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Detection of symmetric homoclinic orbits to saddle-centres in reversible systems. (English) Zbl 1101.37017
Summary: We present a perturbation technique for the detection of symmetric homoclinic orbits to saddle-centre equilibria in reversible systems of ordinary differential equations. We assume that the unperturbed system has primary, symmetric homoclinic orbits, which may be either isolated or appear in a family, and use an idea similar to that of Melnikov’s method to detect homoclinic orbits in their neighbourhood. This technique also allows us to identify bifurcations of unperturbed or perturbed, symmetric homoclinic orbits. Our technique is of importance in applications such as nonlinear optics and water waves since homoclinic orbits to saddle-centre equilibria describe embedded solitons (ESs) in systems of partial differential equations representing physical models, and except for special cases their existence has been previously studied only numerically using shooting methods and continuation techniques. We apply the general theory to two examples, a four-dimensional system describing ESs in nonlinear optical media and a six-dimensional system which can possess a one-parameter family of symmetric homoclinic orbits in the unperturbed case. For these examples, the analysis is compared with numerical computations and an excellent agreement between both results is found.
MSC:
37C10Vector fields, flows, ordinary differential equations
37C29Homoclinic and heteroclinic orbits
37C80Symmetries, equivariant dynamical systems
37G20Hyperbolic singular points with homoclinic trajectories
34C37Homoclinic and heteroclinic solutions of ODE
37K40Soliton theory, asymptotic behavior of solutions