Energy-based theory of autoresonance phenomena: Application to Duffing-like systems

Abstract

A general energy-based theory of autoresonance (self-sustained resonance) in low-dimensional nonautonomous systems is presented. The equations that together govern the autoresonance solutions and excitations are derived with the aid of a variational principle concerning the power functional. These equations provide a feedback autoresonance-controlling mechanism. The theory is applied to Duffing-like systems to obtain exact analytical expressions for autoresonance excitations and solutions which explain all the phenomenological and approximate results arising from a previous (adiabatic) approach to autoresonance phenomena in such systems. The theory is also applied to obtain new, general, and exact properties concerning autoresonance phenomena in a broad class of dissipative and Hamiltonian systems, including (as a particular case) Duffing-like systems.

First-harmonic approximation in nonlinear chirped-driven oscillators

“…In Ref. [17] the term autoresonance is used for a different nonlinear problem where the drive is chosen to fit the state of the oscillator so that a certain functional is minimized.…”

General scaling laws of chaotic escape in dissipative multistable systems subjected to autoresonant excitations

“…Initially studied in the context of a Hamiltonian description, AR phenomena have been well known for about half a century and have been observed in particle accelerators, planetary dynamics, atomic and molecular physics, and nonlinear oscillators [9], to cite a few examples. Regarding dissipative systems, an energy-based AR (EBAR) theory has recently been proposed and applied to the case where the system crosses a separatrix associated with its underlying integrable counterpart [10]. Since in such an escape situation the appearance of transient chaos associated with the occurrence of homoclinic bifurcations is an ubiquitous phenomenon, the question naturally arises: How does AR control the chaotic escape scenario, i.e., the onset and lifetime of transient chaos in generic dissipative multistable systems?…”

“…1), and Ω (t) ≡ ω + α n t n , n = 1, 2, ..., is a time-dependent frequency with α n being the n-th-order sweep rate. One expects [10] γ sin [Ω (t) t] to behave as an effective autoresonant excitation whenever the sweep rate is sufficiently low (adiabatic regime), which is the case considered throughout the present paper in order to apply Melnikov analysis (MA) to autoresonant excitations [11]. Also, the damping and autoresonant excitation terms are taken to be small amplitude perturbations of the underlying integrable system so as to deduce analytical expressions for the scaling laws relating both the onset time t i and lifetime τ of transient chaos with the parameters of the AR excitation from the MA [12] results.…”

“…, is a time-dependent frequency with α n being the nth-order sweep rate. One expects [10] γ sin[ (t)t] to behave as an effective autoresonant excitation whenever the sweep rate is sufficiently low (adiabatic regime), which is the case considered throughout the present communication in order to apply Melnikov analysis (MA) to autoresonant excitations. Note that optimal (exact) AR excitations are generally nonperiodic and unbounded [10], while MA remains valid for nonperiodic temporal perturbations if they are bounded [11].…”

“…One expects [10] γ sin[ (t)t] to behave as an effective autoresonant excitation whenever the sweep rate is sufficiently low (adiabatic regime), which is the case considered throughout the present communication in order to apply Melnikov analysis (MA) to autoresonant excitations. Note that optimal (exact) AR excitations are generally nonperiodic and unbounded [10], while MA remains valid for nonperiodic temporal perturbations if they are bounded [11]. Also, the damping and autoresonant excitation terms are taken to be small amplitude perturbations of the underlying integrable system so as to deduce analytical expressions for the scaling laws relating both the onset time t i and lifetime τ of transient chaos to the parameters of the AR excitation from the MA results [12].…”