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Complex dynamical properties of coupled Van der Pol-Duffing oscillators with balanced loss and gain
Chaotic Dynamics · Submitted 20 Dec 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim It is seen that the ${\cal{PT}}$-symmetry is not an essential requirement for the existence of regular periodic solutions in both the Hamiltonian as well as non-Hamiltonian systems.
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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 ${\cal{PT}}$-symmetry is not an essential requirement for the existence of regular periodic solutions in both the Hamiltonian as well as non-Hamiltonian systems.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider a Hamiltonian system of coupled Van der Pol-Duffing(VdPD) oscillators with balanced loss and gain. | contribution 0.74 | 0.68 | 0.03 | asserted 0.87 |
| s1 | The system is analyzed perturbatively by using Renormalization Group(RG) techniques as well as Multiple Scale Analysis(MSA). | method 0.99 | 0.92 | 0.02 | asserted 1.00 |
| s2 | Both the methods produce identical results in the leading order of the perturbation. | result 0.97 | 0.92 | 0.04 | asserted 1.00 |
| s3 | The RG flow equation is exactly solvable and the slow variation of amplitudes and phases in time can be computed analytically. | result 0.82 | 0.90 | 0.06 | asserted 1.00 |
| s4 | 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. | result 0.78 | 0.95 | 0.06 | asserted 1.00 |
| s5 | 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. | method 0.93 | 0.94 | 0.04 | asserted 0.99 |
| s6 | The Lyapunov exponents are computed numerically. | method 0.93 | 0.94 | 0.03 | asserted 0.96 |
| s7 | 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. | result 0.92 | 0.95 | 0.56 | asserted 1.00 |
| s8 | 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. | contribution 0.88 | 0.94 | 0.04 | asserted 0.99 |
| s9 | The non-Hamiltonian system is analyzed perturbativly as well as numerically and shown to posses regular periodic as well as chaotic solutions. | result 0.82 | 0.96 | 0.09 | asserted 1.00 |
| s10 | It is seen that the ${\cal{PT}}$-symmetry is not an essential requirement for the existence of regular periodic solutions in both the Hamiltonian as well as non-Hamiltonian systems. | interpretation 0.81 | 0.90 | 0.42 | asserted 1.00 |
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Original 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 ${\cal{PT}}$-symmetry is not an essential requirement for the existence of regular periodic solutions in both the Hamiltonian as well as non-Hamiltonian systems.
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We analyze the Hamiltonian system perturbatively by using MSA [ref] and RG [ref] methods.
Introduction to the method of multiple scales · Introduction · 1.00
Cites as background 1 paper
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Various models of coupled Van der Pol oscillators [ref] and coupled VdPD oscillators [ref] have also been considered.
Higher Dimensional Limit Cycles and Coupling Induced Synchronization in Dynamical Systems · Introduction · 0.93
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