A numerical method is proposed for computing time-periodic and relativetime-periodic solutions in dissipative wave systems. In such solutions, thetemporal period, and possibly other additional internal parameters such as thepropagation constant, are unknown priori and need to be determined along withthe solution itself. The main idea of the method is to first express thoseunknown parameters in terms of the solution through quasi-Rayleigh quotients,so that the resulting integro-differential equation is for the time-periodicsolution only. Then this equation is computed in the combined spatiotemporaldomain as a boundary value problem by Newton-conjugate-gradient iterations. Theproposed method applies to both stable and unstable time-periodic solutions;its numerical accuracy is spectral; it is fast-converging; and its coding isshort and simple. As numerical examples, this method is applied to theKuramoto-Sivashinsky equation and the cubic-quintic Ginzburg-Landau equation,whose time-periodic or relative time-periodic solutions with spatially-periodicor spatially-localized profiles are computed. This method also applies tosystems of ordinary differential equations, as is illustrated by its simplecomputation of periodic orbits in the Lorenz equations. MATLAB codes for allnumerical examples are provided in appendices to illustrate the simpleimplementation of the proposed method.
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