The capture of a particle into resonance at potential hole with dissipative perturbation

Abstract

Bibliografische Information der Deutschen NationalbibliothekDie Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.de abrufbar. Universitätsverlag AbstractWe study the capture of a particle into resonance at a potential hole with dissipative perturbation and periodic outside force. The measure of resonance solutions is evaluated. We also derive an asymptotic formula for the parameter range of those solutions which are captured into resonance.

Asymptotic behaviour of measure for captured trajectories into parametric autoresonance

“…In this case the capture is defined by a gap for splitting of separatrises of perturbed pendulum with torque for phase variable of the oscillations. Similar results were obtained for problems of captured trajectories for autoresonance (not parametric) in [6] and for capture into nonlinear resonance of a particle in potential well with dissipation [8]. But the parametric resonance has essentially different from above mentioned problems as an equation for primary term of asymptotic as well as a measure of captured trajectories.…”

“…During one oscillation parameter E changes on a value: Figure 3: The splitting of separatrix for equation (8). The external curve make a loop and tends to the saddle.…”

“…The splitting of separatrix for equation(8). The external curve make a loop and tends to the saddle.…”

Conditions for Phase Locking and Dephasing of Autoresonant Pumping

“…In this work we derive a pendulum-like equation from the equation for nonlinear resonance of order O( 1/3 ) as an asymptotic limit. The asymptotic relation between two of these approaches can be found in [33]. As a result, we obtain the pendulum equations with a special form of the torque which depends on time instead of the small perturbation parameter as in the cited references.…”

“…The trajectories which are captured into resonance pass through the gap between two separatrices close to the saddle, see [33,35].…”