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1.
In this paper, the switching dynamics of linear oscillators with arbitrary discontinuous forcing are investigated through the concept of switching systems, and such switching systems consist of countable prescribed linear oscillators with different external excitations. The traditional treatments are to smoothen the discontinuity at switching points of two subsystems in a switching system, which can provide an approximate solution only. Therefore, an alternative method is presented to obtain an exact solution of the resultant switching linear system. Under periodic piecewise forcing and random forcing, the corresponding exact solutions and stochastic responses of switching linear systems are developed. For any periodic forcing, the periodic responses and stability of the resultant system composed of multiple linear oscillators in different time intervals are presented. In addition, the resultant switching system consisting of two oscillators are discussed, and the corresponding stability analysis is carried out.  相似文献   

2.
The initiation and propagation of reaction-diffusion travellingwaves in two regions, coupled together by a linear diffusiveinterchange across a semipermeable membrane is considered. Twosystems are considered in detail where there are two chemicalspecies present, species A and the autocatalytic species B.The first system is governed by quadratic autocatalysis in bothregions together with a linear decay of the autocatalyst inonly one region, with the coupling taken to be via the exchangeof species A. The second system has the reaction scheme of cubicautocatalysis and linear decay of the autocatalytic speciesB in region I, while region II is taken to be chemically inertcontaining only species A, with the mode of coupling betweenthe two regions again being via A. These two systems are consideredvia an initial-value problem in which the amounts of the autocatalystare introduced locally into a uniform concentration of speciesA. A priori bounds for the initial-value problem are first obtained,and then conditions for the initiation of travelling waves forsmall initial inputs of the autocatalyst are derived. Theseresults are extended for the first system to all initial inputsof the autocatalyst. The effects of strong coupling are thenexamined. Numerical results are presented to demonstrate theeffects of varying the parameters of the two systems as predictedby our analysis. Finally, the authors examine the equationsgoverning the permanent-form travelling waves and derive generalproperties of their solution together with a solution for weakcoupling.  相似文献   

3.
A global symmetric period-1 approximate solution is analytically constructed for the non-resonant periodic response of a periodically excited piecewise nonlinear–linear oscillator. The approximate solutions are found to be in good agreement with the exact solutions that are obtained from the numerical integration of the original equations. In addition, the dynamic behaviour of the oscillator is numerically investigated with the help of bifurcation diagrams, Lyapunov exponents, Poincare maps, phase portraits and basins of attraction. The existence of subharmonic and chaotic motions and the coexistence of four attractors are observed for some combinations of the system parameters.  相似文献   

4.
By using the bifurcation theory of dynamical systems to the generalized Kadomtsov-Petviashvili-Benjamin-Bona-Mahony equation, the existence of solitary wave solutions, compactons solution, non-smooth periodic cusp wave solutions and uncountably infinite many smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.  相似文献   

5.
6.
In this paper, we study a class of set-valued dynamical systems that satisfy maximal monotonicity properties. This class includes linear relay systems, linear complementarity systems, and linear mechanical systems with dry friction under some conditions. We discuss two numerical schemes based on time-stepping methods for the computation of the periodic solutions when these systems are periodically excited. We provide formal mathematical justifications for the numerical schemes in the sense of consistency, which means that the continuous-time interpolations of the numerical solutions converge to the continuous-time periodic solution when the discretization step vanishes. The two time-stepping methods are applied for the computation of the periodic solution exhibited by a power electronic converter and the corresponding methods are compared in terms of approximation accuracy and computation time.  相似文献   

7.
The Bäcklund transformations and abundant explicit exact solutions to the AKNS shallow water wave equation are obtained by combining the extended homogeneous balance method with the extended hyperbolic function method. The solutions obtained admit of multiple arbitrary parameters. These solutions include (a) a compound of the rational fractional function and a linear function, (b) a compound of solitary wave solution and a linear function, (c) a compound of the singular travelling wave solutions and a linear function, and (d) a compound of the periodic wave solutions of triangle function and a linear function. In special cases, we can obtain a series of soliton solutions, singular travelling wave solutions, periodic travelling wave solutions, and rational fractional function solution. In addition to re-deriving some known solutions in a systematic way, some brand-new exact solutions are also established.  相似文献   

8.
By using the bifurcation theory of dynamical systems to modified nonlinear dispersive phi-four equation, we analysis all bifurcations and phase portraits in the parametric space, the existence of solitary wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some explicit exact solution formulas are acquired for some special cases.  相似文献   

9.
Exact traveling-wave solutions of time-dependent nonlinear inhomogeneous PDEs, describing several model systems in geophysical fluid dynamics, are found. The reduced nonlinear ODEs are treated as systems of linear algebraic equations in the derivatives. A variety of solutions are found, depending on the rank of the algebraic systems. The geophysical systems include acoustic gravity waves, inertial waves, and Rossby waves. The solutions describe waves which are, in general, either periodic or monoclinic. The present approach is compared with the earlier one due to Grundland (1974) for finding exact solutions of inhomogeneous systems of nonlinear PDEs.  相似文献   

10.
In this paper, a periodic predator–prey system with distributed time delays and impulsive effect is investigated. By using the Floquet theory of linear periodic impulsive equation, some conditions for the linear stability of trivial periodic solution and semi-trivial periodic solutions are obtained. It is proved that the system can be permanent if all the trivial and semi-trivial periodic solutions are linearly unstable. We improve some results in Guo and Chen (2009) [1].  相似文献   

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