American Institute of Mathematical Sciences

Advanced
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
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, (2009), 757-789.

[2]

P. Bonckaert, V. Naudot, Asymptotic properties of the Dulac map near a hyperbolic saddle in dimension three, Ann. Fac. Sci. Toulouse. Math. 6, (8), (2001), no. 4, 595-617.

[3]

S.N. Chow, B. Deng, B. Fiedler, Homoclinic bifurcation at resonant eigenvalues, Journ. Dynamics and Diff. Eq., 2, (1990), 177-244.

[4]

S. Day, R. Frongillo, R. Treviño, Algorithms for rigorous entropy bounds and symbolic dynamics, SIAM J. Appl. Dyn. Syst. 7, (2008), 1477-1506.

[5]

B. Deng, Homoclinic twisting bifurcation and cusp horseshoe maps, J. Dyn. Diff.Eq. 5, (1993), 417-467.

[6]

S. Day, O. Junge, K. Mischaikow, Towards automated chaos verification, EQUADIFF 2003, 157-162.

[7]

M. Dellnitz, A. Hohmann, O. Junge, M. Rumpf, Exploring invariant sets and invariant measures, Chaos, 7, (1997), 221-228.

[8]

M. Hirsch, C. Pugh, M. Shub, Invariant Manifolds, Lect. Notes Math. 583 Springer 1977.

[9]

A.J. Homburg, Global Aspects of Homoclinic Bifurcations of Vector Fields, Memoirs A.M.S. 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), 667-693.

[11]

W.D. Kalies, K. Mischaikow, R.C.A.M. VanderVorst, An algorithmic approach to chain recurrence, Found. Comput. Math. 5, (2005), 409-449.

[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, (1993), 305-357.

[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, (1996), 1071-1086.

[15]

V. Naudot, Bifurcations homoclines des champs de vecteurs en dimension trois, Thèse de l'Université de Bourgogne, Dijon (1996).

[16]

V. Naudot, J. Yang, Linearization of families of germs of hyperbolic vector fields, Dyn. Syst. 23, (2008), no. 4, 467-489.

[17]

J. Palis, F. Takens., "Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and infinitely many Attractors'', Cambridge University Press 1993.

[18]

J. Palis, W. de Melo, Geometric Theory of Dynamical Systems. An introdcution, Springer Verlag 1982.

[19]

M.R. Rychlik, Lorenz attractors through Shil'nikov-type bifurcation. Part I, Ergod. Th. & Dynam. Syst. 10, (1990), 793-821.

[20]

S. Smale, Differential dynamical systems, Bull. Am. Math. Soc. 73, (1967), 747-817.

show all references

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, (2009), 757-789.

[2]

P. Bonckaert, V. Naudot, Asymptotic properties of the Dulac map near a hyperbolic saddle in dimension three, Ann. Fac. Sci. Toulouse. Math. 6, (8), (2001), no. 4, 595-617.

[3]

S.N. Chow, B. Deng, B. Fiedler, Homoclinic bifurcation at resonant eigenvalues, Journ. Dynamics and Diff. Eq., 2, (1990), 177-244.

[4]

S. Day, R. Frongillo, R. Treviño, Algorithms for rigorous entropy bounds and symbolic dynamics, SIAM J. Appl. Dyn. Syst. 7, (2008), 1477-1506.

[5]

B. Deng, Homoclinic twisting bifurcation and cusp horseshoe maps, J. Dyn. Diff.Eq. 5, (1993), 417-467.

[6]

S. Day, O. Junge, K. Mischaikow, Towards automated chaos verification, EQUADIFF 2003, 157-162.

[7]

M. Dellnitz, A. Hohmann, O. Junge, M. Rumpf, Exploring invariant sets and invariant measures, Chaos, 7, (1997), 221-228.

[8]

M. Hirsch, C. Pugh, M. Shub, Invariant Manifolds, Lect. Notes Math. 583 Springer 1977.

[9]

A.J. Homburg, Global Aspects of Homoclinic Bifurcations of Vector Fields, Memoirs A.M.S. 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), 667-693.

[11]

W.D. Kalies, K. Mischaikow, R.C.A.M. VanderVorst, An algorithmic approach to chain recurrence, Found. Comput. Math. 5, (2005), 409-449.

[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, (1993), 305-357.

[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, (1996), 1071-1086.

[15]

V. Naudot, Bifurcations homoclines des champs de vecteurs en dimension trois, Thèse de l'Université de Bourgogne, Dijon (1996).

[16]

V. Naudot, J. Yang, Linearization of families of germs of hyperbolic vector fields, Dyn. Syst. 23, (2008), no. 4, 467-489.

[17]

J. Palis, F. Takens., "Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and infinitely many Attractors'', Cambridge University Press 1993.

[18]

J. Palis, W. de Melo, Geometric Theory of Dynamical Systems. An introdcution, Springer Verlag 1982.

[19]

M.R. Rychlik, Lorenz attractors through Shil'nikov-type bifurcation. Part I, Ergod. Th. & Dynam. Syst. 10, (1990), 793-821.

[20]

S. Smale, Differential dynamical systems, Bull. Am. Math. Soc. 73, (1967), 747-817.

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