2011, 29(3): 1175-1189. doi: 10.3934/dcds.2011.29.1175

Computational hyperbolicity

1. 

Jagiellonian University, Institute of Mathematics, Reymonta 4, 30-059 Kraków

2. 

Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Computer Science, Grota Roweckiego 6, 30-348 Kraków, Poland

Received  February 2010 Revised  July 2010 Published  November 2010

Using semihyperbolicity as a basic tool, we provide a general computer assisted method for verifying hyperbolicity of a given set. As a consequence we obtain that the Hénon attractor is hyperbolic for some parameter values.
Citation: Marcin Mazur, Jacek Tabor. Computational hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1175-1189. doi: 10.3934/dcds.2011.29.1175
References:
[1]

A. Al-Nayef, P. Diamond, P. Kloeden, V. Kozyakin and A. Pokrovskii, Bi-shadowing and delay equations,, Dynam. Stability Systems, 10 (1996), 121.

[2]

A. A. Al-Nayef, P. E. Kloeden and A. V. Pokrovskii, Semi-hyperbolic mappings, condensing operators, and neutral delay equations,, J. Differential Equations, 137 (1997), 320. doi: 10.1006/jdeq.1997.3262.

[3]

Z. Arai, On Hyperbolic Plateaus of the Hénon Maps,, Experiment. Math., 16 (2007), 181.

[4]

, Boost ver. 1.35.0, \url{http://www.boost.org}., ().

[5]

R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I,, J. Differential Equations, 15 (1974), 429. doi: 10.1016/0022-0396(74)90067-9.

[6]

M. Davis, R. MacKay and A. Sannami, Markov shifts in the Hénon family,, Phys. D, 52 (1991), 171. doi: 10.1016/0167-2789(91)90119-T.

[7]

S. Day, H. Kokubu, S. Luzzatto, K. Mischaikow, H. Oka and P. Pilarczyk, Quantitative hyperbolicity estimates in one-dimensional dynamics,, Nonlinearity, 21 (2008), 1967. doi: 10.1088/0951-7715/21/9/002.

[8]

R. Devaney and Z. Nitecki, Shift automorphisms in the Hénon mapping,, Comm. Math. Phys., 67 (1979), 137. doi: 10.1007/BF01221362.

[9]

P. Diamond, P. Kloeden, V. Kozyakin and A. Pokrovskii, Semihyperbolic mappings,, J. Nonlinear Sci., 5 (1995), 419. doi: 10.1007/BF01212908.

[10]

P. Diamond, P. E. Kloeden, V. S. Kozyakin and A. V. Pokrovskii, "Semi-Hyperbolicity and Bi-Shadowing,'', Manuscript., ().

[11]

Z. Galias and P. Zgliczyński, Infinite-dimensional Krawczyk operator for finding periodic orbits of discrete dynamical systems,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 4261. doi: 10.1142/S0218127407019937.

[12]

S. Hruska, A numerical method for constructing the hyperbolic structure of complex Hénon mappings,, Found. Comput. Math., 6 (2006), 427. doi: 10.1007/s10208-006-0141-2.

[13]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'', Cambridge University Press, (1995).

[14]

M. Mazur, On some useful conditions on hyperbolicity,, Trends in Math., 10 (2008), 57.

[15]

M. Mazur, J. Tabor and P. Kościelniak, Semi-hyperbolicity and hyperbolicity,, Discrete Contin. Dynam. Syst., 20 (2008), 1029. doi: 10.3934/dcds.2008.20.1029.

[16]

M. Mazur, J. Tabor and K. Stolot, Semi-hyperbolicity implies hyperbolicity in the linear case,, Proceedings of the Conference, 36 (1998), 121.

[17]

M. Mazur, J. Tabor, T. Kułaga and P. Kościelniak, Computational hyperbolicity group,, \url{http://www.im.uj.edu.pl/MarcinMazur/comphyp}., ().

[18]

K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: A computer-assisted proof,, Bull. Amer. Math. Soc. (N.S.), 32 (1995), 66.

[19]

S. Newhouse, Cone-fields, domination, and hyperbolicity,, in, (2004), 419.

[20]

K. Palmer, "Shadowing in Dynamical Systems. Theory and Applications,'', Kluwer Academic Publishers, (2000).

[21]

F. Riesz and B. Sz.-Nagy, "Functional Analysis,'', Frederick Ungar Publishing Co., (1955).

show all references

References:
[1]

A. Al-Nayef, P. Diamond, P. Kloeden, V. Kozyakin and A. Pokrovskii, Bi-shadowing and delay equations,, Dynam. Stability Systems, 10 (1996), 121.

[2]

A. A. Al-Nayef, P. E. Kloeden and A. V. Pokrovskii, Semi-hyperbolic mappings, condensing operators, and neutral delay equations,, J. Differential Equations, 137 (1997), 320. doi: 10.1006/jdeq.1997.3262.

[3]

Z. Arai, On Hyperbolic Plateaus of the Hénon Maps,, Experiment. Math., 16 (2007), 181.

[4]

, Boost ver. 1.35.0, \url{http://www.boost.org}., ().

[5]

R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I,, J. Differential Equations, 15 (1974), 429. doi: 10.1016/0022-0396(74)90067-9.

[6]

M. Davis, R. MacKay and A. Sannami, Markov shifts in the Hénon family,, Phys. D, 52 (1991), 171. doi: 10.1016/0167-2789(91)90119-T.

[7]

S. Day, H. Kokubu, S. Luzzatto, K. Mischaikow, H. Oka and P. Pilarczyk, Quantitative hyperbolicity estimates in one-dimensional dynamics,, Nonlinearity, 21 (2008), 1967. doi: 10.1088/0951-7715/21/9/002.

[8]

R. Devaney and Z. Nitecki, Shift automorphisms in the Hénon mapping,, Comm. Math. Phys., 67 (1979), 137. doi: 10.1007/BF01221362.

[9]

P. Diamond, P. Kloeden, V. Kozyakin and A. Pokrovskii, Semihyperbolic mappings,, J. Nonlinear Sci., 5 (1995), 419. doi: 10.1007/BF01212908.

[10]

P. Diamond, P. E. Kloeden, V. S. Kozyakin and A. V. Pokrovskii, "Semi-Hyperbolicity and Bi-Shadowing,'', Manuscript., ().

[11]

Z. Galias and P. Zgliczyński, Infinite-dimensional Krawczyk operator for finding periodic orbits of discrete dynamical systems,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 4261. doi: 10.1142/S0218127407019937.

[12]

S. Hruska, A numerical method for constructing the hyperbolic structure of complex Hénon mappings,, Found. Comput. Math., 6 (2006), 427. doi: 10.1007/s10208-006-0141-2.

[13]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'', Cambridge University Press, (1995).

[14]

M. Mazur, On some useful conditions on hyperbolicity,, Trends in Math., 10 (2008), 57.

[15]

M. Mazur, J. Tabor and P. Kościelniak, Semi-hyperbolicity and hyperbolicity,, Discrete Contin. Dynam. Syst., 20 (2008), 1029. doi: 10.3934/dcds.2008.20.1029.

[16]

M. Mazur, J. Tabor and K. Stolot, Semi-hyperbolicity implies hyperbolicity in the linear case,, Proceedings of the Conference, 36 (1998), 121.

[17]

M. Mazur, J. Tabor, T. Kułaga and P. Kościelniak, Computational hyperbolicity group,, \url{http://www.im.uj.edu.pl/MarcinMazur/comphyp}., ().

[18]

K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: A computer-assisted proof,, Bull. Amer. Math. Soc. (N.S.), 32 (1995), 66.

[19]

S. Newhouse, Cone-fields, domination, and hyperbolicity,, in, (2004), 419.

[20]

K. Palmer, "Shadowing in Dynamical Systems. Theory and Applications,'', Kluwer Academic Publishers, (2000).

[21]

F. Riesz and B. Sz.-Nagy, "Functional Analysis,'', Frederick Ungar Publishing Co., (1955).

[1]

Piotr Zgliczyński. Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof. Journal of Computational Dynamics, 2015, 2 (1) : 95-142. doi: 10.3934/jcd.2015.2.95

[2]

A. Aschwanden, A. Schulze-Halberg, D. Stoffer. Stable periodic solutions for delay equations with positive feedback - a computer-assisted proof. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 721-736. doi: 10.3934/dcds.2006.14.721

[3]

Thomas Wanner. Computer-assisted equilibrium validation for the diblock copolymer model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1075-1107. doi: 10.3934/dcds.2017045

[4]

Marcin Mazur, Jacek Tabor, Piotr Kościelniak. Semi-hyperbolicity and hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1029-1038. doi: 10.3934/dcds.2008.20.1029

[5]

Boris Hasselblatt, Yakov Pesin, Jörg Schmeling. Pointwise hyperbolicity implies uniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2819-2827. doi: 10.3934/dcds.2014.34.2819

[6]

Benjamin Steinberg, Yuqing Wang, Huaxiong Huang, Robert M. Miura. Spatial Buffering Mechanism: Mathematical Model and Computer Simulations. Mathematical Biosciences & Engineering, 2005, 2 (4) : 675-702. doi: 10.3934/mbe.2005.2.675

[7]

Elena Celledoni, Markus Eslitzbichler, Alexander Schmeding. Shape analysis on Lie groups with applications in computer animation. Journal of Geometric Mechanics, 2016, 8 (3) : 273-304. doi: 10.3934/jgm.2016008

[8]

Luis Barreira, Claudia Valls. Growth rates and nonuniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 509-528. doi: 10.3934/dcds.2008.22.509

[9]

Rasul Shafikov, Christian Wolf. Stable sets, hyperbolicity and dimension. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 403-412. doi: 10.3934/dcds.2005.12.403

[10]

Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901

[11]

Arno Berger. On finite-time hyperbolicity. Communications on Pure & Applied Analysis, 2011, 10 (3) : 963-981. doi: 10.3934/cpaa.2011.10.963

[12]

Christian Bonatti, Shaobo Gan, Dawei Yang. On the hyperbolicity of homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1143-1162. doi: 10.3934/dcds.2009.25.1143

[13]

Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048

[14]

Michael Hutchings, Frank Morgan, Manuel Ritore and Antonio Ros. Proof of the double bubble conjecture. Electronic Research Announcements, 2000, 6: 45-49.

[15]

John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367

[16]

Kazuhisa Ichikawa. Synergistic effect of blocking cancer cell invasion revealed by computer simulations. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1189-1202. doi: 10.3934/mbe.2015.12.1189

[17]

Zhenning Cai, Yuwei Fan, Ruo Li. On hyperbolicity of 13-moment system. Kinetic & Related Models, 2014, 7 (3) : 415-432. doi: 10.3934/krm.2014.7.415

[18]

Yakov Pesin. On the work of Dolgopyat on partial and nonuniform hyperbolicity. Journal of Modern Dynamics, 2010, 4 (2) : 227-241. doi: 10.3934/jmd.2010.4.227

[19]

Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Partial hyperbolicity and ergodicity in dimension three. Journal of Modern Dynamics, 2008, 2 (2) : 187-208. doi: 10.3934/jmd.2008.2.187

[20]

Jérôme Buzzi, Todd Fisher. Entropic stability beyond partial hyperbolicity. Journal of Modern Dynamics, 2013, 7 (4) : 527-552. doi: 10.3934/jmd.2013.7.527

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]