Complex dynamical properties of coupled Van der Pol-Duffing oscillators with balanced loss and gain

Abstract

We consider a Hamiltonian system of coupled Van der Pol-Duffing(VdPD) oscillators with balanced loss and gain. The system is analyzed perturbatively by using Renormalization Group(RG) techniques as well as Multiple Scale Analysis(MSA). Both the methods produce identical results in the leading order of the perturbation. The RG flow equation is exactly solvable and the slow variation of amplitudes and phases in time can be computed analytically. The system is analyzed numerically and shown to admit periodic solutions in regions of parameter-space, confirming the results of the linear stability analysis and perturbation methods. The complex dynamical behavior of the system is studied in detail by using time-series, Poincar$\acute{e}$-sections, power-spectra, auto-correlation function and bifurcation diagrams. The Lyapunov exponents are computed numerically. The numerical analysis reveals chaotic behaviour in the system beyond a critical value of the parameter that couples the two VdPD oscillators through linear coupling, thereby providing yet another example of Hamiltonian chaos in a system with balanced loss and gain. Further, we modify the nonlinear terms of the model to make it a non-Hamiltonian system of coupled VdPD oscillators with balanced loss and gain. The non-Hamiltonian system is analyzed perturbativly as well as numerically and shown to posses regular periodic as well as chaotic solutions. It is seen that the $\mathcal{P}\mathcal{T}$-symmetry is not an essential requirement for the existence of regular periodic solutions in both the Hamiltonian as well as non-Hamiltonian systems.

arXiv comment26 pages; 12 multi-panel figures; Added results & discussions on Route to Chaos; To appear in Journal of Physics A: Mathematical & Theoretical

References in corpus (21)

• Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities
• The Renormalization Group and Singular Perturbations: Multiple-Scales, Boundary Layers and Reductive Perturbation Theory
• Quantum synchronization of quantum van der Pol oscillators with trapped ions
• Twofold Transition in PT-Symmetric Coupled Oscillators
• Dynamics of Elastic Excitable Media
• Bifurcation and chaos in the double well Duffing-van der Pol oscillator: Numerical and analytical studies
• Why is CPT fundamental?
• Critical Response of a Quantum van der Pol Oscillator
• PT-Symmetric Dimer of Coupled Nonlinear Oscillators
• New aspects of integrability of force-free Duffing-van der Pol oscillator and related nonlinear systems
• Systems of coupled PT-symmetric oscillators
• Conservative perturbation theory for nonconservative systems
• Extension of the $CPT$ Theorem to non-Hermitian Hamiltonians and Unstable States
• Hypergeometric First Integrals of the Duffing and van der Pol Oscillators
• PT -symmetric rational Calogero model with balanced loss and gain
• On regular and chaotic dynamics of a non-$\mathcal{P}\mathcal{T}$-symmetric Hamiltonian system of a coupled Duffing oscillator with balanced loss and gain
• Hamiltonian formulation of systems with balanced loss-gain and exactly solvable models
• Integrable coupled Li$\acute{e}$nard-type systems with balanced loss and gain
• Taming Hamiltonian systems with balanced loss and gain via Lorentz interaction : General results and a case study with Landau Hamiltonian
• Finding Hannay angle in dissipative oscillatory systems via conservative perturbation theory
• Classical Hamiltonian Systems with Balanced loss and gain