American Institute of Mathematical Sciences

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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

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
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show all references

References:
[1]

SIAM J. Appl. Dyn. Syst. 8, (2009), 757-789.  Google Scholar

[2]

Ann. Fac. Sci. Toulouse. Math. 6, (8), (2001), no. 4, 595-617.  Google Scholar

[3]

Journ. Dynamics and Diff. Eq., 2, (1990), 177-244.  Google Scholar

[4]

SIAM J. Appl. Dyn. Syst. 7, (2008), 1477-1506.  Google Scholar

[5]

J. Dyn. Diff.Eq. 5, (1993), 417-467.  Google Scholar

[6]

EQUADIFF 2003, 157-162.  Google Scholar

[7]

Chaos, 7, (1997), 221-228.  Google Scholar

[8]

Lect. Notes Math. 583 Springer 1977.  Google Scholar

[9]

Memoirs A.M.S. 578, (1996).  Google Scholar

[10]

Ergod. Th. & Dynam. Sys. 14 (1994), 667-693.  Google Scholar

[11]

Found. Comput. Math. 5, (2005), 409-449.  Google Scholar

[12]

Journ. Dynamics and Diff. Eq. 5, (1993), 305-357.  Google Scholar

[13]

Annals of Math. Studies. Princeton University Press, 1973.  Google Scholar

[14]

Ergod. Th. & Dynam. Syst. 16, (1996), 1071-1086.  Google Scholar

[15]

Thèse de l'Université de Bourgogne, Dijon (1996). Google Scholar

[16]

Dyn. Syst. 23, (2008), no. 4, 467-489.  Google Scholar

[17]

Cambridge University Press 1993.  Google Scholar

[18]

Springer Verlag 1982.  Google Scholar

[19]

Ergod. Th. & Dynam. Syst. 10, (1990), 793-821.  Google Scholar

[20]

Bull. Am. Math. Soc. 73, (1967), 747-817.  Google Scholar

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