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Bifurcation of a homoclinic orbit with a saddle-node equilibrium. (English) Zbl 0727.58026
One studies the codimension two unfolding of an orbit $\Gamma\sb 0$ homoclinic to a saddle-node equilibrium in ${\bbfR}\sp d$. The eigenvalues of the equilibrium are supposed to be positive and negative real parts, except for a simple eigenvalue $\lambda\sb 0=0$. Several new methods are employed. The bifurcation diagram is obtained by the bifurcation function which is derived by the method of Lyapunov-Schmidt decomposition. The idea of exponential trichotomy is employed to study the linearized equation around $\Gamma\sb 0.$ The method of cross sections and Poincaré mappings are used to study the bifurcation of the periodic orbits from $\Gamma\sb 0.$ Under the Poincaré mapping, submanifolds of certain type, called the u- slices can be found to be fixed and hence bifurcation of periodic orbits is proved.

37G99Local and nonlocal bifurcation theory
37-99Dynamic systems and ergodic theory (MSC2000)
34C45Invariant manifolds (ODE)
37C70Attractors and repellers, topological structure