BIFURCATION OF TORI FOR PARABOLIC SYSTEMS OF DIFFERENTIAL EQUATIONS WITH WEAK DIFFUSION

Abstract

The aim of the present article is to investigate of some properties of quasiperiodic solutions of nonlinear autonomous parabolic systems with the periodic condition. The research is devoted to the investigation of parabolic systems of differential equations with the help of integral manifolds method in the theory of nonlinear oscillations. We prove the existence of quasiperiodic solutions in autonomous parabolic system of differential equations with weak diffusion on the circle. We study existence and stability of an arbitrarily large finite number of tori for a parabolic system with weak diffusion. The quasiperiodic solution of parabolic system is sought in the form of traveling wave. A representation of the integral manifold is obtained. We seek a solution of parabolic system with the periodic condition in the form of a Fourier series in the complex form and introduce the norm in the space of the coefficients in the Fourier expansion. We use the normal forms method in the general parabolic system of differential equations with weak diffusion. We use bifurcation theory for ordinary differential equations and quasilinear parabolic equations. The existence of quasiperiodic solutions in an autonomous parabolic system of differential equations on the circle with small diffusion is proved. The problems of existence and stability of traveling waves in the parabolic system with weak diffusion are investigated.