Bifurcations in asymptotically autonomous Hamiltonian systems under oscillatory perturbations

Abstract

The effect of decaying oscillatory perturbations on autonomous Hamiltonian systems in the plane with a stable equilibrium is investigated. It is assumed that perturbations preserve the equilibrium and satisfy a resonance condition. The behaviour of the perturbed trajectories in the vicinity of the equilibrium is investigated. Depending on the structure of the perturbations, various asymptotic regimes at infinity in time are possible. In particular, a phase locking and a phase drifting can occur in the systems. The paper investigates the bifurcations associated with a change of Lyapunov stability of the equilibrium in both regimes. The proposed stability analysis is based on a combination of the averaging method and the construction of Lyapunov functions.

Long-term behaviour of asymptotically autonomous Hamiltonian systems with multiplicative noise

“…It is clear that a long-term behaviour of v(t), starting close to zero, depends on the structure of the right-hand side of the first equation in system (11). Note that even for similar deterministic systems without noise, the key role can be played by both primary and secondary terms of the asymptotics as t → ∞ (see, for example, [8]). With this in mind, we consider the following typical assumptions about the structure of these terms:…”

“…1, a). Numerical analysis of system (8) with a = 0 and c = 0 shows that the long-term behaviour of the solutions depends both on the values of the parameters a and c and on the degree of damping of the stochastic perturbation p/q (see Fig. 1, b, c).…”

“…However, in the general case, the long-term behaviour of solutions of asymptotically autonomous systems and the corresponding limiting systems can differ significantly [3,4]. Bifurcations in deterministic asymptotically autonomous systems were discussed in [5][6][7][8][9].…”

“…The influence of time-dependent stochastic perturbations on the long-term behaviour of solutions in scalar systems was discussed in [23][24][25]. Decaying stochastic perturbations that preserve the equilibrium of the limiting Hamiltonian system were considered in [26], where bifurcations associated with a change in the stochastic stability of the equilibrium were discussed. To the best of the author's knowledge, E-mail address: oasultanov@gmail.com.…”

Resonances in asymptotically autonomous systems with a decaying chirped-frequency excitation

“…Asymptotically autonomous systems with oscillatory decreasing perturbations have been studied in several papers, where either linear equations were considered [13,28,41,10,36], or the behaviour of solutions in the vicinity of the equilibrium was discussed [45]. In the present paper, the effect of damped oscillatory disturbances on non-linear systems far from equilibrium is investigated.…”