Multi-stability and basin crisis in synchronized parametrically driven oscillators

Abstract

This paper studies the synchronization dynamics of two linearly coupled parametrically excited oscillators. The Lyapunov stability theory is employed to obtain some sufficient algebraic criteria for global asymptotic stability of the synchronization of the systems, and an estimated critical coupling, k cr , for which synchronization could be observed is determined. The synchronization transition is found to be associated with the boundary crisis of the chaotic attractor. In the bistable states, where two asymmetric T-periodic attractors co-exist, we show that the coupled oscillators can attain multi-stability via a new dynamical transition-the basin crisis wherein two co-existing attractors are destroyed while new co-existing attractors are created. The stability of the steady states is examined and the possible bifurcation routes identified.

Dynamical behaviour of parametrically driven Duffing and externally driven Helmholtz–Duffing oscillators under nonlinear dissipation

“…The results can be possibly improved in higher-order approximations. In 2011, Olusola et al [18] examined the synchronization behaviour of two linearly coupled parametrically excited oscillators. Some necessary and sufficient criteria for global asymptotic stability of synchronized dynamics of the system were derived via Lyapunov stability theory and linear matrix inequality and further obtained an estimated threshold coupling, for which stable synchronous behaviour could be observed.…”