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.

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