Finding Hannay angle in dissipative oscillatory systems via conservative perturbation theory

Abstract

Usage of a Hamiltonian perturbation theory for a nonconservative system is counterintuitive and in general, a technical impossibility by definition. However, the time-independent dual Hamiltonian formalism for the nonconservative systems have opened the door for using various conservative perturbation theories for investigating the dynamics of such systems. Here we demonstrate that the Lie transform Hamiltonian perturbation theory can be adapted to find the perturbative solutions and the frequency corrections for the dissipative oscillatory systems. As a further application, we use the perturbation theory to analytically calculate the Hannay angle for the van der Pol oscillator's limit cycle trajectory when its parameters-the strength of the nonlinearity and the frequency of the linear part-evolve cyclically and adiabatically. For this van der Pol oscillator, we also numerically calculate the corresponding geometric phase and establish its equivalence with the Hannay angle.

12 pages, 1 figure

References in corpus (5)

• Oscillatory Dynamics in Rock-Paper-Scissors Games with Mutations
• Conservative perturbation theory for nonconservative systems
• Hannay angle and geometric phase shifts under adiabatic parameter changes in classical dissipative systems
• The geometric phase of stock trading
• Equivalent linearization finds nonzero frequency corrections beyond first order

Cited by in corpus (3)

• Relaxation oscillations and frequency entrainment in quantum mechanics
• Classical analog of the quantum metric tensor
• Complex dynamical properties of coupled Van der Pol-Duffing oscillators with balanced loss and gain