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Quadratic Volume-Preserving Maps: (Un)stable Manifolds, Hyperbolic Dynamics, and Vortex-Bubble Bifurcations

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Abstract

We implement a semi-analytic scheme for numerically computing high order polynomial approximations of the stable and unstable manifolds associated with the fixed points of the normal form for the family of quadratic volume-preserving diffeomorphisms with quadratic inverse. We use this numerical scheme to study some hyperbolic dynamics associated with an invariant structure called a vortex bubble. The vortex bubble, when present in the system, is the dominant feature in the phase space of the quadratic family, as it encloses all invariant dynamics. Our study focuses on visualizing qualitative features of the vortex bubble such as bifurcations in its geometry, the geometry of some three-dimensional homoclinic tangles associated with the bubble, and the “quasi-capture” of homoclinic orbits by neighboring fixed points. Throughout, we couple our results with previous qualitative numerical studies of the elliptic dynamics within the vortex bubble of the quadratic family.

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Acknowledgements

The author was supported by NSF grant DMS 0354567, by a DARPA FunBio grant, and also by the University of Texas Department of Mathematics Program in Applied and Computational Analysis RTG Fellowship, during the preparation of this work. The author would like to thank Professor Rafael de la Llave for his continued encouragement and support as this manuscript evolved. The images in the manuscript are generated using the PovRay ray-tracing software. Thanks go to Dr. Jason Chambless for many helpful suggestions and insights on the use of the PovRay package. The author spent a week in February 2010 visiting the University of Colorado at Boulder Department of Applied Mathematics, and conversations with Professor James Meiss greatly improved and influenced the content of this work. Thanks again to Professors Meiss and Dullin for granting their permission to reproduce Fig. 3 in the present manuscript. Thanks go to Professors Meiss and Curry and also to Mr. Brock Alan Mosovsky and Ms. Kristine Snyder for their hospitality during that visit. Finally, the author thanks Professors de la Llave and Meiss for carefully reading the manuscript and for their many helpful comments and suggestions.

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    J. D. Mireles James

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Communicated by G. Haller.

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Mireles James, J.D. Quadratic Volume-Preserving Maps: (Un)stable Manifolds, Hyperbolic Dynamics, and Vortex-Bubble Bifurcations. J Nonlinear Sci 23, 585–615 (2013). https://doi.org/10.1007/s00332-012-9162-1

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