Construction Of Periodic Solutions Research Articles

Построение периодических решений модифицированного уравнения Кадомцева–Петвиашвили методом дибар-одевания

Построение периодических решений модифицированного уравнения Кадомцева–Петвиашвили методом дибар-одевания

Modulation theory for the steady forced KdV–Burgers equation and the construction of periodic solutions

We present a multiple-scale perturbation technique for deriving asymptotic solutions to the steady Korteweg–de Vries (KdV) equation, perturbed by external sinusoidal forcing and Burger’s damping term, which models the near resonant forcing of shallow water in a container. The first order solution in the perturbation hierarchy is the modulated cnoidal wave equation. Using the second order equation in the hierarchy, a system of differential equations is found describing the slowly varying properties of the cnoidal wave. We analyse the fixed point solutions of this system, which correspond to periodic solutions to the perturbed KdV equation. These solutions are then compared to the experimental results of Chester and Bones (1968).

Construction of Periodic Solutions of One Class Nonautonomous Systems of Differential Equations

In this article we proposed a method for constructing approximations to periodic solutions of one class nonautonomous system of ordinary differential equations. It is based on successive approximation scheme using parallel symbolic calculations to obtain solutions in analytical form. We showed the convergence of the scheme of successive approximations on the period, and also considered an example of a second order system where the described scheme of calculations can be applied.

Hamiltonian systems as selfdual equations

Hamiltonian systems with various time boundary conditions are formulated as absolute minima of newly devised non-negative action functionals obtained by a generalization of Bogomolnyi’s trick of ‘dcompleting squares’. Reminiscent of the selfdual Yang-Mills equations, they are not derived from the fact that they are critical points (i.e., from the corresponding Euler-Lagrange equations) but from being zeroes of the corresponding non-negative Lagrangians. A general method for resolving such variational problems is also described and applied to the construction of periodic solutions for Hamiltonian systems, but also to study certain Lagrangian intersections.

Investigation of the periodic solutions of nonlinear autonomous systems in the critical case

We analyze the conditions of existence and the numerical-analytic method for the approximate construction of periodic solutions of nonlinear autonomous systems of differential equations in the critical case.

New approach to the investigation of the existence of periodic solutions of systems of differential equations and their construction

Methods developed for the solution of general equations with restrictions are applied to the construction of periodic solutions of systems of differential equations.

Construction of periodic solutions of a nonlinear singularly perturbed system of differential equations with degeneration

Construction of periodic solutions of a nonlinear singularly perturbed system of differential equations with degeneration

Construction of Periodic Solutions of Linear Systems of Partial Differential Equations with Linear Deviations of Arguments

We prove a theorem on the existence of a solution of a system of partial differential equations with linearly transformed arguments that is continuously differentiable and bounded on ℝ2.

Antiphased states in intracavity second-harmonic generation: Stability of the periodic solutions.

We study the self-pulsing solutions which emerge from the steady state via a Hopf bifurcation in a laser with intracavity second-harmonic generation. Our main result is the construction of periodic solutions and the derivation of explicit amplitude equations which rule the stability of these periodic solutions. In this way, we have been able to substantiate the classification into four classes of periodic solutions with simple phase and amplitude relations. Near the Hopf bifurcation, the periodic solutions are stable and the sum over the individual mode intensities is not periodic but constant, a signature of antiphase dynamics in this system. A fifth class of periodic solutions has been found numerically where the sum over all modal intensities is periodic. Finally, a numerical analysis complements these results and shows that the phase relations which characterize antiphase dynamics are maintained throughout the route to chaos and in the first chaotic domain when the pump parameter is increased.

A numerical-analytic method of solution of some singular problems of oscillation theory

Singularities of application of the method of functional expansions in singular problems of oscillation theory are considered, where conditions of boundedness of a solution play the role of boundary conditions. A description is given of a method of numerical realization of a convergence condition of the corresponding functional series. A problem of oscillations of a string that freely hangs in the gravitational field is considered (the expansion of a solution into a power series is used), the problem of oscillations of a flag in a flow of fluid (expansions in half-integer powers of coordinates), and the problem on the construction of periodic solutions of the Mat've equation (expansion into a trigonometric series).

Rotating Waves for Semiconductor Inverter Rings

A simple model for the time-dependent behavior of semiconductor inverter circuits is used for an analysis of the dynamics of inverter rings. VLSI applications of inverter rings include flip-flops and ring oscillators. The analysis concentrates on the construction of periodic solutions in the form of rotating waves. Existence of periodic solutions is established using a Poincaré–Bendixson theorem for monotone cyclic feedback systems. Then three different approaches are used to construct these solutions. First, rotating waves are computed explicitly for an idealized model problem. Second, a Hopf bifurcation theorem for problems with symmetries is employed to study the local behavior of the periodic solutions. Third, software for numerical bifurcation and path following is used to obtain global bifurcation diagrams connecting the different analytical results.

A study of periodic solutions of countable second-order systems

We conduct a numerical and analytic study of the existence and approximate construction of periodic solutions of countable systems of second-order differential equations.

Periodic solutions of nonlinear systems of differential-operator equations with impulse action

Numerical-analytic methods are considered for the investigation of the existence and the approximate construction of periodic solutions of nonlinear differential-operator equations, subjected to an impulse action.

Construction of periodic solutions of nonlinear systems in critical cases

Construction of periodic solutions of nonlinear systems in critical cases

Construction of periodic solutions of second-order differential and integrodifferential wave equations

Construction of periodic solutions of second-order differential and integrodifferential wave equations

A finite-rate theory of resonance in a closed tube: discontinuous solutions of a functional equation

The only solutions to date which describe nonlinear resonant acoustic oscillations are those for which the distortion of the travelling waves is negligible. Many experiments do not comply with this small-rate restriction. A finite-rate theory of resonance for an inviscid gas, in which the intrinsic nonlinearity of the waves is taken into account, necessitates the construction of periodic solutions of a nonlinear functional equation. This is achieved by introducing the notion of a critical point of the functional equation, which corresponds physically to a resonating wavelet. In a finite-rate theory a wave may break in a single cycle in the tube, and thus there may be more than one shock present even at fundamental resonance. Discontinuous solutions of the functional equation are constructed which satisfy the weak shock conditions.

Construction of periodic solutions of integrodifferential equations of viscoelasticity

A theorem on the estimation of the periodic solutions of a linear integrodifferential equation and a theorem of existence and uniqueness of the periodic solution of a nonlinear integrodifferential equation are presented without proof.

On the algorithm for constructing periodic solutions of a system of ordinary differential equations not reduced to the characteristic form of Liapounov's system

As is known the famous Liapounov algorithm (Moiseev, 1969) has been elaborated by its ereator (Liapounov, 1950) for systems of ordinary differential equations of special form which is known as the ‘characteristic’ (Duboshin, 1952, 1964) or ‘canonical’ (Moiseev, 1969) form of Liapounov's system. This algorithm allows the construction of periodic solutions of such systems in the form of infinite series in powers of some arbitrary constant.

Construction of periodic solutions of boundary value problems for parabolic equations by the method of straight lines

Construction of periodic solutions of boundary value problems for parabolic equations by the method of straight lines

On a certain case of the construction of periodic solutions of quasi-linear systems

On a certain case of the construction of periodic solutions of quasi-linear systems