S. Glebov, N. Tarkhanov
We investigate an influence of dissipation on autoresonant threshold for a system of nonlinear oscillators. Exact asymptotic formulas and numerical simulations are presented. These results correspond to an initial interval of autoresonance.
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We investigate a system of two weakly coupled oscillators. It is shown that an external periodic perturbation can lead to the capture into resonance. Asymptotic description and numerical simulations are presented. We have obtained the explicit formula for the threshold value for the amplitude of the perturbation that leads to the autoresonance.
A mathematical model describing the initial stage of the capture into autoresonance for nonlinear oscillating systems with combined parametric and external excitation is considered. The solutions with unboundedly growing amplitude and limited phase mismatch correspond to the autoresonant capture. The paper investigates the existence, stability and bifurcations of such solutions in the presence of a weak dissipation in the system. Our technique is based on the study of particular solutions with power-law asymptotics at infinity and the construction of suitable Lyapunov functions.
We study the autoresonant solution of Duffing's equation in the presence of dissipation. This solution is proved to be an attracting set. We evaluate the maximal amplitude of the autoresonant solution and the time of transition from autoresonant growth of the amplitude to the mode of fast oscillations. Analytical results are illustrated by numerical simulations.
A mathematical model describing the initial stage of the capture into the parametric autoresonance in nonlinear oscillating systems with a dissipation is considered. Solutions with unboundedly growing energy in time at infinity are associated with the autoresonance phenomenon. Stability of such solutions is investigated. We describe classes of admissible deterministic and random perturbations such that the stability of autoresonance is preserved on an asymptotically large interval.
A mathematical model describing the initial stage of the capture of oscillatory systems into autoresonance under the action of slowly varying pumping is considered. Solutions with an infinitely growing amplitude are associated with the autoresonance phenomenon. Stability of such solutions is of great importance because only stable solutions correspond to physically observable motions. We study the stabilizing problem and we show that the adiabatically varying parametric perturbation with decreasing amplitude in time can stabilize the unstable autoresonant modes.
It is known that a self-adjoint, time-independent hamiltonian can be defined for the quantum damped harmonic oscillator. We show here that the two vacua naturally associated to this operator, when expressed in terms of pseudo-bosonic lowering and raising operators, appear to be non square-integrable. This fact is interpreted as the evidence of the dissipation effect of the classical oscillator at a purely quantum level.
We study a system of stochastically forced infinite-dimensional coupled harmonic oscillators. Although this system formally conserves energy and is not explicitly dissipative, we show that it has a nontrivial invariant probability measure. This phenomenon, which has no finite dimensional equivalent, is due to the appearance of some anomalous dissipation mechanism which transports energy to infinity. This prevents the energy from building up locally and allows the system to converge to the invariant measure. The invariant measure is constructed explicitly and some of its properties are analyzed…
Dissipation functions and Brownian oscillators
该研究以耗散函数为核心构建 Fokker-Planck 扩散过程的响应理论框架，并借助 Brownian oscillator 模型给出过阻尼与欠阻尼动力学的精确响应公式，同时揭示模型约化与响应提取在特定条件下的可交换性。
We study the dynamic behavior at high energies of a chain of anharmonic oscillators coupled at its ends to heat baths at possibly different temperatures. In our setup, each oscillator is subject to a homogeneous anharmonic pinning potential $V_1(q_i) =|q_i|^{2k}/2k$ and harmonic coupling potentials $V_2(q_i- q_{i-1}) = (q_i- q_{i-1})^2/2$ between itself and its nearest neighbors. We consider the case $k > 1$ when the pinning potential is stronger then the coupling potential. At high energy, when a large fraction of the energy is located in the bulk of the chain, breathers appear and block the …
In this chapter we treat the quantum damped harmonic oscillator, and study mathematical structure of the model, and construct general solution with any initial condition, and give a quantum counterpart in the case of taking coherent state as an initial condition. This is a simple and good model of Quantum Mechanics with dissipation which is important to understand real world, and readers will get a powerful weapon for Quantum Physics.
We study the quantum counterpart of the theorem on energy equipartition for classical systems. We consider a free quantum Brownian particle modelled in terms of the Caldeira-Leggett framework: a system plus thermostat consisting of an infinite number of harmonic oscillators. By virtue of the theorem on the averaged kinetic energy $E_k$ of the quantum particle, it is expressed as $E_k = \langle \mathcal E_k \rangle$, where $\mathcal E_k$ is thermal kinetic energy of the thermostat per one degree of freedom and $\langle ...\rangle$ denotes averaging over frequencies $ω$ of thermostat oscillators…
We consider a resonantly perturbed system of coupled nonlinear oscillators with small dissipation and outer periodic perturbation. We show that for the large time $t \sim ε^{-2}$ one component of the system is described in the main by the inhomogeneous Mathieu equation while the other component represents pulsation of large amplitude. A Hamiltonian system is obtained which describes the behaviour of the envelope in the main. The analytic results agree to numerical simulations.
Model reduction of Brownian oscillators: quantification of errors and long-time behaviour
针对基于不变流形与涨落–耗散关系的随机常微分方程模型约化，本文以Wasserstein距离给出有无时间尺度分离时及长时间情形下的显式误差公式与估计。
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