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Volume 7, Number 1, 2017, Pages 172-188                                                                DOI:10.11948/2017012
Codiagonalization of Matrices and Existence of Multiple Homoclinic Solutions
Xiaobiao Lin,Changrong Zhu
Keywords:Degenerate homoclinic bifurcation, Lyapunov-Schmidt reduction, Lagrange's method, codiagonalization of quadratic forms.
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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