Пространственная динамика в семействе дифференциальных уравнений шестого порядка из теории структурообразования

Известия высших учебных заведений. Прикладная нелинейная динамика. 2024. Т. 32. № 6. С. 878–896.

Кулагин Н. Е., Лерман Л.М.

Bounded stationary (i.e. independent in time) spatially one-dimensional solutions of a quasilinear parabolic PDE are studied on the whole real line. Its stationary solutions are described by a nonlinear ODE of the sixth order of the Euler–Lagrange–Poisson type and therefore can be transformed to the Hamiltonian system with three degrees
of freedom being in addition reversible with respect two linear involutions. The system has three symmetric equilibria, two of them are hyperbolic in some region of the parameter plane. Goal of the paper. In this paper we, combining methods of dynamical systems theory and numerical simulations, investigate the orbit behavior near the symmetric heteroclinic connection based on these equilibria. It was found both simple (periodic) and complicated orbit behavior. To this end we use the theorem on a global center manifold near the heteroclinic connection. For the third symmetric equilibrium at the origin we found the region in the parameter plane where this equilibrium is of the saddle-focus-center type and found the existence of its homoclinic orbits, long-periodic orbits near homoclinic orbits and orbits with complicated structure. 

Keywords: гомоклиническая траектория, стационарные решения, гамильтонова система, гетероклинический контур, равнение Эйлера–Лагранжа–Пуассона,, седло-фокус-центр, глобальное центральное многообразие, сложная динамика.

Publication based on the results of: The development of analytical and numerical methods for studying multidimensional dynamical systems: second stage (2024)