American Institute of Mathematical Sciences

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

Computational hyperbolicity

Marcin Mazur 1, and  Jacek Tabor 2,

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.
Keywords: computer assisted proof., semihyperbolicity, Hyperbolicity.
Mathematics Subject Classification: Primary: 37D05, 37D20, Secondary: 37M9.
Citation: Marcin Mazur, Jacek Tabor. Computational hyperbolicity. Discrete & Continuous Dynamical Systems, 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), 121-134.  Google Scholar

[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), 320-339. doi: 10.1006/jdeq.1997.3262.  Google Scholar

[3]

Z. Arai, On Hyperbolic Plateaus of the Hénon Maps, Experiment. Math., 16 (2007), 181-188.  Google Scholar

[4]

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

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R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations, 15 (1974), 429-458. doi: 10.1016/0022-0396(74)90067-9.  Google Scholar

[6]

M. Davis, R. MacKay and A. Sannami, Markov shifts in the Hénon family, Phys. D, 52 (1991), 171-178. doi: 10.1016/0167-2789(91)90119-T.  Google Scholar

[7]

S. Day, H. Kokubu, S. Luzzatto, K. Mischaikow, H. Oka and P. Pilarczyk, Quantitative hyperbolicity estimates in one-dimensional dynamics, Nonlinearity, 21 (2008), 1967-1987. doi: 10.1088/0951-7715/21/9/002.  Google Scholar

[8]

R. Devaney and Z. Nitecki, Shift automorphisms in the Hénon mapping, Comm. Math. Phys., 67 (1979), 137-146. doi: 10.1007/BF01221362.  Google Scholar

[9]

P. Diamond, P. Kloeden, V. Kozyakin and A. Pokrovskii, Semihyperbolic mappings, J. Nonlinear Sci., 5 (1995), 419-431. doi: 10.1007/BF01212908.  Google Scholar

[10]

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

[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), 4261-4272. doi: 10.1142/S0218127407019937.  Google Scholar

[12]

S. Hruska, A numerical method for constructing the hyperbolic structure of complex Hénon mappings, Found. Comput. Math., 6 (2006), 427-455. doi: 10.1007/s10208-006-0141-2.  Google Scholar

[13]

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

[14]

M. Mazur, On some useful conditions on hyperbolicity, Trends in Math., 10 (2008), 57-64. Google Scholar

[15]

M. Mazur, J. Tabor and P. Kościelniak, Semi-hyperbolicity and hyperbolicity, Discrete Contin. Dynam. Syst., 20 (2008), 1029-1038. doi: 10.3934/dcds.2008.20.1029.  Google Scholar

[16]

M. Mazur, J. Tabor and K. Stolot, Semi-hyperbolicity implies hyperbolicity in the linear case, Proceedings of the Conference "Topological Methods in Differential Equations and Dynamical Systems'' (Krakow-Przegorzaly, 1996), Univ. Iagell. Acta Math., 36 (1998), 121-126.  Google Scholar

[17]

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

[18]

K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: A computer-assisted proof, Bull. Amer. Math. Soc. (N.S.), 32 (1995), 66-72.  Google Scholar

[19]

S. Newhouse, Cone-fields, domination, and hyperbolicity, in "Modern Dynamical Systems and Applications,'' Cambridge Univ. Press, Cambridge, (2004), 419-432. Google Scholar

[20]

K. Palmer, "Shadowing in Dynamical Systems. Theory and Applications,'' Kluwer Academic Publishers, Dordrecht, 2000.  Google Scholar

[21]

F. Riesz and B. Sz.-Nagy, "Functional Analysis,'' Frederick Ungar Publishing Co., New York, 1955.  Google Scholar

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), 121-134.  Google Scholar

[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), 320-339. doi: 10.1006/jdeq.1997.3262.  Google Scholar

[3]

Z. Arai, On Hyperbolic Plateaus of the Hénon Maps, Experiment. Math., 16 (2007), 181-188.  Google Scholar

[4]

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

[5]

R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems. I, J. Differential Equations, 15 (1974), 429-458. doi: 10.1016/0022-0396(74)90067-9.  Google Scholar

[6]

M. Davis, R. MacKay and A. Sannami, Markov shifts in the Hénon family, Phys. D, 52 (1991), 171-178. doi: 10.1016/0167-2789(91)90119-T.  Google Scholar

[7]

S. Day, H. Kokubu, S. Luzzatto, K. Mischaikow, H. Oka and P. Pilarczyk, Quantitative hyperbolicity estimates in one-dimensional dynamics, Nonlinearity, 21 (2008), 1967-1987. doi: 10.1088/0951-7715/21/9/002.  Google Scholar

[8]

R. Devaney and Z. Nitecki, Shift automorphisms in the Hénon mapping, Comm. Math. Phys., 67 (1979), 137-146. doi: 10.1007/BF01221362.  Google Scholar

[9]

P. Diamond, P. Kloeden, V. Kozyakin and A. Pokrovskii, Semihyperbolic mappings, J. Nonlinear Sci., 5 (1995), 419-431. doi: 10.1007/BF01212908.  Google Scholar

[10]

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

[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), 4261-4272. doi: 10.1142/S0218127407019937.  Google Scholar

[12]

S. Hruska, A numerical method for constructing the hyperbolic structure of complex Hénon mappings, Found. Comput. Math., 6 (2006), 427-455. doi: 10.1007/s10208-006-0141-2.  Google Scholar

[13]

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

[14]

M. Mazur, On some useful conditions on hyperbolicity, Trends in Math., 10 (2008), 57-64. Google Scholar

[15]

M. Mazur, J. Tabor and P. Kościelniak, Semi-hyperbolicity and hyperbolicity, Discrete Contin. Dynam. Syst., 20 (2008), 1029-1038. doi: 10.3934/dcds.2008.20.1029.  Google Scholar

[16]

M. Mazur, J. Tabor and K. Stolot, Semi-hyperbolicity implies hyperbolicity in the linear case, Proceedings of the Conference "Topological Methods in Differential Equations and Dynamical Systems'' (Krakow-Przegorzaly, 1996), Univ. Iagell. Acta Math., 36 (1998), 121-126.  Google Scholar

[17]

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

[18]

K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: A computer-assisted proof, Bull. Amer. Math. Soc. (N.S.), 32 (1995), 66-72.  Google Scholar

[19]

S. Newhouse, Cone-fields, domination, and hyperbolicity, in "Modern Dynamical Systems and Applications,'' Cambridge Univ. Press, Cambridge, (2004), 419-432. Google Scholar

[20]

K. Palmer, "Shadowing in Dynamical Systems. Theory and Applications,'' Kluwer Academic Publishers, Dordrecht, 2000.  Google Scholar

[21]

F. Riesz and B. Sz.-Nagy, "Functional Analysis,'' Frederick Ungar Publishing Co., New York, 1955.  Google Scholar

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