Codiagonalization of Matrices and Existence of Multiple Homoclinic Solutions |
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Authors: | Xiaobiao Lin and Changrong Zhu |
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Institution: | Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA and School of Mathematics and Statistics, Chongqing University, Chongqing, 401331 China |
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Abstract: | The purpose of this paper is twofold. First, we use Lagrange''s method and the generalized eigenvalue problem to study systems of two quadratic equations. We find exact conditions so the system can be codiagonalized and can have up to $4$ solutions. Second, we use this result to study homoclinic bifurcations for a periodically perturbed system. The homoclinic bifurcation is determined by $3$ bifurcation equations. To the lowest order, they are $3$ quadratic equations, which can be simplified by the codiagonalization of quadratic forms. We find that up to $4$ transverse homoclinic orbits can be created near the degenerate homoclinic orbit. |
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Keywords: | Degenerate homoclinic bifurcation Lyapunov-Schmidt reduction Lagrange''s method codiagonalization of quadratic forms |
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