Abstract
Abstract We consider a heteroclinic network in the framework of winnerless competition, realized by generalized Lotka-Volterra equations. By an appropriate choice of predation rates we impose a structural hierarchy so that the network consists of a heteroclinic cycle of three heteroclinic cycles which connect saddles on the basic level. As we have demonstrated in previous work, the structural hierarchy can induce a hierarchy in time scales such that slow oscillations modulate fast oscillations of species concentrations. Here we derive a Poincaré map to determine analytically the number of revolutions of the trajectory within one heteroclinic cycle on the basic level, before it switches to the heteroclinic connection on the second level. This provides an understanding of which parameters control the separation of time scales and determine the decisions of the trajectory at branching points of this network.
Highlights
Nonlinear dynamics of heteroclinic networks is frequently found in ordinary differential equations under certain constraints like symmetries [1] or delay [2]
In our previous work [18] we considered a hierarchical heteroclinic network composed of a heteroclinic cycle (LHC) of three heteroclinic cycles (SHCs)
We worked out the map in detail for predicting the number of revolutions within a small heteroclinic cycle, before the trajectory turns to the heteroclinic connection in the large cycle
Summary
Nonlinear dynamics of heteroclinic networks is frequently found in ordinary differential equations under certain constraints like symmetries [1] or delay [2]. It is predicted in models of coupled phase oscillators [2, 3], vector models [2], pulse-coupled oscillators [4] and models of winnerless competition [5] that we consider in the following. In our previous work [18] we considered a hierarchical heteroclinic network composed of a (large, superordinated) heteroclinic cycle (LHC) of three (small) heteroclinic cycles (SHCs).
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