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Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics. (English) Zbl 1194.37154
Summary: This survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous-time dynamical systems, where the homoclinic orbit and the equilibrium are both reversible. The focus is on even-order reversible systems in four or more dimensions. Local theory, generic argument, and global existence theories are examined for each qualitatively distinct linearisation. Several recent results, such as coalescence caused by non-transversality and the reversible orbit-flip bifurcation are covered. A number of open problems are highlighted. Applications are reviewed to systems arising from a variety of disciplines. With the aid of numerical methods, three examples are presented in detail, one of which is infinite dimensional.
MSC:
37N10Dynamical systems in fluid mechanics, oceanography and meteorology
37N20Dynamical systems in other branches of physics
34C37Homoclinic and heteroclinic solutions of ODE
76E30Nonlinear effects (fluid mechanics)
78A99Miscellaneous topics in optics and electromagnetic theory