2014, 19(2): 523-541. doi: 10.3934/dcdsb.2014.19.523

Expanding Baker Maps as models for the dynamics emerging from 3D-homoclinic bifurcations

1. 

Departamento de Matemáticas, Universidad de Oviedo, Calvo Sotelo s/n, 33007 Oviedo

2. 

Dep. de Matemáticas, Universidad de Oviedo, Calvo Sotelo s/n, 33007, Oviedo, Spain, Spain

3. 

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via, 585, 08080 Barcelona, Spain

Received  March 2013 Revised  December 2013 Published  February 2014

For certain 3D-homoclinic tangencies where the unstable manifold of the saddle point involved in the homoclinic tangency has dimension two, many different strange attractors have been numerically observed for the corresponding family of limit return maps. Moreover, for some special value of the parameter, the respective limit return map is conjugate to what was called bidimensional tent map. This piecewise affine map is an example of what we call now Expanding Baker Map, and the main objective of this paper is to show how many of the different attractors exhibited for the limit return maps resemble the ones observed for Expanding Baker Maps.
Citation: Antonio Pumariño, José Ángel Rodríguez, Joan Carles Tatjer, Enrique Vigil. Expanding Baker Maps as models for the dynamics emerging from 3D-homoclinic bifurcations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 523-541. doi: 10.3934/dcdsb.2014.19.523
References:
[1]

V. I. Arnold and A. Avez, Problemes Ergodiques De La Mecanique Classique,, Paris: Gauthier-Villars, (1967).

[2]

M. Benedicks and L. Carleson, On iterations of $1-ax^2$ on $(0,1)$,, Ann. Math., 122 (1985), 1. doi: 10.2307/1971367.

[3]

M. Benedicks and L. Carleson, The dynamics of the Hénon map,, Ann. Math., 133 (1991), 73. doi: 10.2307/2944326.

[4]

J. Buzzi, Absolutely continuous invariant measures for generic multi-dimensional piecewise affine expanding maps,, Inter. Jour. Bif. and Chaos, 9 (1999), 1743. doi: 10.1142/S021812749900122X.

[5]

P. Collet and J. P. Eckmann, Iterated Maps of the Interval as Dynamical Systems,, Birkhauser, (1980).

[6]

E. Colli, Infinitely many coexisting strange attractors,, Ann. Inst. H. Poincaré, 15 (1998), 539. doi: 10.1016/S0294-1449(98)80001-2.

[7]

S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, Dynamical phenomena in multidimensional systems with a non-rough Poincare homoclinic curve,, Russ. Acad. Sci. Dokl. Math., 47 (1993), 410.

[8]

S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, On the existence of Newhouse regions near systems with non-rough Poincare homoclinic curve (multidimensional case),, Russian Acad. Sci. Dokl. Math., 47 (1993), 268.

[9]

S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, On dynamical properties of diffeomorphisms with homoclinic tangencies,, J. Math. Sci., 126 (2005), 1317.

[10]

S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, On dynamical properties of multidimensional diffeomorphisms from Newhouse regions. I,, Nonlinearity, 21 (2008), 923. doi: 10.1088/0951-7715/21/5/003.

[11]

S. V. Gonchenko, V. S. Gonchenko and J. C. Tatjer, Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Hénon maps,, Regular and Chaotic Dynamics, 12 (2007), 233. doi: 10.1134/S156035470703001X.

[12]

M. R. Herman, Sur la conjugaison des difféomorphismes du cercle à des rotations,, (French), 46 (1976), 181.

[13]

M. V. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps,, Comm. Math. Phys., 81 (1981), 39. doi: 10.1007/BF01941800.

[14]

A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations,, Trans. Am. Math. Soc., 186 (1973), 481. doi: 10.1090/S0002-9947-1973-0335758-1.

[15]

W. de Melo and S. van Strien, One Dimensional Dynamics,, Berlin, (1993).

[16]

J. Milnor and P. Thurston, On iterated maps of the interval,, Lecture Notes in Math., 1342 (1988), 465. doi: 10.1007/BFb0082847.

[17]

C. Mira, L. Gardini, A. Barugola and J. C. Cathala, Chaotic Dynamics in Two-Dimensional Noninvertible Maps,, World Scientific, (1996). doi: 10.1142/9789812798732.

[18]

L. Mora and M. Viana, Abundance of strange attractors,, Acta Math., 171 (1993), 1. doi: 10.1007/BF02392766.

[19]

S. Newhouse, Non-density of Axiom A (a) on $S^2$,, Proc. Amer. Math. Soc. Sympos. Pure Math., 14 (1970), 191.

[20]

S. Newhouse, Diffeomorphisms with infinitely many sinks,, Topology, 13 (1974), 9. doi: 10.1016/0040-9383(74)90034-2.

[21]

S. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms,, Publ. Math. I.H.E.S. 50 (1979), 50 (1979), 101.

[22]

J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations,, Cambridge University Press, (1993).

[23]

J. Palis and J. C. Yoccoz, Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension,, Acta Math., 172 (1994), 91. doi: 10.1007/BF02392792.

[24]

J. Palis and M. Viana, High dimension diffeomorphisms displaying infinitely many sinks,, Ann. Math., 140 (1994), 207. doi: 10.2307/2118546.

[25]

W. Parry, Symbolic dynamics and transformations of the unit interval,, Trans. AMS, 122 (1966), 368. doi: 10.1090/S0002-9947-1966-0197683-5.

[26]

A. Pumariño and J. C. Tatjer, Dynamics near homoclinic bifurcations of three-dimensional dissipative diffeomorphisms,, Nonlinearity, 19 (2006), 2833. doi: 10.1088/0951-7715/19/12/006.

[27]

A. Pumariño and J. C. Tatjer, Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphism,, Discrete and continuous dynamical systems, (2007), 971. doi: 10.3934/dcdsb.2007.8.971.

[28]

N. Romero, Persistence of homoclinic tangencies in higher dimensions,, Ergod. Th. Dyn. Sys., 15 (1995), 735. doi: 10.1017/S0143385700008634.

[29]

J. C. Tatjer, Three-dimensional dissipative diffeomorphisms with homoclinic tangencies,, Ergodic Theory and Dynamical Systems, 21 (2001), 249. doi: 10.1017/S0143385701001146.

[30]

L. Tedeschini-Lalli and J. A. Yorke, How often do simple dynamical processes have infinitely many coexisting sinks?, Comm. Math. Phys., 106 (1986), 635. doi: 10.1007/BF01463400.

[31]

M. Tsujii, Absolutely continuous invariant measures for expanding piecewise linear maps,, Invent. Math., 143 (2001), 349. doi: 10.1007/PL00005797.

[32]

M. Viana, Strange attractors in higher dimensions,, Bull. Braz. Math. Soc., 24 (1993), 13. doi: 10.1007/BF01231695.

[33]

M. Viana, Homoclinic bifurcations and strange attractors,, Available from: , (). doi: 10.1007/978-94-015-8439-5_10.

[34]

J. A. Yorke and K. T. Alligood, Cascades of period doubling bifurcations: A prerequisite for horseshoes,, Bull. A.M.S. 9 (1983), 9 (1983), 319. doi: 10.1090/S0273-0979-1983-15191-1.

show all references

References:
[1]

V. I. Arnold and A. Avez, Problemes Ergodiques De La Mecanique Classique,, Paris: Gauthier-Villars, (1967).

[2]

M. Benedicks and L. Carleson, On iterations of $1-ax^2$ on $(0,1)$,, Ann. Math., 122 (1985), 1. doi: 10.2307/1971367.

[3]

M. Benedicks and L. Carleson, The dynamics of the Hénon map,, Ann. Math., 133 (1991), 73. doi: 10.2307/2944326.

[4]

J. Buzzi, Absolutely continuous invariant measures for generic multi-dimensional piecewise affine expanding maps,, Inter. Jour. Bif. and Chaos, 9 (1999), 1743. doi: 10.1142/S021812749900122X.

[5]

P. Collet and J. P. Eckmann, Iterated Maps of the Interval as Dynamical Systems,, Birkhauser, (1980).

[6]

E. Colli, Infinitely many coexisting strange attractors,, Ann. Inst. H. Poincaré, 15 (1998), 539. doi: 10.1016/S0294-1449(98)80001-2.

[7]

S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, Dynamical phenomena in multidimensional systems with a non-rough Poincare homoclinic curve,, Russ. Acad. Sci. Dokl. Math., 47 (1993), 410.

[8]

S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, On the existence of Newhouse regions near systems with non-rough Poincare homoclinic curve (multidimensional case),, Russian Acad. Sci. Dokl. Math., 47 (1993), 268.

[9]

S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, On dynamical properties of diffeomorphisms with homoclinic tangencies,, J. Math. Sci., 126 (2005), 1317.

[10]

S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, On dynamical properties of multidimensional diffeomorphisms from Newhouse regions. I,, Nonlinearity, 21 (2008), 923. doi: 10.1088/0951-7715/21/5/003.

[11]

S. V. Gonchenko, V. S. Gonchenko and J. C. Tatjer, Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Hénon maps,, Regular and Chaotic Dynamics, 12 (2007), 233. doi: 10.1134/S156035470703001X.

[12]

M. R. Herman, Sur la conjugaison des difféomorphismes du cercle à des rotations,, (French), 46 (1976), 181.

[13]

M. V. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps,, Comm. Math. Phys., 81 (1981), 39. doi: 10.1007/BF01941800.

[14]

A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations,, Trans. Am. Math. Soc., 186 (1973), 481. doi: 10.1090/S0002-9947-1973-0335758-1.

[15]

W. de Melo and S. van Strien, One Dimensional Dynamics,, Berlin, (1993).

[16]

J. Milnor and P. Thurston, On iterated maps of the interval,, Lecture Notes in Math., 1342 (1988), 465. doi: 10.1007/BFb0082847.

[17]

C. Mira, L. Gardini, A. Barugola and J. C. Cathala, Chaotic Dynamics in Two-Dimensional Noninvertible Maps,, World Scientific, (1996). doi: 10.1142/9789812798732.

[18]

L. Mora and M. Viana, Abundance of strange attractors,, Acta Math., 171 (1993), 1. doi: 10.1007/BF02392766.

[19]

S. Newhouse, Non-density of Axiom A (a) on $S^2$,, Proc. Amer. Math. Soc. Sympos. Pure Math., 14 (1970), 191.

[20]

S. Newhouse, Diffeomorphisms with infinitely many sinks,, Topology, 13 (1974), 9. doi: 10.1016/0040-9383(74)90034-2.

[21]

S. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms,, Publ. Math. I.H.E.S. 50 (1979), 50 (1979), 101.

[22]

J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations,, Cambridge University Press, (1993).

[23]

J. Palis and J. C. Yoccoz, Homoclinic tangencies for hyperbolic sets of large Hausdorff dimension,, Acta Math., 172 (1994), 91. doi: 10.1007/BF02392792.

[24]

J. Palis and M. Viana, High dimension diffeomorphisms displaying infinitely many sinks,, Ann. Math., 140 (1994), 207. doi: 10.2307/2118546.

[25]

W. Parry, Symbolic dynamics and transformations of the unit interval,, Trans. AMS, 122 (1966), 368. doi: 10.1090/S0002-9947-1966-0197683-5.

[26]

A. Pumariño and J. C. Tatjer, Dynamics near homoclinic bifurcations of three-dimensional dissipative diffeomorphisms,, Nonlinearity, 19 (2006), 2833. doi: 10.1088/0951-7715/19/12/006.

[27]

A. Pumariño and J. C. Tatjer, Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphism,, Discrete and continuous dynamical systems, (2007), 971. doi: 10.3934/dcdsb.2007.8.971.

[28]

N. Romero, Persistence of homoclinic tangencies in higher dimensions,, Ergod. Th. Dyn. Sys., 15 (1995), 735. doi: 10.1017/S0143385700008634.

[29]

J. C. Tatjer, Three-dimensional dissipative diffeomorphisms with homoclinic tangencies,, Ergodic Theory and Dynamical Systems, 21 (2001), 249. doi: 10.1017/S0143385701001146.

[30]

L. Tedeschini-Lalli and J. A. Yorke, How often do simple dynamical processes have infinitely many coexisting sinks?, Comm. Math. Phys., 106 (1986), 635. doi: 10.1007/BF01463400.

[31]

M. Tsujii, Absolutely continuous invariant measures for expanding piecewise linear maps,, Invent. Math., 143 (2001), 349. doi: 10.1007/PL00005797.

[32]

M. Viana, Strange attractors in higher dimensions,, Bull. Braz. Math. Soc., 24 (1993), 13. doi: 10.1007/BF01231695.

[33]

M. Viana, Homoclinic bifurcations and strange attractors,, Available from: , (). doi: 10.1007/978-94-015-8439-5_10.

[34]

J. A. Yorke and K. T. Alligood, Cascades of period doubling bifurcations: A prerequisite for horseshoes,, Bull. A.M.S. 9 (1983), 9 (1983), 319. doi: 10.1090/S0273-0979-1983-15191-1.

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