Homoclinic Solutions and Chaos in Ordinary Differential Equations with Singular Perturbations |
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Authors: | Joseph Gruendler |
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Institution: | Department of Mathematics, North Carolina A&T State University, Greensboro, North Carolina 27411 |
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Abstract: | Ordinary differential equations are considered which contain a singular perturbation. It is assumed that when the perturbation parameter is zero, the equation has a hyperbolic equilibrium and homoclinic solution. No restriction is placed on the dimension of the phase space or on the dimension of intersection of the stable and unstable manifolds. A bifurcation function is established which determines nonzero values of the perturbation parameter for which the homoclinic solution persists. It is further shown that when the vector field is periodic and a transversality condition is satisfied, the homoclinic solution to the perturbed equation produces a transverse homoclinic orbit in the period map. The techniques used are those of exponential dichotomies, Lyapunov-Schmidt reduction and scales of Banach spaces. A much simplified version of this latter theory is developed suitable for the present case. This work generalizes some recent results of Battelli and Palmer. |
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Keywords: | Ordinary differential equations homoclinic solutions bifurcations singular perturbations |
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