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April  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

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.

Keywords: Intermingled basins, chaotic attractors, skew product system, stability index, uncertainty exponent, thermodynamic formalism, large deviations.
Mathematics Subject Classification: 37D35, 37D45, 60F10.
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 and I. Kan, Riddled Basins, International Journal of Bifurcation and Chaos, 2 (1992), 795-813.  doi: 10.1142/S0218127492000446.  Google Scholar

[2]

V. Anagnostopoulou and 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.  Google Scholar

[3]

V. Baladi, Positive Transfer Operators and Decay of Correlations volume 16 of Advanced Series in Nonlinear Dynamics, World Scientific, 2000. doi: 10.1142/9789812813633.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. Bowen, Invariant measures for Markov maps of the interval, Commun. Math. Phys., 69 (1979), 1-17.  doi: 10.1007/BF01941319.  Google Scholar

[7]

I. Cornfeld, S. Fomin and Y. Sinai, Ergodic Theory Springer Verlag, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[8]

W. de Melo and S. van Strien, One-Dimensional Dynamics Springer, 1993. doi: 10.1007/978-3-642-78043-1.  Google Scholar

[9]

A. Dembo and 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.  Google Scholar

[10]

A. Dembo and O. Zeitouni, Large Deviations, Techniques and Applications Springer, second edition, 1998. doi: 10.1007/978-1-4612-5320-4.  Google Scholar

[11]

K. Duffy and M. Rodgers-Lee, Some useful functions for functional large deviations, Stochastics and Stochastic Reports, 76 (2004), 267-279.  doi: 10.1080/10451120410001720434.  Google Scholar

[12]

A. Ganesh and N. O'Connell, A large deviation principle with queueing applications, Stochastics and Stochastic Reports, 73 (2002), 25-35.  doi: 10.1080/10451120212871.  Google Scholar

[13]

C. Grebogi, S. W. McDonald, E. Ott and J. A. Yorke, Exterior dimension of fat fractals, Phys. Lett. A, 110 (1985), 1-4.  doi: 10.1016/0375-9601(85)90220-8.  Google Scholar

[14]

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

[15]

T. Jäger, Quasiperiodically forced interval maps with negative Schwarzian derivative, Nonlinearity, 16 (2003), 1239-1255.  doi: 10.1088/0951-7715/16/4/303.  Google Scholar

[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.  Google Scholar

[17]

G. Keller, Equilibrium States in Ergodic Theory volume 42 of LMS Student Texts, Cambridge University Press, 1998. doi: 10.1017/CBO9781107359987.  Google Scholar

[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. Google Scholar

[19]

G. Keller, Stability index for chaotically driven concave maps, J. London Math. Soc.(2), 89 (2014), 603-622.  doi: 10.1112/jlms/jdt070.  Google Scholar

[20]

U. A. Mohd Roslan, Stability Index for Riddled Basins of Attraction with Applications to Skew Product Systems PhD thesis, University of Exeter, 2015. Google Scholar

[21]

T. Nowicki and D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Inventiones Mathematicae, 132 (1998), 633-680.  doi: 10.1007/s002220050236.  Google Scholar

[22]

E. Ott, J. Alexander, I. Kan, J. Sommerer and 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.  Google Scholar

[23]

E. Ott, J. Sommerer, J. Alexander, I. Kan and J. Yorke, Scaling behavior of chaotic systems with riddled basins, Physical Review Letters, 71 (1993), 4134-4137.  doi: 10.1103/PhysRevLett.71.4134.  Google Scholar

[24]

W. Ott, M. Stenlund and L. Young, Memory loss for time-dependent dynamical systems, Math. Research Letters, 16 (2009), 463-475.  doi: 10.4310/MRL.2009.v16.n3.a7.  Google Scholar

[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.  Google Scholar

[26]

R. F. Pereira, S. Camargo, S. E. De, S. R. Lopes and 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.  Google Scholar

[27]

Y. Pesin, Dimension Theory in Dynamical Systems The University of Chicago Press, 1997. doi: 10.7208/chicago/9780226662237.001.0001.  Google Scholar

[28]

D. Plachky and 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.  Google Scholar

[29]

O. Podvigina and 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.  Google Scholar

[30]

J. C. Sommerer and E. Ott, A physical system with qualitatively uncertain dynamics, Nature, 365 (1993), 138-140.  doi: 10.1038/365138a0.  Google Scholar

show all references

References:
[1]

J. Alexander, J. A. Yorke, Z. You and I. Kan, Riddled Basins, International Journal of Bifurcation and Chaos, 2 (1992), 795-813.  doi: 10.1142/S0218127492000446.  Google Scholar

[2]

V. Anagnostopoulou and 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.  Google Scholar

[3]

V. Baladi, Positive Transfer Operators and Decay of Correlations volume 16 of Advanced Series in Nonlinear Dynamics, World Scientific, 2000. doi: 10.1142/9789812813633.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. Bowen, Invariant measures for Markov maps of the interval, Commun. Math. Phys., 69 (1979), 1-17.  doi: 10.1007/BF01941319.  Google Scholar

[7]

I. Cornfeld, S. Fomin and Y. Sinai, Ergodic Theory Springer Verlag, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[8]

W. de Melo and S. van Strien, One-Dimensional Dynamics Springer, 1993. doi: 10.1007/978-3-642-78043-1.  Google Scholar

[9]

A. Dembo and 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.  Google Scholar

[10]

A. Dembo and O. Zeitouni, Large Deviations, Techniques and Applications Springer, second edition, 1998. doi: 10.1007/978-1-4612-5320-4.  Google Scholar

[11]

K. Duffy and M. Rodgers-Lee, Some useful functions for functional large deviations, Stochastics and Stochastic Reports, 76 (2004), 267-279.  doi: 10.1080/10451120410001720434.  Google Scholar

[12]

A. Ganesh and N. O'Connell, A large deviation principle with queueing applications, Stochastics and Stochastic Reports, 73 (2002), 25-35.  doi: 10.1080/10451120212871.  Google Scholar

[13]

C. Grebogi, S. W. McDonald, E. Ott and J. A. Yorke, Exterior dimension of fat fractals, Phys. Lett. A, 110 (1985), 1-4.  doi: 10.1016/0375-9601(85)90220-8.  Google Scholar

[14]

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

[15]

T. Jäger, Quasiperiodically forced interval maps with negative Schwarzian derivative, Nonlinearity, 16 (2003), 1239-1255.  doi: 10.1088/0951-7715/16/4/303.  Google Scholar

[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.  Google Scholar

[17]

G. Keller, Equilibrium States in Ergodic Theory volume 42 of LMS Student Texts, Cambridge University Press, 1998. doi: 10.1017/CBO9781107359987.  Google Scholar

[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. Google Scholar

[19]

G. Keller, Stability index for chaotically driven concave maps, J. London Math. Soc.(2), 89 (2014), 603-622.  doi: 10.1112/jlms/jdt070.  Google Scholar

[20]

U. A. Mohd Roslan, Stability Index for Riddled Basins of Attraction with Applications to Skew Product Systems PhD thesis, University of Exeter, 2015. Google Scholar

[21]

T. Nowicki and D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Inventiones Mathematicae, 132 (1998), 633-680.  doi: 10.1007/s002220050236.  Google Scholar

[22]

E. Ott, J. Alexander, I. Kan, J. Sommerer and 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.  Google Scholar

[23]

E. Ott, J. Sommerer, J. Alexander, I. Kan and J. Yorke, Scaling behavior of chaotic systems with riddled basins, Physical Review Letters, 71 (1993), 4134-4137.  doi: 10.1103/PhysRevLett.71.4134.  Google Scholar

[24]

W. Ott, M. Stenlund and L. Young, Memory loss for time-dependent dynamical systems, Math. Research Letters, 16 (2009), 463-475.  doi: 10.4310/MRL.2009.v16.n3.a7.  Google Scholar

[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.  Google Scholar

[26]

R. F. Pereira, S. Camargo, S. E. De, S. R. Lopes and 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.  Google Scholar

[27]

Y. Pesin, Dimension Theory in Dynamical Systems The University of Chicago Press, 1997. doi: 10.7208/chicago/9780226662237.001.0001.  Google Scholar

[28]

D. Plachky and 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.  Google Scholar

[29]

O. Podvigina and 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.  Google Scholar

[30]

J. C. Sommerer and E. Ott, A physical system with qualitatively uncertain dynamics, Nature, 365 (1993), 138-140.  doi: 10.1038/365138a0.  Google Scholar

Figure 1.  The critical graph $\varphi_c$ for various choices of the parameter $a$ in Example 2.3
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