Agnessa Kovaleva

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.

Approximation of Escape Time for Lagrangian Systems With Fast Noise

This note is concerned with the large deviations asymptotics estimate of the mean escape time for Lagrangian systems subjected to fast Gaussian noise. The solution to the Hamilton-Jacobi equation of the associated variational problem is derived as a sum of two terms dependent on kinetic and potential energy of the system, respectively. A closed-form solution for classes of linear and nonlinear systems is obtained. An application to a controlled pointing problem is discussed as an example.

Large Deviations Estimates of Escape Time for Lagrangian Systems

This paper is concerned with analysis of the asymptotic behavior of a Lagranian system with small noise effects. The domain of the system operation is supposed to be within the domain of attraction of an asymptotically stable point of the unperturbed system. If noise is weak, escape from the reference domain is a rare event associated with large deviations in the system. This paper uses an extension of large deviations theory to the degenerate systems to develop the escape time asymptotics for a weakly perturbed Lagrangian system. Estimation of the statistical quantities is reduced to minimization of an associated action functional. It is shown that, in the case of the Lagrangian system, the solution of the associated variational problem can be found in a closed form, as a function of the system and noise parameters. As an example, motion of a 2n-dimensional linear system in an ellipsoidal domain is studied. Application of the theory to the nonlinear systems is illustrated by estimation of the lifetime of the Henon-Heiles system.

Random Rocking Dynamics of a Multidimensional Structure

In this paper we investigate the effect of structural flexibility on rocking motion of a system consisting of a free standing rigid block with an attached chain of uniaxially moving point masses. Motion is excited by random acceleration of the ground; instability is directly associated with overturning of the overall structure. The condition of instability is constructed by the stochastic Melnikov method. As an example, the dynamics of a system with a single-mass secondary structure is discussed. The paper is restricted to the consideration of seismic vulnerability of the structure. A similar approach can be applied to systems with wind or wave loading.

Risk-Sensitive Control for Nonlinear Oscillatory Systems with Small Noise

Abstract The problem of controlling a near-Hamiltonian noisy system so as to prevent it from overcoming the potential barrier is considered. An exponential risk-sensitive residence time criterion is examined as a solution of a related HJB equation. The averaged HJB equation is constructed as a first order PDE with the coefficients dependent on the noise intensity in the leading order term, though this intensity tends to zero in the original system. The leading order nearly optimal control is constructed as a stationary feedback with parameters dependent on the noise intensity.

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. It is shown that weak dissipation may be a key factor preventing the emergence of resonance in the entire chain, even if its nondissipative analog is entirely captured into resonance. The resulting process in the dissipative oscillator array represents large-amplitude resonant oscillations in a part of the chain adjacent to the actuator and small-amplitude oscillations in the distant part of the chain. The conditions of the emergence of resonance as well as the conditions of energy localization are derived. An agreement between the obtained analytical results and numerical simulations is demonstrated.

Targeted Energy Transfer

This chapter presents the analytical and numerical study of energy transport in a system of n linear impulsively loaded oscillators (a primary linear system), in which the nth oscillator is coupled with an essentially nonlinear attachment—the nonlinear energy sink (NES).

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.

Noise-Induced Synchronization and Stochastic Resonance in a Bistable System

We determine stochastic resonance and locking conditions for noise-induced interwell jumps in a bistable system. We demonstrate that the phenomena of stochastic resonance and synchronization are not contradictory and can be interpreted as the limit cases of hopping dynamics modulated by a weak signal. The boundary between the domains of synchronization and stochastic resonance is found as a function of the system parameters.

Control of Structures by Means of High-Frequency Vibration

In recent years, there has been increasing interest in applications of vibrational control theory [4], [6], [10] to problems of stabilization in mechanical structures. It has been found that high-frequency periodic or quasi-periodic excitation (“vibrational control”) can stabilize an unstable equilibrium position (“vibrational stabilization”). The effect of high-frequency vibration is similar to modification of the “effective potential” of the system and can be achieved either by high-frequency programme control, or by nonlinear feedback control. However, feedback control requires measurement of the system state and costly signal processing, whereas vibrational control can be applied as a pregiven programme. This makes it a powerful tool for stabilization of complicated mechanical systems.