Abstract
Time-delay effects on synchronization features of delay-coupled slow-fast van der Pol systems are investigated in the present paper. The synchronization mechanism of “slow-manifold adjustment” is firstly described on the basis of geometric singular perturbation theory. Then, the impact of time delay on the structure of the slow manifold of synchronized system is obtained by using the method of stability switch, and thus, time-delay effects on synchronization features are stated. It is shown the time delay cannot qualitatively affect the synchronization mechanism, however, it can result in the drift of the optimal coupling strength.
Highlights
Real system is often with two kinds of different dynamical variables, evolving on very different timescales [1] [2], this system is called slow-fast system, and the rapid evolving and slowly evolving variables are called fast and slow variables respectively. van der Pol system with one slow variable and one fast variable is one of the typical slow-fast system [3]
Time-delay effects on synchronization features of delay-coupled slow-fast van der Pol systems are investigated in the present paper
This theory defines a slow manifold of slow-fast system, which is an approximation of the invariant manifold, and it indicates the evolution of solution trajectory is governed by the structure of the slow manifold including the shape, stability and bifurcation points of the slow manifold, and the solution trajectory will be attracted by stable slow manifold and repelled by unstable slow manifold
Summary
Real system is often with two kinds of different dynamical variables, evolving on very different timescales [1] [2], this system is called slow-fast system, and the rapid evolving and slowly evolving variables are called fast and slow variables respectively. van der Pol system with one slow variable and one fast variable is one of the typical slow-fast system [3]. The chains of relaxation-type neural oscillators with local excitatory coupling is studied by using the phase reduction and fast threshold modulation theories [28], it is shown the chains undergo the transition from waves to synchronization when the system approaches the relaxation limit. This paper is devoted to further investigate time-delay effect on synchronization features of delay-coupled slow-fast van der Pol systems on behalf of the method of stability switch [42] and Geometric singular perturbation theory [13] [14] [15] [16].
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