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

Abstract

We propose a generalized parity-time (PT ) -symmetric Liénard oscillator with two different orders of nonlinear position-dependent dissipation. We study the stability of the stationary states by using the eigenvalues of 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 blow-up dynamics for certain choice of initial conditions and control parameters. Employing an external driving force on the loss oscillator, we find that the blow-up dynamics 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 quasi-periodic 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.

Discrete solitons dynamics in $$\mathscr {PT}$$-symmetric oligomers with complex-valued couplings

“…There are also works on nonlocal models that involve PT -symmetry, such as the nonlocal NLS equation [24][25][26][27][28] and nonlocal Davey-Stewartson equations [29][30][31]. Other models outside the NLS realm where PT -symmetry plays a role include a nonlocal nonlinear Hirota equation [32], a nonlocal long-wave-short wave interaction equation [33], double-ring optical power resonator [34], quantum master Lindblad equation [35], and dampled harmonic Liénard oscillator equations [36].…”

KAM theorem with large twist and finite smooth large perturbation

“…This equation can be regarded as one coupled by two Duffing oscillators. See [17][18][19][20][21][22][23][24] for more details.…”

“…Thus, the coordinates change Φm+1 is symplectic if it exists. Moreover, we get the changed Hamiltonian H 24) where θ = θ(φ, t, ρ) is implicitly defined by (5.23). By Taylor formula, we have…”

Modeling and analysis of schottky diode bridge and JFET based liénard oscillator circuit

“…In [21], the quantum versions of Rayleigh, Van der Pol, and several other variants of Liénard Oscillators are derived as special cases. In [22], a PT-symmetric Liénard Oscillator configuration is investigated for two nonlinear power consumption cases. Due to these characteristics, modeling ability, and possible application areas such as synchronization and chaos, it is important to make and examine new types of VDPO and Liénard oscillators which can be possibly used in oscillator and chaos studies.…”