Bifurcations in distributed kinetic systems with aperiodic instability

Abstract

Solutions to nonlinear parabolic partial differential equations which describe non-equilibrium systems of different physical nature, arising after the trivial solution has become unstable, are considered. It is demonstrated that in the case of the short-wave instability of the trivial state the primary bifurcation results in the appearance of spatially periodic quasiharmonic solutions, their stability being determined by the universal criterion. With further growth of the bifurcation parameter, two higher (secondary) bifurcations are revealed, one transforming the stationary solution into a travelling wave, the other one giving rise to “ripples” on its “crest”. In the case of the long-wave instability, stationary periodic solutions also arise, but, generally speaking, they are not quasiharmonic, and their stability criterion cannot be expressed in a universal form.