The motion of a nonlinearly oscillating particle under the influence of a periodic sequence of short impulses is investigated. We analyze the Schrödinger equation for the universal Hamiltonian. It is shown that the quantum criterion of overlapping of resonances is of the form lambdaK>or=1, where K is the classical coefficient of stochasticity and lambda is the functional defined with the use of Mathieu functions. The area of the maximal values of lambda is determined. The idea about the emerging of quantum chaos due to the adiabatic motion along the curves of Mathieu characteristics at multiple passages through the points of branching is advanced.
The pendulum is the simplest system having all the basic properties inherent in dynamic stochastic systems. In the present paper we investigate the pendulum with the aim to reveal the properties of a quantum analogue of dynamic stochasticity or, in other words, to obtain the basic properties of quantum chaos. It is shown that a periodic perturbation of the quantum pendulum (similarly to the classical one) in the neighborhood of the separatrix can bring about irreversible phenomena. As a result of recurrent passages between degenerate states, the system gets self-chaotized and passes from the pure state to the mixed one. Chaotization involves the states, the branch points of whose levels participate in a slow "drift" of the system along the Mathieu characteristics this "drift" being caused by a slowly changing variable field. Recurrent relations are obtained for populations of levels participating in the irreversible evolution process. It is shown that the entropy of the system first grows and, after reaching the equilibrium state, acquires a constant value.
The quantum-mechanical investigation of nonlinear resonance in terms of approximation to moderate nonlinearity is reduced to the investigation of eigenfunctions and eigenvalues of the Mathieu-Schrodinger equation. The eigenstates of the Mathieu-Schrodinger equation are nondegenerate in a certain area of pumping amplitude values in the neighborhood of the classical separatrix. Outside this area, the system finds itself in a degenerate state for both small and large pumping amplitude values. Degenerate energy terms arise as a result of merging and branching of pairs of nondegenerate energy terms. Equations are obtained for finding the merging points of energy terms. These equations are solved by numerical methods. The main objective of this paper is to establish a quantum analog of the classical stochastic layer formed in the separatrix area. With this end in view, we consider a nonstationary quantum-mechanical problem of perturbation of the state of the Mathieu-Schrodinger equation. It is shown that in passing through the branching point the system may pass from the pure state to the mixed one. At multiple passages through branching points there develops the irreversible process of "creeping" of the system to quantum states. In that case, the observed population of a certain number of levels can be considered, in our opinion, to be a quantum analog of the stochastic layer. The number of populated levels is defined by a perturbation amplitude.
Nonlinear gyrotropic medium is a medium whose natural optical activity depends on the intensity of the incident light wave. The Kuhn's model is used to study nonlinear gyrotropic medium with great success. The Kuhn's model presents itself as a model of nonlinear coupled oscillators. In the study of the Kuhn's nonlinear model, classical dynamics in the case of weak as well as strong nonlinearity is analysed. In the case of weak nonlinearity, analytical solutions which are in good agreement with the numerical solutions are obtained. In the case of strong nonlinearity, the values of those parameters for which chaos is formed in the system under study have been determined. The subject of interest is also the question of the Kuhn's model integrability. It is seen that at certain values of the interaction potential, this model is exactly integrable and under certain conditions, it is reduced to the so-called universal Hamiltonian. In the case of quantum-mechanical consideration, the possibility of stochastic absorption of external field energy by nonlinear gyrotropic medium is shown. Finally, further generalization of the Kuhn's model for an infinite chain of interacting oscillators is offered.
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