A reinjected cuspidal horseshoe

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2013, 2013(special): 227-236. doi: 10.3934/proc.2013.2013.227

A reinjected cuspidal horseshoe

Marcus Fontaine ^1, , William D. Kalies ^1, and Vincent Naudot ^1,

1.	Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, 33431 Boca Raton, United States, United States

Received September 2012 Revised July 2013 Published November 2013

Abstract
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Horseshoes play a central role in dynamical systems and are observed in many chaotic systems. However most points in a neighborhood of the horseshoe escape after finite iterations. In this work we construct a model that possesses an attracting set that contains a cuspidal horseshoe with positive entropy. This model is obtained by reinjecting the points that escape the horseshoe and can be realized in a 3-dimensional vector field.

Keywords: inclination-flip homoclinic orbit, topological entropy, Cuspidal horseshoe, Poincaré return map, invariant foliation., hyperbolicity.

Mathematics Subject Classification: Primary: 37D45, 37C29; Secondary: 37B1.

Citation: Marcus Fontaine, William D. Kalies, Vincent Naudot. A reinjected cuspidal horseshoe. Conference Publications, 2013, 2013 (special) : 227-236. doi: 10.3934/proc.2013.2013.227

References:

[1]	Z. Arai, W.D. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilarczyk, A database schema for the analysis of global dynamics of multiparameter systems,, SIAM J. Appl. Dyn. Syst. 8, 8 (2009), 757.
[2]	P. Bonckaert, V. Naudot, Asymptotic properties of the Dulac map near a hyperbolic saddle in dimension three,, Ann. Fac. Sci. Toulouse. Math. 6, 6 (2001), 595.
[3]	S.N. Chow, B. Deng, B. Fiedler, Homoclinic bifurcation at resonant eigenvalues,, Journ. Dynamics and Diff. Eq., 2 (1990), 177.
[4]	S. Day, R. Frongillo, R. Treviño, Algorithms for rigorous entropy bounds and symbolic dynamics,, SIAM J. Appl. Dyn. Syst. 7, 7 (2008), 1477.
[5]	B. Deng, Homoclinic twisting bifurcation and cusp horseshoe maps,, J. Dyn. Diff.Eq. 5, 5 (1993), 417.
[6]	S. Day, O. Junge, K. Mischaikow, Towards automated chaos verification,, EQUADIFF 2003, (2003), 157.
[7]	M. Dellnitz, A. Hohmann, O. Junge, M. Rumpf, Exploring invariant sets and invariant measures,, Chaos, 7 (1997), 221.
[8]	M. Hirsch, C. Pugh, M. Shub, Invariant Manifolds,, Lect. Notes Math. 583* Springer 1977., 583* (1977).
[9]	A.J. Homburg, Global Aspects of Homoclinic Bifurcations of Vector Fields,, Memoirs A.M.S. 578, 578 (1996).
[10]	A.J. Homburg, H. Kokubu, M. Krupa, The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit,, Ergod. Th. & Dynam. Sys. 14 (1994), 14 (1994), 667.
[11]	W.D. Kalies, K. Mischaikow, R.C.A.M. VanderVorst, An algorithmic approach to chain recurrence,, Found. Comput. Math. 5, 5 (2005), 409.
[12]	M. Kisaka, H. Kokubu, K. Oka, Bifurcations to N-homoclinic orbits and N-periodic orbits in vector fields,, Journ. Dynamics and Diff. Eq. 5, 5 (1993), 305.
[13]	J. Moser., Stable and Random Motions in Dynamical Systems,, Annals of Math. Studies. Princeton University Press, (1973).
[14]	V. Naudot, Strange attractor in the unfolding of an inclination-flip homoclinic orbit,, Ergod. Th. & Dynam. Syst. 16, 16 (1996), 1071.
[15]	V. Naudot, Bifurcations homoclines des champs de vecteurs en dimension trois,, Thèse de l'Université de Bourgogne, (1996).
[16]	V. Naudot, J. Yang, Linearization of families of germs of hyperbolic vector fields,, Dyn. Syst. 23, 23 (2008), 467.
[17]	J. Palis, F. Takens., "Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and infinitely many Attractors'',, Cambridge University Press 1993., (1993).
[18]	J. Palis, W. de Melo, Geometric Theory of Dynamical Systems. An introdcution,, Springer Verlag 1982., (1982).
[19]	M.R. Rychlik, Lorenz attractors through Shil'nikov-type bifurcation. Part I,, Ergod. Th. & Dynam. Syst. 10, 10 (1990), 793.
[20]	S. Smale, Differential dynamical systems,, Bull. Am. Math. Soc. 73, 73 (1967), 747.

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References:

[1]	Z. Arai, W.D. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilarczyk, A database schema for the analysis of global dynamics of multiparameter systems,, SIAM J. Appl. Dyn. Syst. 8, 8 (2009), 757.
[2]	P. Bonckaert, V. Naudot, Asymptotic properties of the Dulac map near a hyperbolic saddle in dimension three,, Ann. Fac. Sci. Toulouse. Math. 6, 6 (2001), 595.
[3]	S.N. Chow, B. Deng, B. Fiedler, Homoclinic bifurcation at resonant eigenvalues,, Journ. Dynamics and Diff. Eq., 2 (1990), 177.
[4]	S. Day, R. Frongillo, R. Treviño, Algorithms for rigorous entropy bounds and symbolic dynamics,, SIAM J. Appl. Dyn. Syst. 7, 7 (2008), 1477.
[5]	B. Deng, Homoclinic twisting bifurcation and cusp horseshoe maps,, J. Dyn. Diff.Eq. 5, 5 (1993), 417.
[6]	S. Day, O. Junge, K. Mischaikow, Towards automated chaos verification,, EQUADIFF 2003, (2003), 157.
[7]	M. Dellnitz, A. Hohmann, O. Junge, M. Rumpf, Exploring invariant sets and invariant measures,, Chaos, 7 (1997), 221.
[8]	M. Hirsch, C. Pugh, M. Shub, Invariant Manifolds,, Lect. Notes Math. 583* Springer 1977., 583* (1977).
[9]	A.J. Homburg, Global Aspects of Homoclinic Bifurcations of Vector Fields,, Memoirs A.M.S. 578, 578 (1996).
[10]	A.J. Homburg, H. Kokubu, M. Krupa, The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit,, Ergod. Th. & Dynam. Sys. 14 (1994), 14 (1994), 667.
[11]	W.D. Kalies, K. Mischaikow, R.C.A.M. VanderVorst, An algorithmic approach to chain recurrence,, Found. Comput. Math. 5, 5 (2005), 409.
[12]	M. Kisaka, H. Kokubu, K. Oka, Bifurcations to N-homoclinic orbits and N-periodic orbits in vector fields,, Journ. Dynamics and Diff. Eq. 5, 5 (1993), 305.
[13]	J. Moser., Stable and Random Motions in Dynamical Systems,, Annals of Math. Studies. Princeton University Press, (1973).
[14]	V. Naudot, Strange attractor in the unfolding of an inclination-flip homoclinic orbit,, Ergod. Th. & Dynam. Syst. 16, 16 (1996), 1071.
[15]	V. Naudot, Bifurcations homoclines des champs de vecteurs en dimension trois,, Thèse de l'Université de Bourgogne, (1996).
[16]	V. Naudot, J. Yang, Linearization of families of germs of hyperbolic vector fields,, Dyn. Syst. 23, 23 (2008), 467.
[17]	J. Palis, F. Takens., "Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and infinitely many Attractors'',, Cambridge University Press 1993., (1993).
[18]	J. Palis, W. de Melo, Geometric Theory of Dynamical Systems. An introdcution,, Springer Verlag 1982., (1982).
[19]	M.R. Rychlik, Lorenz attractors through Shil'nikov-type bifurcation. Part I,, Ergod. Th. & Dynam. Syst. 10, 10 (1990), 793.
[20]	S. Smale, Differential dynamical systems,, Bull. Am. Math. Soc. 73, 73 (1967), 747.

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