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• W2062309332 abstract "An autonomous differential equation in $mathcal{R}^n $ with two parameters, $mu _1 $, $mu _2 $, is considered and it is assumed that when both parameter values are zero the equation has a hyperbolic equilibrium and a homoclinic solution. Curves are sought through the origin in the $mu _1 $-$mu _2 $ plane along which the homoclinic solution persists for nonzero parameter values. By using the method of Lyapunov–Schmidt, a function, H, is obtained between two finite-dimensional spaces where the zeros of H represent homoclinic solutions for nonzero parameter values. The implicit function theorem is applied to H in various cases. For $n = 2$ a single curve is obtained as in the work of Melnikov. When $n > 2$ and the stable and unstable manifolds of the hyperbolic equilibrium have an intersection of dimension one, a result of Palmer is achieved which, again, yields a single curve. When this dimension of intersection is greater than one, original results give multiple curves. The various cases of the theory are illustrated by seven examples: one in $mathcal{R}^2 $, one in $mathcal{R}^3 $, four in $mathcal{R}^4 $, and one in $mathcal{R}^6 $." @default.
• W2062309332 created "2016-06-24" @default.
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• W2062309332 date "1992-05-01" @default.
• W2062309332 modified "2023-09-30" @default.
• W2062309332 title "Homoclinic Solutions for Autonomous Dynamical Systems in Arbitrary Dimension" @default.
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• W2062309332 doi "https://doi.org/10.1137/0523036" @default.
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