Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators

Abstract

We study the bifurcation and dynamical behaviour of the system of N globally coupled identical phase oscillators introduced by Hansel, Mato and Meunier, in the cases N = 3 and N = 4. This model has been found to exhibit robust 'slow switching' oscillations that are caused by the presence of robust heteroclinic attractors. This paper presents a bifurcation analysis of the system in an attempt to better understand the creation of such attractors. We consider bifurcations that occur in a system of identical oscillators on varying parameters in the coupling function. These bifurcations preserve the permutation symmetry of the system. We then investigate implications of these bifurcations for the sensitivity to detuning (i.e. the size of the smallest perturbations that give rise to loss of frequency locking).For N = 3 we find three types of heteroclinic bifurcation that are codimensionone with symmetry. On varying two parameters in the coupling function we find three curves giving (a) an S 3 -transcritical homoclinic bifurcation, (b) a saddle-node/heteroclinic bifurcation and (c) a Z 3 -heteroclinic bifurcation. We also identify several global bifurcations with symmetry that organize the bifurcation diagram; these are codimension-two with symmetry.

On Bifurcations in Nonlinear Consensus Networks

“…As the value of c is increased from zero, the reduced system has seven equilibrium points, three of which are sinks, three are saddle points, and one is a source. As the value of c is further increased the three saddle points move towards the source, reaching it at c = √ 3γ at an S 3 -symmetric transcritical bifurcation [2], [8]. As the saddle points cross the source, i.e., for c > √ 3γ, the source becomes a sink, and the three saddle points move towards the other three sinks.…”

Design of Oscillator Networks for Generating Signal with Prescribed Statistical Property

“…(1). Is is known that when the α > α c = 0.25, the oscillator networks shows chaotic dynamics [5,6]. The connections between oscillators are asymmetrical and weighted, and A i,j takes any of 2L + 1 number of values, i.e.,…”

Heteroclinic Networks in Homogeneous and Heterogeneous Identical Cell Systems

“…We refer to (Aguiar et al 2011, [ §5]) and the next section. We remark that robust heteroclinic networks can occur in systems of at least four all-to-all symmetrically coupled phase oscillators (see Ashwin et al 2008;Ashwin et al 2010, but note the restrictions placed on coupling functions). On account of eigenvalue multiplicities that are often absent in systems with asymmetric inputs, we concentrate on realizing heteroclinic networks in networks of identical cells with asymmetric inputs.…”