Finding the Hannay angle in dissipative oscillatory systems via conservative perturbation theory

Chattopadhyay, Rohitashwa; Shah, Tirth; Chakraborty, Sagar

doi:10.1103/physreve.97.062209

Cited by 7 publications

(4 citation statements)

References 64 publications

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“…For classical integrable systems, it is an extra angle shift picked up by the angle variables of the system when the parameters undergo a closed adiabatic excursion in the parameter space. This classical angle was investigated in a variety of systems [18][19][20][21], and the semiclassical relation between it and Berry's phase was established in Ref. [22] and has been verified in many systems [22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

Classical analog of the quantum metric tensor

2019

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We present a classical analog of the quantum metric tensor, which is defined for classical integrable systems that undergo an adiabatic evolution governed by slowly varying parameters. This classical metric measures the distance, on the parameter space, between two infinitesimally different points in phase space, whereas the quantum metric tensor measures the distance between two infinitesimally different quantum states. We discuss the properties of this metric and calculate its components, exactly in the cases of the generalized harmonic oscillator, the generalized harmonic oscillator with a linear term, and perturbatively for the quartic anharmonic oscillator. Finally, we propose alternative expressions for the quantum metric tensor and Berry's connection in terms of quantum operators.

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Section: Introductionmentioning

confidence: 99%

Classical analog of the quantum metric tensor

2019

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“…The standard techniques for analyzing a Hamiltonian system like canonical perturbation theory, quantization, KAM theory, integrability etc may be used to study VdPD oscillator. The canonical perturbation theory has been used successfully [36,37] for the standard Van der Pol oscillator, i.e., s = β 1 = g = 0.…”

Section: Hamiltonian Systemmentioning

confidence: 99%

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

Roy¹,

Ghosh²

2022

J. Phys. A: Math. Theor.

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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.

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“…may be used to study VdPD oscillator. The canonical perturbation theory has been used successfully [36,37] for the standard Van der Pol oscillator, i.e, s = β 1 = g = 0.…”

Section: Hamiltonian Systemmentioning

confidence: 99%

“…In the limit of small θ A , θ B , the amplitudes A 0 , B 0 and the phase θ A are determined by the Eqs. (35,36) and the first equation of (37). The equation for θ B is different and given by,…”

Section: Non-pt -Symmetric Non-hamiltonian Systemmentioning

confidence: 99%

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

Roy,

Ghosh

2021

Preprint

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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, Poincaré-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 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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Finding the Hannay angle in dissipative oscillatory systems via conservative perturbation theory

Cited by 7 publications

References 64 publications

Classical analog of the quantum metric tensor

Classical analog of the quantum metric tensor

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

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

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