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Russian
Heading: 
Bifurcation in Dynamical Systems. Deterministic Chaos. Quantum Chaos.
Article type: 
Article
UDC: 
517.925+517.93
DOI: 
10.18500/0869-6632-003137
EDN: 
ONIPZY

Spatial dynamics in the family of sixth-order differential equations from the theory of partial formation

Autors: 
Kulagin Nikolaj Evgenevich, P.L. Kapitza Institute for Physical Problems of Russian Academy of Sciences
Lerman Lev Mikhailovich, National Research University "Higher School of Economics"
Abstract: 

Topic of the paper. 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.

Key words: 
stationary solutions
Euler–Lagrange–Poisson equation
Hamiltonian system
equilibrium state
saddle
saddlefocus-center
heteroclinic connection
homoclinic orbit
global center invariant manifold
complicated dynamics
Acknowledgments: 
The work by N. E. Kulagin was supported by the RSF grant 22-11-00027 (simulations). L. M. Lerman acknowledges RSF for a partial support under the grant 24-11-00339 (theoretical studies near the loop of a saddle-focus-center). The authors are thankful to D. V. Turaev for the useful discussions.
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Received: 
22.08.2024
Accepted: 
21.10.2024
Available online: 
08.11.2024
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