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2017, 10(2): 313-334. doi: 10.3934/dcdss.2017015

Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors

 Gerhard Keller, Department Mathematik, Univ. Erlangen-Nuremberg Cauerstr. 11, D-91058 Erlangen, Germany

Received  November 2015 Revised  November 2016 Published  January 2017

Fund Project: This work was funded by DFG grant Ke 514/8-1. It also profited from the activities of the DFG Scientific Network "Skew Product Dynamics and Multifractal Analysis" organized by Tobias Oertel-Jäger

Skew product systems with monotone one-dimensional fibre maps driven by piecewise expanding Markov interval maps may show the phenomenon of intermingled basins [1, 5, 16, 30]. To quantify the degree of intermingledness the uncertainty exponent [23] and the stability index [29, 20] were suggested and characterized (partially). Here we present an approach to evaluate/estimate these two quantities rigorously using thermodynamic formalism for the driving Markov map.

Citation: Gerhard Keller. Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 313-334. doi: 10.3934/dcdss.2017015
References:
 [1] J. Alexander, J. A. Yorke, Z. You, I. Kan, Riddled Basins, International Journal of Bifurcation and Chaos, 2 (1992), 795-813. doi: 10.1142/S0218127492000446. [2] V. Anagnostopoulou, T. Jäger, Nonautonomous saddle-node bifurcations: Random and deterministic forcing, Journal of Differential Equations, 253 (2012), 379-399. doi: 10.1016/j.jde.2012.03.016. [3] V. Baladi, Positive Transfer Operators and Decay of Correlations volume 16 of Advanced Series in Nonlinear Dynamics, World Scientific, 2000. [4] T. Bedford, The box dimension of self-affine graphs and repellers, Nonlinearity, 2 (1999), 53-71. doi: 10.1088/0951-7715/2/1/005. [5] A. Bonifant and J. Milnor, Schwarzian derivatives and cylinder maps, In M. Lyubich and Yampolsky, editors, Fields Institute Communications: Holomorphic Dynamics and Renormalization, 53 (2008), 1-21. [6] R. Bowen, Invariant measures for Markov maps of the interval, Commun. Math. Phys., 69 (1979), 1-17. doi: 10.1007/BF01941319. [7] I. Cornfeld, S. Fomin and Y. Sinai, Ergodic Theory Springer Verlag, 1982. [8] W. de Melo and S. van Strien, One-Dimensional Dynamics Springer, 1993. [9] A. Dembo, T. Zajic, Large deviations: From empirical mean and measure to partial sums process, Stochastic Processes and their Applications, 57 (1995), 191-224. doi: 10.1016/0304-4149(94)00048-X. [10] A. Dembo and O. Zeitouni, Large Deviations, Techniques and Applications Springer, second edition, 1998. [11] K. Duffy, M. Rodgers-Lee, Some useful functions for functional large deviations, Stochastics and Stochastic Reports, 76 (2004), 267-279. doi: 10.1080/10451120410001720434. [12] A. Ganesh, N. O'Connell, A large deviation principle with queueing applications, Stochastics and Stochastic Reports, 73 (2002), 25-35. doi: 10.1080/10451120212871. [13] C. Grebogi, S. W. McDonald, E. Ott, J. A. Yorke, Exterior dimension of fat fractals, Phys. Lett. A, 110 (1985), 1-4. doi: 10.1016/0375-9601(85)90220-8. [14] F. Hofbauer, J. Hofbauer, P. Raith, T. Steinberger, Intermingled basins in a two species system, Journal of Mathematical Biology, 49 (2004), 293-309. doi: 10.1007/s00285-003-0253-3. [15] T. Jäger, Quasiperiodically forced interval maps with negative Schwarzian derivative, Nonlinearity, 16 (2003), 1239-1255. doi: 10.1088/0951-7715/16/4/303. [16] I. Kan, Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin, Bulletin of the American Mathematical Society, 31 (1994), 68-74. doi: 10.1090/S0273-0979-1994-00507-5. [17] G. Keller, Equilibrium States in Ergodic Theory volume 42 of LMS Student Texts, Cambridge University Press, 1998. [18] G. Keller, An elementary proof for the dimension of the graph of the classical Weierstrass function, http://arxiv.org/abs/1406.3571v4(to appear in Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques), 2014. [19] G. Keller, Stability index for chaotically driven concave maps, J. London Math. Soc., 89 (2014), 603-622. doi: 10.1112/jlms/jdt070. [20] U. A. Mohd Roslan, Stability Index for Riddled Basins of Attraction with Applications to Skew Product Systems PhD thesis, University of Exeter, 2015. [21] T. Nowicki, D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Inventiones Mathematicae, 132 (1998), 633-680. doi: 10.1007/s002220050236. [22] E. Ott, J. Alexander, I. Kan, J. Sommerer, J. Yorke, The transition to chaotic attractors with riddled basins, Physica D: Nonlinear Phenomena, 76 (1994), 384-410. doi: 10.1016/0167-2789(94)90047-7. [23] E. Ott, J. Sommerer, J. Alexander, I. Kan, J. Yorke, Scaling behavior of chaotic systems with riddled basins, Physical Review Letters, 71 (1993), 4134-4137. doi: 10.1103/PhysRevLett.71.4134. [24] W. Ott, M. Stenlund, L. Young, Memory loss for time-dependent dynamical systems, Math. Research Letters, 16 (2009), 463-475. doi: 10.4310/MRL.2009.v16.n3.a7. [25] W. Parry and M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics volume 187-188 of Astérisque, Société Mathématique de France, 1990. [26] R. F. Pereira, S. Camargo, S. E. De, S. R. Lopes, R. L. Viana, Periodic-orbit analysis and scaling laws of intermingled basins of attraction in an ecological dynamical system, Physical Review E -Statistical, Nonlinear, and Soft Matter Physics, 78 (2008), 1-10. doi: 10.1103/PhysRevE.78.056214. [27] Y. Pesin, Dimension Theory in Dynamical Systems The University of Chicago Press, 1997. [28] D. Plachky, J. Steinebach, A theorem about probabilities of large deviations with an application to queuing theory, Periodica Mathematica Hungarica, 6 (1975), 343-345. doi: 10.1007/BF02017929. [29] O. Podvigina, P. Ashwin, On local attraction properties and a stability index for heteroclinic connections, Nonlinearity, 24 (2011), 887-929. doi: 10.1088/0951-7715/24/3/009. [30] J. C. Sommerer, E. Ott, A physical system with qualitatively uncertain dynamics, Nature, 365 (1993), 138-140. doi: 10.1038/365138a0.

show all references

References:
 [1] J. Alexander, J. A. Yorke, Z. You, I. Kan, Riddled Basins, International Journal of Bifurcation and Chaos, 2 (1992), 795-813. doi: 10.1142/S0218127492000446. [2] V. Anagnostopoulou, T. Jäger, Nonautonomous saddle-node bifurcations: Random and deterministic forcing, Journal of Differential Equations, 253 (2012), 379-399. doi: 10.1016/j.jde.2012.03.016. [3] V. Baladi, Positive Transfer Operators and Decay of Correlations volume 16 of Advanced Series in Nonlinear Dynamics, World Scientific, 2000. [4] T. Bedford, The box dimension of self-affine graphs and repellers, Nonlinearity, 2 (1999), 53-71. doi: 10.1088/0951-7715/2/1/005. [5] A. Bonifant and J. Milnor, Schwarzian derivatives and cylinder maps, In M. Lyubich and Yampolsky, editors, Fields Institute Communications: Holomorphic Dynamics and Renormalization, 53 (2008), 1-21. [6] R. Bowen, Invariant measures for Markov maps of the interval, Commun. Math. Phys., 69 (1979), 1-17. doi: 10.1007/BF01941319. [7] I. Cornfeld, S. Fomin and Y. Sinai, Ergodic Theory Springer Verlag, 1982. [8] W. de Melo and S. van Strien, One-Dimensional Dynamics Springer, 1993. [9] A. Dembo, T. Zajic, Large deviations: From empirical mean and measure to partial sums process, Stochastic Processes and their Applications, 57 (1995), 191-224. doi: 10.1016/0304-4149(94)00048-X. [10] A. Dembo and O. Zeitouni, Large Deviations, Techniques and Applications Springer, second edition, 1998. [11] K. Duffy, M. Rodgers-Lee, Some useful functions for functional large deviations, Stochastics and Stochastic Reports, 76 (2004), 267-279. doi: 10.1080/10451120410001720434. [12] A. Ganesh, N. O'Connell, A large deviation principle with queueing applications, Stochastics and Stochastic Reports, 73 (2002), 25-35. doi: 10.1080/10451120212871. [13] C. Grebogi, S. W. McDonald, E. Ott, J. A. Yorke, Exterior dimension of fat fractals, Phys. Lett. A, 110 (1985), 1-4. doi: 10.1016/0375-9601(85)90220-8. [14] F. Hofbauer, J. Hofbauer, P. Raith, T. Steinberger, Intermingled basins in a two species system, Journal of Mathematical Biology, 49 (2004), 293-309. doi: 10.1007/s00285-003-0253-3. [15] T. Jäger, Quasiperiodically forced interval maps with negative Schwarzian derivative, Nonlinearity, 16 (2003), 1239-1255. doi: 10.1088/0951-7715/16/4/303. [16] I. Kan, Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin, Bulletin of the American Mathematical Society, 31 (1994), 68-74. doi: 10.1090/S0273-0979-1994-00507-5. [17] G. Keller, Equilibrium States in Ergodic Theory volume 42 of LMS Student Texts, Cambridge University Press, 1998. [18] G. Keller, An elementary proof for the dimension of the graph of the classical Weierstrass function, http://arxiv.org/abs/1406.3571v4(to appear in Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques), 2014. [19] G. Keller, Stability index for chaotically driven concave maps, J. London Math. Soc., 89 (2014), 603-622. doi: 10.1112/jlms/jdt070. [20] U. A. Mohd Roslan, Stability Index for Riddled Basins of Attraction with Applications to Skew Product Systems PhD thesis, University of Exeter, 2015. [21] T. Nowicki, D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Inventiones Mathematicae, 132 (1998), 633-680. doi: 10.1007/s002220050236. [22] E. Ott, J. Alexander, I. Kan, J. Sommerer, J. Yorke, The transition to chaotic attractors with riddled basins, Physica D: Nonlinear Phenomena, 76 (1994), 384-410. doi: 10.1016/0167-2789(94)90047-7. [23] E. Ott, J. Sommerer, J. Alexander, I. Kan, J. Yorke, Scaling behavior of chaotic systems with riddled basins, Physical Review Letters, 71 (1993), 4134-4137. doi: 10.1103/PhysRevLett.71.4134. [24] W. Ott, M. Stenlund, L. Young, Memory loss for time-dependent dynamical systems, Math. Research Letters, 16 (2009), 463-475. doi: 10.4310/MRL.2009.v16.n3.a7. [25] W. Parry and M. Pollicott, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics volume 187-188 of Astérisque, Société Mathématique de France, 1990. [26] R. F. Pereira, S. Camargo, S. E. De, S. R. Lopes, R. L. Viana, Periodic-orbit analysis and scaling laws of intermingled basins of attraction in an ecological dynamical system, Physical Review E -Statistical, Nonlinear, and Soft Matter Physics, 78 (2008), 1-10. doi: 10.1103/PhysRevE.78.056214. [27] Y. Pesin, Dimension Theory in Dynamical Systems The University of Chicago Press, 1997. [28] D. Plachky, J. Steinebach, A theorem about probabilities of large deviations with an application to queuing theory, Periodica Mathematica Hungarica, 6 (1975), 343-345. doi: 10.1007/BF02017929. [29] O. Podvigina, P. Ashwin, On local attraction properties and a stability index for heteroclinic connections, Nonlinearity, 24 (2011), 887-929. doi: 10.1088/0951-7715/24/3/009. [30] J. C. Sommerer, E. Ott, A physical system with qualitatively uncertain dynamics, Nature, 365 (1993), 138-140. doi: 10.1038/365138a0.
The critical graph $\varphi_c$ for various choices of the parameter $a$ in Example 2.3
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