CODIAGONALIZATION OF MATRICES AND EXISTENCE OF MULTIPLE HOMOCLINIC SOLUTIONS

Xiaobiao Lin; Changrong Zhu

doi:10.11948/2017012

2017 Volume 7 Issue 1

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Xiaobiao Lin, Changrong Zhu. CODIAGONALIZATION OF MATRICES AND EXISTENCE OF MULTIPLE HOMOCLINIC SOLUTIONS[J]. Journal of Applied Analysis & Computation, 2017, 7(1): 172-188. doi: 10.11948/2017012

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Xiaobiao Lin, Changrong Zhu. CODIAGONALIZATION OF MATRICES AND EXISTENCE OF MULTIPLE HOMOCLINIC SOLUTIONS[J]. Journal of Applied Analysis & Computation, 2017, 7(1): 172-188. doi: 10.11948/2017012

CODIAGONALIZATION OF MATRICES AND EXISTENCE OF MULTIPLE HOMOCLINIC SOLUTIONS

Xiaobiao Lin¹,
Changrong Zhu²

1 Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA;
2 School of Mathematics and Statistics, Chongqing University, Chongqing, 401331 China

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Abstract

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.
- Degenerate homoclinic bifurcation /
- Lyapunov-Schmidt reduction /
- Lagrange's method /
- codiagonalization of quadratic forms
MSC: 15B99;34C23;37G25

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References

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