
-
Previous Article
A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory
- DCDS-S Home
- This Issue
-
Next Article
A family of self-avoiding random walks interpolating the loop-erased random walk and a self-avoiding walk on the Sierpiński gasket
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 |
Skew product systems with monotone one-dimensional fibre maps driven by piecewise expanding Markov interval maps may show the phenomenon of intermingled basins [
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. |
[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. |
[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. |
[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.
doi: 10.1007/978-1-4615-6927-5. |
[8] |
W. de Melo and S. van Strien,
One-Dimensional Dynamics Springer, 1993.
doi: 10.1007/978-3-642-78043-1. |
[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. |
[10] |
A. Dembo and O. Zeitouni,
Large Deviations, Techniques and Applications Springer, second edition, 1998.
doi: 10.1007/978-1-4612-5320-4. |
[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. |
[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. |
[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. |
[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. |
[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.
doi: 10.1017/CBO9781107359987. |
[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.(2), 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 and 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 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. |
[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. |
[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. |
[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 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. |
[27] |
Y. Pesin,
Dimension Theory in Dynamical Systems The University of Chicago Press, 1997.
doi: 10.7208/chicago/9780226662237.001.0001. |
[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. |
[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. |
[30] |
J. C. Sommerer and 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 and I. Kan,
Riddled Basins, International Journal of Bifurcation and Chaos, 2 (1992), 795-813.
doi: 10.1142/S0218127492000446. |
[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. |
[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. |
[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.
doi: 10.1007/978-1-4615-6927-5. |
[8] |
W. de Melo and S. van Strien,
One-Dimensional Dynamics Springer, 1993.
doi: 10.1007/978-3-642-78043-1. |
[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. |
[10] |
A. Dembo and O. Zeitouni,
Large Deviations, Techniques and Applications Springer, second edition, 1998.
doi: 10.1007/978-1-4612-5320-4. |
[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. |
[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. |
[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. |
[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. |
[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.
doi: 10.1017/CBO9781107359987. |
[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.(2), 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 and 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 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. |
[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. |
[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. |
[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 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. |
[27] |
Y. Pesin,
Dimension Theory in Dynamical Systems The University of Chicago Press, 1997.
doi: 10.7208/chicago/9780226662237.001.0001. |
[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. |
[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. |
[30] |
J. C. Sommerer and E. Ott,
A physical system with qualitatively uncertain dynamics, Nature, 365 (1993), 138-140.
doi: 10.1038/365138a0. |
[1] |
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 |
[2] |
P.E. Kloeden, Victor S. Kozyakin. The perturbation of attractors of skew-product flows with a shadowing driving system. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 883-893. doi: 10.3934/dcds.2001.7.883 |
[3] |
Mathias Staudigl, Srinivas Arigapudi, William H. Sandholm. Large deviations and Stochastic stability in Population Games. Journal of Dynamics and Games, 2021 doi: 10.3934/jdg.2021021 |
[4] |
Tomás Caraballo, Alexandre N. Carvalho, Henrique B. da Costa, José A. Langa. Equi-attraction and continuity of attractors for skew-product semiflows. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 2949-2967. doi: 10.3934/dcdsb.2016081 |
[5] |
Vaughn Climenhaga. A note on two approaches to the thermodynamic formalism. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 995-1005. doi: 10.3934/dcds.2010.27.995 |
[6] |
Michael Jakobson, Lucia D. Simonelli. Countable Markov partitions suitable for thermodynamic formalism. Journal of Modern Dynamics, 2018, 13: 199-219. doi: 10.3934/jmd.2018018 |
[7] |
Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Thermodynamic formalism for random countable Markov shifts. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 131-164. doi: 10.3934/dcds.2008.22.131 |
[8] |
Yongluo Cao, De-Jun Feng, Wen Huang. The thermodynamic formalism for sub-additive potentials. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 639-657. doi: 10.3934/dcds.2008.20.639 |
[9] |
Anna Mummert. The thermodynamic formalism for almost-additive sequences. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 435-454. doi: 10.3934/dcds.2006.16.435 |
[10] |
Luis Barreira. Nonadditive thermodynamic formalism: Equilibrium and Gibbs measures. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 279-305. doi: 10.3934/dcds.2006.16.279 |
[11] |
Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Corrigendum to: Thermodynamic formalism for random countable Markov shifts. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 593-594. doi: 10.3934/dcds.2015.35.593 |
[12] |
Yakov Pesin. On the work of Sarig on countable Markov chains and thermodynamic formalism. Journal of Modern Dynamics, 2014, 8 (1) : 1-14. doi: 10.3934/jmd.2014.8.1 |
[13] |
Miguel Abadi, Sandro Vaienti. Large deviations for short recurrence. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 729-747. doi: 10.3934/dcds.2008.21.729 |
[14] |
Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113 |
[15] |
Salah-Eldin A. Mohammed, Tusheng Zhang. Large deviations for stochastic systems with memory. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 881-893. doi: 10.3934/dcdsb.2006.6.881 |
[16] |
L. Cioletti, E. Silva, M. Stadlbauer. Thermodynamic formalism for topological Markov chains on standard Borel spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6277-6298. doi: 10.3934/dcds.2019274 |
[17] |
Renaud Leplaideur. From local to global equilibrium states: Thermodynamic formalism via an inducing scheme. Electronic Research Announcements, 2014, 21: 72-79. doi: 10.3934/era.2014.21.72 |
[18] |
Eugen Mihailescu. Applications of thermodynamic formalism in complex dynamics on $\mathbb{P}^2$. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 821-836. doi: 10.3934/dcds.2001.7.821 |
[19] |
Peng Sun. Measures of intermediate entropies for skew product diffeomorphisms. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1219-1231. doi: 10.3934/dcds.2010.27.1219 |
[20] |
Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2369-2384. doi: 10.3934/cpaa.2020103 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]