# American Institute of Mathematical Sciences

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

## A reinjected cuspidal horseshoe

 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.
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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