Multifaceted nonlinear dynamics in $$\mathcal {PT}$$-symmetric coupled Liénard oscillators

Abstract

We propose a generalized parity-time ($$\mathcal {PT}$$)-symmetric Lienard oscillator with two different orders of nonlinear position-dependent dissipation. We study the stability of the stationary states by using the eigenvalues of the Jacobian and evaluate the stability threshold thereafter. In the first-order nonlinear damping model, we discover that the temporal evolution of both gain and lossy oscillators attains a complete convergence towards the stable stationary state leading to the emergence of oscillation and amplitude deaths. Also, the system displays a remarkable manifestation of transient chaos in the lossy oscillator while the gain counterpart exhibits growing oscillations for certain choice of initial conditions and control parameters. Employing an external driving force on the loss oscillator, we find that the growing temporal evolution can be controlled and a pure aperiodic state is achievable. On the other hand, the second-order nonlinear damping model yields a completely different dynamics on contrary to the first order where the former reveals a conventional quasiperiodic route to chaos upon decreasing the natural frequency of both gain and loss oscillators. An electronic circuit scheme for the experimental realization of the proposed system has also been put forward.