Abstract
An iterative method for solution of Cauchy problem for one-dimensional nonlinear hyperbolic differential equation is proposed in this paper. The method is based on continuous method for solution of nonlinear operator equations. The keystone idea of the method consists in transition from the original problem to a nonlinear integral equation and its successive solution via construction of an auxiliary system of nonlinear differential equations that can be solved with the help of different numerical methods. The result is presented as a mesh function that consists of approximate values of the solution of stated problem and is constructed on a uniform mesh in a bounded domain of two-dimensional space. The advantages of the method are its simplicity and also its universality in the sense that the method can be applied for solving problems with a wide range of nonlinearities. Finally it should be mentioned that one of the important advantages of the proposed method is its stability to perturbations of initial data that is substantiated by methods for analysis of stability of solutions of systems of ordinary differential equations. Solving several model problems shows effectiveness of the proposed method.
Highlights
We consider Duffing equation with small perturbation consisting of time-independent nonconservative part similar to Van der Pol equation and quasiperiodic two-frequency part with irrational frequency ratio
We establish that the number of "partly passable"resonance levels is finite and qualitative behavior of solutions near other resonance levels is determined by the autonomous part of perturbation
We study solutions corresponding to a limit cycle generated by the autonomous part of the perturbation
Summary
Cлучай периодических возмущений рассмотрен наиболее полно (см., например, [7,8], где приведены решения задачи о глобальном поведении решений). Рассмотрим поведение решений системы (1.2) с начальными условиями из области D, где: а) D – компактная часть плоскости (x, y), из которой выброшена окрестность состояния равновесия при α = 1; б) D – компактная область вне «восьмерки», из которой выброшена окрестность сепаратрисы или области внутри «восьмерки» с выброшенной окрестностью состояний равновесия и окрестностью сепаратрисы. Глобальное исследование системы (1.2) связано с рассмотрением поведения решений как в области D, так и в окрестности невозмущенных сепаратрис. Исследование системы (1.2) в окрестностях индивидуальных резонансных уровней H(x, y) = hres было проведено в работе [5]. Если все уровни H(x, y) = h являются проходимыми, то качественное поведение решений системы (1.2) в ячейке D определяется автономной системой. В работе [5] были намечены подходы к решению задачи о глобальном поведении решений. Прежде всего опишем общую схему глобального исследования, затем перейдем к вычислительной части
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