April  2013, 33(4): 1545-1562. doi: 10.3934/dcds.2013.33.1545

SRB attractors with intermingled basins for non-hyperbolic diffeomorphisms

1. 

School of Statistics, Capital University of Economics and Business, Beijing 100070, China

Received  January 2011 Revised  August 2012 Published  October 2012

We investigate a class of non-hyperbolic diffeomorphisms defined on the product space. By using the Pesin theory combined with the general theory of differentiable dynamical systems, we prove that there are exactly two SRB attractors, and their basins cover a full measure subset of the ambient manifold. Furthermore, we prove that the basins of SRB attractors have the strange intermingled phenomenon, i.e. they are measure-theoretically dense in each other. The intermingled phenomena have been observed in many physical systems by numerical experiments, and considered to be important to some fundamental problems in physical, biology and computer science etc. Finally, we describe a concrete example for application.
Citation: Zhicong Liu. SRB attractors with intermingled basins for non-hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1545-1562. doi: 10.3934/dcds.2013.33.1545
References:
[1]

J. C. Alexander, B. Hunt, I. Kan and J. A. Yorke, Intermingled basins for the triangle map, Ergodic Theory and Dynamical Systems, 16 (1996), 651-662. doi: 10.1017/S0143385700009020.

[2]

J. C. Alexander, I. Kan, J. A. Yorke and Z. You, Riddled basins, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 2 (1992), 795-813.

[3]

D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory, Ergodic diffeomorphisms, Trudy Moskov. Mat. Obšč, 23 (1970), 3-36.

[4]

P. Ashwin, J. Buescu and I. Stewart, Bubbling of attractors and synchronisation of chaotic oscillators, Phys. Lett. A, 193 (1994), 126-139. doi: 10.1016/0375-9601(94)90947-4.

[5]

L. Barreira and Y. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory," University Lecture Series 23, American Mathematical Society, Providence, RI, 2002.

[6]

G. D. Birkhoff, Probability and physical systems, Bull. Amer. Math. Soc., 38 (1932), 361-379. doi: 10.1090/S0002-9904-1932-05389-7.

[7]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel Journal of Mathematics, 115 (2000), 157-193. doi: 10.1007/BF02810585.

[8]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975.

[9]

M. Brin and G. Stuck, "Introduction to Dynamical Systems," Cambridge University Press, Cambridge, 2002.

[10]

K. Burns, C. Pugh, M. Shub and A. Wilkinson, Recent results about stable ergodicity, in "Smooth ergodic theory and its applications (Seattle, WA, 1999)", 327-366, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001.

[11]

B. Fayad, Topologically mixing flows with pure point spectrum, in "Dynamical Systems, part II", Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Normale Superiore, Pisa, 2003, 113-136.

[12]

P. Grete and M. Markus, Residence time distributions for double-scroll attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 1007-1015. doi: 10.1142/S0218127407017720.

[13]

F. Hofbauer, J. Hofbauer, P. Raith and T. Steinberger, Intermingled basins in a two species system, J. Math. Biol., 49 (2004), 293-309. doi: 10.1007/s00285-003-0253-3.

[14]

I. Kan, Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin, Bull. Amer. Math. Soc., 31 (1994), 68-74. doi: 10.1090/S0273-0979-1994-00507-5.

[15]

T. Kapitaniak, Uncertainty in coupled chaotic systems: Locally intermingled basins of attraction, Phys. Rev. E(3), 53 (1996), part B, 6555-6557. doi: 10.1103/PhysRevE.53.6555.

[16]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems, With a Supplementary Chapter by Katok and Leonardo Mendoza," Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.

[17]

Y. C. Lai, C. Grebogi and J. A. Yorke, Intermingled basins and riddling bifurcation in chaotic dynamical systems, Differential equations and applications, 138-163, Int. Press, Cambridge, MA, 1996.

[18]

I. Melbourne and A. Windsor, A $C^\infty$ diffeomorphism with infinitely many intermingled basins, Ergodic Theory Dynamical Systems, 25 (2005), 1951-1959. doi: 10.1017/S0143385705000325.

[19]

Hiroyuki Nakajima and Yoshisuke Ueda, Riddled basins of the optimal states in learning dynamical systems, Phys. D, 99 (1996), 35-44. doi: 10.1016/S0167-2789(96)00131-5.

[20]

E. Ott, J. C. Alexander, I. Kan, J. C. Sommerer and J. A. Yorke, The transition to chaotic attractors with riddled basins, Phys. D, 76 (1994), 384-410. doi: 10.1016/0167-2789(94)90047-7.

[21]

E. Ott, J. C. Sommerer, J. C. Alexander, I. Kan and J. A. Yorke, Scaling behavior of chaotic systems with riddled basins, Phys. Rev. Lett., 71 (1993), 4134-4137. doi: 10.1103/PhysRevLett.71.4134.

[22]

J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Géométrie complexe et systémes dynamiques (Orsay, 1995), Astérisque No. 261 (2000), xiii-xiv, 335-347.

[23]

J. Palis and W. De Melo, "Geometric Theory of Dynamical Systems: An Introduction," Translated from the Portuguese by A. K. Manning, Springer-Verlag, New York-Berlin, 1982.

[24]

T. N. Palmer, A local deterministic model of quantum spin measurement, Proc. Roy. Soc. London Ser. A, 451 (1995), 585-608. doi: 10.1098/rspa.1995.0145.

[25]

M. W.Parker, Undecidability in $R^n$: riddled basins, the KAM tori, and the stability of the solar system, Philos. Sci., 70 (2003), 359-382. doi: 10.1086/375472.

[26]

Ja. B. Pesin, Characteristic Ljapunov exponents, and ergodic properties of smooth dynamical systems with invariant measure, (Russian) Dokl. Akad. Nauk SSSR, 226 (1976), 774-777.

[27]

Ja. B. Pesin, Families of invariant manifolds that correspond to nonzero characteristic exponents, (Russian) Izv. Akad. Nauk SSSR Ser. Mat, 40 (1976), 1332-1379, 1440.

[28]

Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, (Russian) Uspehi Mat. Nauk, 32 (1977), 55-112, 287.

[29]

C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc., 312 (1989), 1-54. doi: 10.1090/S0002-9947-1989-0983869-1.

[30]

A. Saito and K. Kaneko, Inaccessibility in decision procedures, in "Unconventional models of computation", UMC'2K (Brussels, 2000), 215-233, Discrete Math. Theor. Comput. Sci. Springer, London, 2001.

[31]

A. Saito and K. Kaneko, Inaccessibility and undecidability in computation, geometry, and dynamical systems, Phys. D, 155 (2001), 1-33. doi: 10.1016/S0167-2789(01)00232-9.

[32]

J. C. Sommerer and E. Ott, Intermingled basins of attraction: uncomputability in a simple physical system, Phys. Lett. A, 214 (1996), 243-251. doi: 10.1016/0375-9601(96)00165-X.

[33]

S. van Strien, Transitive maps which are not ergodic with respect to Lebesgue measure, Ergodic Theory Dynam. Systems, 16 (1996), 833-848.

[34]

P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[35]

A. Windsor, Minimal but not uniquely ergodic diffeomorphisms, in "Smooth Ergodic Theory and its Applications (Proc. Symp. Pure Math., 69)", (Ed. A. Katok et al.), American Mathematical Society, Providence, RI, 1999, 809-824.

[36]

A. Yakubu and C. Carlos, Interplay between local dynamics and dispersal in discrete-time metapopulation models, Journal of Theoretical Biology, 218 (2002), 273-288.

show all references

References:
[1]

J. C. Alexander, B. Hunt, I. Kan and J. A. Yorke, Intermingled basins for the triangle map, Ergodic Theory and Dynamical Systems, 16 (1996), 651-662. doi: 10.1017/S0143385700009020.

[2]

J. C. Alexander, I. Kan, J. A. Yorke and Z. You, Riddled basins, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 2 (1992), 795-813.

[3]

D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory, Ergodic diffeomorphisms, Trudy Moskov. Mat. Obšč, 23 (1970), 3-36.

[4]

P. Ashwin, J. Buescu and I. Stewart, Bubbling of attractors and synchronisation of chaotic oscillators, Phys. Lett. A, 193 (1994), 126-139. doi: 10.1016/0375-9601(94)90947-4.

[5]

L. Barreira and Y. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory," University Lecture Series 23, American Mathematical Society, Providence, RI, 2002.

[6]

G. D. Birkhoff, Probability and physical systems, Bull. Amer. Math. Soc., 38 (1932), 361-379. doi: 10.1090/S0002-9904-1932-05389-7.

[7]

C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel Journal of Mathematics, 115 (2000), 157-193. doi: 10.1007/BF02810585.

[8]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975.

[9]

M. Brin and G. Stuck, "Introduction to Dynamical Systems," Cambridge University Press, Cambridge, 2002.

[10]

K. Burns, C. Pugh, M. Shub and A. Wilkinson, Recent results about stable ergodicity, in "Smooth ergodic theory and its applications (Seattle, WA, 1999)", 327-366, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001.

[11]

B. Fayad, Topologically mixing flows with pure point spectrum, in "Dynamical Systems, part II", Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Normale Superiore, Pisa, 2003, 113-136.

[12]

P. Grete and M. Markus, Residence time distributions for double-scroll attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 1007-1015. doi: 10.1142/S0218127407017720.

[13]

F. Hofbauer, J. Hofbauer, P. Raith and T. Steinberger, Intermingled basins in a two species system, J. Math. Biol., 49 (2004), 293-309. doi: 10.1007/s00285-003-0253-3.

[14]

I. Kan, Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin, Bull. Amer. Math. Soc., 31 (1994), 68-74. doi: 10.1090/S0273-0979-1994-00507-5.

[15]

T. Kapitaniak, Uncertainty in coupled chaotic systems: Locally intermingled basins of attraction, Phys. Rev. E(3), 53 (1996), part B, 6555-6557. doi: 10.1103/PhysRevE.53.6555.

[16]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems, With a Supplementary Chapter by Katok and Leonardo Mendoza," Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.

[17]

Y. C. Lai, C. Grebogi and J. A. Yorke, Intermingled basins and riddling bifurcation in chaotic dynamical systems, Differential equations and applications, 138-163, Int. Press, Cambridge, MA, 1996.

[18]

I. Melbourne and A. Windsor, A $C^\infty$ diffeomorphism with infinitely many intermingled basins, Ergodic Theory Dynamical Systems, 25 (2005), 1951-1959. doi: 10.1017/S0143385705000325.

[19]

Hiroyuki Nakajima and Yoshisuke Ueda, Riddled basins of the optimal states in learning dynamical systems, Phys. D, 99 (1996), 35-44. doi: 10.1016/S0167-2789(96)00131-5.

[20]

E. Ott, J. C. Alexander, I. Kan, J. C. Sommerer and J. A. Yorke, The transition to chaotic attractors with riddled basins, Phys. D, 76 (1994), 384-410. doi: 10.1016/0167-2789(94)90047-7.

[21]

E. Ott, J. C. Sommerer, J. C. Alexander, I. Kan and J. A. Yorke, Scaling behavior of chaotic systems with riddled basins, Phys. Rev. Lett., 71 (1993), 4134-4137. doi: 10.1103/PhysRevLett.71.4134.

[22]

J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Géométrie complexe et systémes dynamiques (Orsay, 1995), Astérisque No. 261 (2000), xiii-xiv, 335-347.

[23]

J. Palis and W. De Melo, "Geometric Theory of Dynamical Systems: An Introduction," Translated from the Portuguese by A. K. Manning, Springer-Verlag, New York-Berlin, 1982.

[24]

T. N. Palmer, A local deterministic model of quantum spin measurement, Proc. Roy. Soc. London Ser. A, 451 (1995), 585-608. doi: 10.1098/rspa.1995.0145.

[25]

M. W.Parker, Undecidability in $R^n$: riddled basins, the KAM tori, and the stability of the solar system, Philos. Sci., 70 (2003), 359-382. doi: 10.1086/375472.

[26]

Ja. B. Pesin, Characteristic Ljapunov exponents, and ergodic properties of smooth dynamical systems with invariant measure, (Russian) Dokl. Akad. Nauk SSSR, 226 (1976), 774-777.

[27]

Ja. B. Pesin, Families of invariant manifolds that correspond to nonzero characteristic exponents, (Russian) Izv. Akad. Nauk SSSR Ser. Mat, 40 (1976), 1332-1379, 1440.

[28]

Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, (Russian) Uspehi Mat. Nauk, 32 (1977), 55-112, 287.

[29]

C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc., 312 (1989), 1-54. doi: 10.1090/S0002-9947-1989-0983869-1.

[30]

A. Saito and K. Kaneko, Inaccessibility in decision procedures, in "Unconventional models of computation", UMC'2K (Brussels, 2000), 215-233, Discrete Math. Theor. Comput. Sci. Springer, London, 2001.

[31]

A. Saito and K. Kaneko, Inaccessibility and undecidability in computation, geometry, and dynamical systems, Phys. D, 155 (2001), 1-33. doi: 10.1016/S0167-2789(01)00232-9.

[32]

J. C. Sommerer and E. Ott, Intermingled basins of attraction: uncomputability in a simple physical system, Phys. Lett. A, 214 (1996), 243-251. doi: 10.1016/0375-9601(96)00165-X.

[33]

S. van Strien, Transitive maps which are not ergodic with respect to Lebesgue measure, Ergodic Theory Dynam. Systems, 16 (1996), 833-848.

[34]

P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[35]

A. Windsor, Minimal but not uniquely ergodic diffeomorphisms, in "Smooth Ergodic Theory and its Applications (Proc. Symp. Pure Math., 69)", (Ed. A. Katok et al.), American Mathematical Society, Providence, RI, 1999, 809-824.

[36]

A. Yakubu and C. Carlos, Interplay between local dynamics and dispersal in discrete-time metapopulation models, Journal of Theoretical Biology, 218 (2002), 273-288.

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