and becomes active only when there
is a transition between vertices. We exhibit open sets of parameter values such
that these dynamical networks exist and demonstrate via numerical simulation
that they can be attractors for suitably chosen parameters
Spiralling dynamics near heteroclinic networks
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
\EU^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 phenonema}. The
vector field is the restriction to \EU^3 of a polynomial vector field in
\RR^4. In this article, we also identify global bifurcations that induce
chaotic dynamics of different types.Comment: change in one figur
Resonance bifurcations of robust heteroclinic networks
Robust heteroclinic cycles are known to change stability in resonance
bifurcations, which occur when an algebraic condition on the eigenvalues of the
system is satisfied and which typically result in the creation or destruction
of a long-period periodic orbit. Resonance bifurcations for heteroclinic
networks are more complicated because different subcycles in the network can
undergo resonance at different parameter values, but have, until now, not been
systematically studied. In this article we present the first investigation of
resonance bifurcations in heteroclinic networks. Specifically, we study two
heteroclinic networks in and consider the dynamics that occurs as
various subcycles in each network change stability. The two cases are
distinguished by whether or not one of the equilibria in the network has real
or complex contracting eigenvalues. We construct two-dimensional Poincare
return maps and use these to investigate the dynamics of trajectories near the
network. At least one equilibrium solution in each network has a
two-dimensional unstable manifold, and we use the technique developed in [18]
to keep track of all trajectories within these manifolds. In the case with real
eigenvalues, we show that the asymptotically stable network loses stability
first when one of two distinguished cycles in the network goes through
resonance and two or six periodic orbits appear. In the complex case, we show
that an infinite number of stable and unstable periodic orbits are created at
resonance, and these may coexist with a chaotic attractor. There is a further
resonance, for which the eigenvalue combination is a property of the entire
network, after which the periodic orbits which originated from the individual
resonances may interact. We illustrate some of our results with a numerical
example.Comment: 46 pages, 20 figures. Supplementary material (two animated gifs) can
be found on
http://www.maths.leeds.ac.uk/~alastair/papers/KPR_res_net_abs.htm
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