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A nonlinear oscillating system with small perturbation is considered. It is assumed that the perturbation corresponds to external pumping which has a slow varying frequency. An asymptotics with respect to a small parameter is constructed for the solutions capturing in resonance in a neighborhood of stable equilibrium of the unperturbed system. The time interval of resonance is defined.

We consider a system of differential equations that describes the interaction of two weakly coupled nonlinear oscillators. The initial data are such that, in the absence of coupling, the first oscillator is far from equilibrium, the second oscillator is near equilibrium, and their natural frequencies are close to each other. We study the resonance capture phenomenon, when the frequencies of the coupled oscillators remain close and the amplitudes of their oscillations undergo significant time variations; in particular, the second oscillator moves far away from the equilibrium. We find that the initial stage of the resonance capture is described by a solution of the second Painleve equation. The description is obtained under an asymptotic approximation with respect to a small parameter corresponding to the coupling factor.

We consider a system of differential equations describing the interaction of two weakly coupled nonlinear oscillators. We assume that one oscillator is initially far from equilibrium, the other is near equilibrium, and their frequencies are close. We study the effect of resonance capture, when the frequencies of the coupled oscillators remain close while the oscillation energies change in time significantly; in particular, the second oscillator goes far from equilibrium. We find that the initial stage of resonance capture is described by the second Painlev´e equation. We obtain such a description in the asymptotic approximation in a small parameter corresponding to the coupling constant.

The system of ordinary differential equations that in a specific case describes the cyclotron motion of a charged particle in an electromagnetic wave is considered. The capture of the particle into autoresonance when its energy undergoes a significant change is studied. The main result is a description of the capture domain, which is the set of initial points in the phase plane where the resonance trajectories start. This description is obtained in the asymptotic approximation with respect to the small parameter that in this problem corresponds to the amplitude of the electromagnetic wave.

We consider a system of three ordinary first-order differential equations known in the theory of nuclear magnetism as the Bloch equations. The system contains four dimensionless parameters as coefficients. Equilibrium states and the dependence of their stability on these parameters are investigated. The possibility of the appearance of two stable equilibrium states is discovered. The equations are integrable in the absence of dissipation. For the problem with small dissipation far from equilibrium, approximate solutions are constructed by the method of averaging.

We study a system of two first-order differential equations arising in averaging nonlinear systems over fast single-frequency oscillations. We consider the situation where the original system contains weak dissipative terms. We construct the asymptotic form of a two-parameter solution with an unbounded increasing amplitude. This result gives a key for understanding autoresonance in weak dissipative systems as a phenomenon of significant increase in the forced nonlinear oscillation initiated by a small external pumping.

The asymptotics with respect to a small parameter for solutions of a system of Landau-Lifshitz equations with slowly varying coefficients and small dissipative terms is investigated. These equations are a mathematical model of a uniaxial ferromagnet in a time-dependent magnetic field. The asymptotics constructed make it possible to describe the magnetization reversal effect and to reveal the influence of the parameters of the external magnetic field and dissipation on the stability of this process.

Dynamics of localized magnetic inhomogeneities in the form of breathers has been studied in an easy-plane ferromagnet in an applied ac magnetic field of a variable frequency. Conditions have been determined for an autoresonance parametric excitation of a breather in the presence of dissipation with its amplitude growing in time from small values to approximately unity. The possibility has been shown of not only its resonance growth but also of a resonance depression of nonlinear oscillations of localized magnetic inhomogeneities.

A mathematical model of the magnetodynamics of a weak ferromagnetic in an external magnetic field with variable frequency is studied. Conditions for the appearance of autoresonance are found under which the amplitude of the local magnetic inhomogeneities is considerably increased. Mathematically, the problem is to analyze the solutions to the sine-Gordon equation subject to a specific nonautonomous perturbation. For the perturbed equation, asymptotic solutions in the form of a breather with a slowly varying amplitude and a phase shift are constructed. The solutions whose amplitude increases with time from small values to quantities of order one are associated with the autoresonance phenomenon.

We study the problem of forced oscillations near a stable equilibrium of a two-dimensional nonlinear Hamiltonian system of equations. A given exciting force is represented as rapid oscillations with a small amplitude and a slowly varying frequency. We study the conditions under which such a perturbation makes the phase trajectory of the system recede from the original equilibrium point to a distance of the order of unity. To study the problem, we construct asymptotic solutions using a small amplitude parameter. We present the solution for not-too-small values of time outside the original boundary layer.

Nonlinear dynamics of domain walls in a ferromagnetic plate with a stripe domain structure in an external field with slowly varying amplitude and frequency has been theoretically investigated. The conditions for guided autoresonant excitation of nonlinear domain-wall oscillations in the presence of dissipation have been determined.

Weak perturbation of the KdV equation with multisoliton initial data is studied. The ordinary differential equations for the slow modulation of the soliton parameters are derived. It is found that the phase shift of each soliton depends on the slowly varying amplitudes of all fore solitons. The first correction of the asymptotics is obtained. The leading term outside of the soliton sectors is given, which is called the soliton tail.