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In the study of nonlinear oscillations, the question on the dependence of the frequency or the period on the energy often arises. In this paper, we find conditions under which the frequency depends on the energy monotonically. In addition, for oscillations near separatrix trajectories, an asymptotics of the period with respect to the energy is constructed.

We study the boundary value problem on a segment of length L for a nonlinear second order differential equation of pendulum type with natural boundary conditions generated by the variational problem of minimizing the total energy functional. We show that the number of solutions depends on L and unboundedly grows as L → ∞. Bibliography: 7 titles. Illustrations: 3 figures.

A second-order semilinear differential equation with slowly varying parameters is considered. With frozen parameters, the corresponding autonomous equation has fixed points: a saddle point and stable nodes. Upon deformation of the parameters, the saddle–node pair merges. An asymptotic solution near such a dynamic bifurcation is constructed. It is found that, in a narrow transition layer, the principal terms of the asymptotics are described by the Riccati and Kolmogorov–Petrovsky–Piskunov equations. An important result is finding the dragging out of the stability: the moment of disruption significantly shifts from the moment of bifurcation. The exact assertions are illustrated by the results of numerical experiments.

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.

A nonlinear second-order differential equation with a small parameter at the derivatives is considered in the case where the limit algebraic equation has a multiple root. The matching method is applied to construct an asymptotic expansion of the solution of the boundary value problem. Two boundary layer variables with different scales are used to describe the asymptotic solution near the boundary.

Aspecific problem is used to illustrate the limits of the approach resulting in an adiabatic approximation. The system of differential equations modeling the cyclotron motion of a charged relativistic particle in the field of an electromagnetic wave is considered. The problem of resonance capture of a particle with significantly varying energy is studied. The main result is the description of the capture area, i.e., the set of initial points in the phase plane from which the resonance trajectories issue. Such a description is obtained by the method of asymptotic approximation in a small parameter which corresponds to the rate of variation in the magnetic field. It is discovered that such an approximation is inapplicable in the case of small amplitudes of the electromagnetic wave.

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.

A system of ordinary differential equations with a local asymptotically stable equilibrium is considered. The problem of stability with respect to a persistent perturbation of the white noise type is discussed. Stability with given bounds is proved on a large time interval with length of the order of the squared inverse perturbation amplitude. The proof is based on the construction of a barrier function for the parabolic Kolmogorov equation associated with the perturbed dynamical system.

The autoresonance problem is to distinguish solutions with unboundedly increasing amplitude for model equations of principal resonance. At the level of formal constructions, the problem is solved by constructing an asymptotic solution in the form of a power series with constant coefficients. As is known, such a series represents the asymptotic behavior of the exact solution. However, for this solution to be related to the description of a physical phenomenon, the stability of the solution is required both with respect to perturbations of the initial data and with respect to the relatively constantly acting perturbations. These two properties are established using the Lyapunov function.

A formal asymptotic solution is considered for a nonlinear system of ordinary differential equations in a neighborhood of a singular point. The problem of existence of an exact solution with such an asymptotics and the problem of stability of this solution are solved. The main tool in these studies is the Lyapunov function for a system linearized on a formal solution.

We study a model system of nonautonomous nonlinear differential equations arising in magnetodynamics theory. We find constraints on the parameters such that Lyapunov-stable solutions with a stabilized phase exist. These solutions describe the synchronization phenomenon in a nonisochronous system with slowly varying parameters.

The Cauchy problem for a parabolic equation with a small parameter multiplying the highest derivatives is considered. The dynamical system corresponding to the limit equation of the first order has an asymptotically stable equilibrium. A Lyapunov function known in a neighborhood of this equilibrium is used to construct a barrier in the Cauchy problem for the original parabolic equation. This result is applied to study the dynamical system with respect to random perturbations of the “white noise” type.

In this paper, we study systems of nonlinear, nonautonomous, ordinary differential equations that appear in the theory of averaging of nonlinear oscillations. We prove existence theorems for them and obtain conditions under which variables of the type of amplitude (or energy) are uniformly bounded with respect to time.

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.

The dehydrogenation of graphane due to external harmonic driving is studied by molecular dynamics simulation. It is found that the dehydrogenation can be initiated by harmonic driving of carbon atoms with as small amplitude as 0.02 A, if the driving frequency is somewhat above the small-amplitude vibration frequency of the C-H bond. Thus, a novel mechanism for the dehydrogenation of graphane is reported.

A model system of two first-order differential equations that arises in the synchronization theory of nonlinear oscillations is investigated. Constraints on the parameters of the equations under which the synchronization is realized on every solution are found. A domain of the parameters in which the synchronization occurs only for a part of the solution set is determined.

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.

A nonlinear non-autonomous system of two ordinary differential equations is considered. It is assumed that the equations corresponding to the principal part in the asymptotics at infinity with respect to the independent variable are integrable and written in the action-angle variables. In the case where the lower terms in the equation periodically depend on the angle, the asymptotic expansion at infinity of the two-parametric family of solutions is constructed.

The equations obtained by averaging a system of three weakly nonlinear oscillators are considered. Distinctive properties of asymptotics at infinity for the solutions of averaged equations for both constant and variable proper frequencies are studied.

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.

The system of two first-order differential equations that arises in averaging nonlinear systems over fast single-frequency oscillations is investigated. The averaging is performed in the neighborhood of the critical free frequency of a nonlinear system. In this case, the original equations differ from the principal resonance equations in the general case. The main result is the construction of the asymptotics of a two-parameter family of solutions in the neighborhood of a solution with unboundedly increasing amplitude. The results, in particular, provide a key to understanding the particle acceleration process in relativistic accelerators near the critical free frequency.

We investigate a system of six nonlinear nonautonomous differential equations resulting from averaging fast driven oscillations and construct resonant asymptotic solutions with a linear time increase of amplitudes.

We consider a system of six nonlinear differential equations obtained by averaging fast forced oscillations. The main result consist in the construction of the asymptotics at infinity for the general solution with bounded amplitudes. We show that the structure of asymptotic series depends on the parameters so that the coefficients of the series vary in jumps on the resonance set.

Autoresonance is one of the most interesting phenomena in nonlinear oscillations. The energy of forced oscillations of a nonlinear system substantially increases in the course of time, although the driving force remains small. This phenomenon, observed in various physical systems [1–8], is used, for example, for accelerating relativistic particles [9–12]. Numerical experiments based on various mathematical models have shown that, for autoresonance to appear, it is necessary that the pump amplitude exceed some threshold value [13, 14]. In the present paper, we obtain a result of this sort analytically by an asymptotic analysis of the equations

with a small coefficient e of the external driving force, which oscillates rapidly: φ = Φ(τ)/e . Here f , g, Φ(τ) are given smooth functions of the slow time τ = et . It is assumed that, for e = 0, the nonperturbed system has a stable fixed “center” type point in general position at the origin; this point is taken to be the initial point. In this paper, we construct an asymptotics of the solution u, v(t ; e) as e → 0, which remains valid for large times t = O(e−1) . The goal of these constructions is to find conditions under which autoresonance occurs. This phenomenon, which is similar to the resonance in linear systems, is characterized by an increase in the energy H(u, v) + e[vf − ug] cos(φ) until it becomes a quantity of order unity, while the driving force (the pumping) is small [1–3]

We investigate a system of two first-order differential equations that appears when averaging nonlinear systems over fast one-frequency oscillations. The main result is the asymptotic behavior of a two-parameter family of solutions with an infinitely growing amplitude. In addition, we find the asymptotic behavior of another two-parameter family of solutions with a bounded amplitude. In particular, these results provide the key to understanding autoresonance as the phenomenon of a considerable growth of forced nonlinear oscillations initiated by a small external pumping.

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.

We study the Cauchy problem for a nonlinear second-order differential equation with a small parameter in the case where the exact solution has a power singularity depending on a small parameter. We propose an asymptotic method similar to the Krylov–Bogoliubov method for localizing the singularity up to the accuracy of any order and construct an asymptotic expansion of the solution in the domain of regular behavior.

We consider the Cauchy problem for a perturbed Liouville equation. An asymptotic solution is constructed with respect to the perturbation parameter by the two-scale expansion method; this construction can be applied over long time intervals. The main result is the definition of a deformation of the leading term of the asymptotic expansion within a slow time scale.

The stability of steady regimes of filtration of a gas—liquid mixture at pressure lower than the saturation pressure is studied for the case of a nonmonotonic dependence of the relative phase permeability of the liquid on the gas saturation. It is shown that periodic self–oscillations can appear, and their evolution leads to deterministic chaos due to the appearance and destruction of quasiperiodic motions.

We construct an asymptotic (with respect to a small parameter) solution of the Cauchy problem for the perturbed Liouville equation in the case where the unperturbed solution has singularities on timelike lines. We propose a modification of the Krylov-Bogoliubov method that, in particular, allows us to find the asymptotic corrections to the singularity lines.

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.

The Cauchy problem for the perturbed Korteveg-de Vries equation with the two-soliton initial data is considered. Differential equations for the slow deformation of the parameters—amplitudes and phase shifts—are derived. It is shown that the phase shift of the slow soliton depends on the deformation of the fast solition.

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.

For an integro-differential equation with rapidly oscillating kernel of Cauchy type it is proved that a solution exists and is uniformly bounded in the Holder norm with respect to a small parameter . This equation describes the essential part of the first correction in soliton perturbation theory. An asymptotic estimate is obtained for this correction, which implies that the soliton structure of the solution of an equation close to the integral equation is preserved for times .

We consider a triple Fourier-type integral that represents a solution to the KdV equation linearized on anN-soliton potential. Assuming that the parameters of the potential depend on the slow timeet, we construct an asymptotics of this integral ase → 0 uniform with respect tox, t up to large timet ∼ e−1.

A new method for analyzing the problems of chemical kinetics is elaborated involving the technique of mathematical modeling. Namely, the matching method of the asymptotic expansion is applied to analyzing the inhibition mechanism of oxidation. The proposed approach is an extension of the well-known method of quasi-stationary concentrations and may be applied to study a series of problems in the field of chemical kinetics. Three different time scales were established for the mechanism of inhibited oxidation under restrictions k7[InH]0/(2k6Wi)1/2 ⩽ 1 and k8 ≫ 2k6 ≫ k7. At the first time scale (that is very fast and is measured in second fractions) the concentration of radicals In only changes while [RO2] ≃ [RO2]0, [In H] ≃ [In H]0 are constants. At the second time scale (s), [RO2] changes while [In] ≃ [In]st, [In H] ≃ [In H]0 are constants. At the third time scale (min), [In H] changes. An asymptotic analysis of the differential equations allows us to find out both the time duration of each step and the variation of the component which changes at this step. After that the rate constants k8, 2k6, k7 are determined from comparison with the experimental measurements of [In], [RO2], and [In H]. Due to the simplicity and efficiency of the asymptotic method, one may be applied to treating the complex multicenter radical chain processes such as conjugated oxidation, radical copolymerization, sulfoxidation, etc. © 1993 John Wiley & Sons, Inc.

For an arbitrary perturbation operator, equations for the modulation of the parameters of the KdV soliton are obtained. The asymptotic behavior of the first correction is investigated, and the influence of the leading term of this asymptotic behavior on the soliton phase shift is demonstrated.

The aim of this paper is to construct the nondecreasing terms of the asymptotic expansion (AE) of the integral J (y, t, ~, s) for ~ + 0, t + ~, uniformly in y. On the one hand, such an asymptotic is of interest in its own right, since it allows one to discover the so-called tail in the form of nondecreasing terms of the AE in the sector behind the soliton. On the other hand, information about the first correction turns out to be necessary in find0 1 ing the functions s(T), ~ (T) in the leading term of the AE. Note that the problem of the perturbed KdV equation was investigated by many authors [1-13]. However, no complete solution was known until now for the problem of finding the total phase shift