Resonance-induced energy localization in a weakly dissipative nonlinear chain.

This work investigates the emergence of resonance and resonance-induced localization in a weakly dissipative chain of coupled anharmonic oscillators under the influence of a harmonic force applied at one end of the chain. The dynamics of the chain is studied assuming 1:1 (fundamental) resonance, when the response of each nonlinear oscillator has a dominant harmonic component with the frequency close to the frequency of the external excitation. It is shown that weak dissipation in a strongly nonlinear chain may be a key factor preventing large-amplitude resonance.

Internal autoresonance in coupled oscillators with slowly decaying frequency.

In this work, we study resonance energy transfer from an impulsively loaded strongly nonlinear oscillator to a weakly coupled linear attachment with a slowly time-decaying stiffness. It is shown that even in the absence of external periodic forcing both oscillators may exhibit the resonance phenomenon, with the permanent response enhancement of the linear oscillator and the corresponding response reduction of the nonlinear actuator. This effect is said to be internal autoresonance.

Energy localization in weakly dissipative resonant chains.

Localization of energy in oscillator arrays has been of interest for a number of years, with special attention paid to the role of nonlinearity and discreteness in the formation of localized structures. This work examines a different type of energy localization arising due to the presence of dissipation in nonlinear resonance arrays. As a basic model, we consider a Klein-Gordon chain of finite length subjected to a harmonic excitation applied at an edge of the chain.

Capture into resonance of coupled Duffing oscillators.

In this paper we investigate capture into resonance of a pair of coupled Duffing oscillators, one of which is excited by periodic forcing with a slowly varying frequency. Previous studies have shown that, under certain conditions, a single oscillator can be captured into persistent resonance with a permanently growing amplitude of oscillations (autoresonance). This paper demonstrates that the emergence of autoresonance in the forced oscillator may be insufficient to generate oscillations with increasing amplitude in the attachment.

Limiting phase trajectories and emergence of autoresonance in nonlinear oscillators.

Existence of stable autoresonance (AR) with continuously growing energy is directly connected with the inherent property of nonlinear systems to remain in resonance when the driving frequency varies in time. However, the physical mechanism underlying the transformation of bounded oscillations into AR remains unclear. As this paper demonstrates, the emergence of AR from stable bounded oscillations is basically analogous to the transition from quasilinear to nonlinear oscillations in the time-invariant oscillator driven by an external harmonic excitation with constant frequency, and AR can occur as a result of the loss of stability of the so-called limiting phase trajectory.

Resonance energy transport and exchange in oscillator arrays.

It is well known that complete energy transfer between two weakly coupled linear oscillators occurs only at resonance. If the oscillators are nonlinear, the amplitude dependence of their frequencies may destroy, in general, any eventual resonance. This means that no substantial energy transfer may occur unless, exceptionally, resonance persists during the transfer.

Nonlinear energy transfer in classical and quantum systems.

In this paper we investigate the effect of slowly-varying parameters on the energy transfer in a weakly coupled system. For definiteness, we consider a system of two nonlinear oscillators, in which the directly excited first oscillator with constant parameter is attached to the oscillator with slowly time-varying frequency. It is proved that the equations of the slow passage through resonance in this system are identical to the equations of nonlinear Landau-Zener (LZ) tunneling.

Classical analog of quasilinear Landau-Zener tunneling.

In this paper we develop an analytical framework to study the effect of nonlinearity on irreversible energy transfer in a system of two weakly coupled oscillators with time-dependent parameters, with special attention to an analogy between classical energy transfer and nonadiabatic quantum tunneling. For preciseness, we suppose that a linear oscillator with constant parameters is excited by an initial impulse but a coupled quasilinear oscillator with slowly varying parameters is initially at rest. It is shown that the equations of the slow passage through resonance in this system are identical to quasilinear equations of nonadiabatic Landau-Zener tunneling.

Fresnel integrals and irreversible energy transfer in an oscillatory system with time-dependent parameters.

We demonstrate that in significant limiting cases the problem of irreversible energy transfer in an oscillatory system with time-dependent parameters can be efficiently solved in terms of the Fresnel integrals. For definiteness, we consider a system of two weakly coupled linear oscillators in which the first oscillator with constant parameters is excited by an initial impulse, whereas the coupled oscillator with a slowly varying frequency is initially at rest but then acts as an energy trap. We show that the evolution equations of the slow passage through resonance are identical to the equations of the Landau-Zener tunneling problem, and therefore, the suggested asymptotic solution of the classical problem provides a simple analytic description of the quantum Landau-Zener tunneling with arbitrary initial conditions over a finite time interval.

Intense energy transfer and superharmonic resonance in a system of two coupled oscillators.

The paper presents the analytic study of energy exchange in a system of coupled nonlinear oscillators subject to superharmonic resonance. The attention is given to complete irreversible energy transfer that occurs in a system with definite initial conditions corresponding to a so-called limiting phase trajectory (LPT). We show that the energy imparted in the system is partitioned among the principal and superharmonic modes but energy exchange can be due to superharmonic oscillations.

Subharmonic transitions in a bistable oscillator with bimodal periodic excitation.

We analyze the phenomenon of low-frequency signal enhancement in a bistable system excited by a sum of low-frequency and high-frequency harmonic signals. A mechanism alternate to chaotic resonance is discussed. It is shown that a high-frequency signal may generate interwell transitions of subharmonic frequency.

Upper and lower bounds of stochastic resonance and noise-induced synchronization in a bistable oscillator.

This paper discusses concepts of stochastic resonance and noise-induced synchronization in a bistable oscillator subject to both periodic signal and noise. We demonstrate that stochastic resonance is not directly correlated with the matching of the signal frequency and the switching rate. The phenomena of stochastic resonance and noise-induced synchronization are the limiting cases of noise-induced transitions, and the spectral response heavily depends on the input signal-to-noise ratio.