Classical analog of quasilinear Landau-Zener tunneling

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. The influence of the system parameters on the emergence and stability of autoresonance is investigated both analytically and numerically.

Consecutive transitions from localized to delocalized transport states in the anharmonic chain of three coupled oscillators

In the present paper, we study the mechanism of formation and bifurcations of highly nonstationary regimes manifested by different energy transport intensities, emerging in an anharmonic trimer model. The basic model under investigation comprises a chain of three coupled anharmonic oscillators subject to localized excitation, where the initial energy is imparted to the first oscillator only. We report the formation of three basic nonstationary transport states traversed by locally excited regimes. These states differ by spatial energy distribution, as well as by the intensity of energy transport along the chain. In the current study, we focus on numerical and analytical investigation of the intricate resonant mechanism governing the inter-state transitions of locally excited regimes. Results of the analytical study are in good agreement with the numerical simulations of the trimer model.

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. A parametric domain, in which both oscillators can be captured into resonance, is determined. The quasisteady states determining the growth of amplitudes are found. An agreement between the theoretical and numerical results is demonstrated.

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. We obtain the parametric threshold, which determines the transition from bounded oscillations to AR in the time-dependent system. The accuracy of the obtained approximations is confirmed by numerical simulations.

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. In this paper, the self-sustained resonance is considered for an oscillator array consisting of n coupled linear oscillators (a primary system) initially excited by impulse loading and connected to an essentially nonlinear attachment (NLA). Under the condition of resonance, initial energy is transferred to the NLA and then travels back and forth between the linear and nonlinear oscillators. It is shown that the general mechanism of the energy transport is similar to that in the previously studied system of two coupled oscillators but, in contrast to the two degree-of-freedom case, the multidimensional system requires a proper tuning not only for the NLA but for the entire array. In this work, we develop an order-reduction procedure, which allows the separated dynamical analysis for the pair of nonlinearly coupled oscillators and the remaining (n-1) linear oscillators. Using simplifications based on the low-order reduced model, we detect an admissible domain of parameters ensuring resonance interaction and then derive a closed-form approximate solution adequately describing the transient processes in the entire system. We obtain explicit approximate solutions for both conservative (complete energy exchange) and dissipative (irreversible energy transfer) systems and then illustrate the theoretical results by an example of the four degree-of-freedom system. Analytical results are confirmed by numerical simulations.

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. Three types of dynamical behavior are distinguished, namely, quasilinear, moderately nonlinear, and strongly nonlinear ones. Quasilinear systems exhibit a gradual energy transfer from the excited to the attached oscillator, while moderately nonlinear systems are characterized by an abrupt transition from the energy localization on the excited oscillator to the localization on the attached oscillator. In strongly nonlinear systems, the transition from the energy localization to strong energy exchange between the oscillators is revealed. Explicit approximate solutions describing the transient processes in moderately and strongly nonlinear systems are suggested. Correctness of the constructed approximations is confirmed by numerical results. The results presented in this paper, in addition to providing an analytical framework for understanding the transient dynamics, suggest an approximate procedure for solving the nonlinear LZ problem with arbitrary initial conditions over a finite time-interval.