# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [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.  Google Scholar [3] D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory, Ergodic diffeomorphisms, Trudy Moskov. Mat. Obšč, 23 (1970), 3-36.  Google Scholar [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.  Google Scholar [5] L. Barreira and Y. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory," University Lecture Series 23, American Mathematical Society, Providence, RI, 2002.  Google Scholar [6] G. D. Birkhoff, Probability and physical systems, Bull. Amer. Math. Soc., 38 (1932), 361-379. doi: 10.1090/S0002-9904-1932-05389-7.  Google Scholar [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.  Google Scholar [8] R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975.  Google Scholar [9] M. Brin and G. Stuck, "Introduction to Dynamical Systems," Cambridge University Press, Cambridge, 2002.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, (Russian) Uspehi Mat. Nauk, 32 (1977), 55-112, 287.  Google Scholar [29] C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc., 312 (1989), 1-54. doi: 10.1090/S0002-9947-1989-0983869-1.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] S. van Strien, Transitive maps which are not ergodic with respect to Lebesgue measure, Ergodic Theory Dynam. Systems, 16 (1996), 833-848.  Google Scholar [34] P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [3] D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory, Ergodic diffeomorphisms, Trudy Moskov. Mat. Obšč, 23 (1970), 3-36.  Google Scholar [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.  Google Scholar [5] L. Barreira and Y. B. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory," University Lecture Series 23, American Mathematical Society, Providence, RI, 2002.  Google Scholar [6] G. D. Birkhoff, Probability and physical systems, Bull. Amer. Math. Soc., 38 (1932), 361-379. doi: 10.1090/S0002-9904-1932-05389-7.  Google Scholar [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.  Google Scholar [8] R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975.  Google Scholar [9] M. Brin and G. Stuck, "Introduction to Dynamical Systems," Cambridge University Press, Cambridge, 2002.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, (Russian) Uspehi Mat. Nauk, 32 (1977), 55-112, 287.  Google Scholar [29] C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc., 312 (1989), 1-54. doi: 10.1090/S0002-9947-1989-0983869-1.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [33] S. van Strien, Transitive maps which are not ergodic with respect to Lebesgue measure, Ergodic Theory Dynam. Systems, 16 (1996), 833-848.  Google Scholar [34] P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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