Logarithmic correction to the probability of capture for dissipatively perturbed Hamiltonian systems

Abstract

Hamiltonian systems are analyzed with a double homoclinic orbit connecting a saddle to itself. Competing centers exist. A small dissipative perturbation causes the stable and unstable manifolds of the saddle point to break apart. The stable manifolds of the saddle point are the boundaries of the basin of attraction for the competing attractors. With small dissipation, the boundaries of the basins of attraction are known to be tightly wound and spiral-like. Small changes in the initial condition can alter the equilibrium to which the solution is attracted. Near the unperturbed homoclinic orbit, the boundary of the basin of attraction consists of a large sequence of nearly homoclinic orbits surrounded by close approaches to the saddle point. The slow passage through an unperturbed homoclinic orbit (separatrix) is determined by the change in the value of the Hamiltonian from one saddle approach to the next. The probability of capture can be asymptotically approximated using this change in the Hamiltonian. The well-known leading-order change of the Hamiltonian from one saddle approach to the next is due to the effect of the perturbation on the homoclinic orbit. A logarithmic correction to this change of the Hamiltonian is shown to be due to the effect of the perturbation on the saddle point itself. It is shown that the probability of capture can be significantly altered from the well-known leading-order probability for Hamiltonian systems with double homoclinic orbits of the twisted type, an example of which is the Hamiltonian system corresponding to primary resonance. Numerical integration of the perturbed Hamiltonian system is used to verify the accuracy of the analytic formulas for the change in the Hamiltonian from one saddle approach to the next. (c) 1995 American Institute of Physics.

Classical and quantum dissipative dynamics in Josephson junctions: An Arnold problem, bifurcation, and capture into resonance

“…This problem was studied earlier in a charged particle motion under an electromagnetic field [18]. It was rigorously framed into mathematical mechanics by V. I. Arnold [19] and is known as an Arnold problem in nonlinear dynamics [20][21][22]. The indeterministic nature of the branching on damping in the phase Josephson junctions was reported experimentally by others, where the Josephson junctions were operated in the classical regime [23,24].…”

“…If the particle initiates its motion from a phase point randomly chosen outside the separatrix, it will eventually cross the separatrix after a long time by dissipation to stochastically enter one of the lobes. It is well known that the probability of capture into each basin of attraction is proportional to the change in the Hamiltonian during one cycle of the corresponding homoclinic trajectory [21]. Accordingly, the probability P α of the particle falling onto either the left (L) or right (R) equilibrium states would be proportional to the bounded area of each lobe, S L and S R , respectively.…”

Higher-order change of the Hamiltonian (energy) for nearly homoclinic orbits

“…Thus, if we consider a sequence of nearly homoclinic orbits, then this correction is different for each nearly homoclinic orbit. This recursive effect has been analyzed in Haberman and Ho (1995b), where a calculation was performed on the boundary of the basin of attraction which requires a sequence of nearly homoclinic orbits.…”

“…For general perturbed Hamiltonian systems, Ho (1993) shows that (1.1) is still valid (with appropriate modifications for the definitions of f3 and k). However, Haberman and Ho (1995b) show that the probability of capture is changed by 0(6hie) from the usual Melnikov probabilities when the topology of the double homoclinic orbit is twisted as in the Hamiltonian system associated with primary resonance.…”

“…In (1.1), (H+ + H-)/2e = 0(1) since the Hamiltonians are O(e). For double-well potentials, the probability of capture into the left and right wells is determined from (1.1), and Haberman and Ho (1995b) show that the probability of capture is not affected by this 0(£ 2 Ins) energy correction. For general perturbed Hamiltonian systems, Ho (1993) shows that (1.1) is still valid (with appropriate modifications for the definitions of f3 and k).…”