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Constructing dynamical systems having homoclinic bifurcation points of codimension two

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Abstract

A procedure is derived which allows for a systematic construction of three-dimensional ordinary differential equations having homoclinic solutions. The equations are proved to exhibit codimension-two homoclinic bifurcation points. Examples include the non-orientable resonant bifurcation, the inclination-flip, and the orbit-flip. In addition, an equation is constructed which has a homoclinic orbit converging to a saddle-focus satisfying Shilnikov's condition. The vector fields are polynomial and non-stiff in that the eigenvalues are of moderate size.

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Sandstede, B. Constructing dynamical systems having homoclinic bifurcation points of codimension two. J Dyn Diff Equat 9, 269–288 (1997). https://doi.org/10.1007/BF02219223

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  • DOI: https://doi.org/10.1007/BF02219223

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