Stability Of Heteroclinic Cycles Research Articles

Correction to: Stability of Heteroclinic Cycles: A New Approach Based on a Replicator Equation

Correction to: Stability of Heteroclinic Cycles: A New Approach Based on a Replicator Equation

Stability of Heteroclinic Cycles: A New Approach Based on a Replicator Equation

Stability of Heteroclinic Cycles: A New Approach Based on a Replicator Equation

Stability of heteroclinic cycles in ring graphs.

Networks of interacting nodes connected by edges arise in almost every branch of scientific inquiry. The connectivity structure of the network can force the existence of invariant subspaces, which would not arise in generic dynamical systems. These invariant subspaces can result in the appearance of robust heteroclinic cycles, which would otherwise be structurally unstable. Typically, the dynamics near a stable heteroclinic cycle is non-ergodic: mean residence times near the fixed points in the cycle are undefined, and there is a persistent slowing down. In this paper, we examine ring graphs with nearest-neighbor or nearest- m-neighbor coupling and show that there exist classes of heteroclinic cycles in the phase space of the dynamics. We show that there is always at least one heteroclinic cycle that can be asymptotically stable, and, thus, the attracting dynamics of the network are expected to be non-ergodic. We conjecture that much of this behavior persists in less structured networks and as such, non-ergodic behavior is somehow typical.

Heteroclinic Dynamics of Localized Frequency Synchrony: Stability of Heteroclinic Cycles and Networks

Coupled populations of identical phase oscillators with higher-order interactions can give rise to heteroclinic cycles between invariant sets where populations show distinct frequencies. For these heteroclinic cycles to be observable, they have to have some stability properties. In this paper, we complement the existence results for heteroclinic cycles given in a companion paper by proving stability results for heteroclinic cycles for coupled oscillator populations consisting of two oscillators each. Moreover, we show that for systems with four coupled phase oscillator populations, there are distinct heteroclinic cycles that form a heteroclinic network. While such networks cannot be asymptotically stable, the local attraction properties of each cycle in the network can be quantified by stability indices. We calculate these stability indices in terms of the coupling parameters between oscillator populations. Hence, our results elucidate how oscillator coupling influences sequential transitions along a heteroclinic network where individual oscillator populations switch sequentially between a high and a low frequency regime; such dynamics appear relevant for the functionality of neural oscillators.

Stability of quasi-simple heteroclinic cycles

ABSTRACTThe stability of heteroclinic cycles may be obtained from the value of the local stability index along each connection of the cycle. We establish a way of calculating the local stability index for quasi-simple cycles: cycles whose connections are one-dimensional and contained in flow-invariant spaces of equal dimension. These heteroclinic cycles exist both in symmetric and non-symmetric contexts. We make one assumption on the dynamics along the connections to ensure that the transition matrices have a convenient form. Our method applies to all simple heteroclinic cycles of type Z and to various heteroclinic cycles arising in population dynamics, namely non-simple heteroclinic cycles, as well as to cycles that are part of a heteroclinic network. We illustrate our results with a non-simple cycle present in a heteroclinic network of the Rock–Scissors–Paper game.

On the equivalent classification of three-dimensional competitive Leslie/Gower models via the boundary dynamics on the carrying simplex.

We study the asymptotic behavior of the competitive Leslie/Gower model (map) [Formula: see text]It is shown that T unconditionally admits a globally attracting 1-codimensional invariant hypersurface [Formula: see text], called carrying simplex, such that every nontrivial orbit is asymptotic to one in [Formula: see text]. More general and easily checked conditions to guarantee the existence of carrying simplex for competitive maps are provided. An equivalence relation is defined relative to local stability of fixed points on [Formula: see text] (the boundary of [Formula: see text]) on the space of all three-dimensional Leslie/Gower models. Using a formula on the sum of the indices of all fixed points on the carrying simplex for three-dimensional maps, we list the 33 stable equivalence classes in terms of simple inequalities on the parameters [Formula: see text] and [Formula: see text] and draw their orbits on [Formula: see text]. In classes 1-18, every nontrivial orbit tends to a fixed point on [Formula: see text]. In classes 19-25, each map possesses a unique positive fixed point which is a saddle on [Formula: see text], and hence Neimark-Sacker bifurcations do not occur. Neimark-Sacker bifurcation does occur within each of classes 26-31, while it does not occur in class 32. Each map from class 27 admits a heteroclinic cycle, which forms the boundary of [Formula: see text]. The criteria on the stability of heteroclinic cycles are also given. This classification makes it possible to further investigate various dynamical properties in respective class.

A Hamiltonian approach to model and analyse networks of nonlinear oscillators: Applications to gyroscopes and energy harvesters

Over the past twelve years, ideas and methods from nonlinear dynamics system theory, in particular, group theoretical methods in bifurcation theory, have been used to study, design, and fabricate novel engineering technologies. For instance, the existence and stability of heteroclinic cycles in coupled bistable systems has been exploited to develop and deploy highly sensitive, low-power, magnetic and electric field sensors. Also, patterns of behaviour in networks of oscillators with certain symmetry groups have been extensively studied and the results have been applied to conceptualize a multifrequency up /down converter, a channelizer to lock into incoming signals, and a microwave signal generator at the nanoscale. In this manuscript, a review of the most recent work on modelling and analysis of two seemingly different systems, an array of gyroscopes and an array of energy harvesters, is presented. Empirical values of operational parameters suggest that damping and external forcing occur at a lower scale compared to other parameters, so that the individual units can be treated as Hamiltonian systems. Casting the governing equations in Hamiltonian form leads to a common approach to study both arrays. More importantly, the approach yields analytical expressions for the onset of bifurcations to synchronized oscillations. The expressions are valid for arrays of any size and the ensuing synchronized oscillations are critical to enhance performance.

Stability of heteroclinic cycles in transverse bifurcations

Heteroclinic cycles and networks exist robustly in dynamical systems with symmetry. They can be asymptotically stable, and gradually lose this stability through a variety of bifurcations, displaying different forms of non-asymptotic stability along the way. We analyse the stability change in a transverse bifurcation for different types of simple cycles in R4. This is done by first showing how stability of the cycle or network as a whole is related to stability indices along its connections — in particular, essential asymptotic stability is equivalent to all local stability indices being positive. Then we study the change of the stability indices. We find that all cycles of types B and C are generically essentially asymptotically stable after a transverse bifurcation, and that no type B cycle can be almost completely unstable (unlike type C cycles).

Symmetry Induced Heteroclinic Cycles in Coupled Sensor Devices

In this paper we discuss the existence and stability of heteroclinic cycles in coupled systems and show how they canbeexploitedto designandfabricate anew generationof highly-sensitive,low-powered, sensor devices. More specifically, we present theoretical and experimental proof of concept that coupling-induced oscillations located near the bifurcation point of a heteroclinic cycle can significantly enhance the sensitivity of an array of magnetic sensors. In particular, we consider arrays made up of fluxgate magnetometers inductively coupled through electronic circuits.

Shapley Polygons in 4 x 4 Games

We study 4 x 4 games for which the best response dynamics contain a cycle. We give examples in which multiple Shapley polygons occur for these kinds of games. We derive conditions under which Shapley polygons exist and conditions for the stability of these polygons. It turns out that there is a very strong connection between the stability of heteroclinic cycles for the replicator equation and Shapley polygons for the best response dynamics. It is also shown that chaotic behaviour can not occur in this kind of game.

Heteroclinic cycles and wreath product symmetries

We consider the existence and stability of heteroclinic cycles arising by local bifurcation in dynamical systems with wreath product symmetry = Z 2 G, where Z 2 acts by - 1 on R and G is a transitive subgroup of the permutation group S N (thus G has degree N). The group acts absolutely irreducibly on R N . We consider primary (codimension one) bifurcations from an equilibrium to heteroclinic cycles as real eigenvalues pass through zero. We relate the possibility of such cycles to the existence of non-gradient equivariant vector fields of cubic order. Using Hilbert series and the software package MAGMA we show that apart from the cyclic groups G (previously studied by other authors) only five groups G of degree h 7 are candidates for the existence of heteroclinic cycles. We establish the existence of certain types of heteroclinic cycle in these cases by making use of the concept of a subcycle. We also discusss edge cycles, and a generalization of heteroclinic cycles which we call a heteroclinic web. We apply our method to three examples.