Stability and bifurcation phenomena in asymptotically Hamiltonian systems

Abstract

The influence of time-dependent perturbations on an autonomous Hamiltonian system with an equilibrium of center type is considered. It is assumed that the perturbations decay at infinity in time and vanish at the equilibrium of the unperturbed system. In this case the stability and the long-term behaviour of trajectories depend on nonlinear and non-autonomous terms of the equations. The paper investigates bifurcations associated with a change of Lyapunov stability of the equilibrium and the emergence of new attracting or repelling states in the perturbed asymptotically autonomous system. The dependence of bifurcations on the structure of decaying perturbations is discussed.

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

“…as ∆ → 0 and τ → ∞. Hence, there exist 0 < ∆ 2 ≤ ∆ 1 and τ 2 ≥ τ 1 such that (40) with respect to τ yields…”

“…Hence, the fixed point (0, 0) is a centre in the asymptotic limit, and the linear analysis fails to determine the stability of the solution ̺ * (τ ), ϕ * (τ ) in the full nonlinear system (see, for example, [40]).…”

Damped perturbations of systems with centre-saddle bifurcation

“…and ℜe ± (t) = O(t −n/q ) as t → ∞. Hence, the fixed point (0, 0) of system ( 16) is a centre in the asymptotic limit, and the linear stability analysis fails (see, for example, [26]).…”

“…Integrating the second inequality in (27) with respect to t and taking into account (26), we obtain the following:…”

“…Note that from (26) and the second inequality in (27) it follows that the trivial solution X(t) ≡ 0, Y (t) ≡ 0 of system ( 25) is unstable if n = q and α n,m > 0. Combining this with (28), we see that the particular solution x * (t), y * (t) with asymptotics (9), x n+m = µ is unstable with weights: there exists ε > 0 such that for all δ > 0 there is a solution of system (1)…”