December  2011, 15(3): 739-767. doi: 10.3934/dcdsb.2011.15.739

Computation of symbolic dynamics for two-dimensional piecewise-affine maps

1. 

Niels Bohrweg 1, Leiden, 2333 CA, Netherlands

2. 

Bouillonstraat 8-10, 6211 LH Maastricht, Netherlands

Received  June 2009 Revised  June 2010 Published  February 2011

In this paper we design and implement an algorithm for computing symbolic dynamics for two dimensional piecewise-affine maps. The algorithm is based on detection of periodic orbits using the Conley index and Szymczak decomposition of Conley index pair. The algorithm is also extended to deal with discontinuous maps. We compare the algorithm with the algorithm based on tangle of fixed points. We apply the algorithms to compute the symbolic dynamics and entropy bounds for the Lozi map.
Citation: Lorenzo Sella, Pieter Collins. Computation of symbolic dynamics for two-dimensional piecewise-affine maps. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 739-767. doi: 10.3934/dcdsb.2011.15.739
References:
[1]

D. Lind and B. Marcus, "An Introduction To Symbolic Dynamics And Coding,", Cambridge University Press, (1995). doi: 10.1017/CBO9780511626302. Google Scholar

[2]

J. Milnor and W. Thurston, On iterated maps of the interval,, in, (1342), 1986. Google Scholar

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J. P. Lampreia and S. Ramos, Trimodal maps,, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 3 (1993), 1607. doi: 10.1142/S0218127493001276. Google Scholar

[4]

J. P. Lampreia and S. Ramos, Kneading theory for tree maps,, Ergodic Theory and Dynamical Systems, 24 (2004), 957. doi: 10.1017/S014338570400015X. Google Scholar

[5]

J. L. Rocha and S. Ramos, On iterated maps of the interval with holes,, Journal of Difference Equations and Applications, 9 (2003), 319. doi: 10.1080/1023619021000047752. Google Scholar

[6]

L. Sella and P. Collins, "Discrete Dynamics of Two-Dimensional Nonlinear Hybrid Automata,", Hybrid Systems: Computation and Control, (2008). Google Scholar

[7]

P. Collins, Symbolic dynamics from homoclinic tangles,, HInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 12 (2002), 605. doi: 10.1142/S0218127402004565. Google Scholar

[8]

T. Kaczynski, K. Mischaikow and M. Mrozek, "Computational Homology,", Applied Mathematical Sciences, (). Google Scholar

[9]

S. Day, O. Junge and M. Konstantin, Towards automated chaos verification,, EQUADIFF, (2003), 157. Google Scholar

[10]

Z. Galias and P. Zgliczyński, Abundance of homoclinic and heteroclinic orbits and rigorous bounds for the topological entropy for the Hénon map,, Nonlinearity, 14 (2001), 909. doi: 10.1088/0951-7715/14/5/301. Google Scholar

[11]

A. Szymczak, The Conley index for decompositions of isolated invariant sets,, Fundamenta Mathematicae, 148 (1995), 71. Google Scholar

[12]

P. Collins, Dynamics of surface diffeomorphisms relative to homoclinic and heteroclinic orbits,, Dynamical Systems, 19 (2004), 1. doi: 10.1080/14689360310001623421. Google Scholar

[13]

M. Misiurewicz, Strange attractors for the Lozi mappings,, Nonlinear Dynamics (Internat. Conf., (1979), 348. Google Scholar

[14]

A. Hatcher, "Algebraic Topology,", Cambridge University Press, (2002). Google Scholar

[15]

J. Munkres, "Elements of Algebraic Topology,", Addison-Wesley Publishing Company, (2002). Google Scholar

[16]

R. Gilmore and M. Lefranc, "The Topology of Chaos," Alice in Stretch and Squeezeland,, Wiley-Interscience [John Wiley & Sons], (1984). Google Scholar

[17]

D. Sand, Numerical computations on Lozi maps,, \url{http://topo.math.u-psud.fr/ sands/Programs/Lozi/index.html}., (). Google Scholar

show all references

References:
[1]

D. Lind and B. Marcus, "An Introduction To Symbolic Dynamics And Coding,", Cambridge University Press, (1995). doi: 10.1017/CBO9780511626302. Google Scholar

[2]

J. Milnor and W. Thurston, On iterated maps of the interval,, in, (1342), 1986. Google Scholar

[3]

J. P. Lampreia and S. Ramos, Trimodal maps,, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 3 (1993), 1607. doi: 10.1142/S0218127493001276. Google Scholar

[4]

J. P. Lampreia and S. Ramos, Kneading theory for tree maps,, Ergodic Theory and Dynamical Systems, 24 (2004), 957. doi: 10.1017/S014338570400015X. Google Scholar

[5]

J. L. Rocha and S. Ramos, On iterated maps of the interval with holes,, Journal of Difference Equations and Applications, 9 (2003), 319. doi: 10.1080/1023619021000047752. Google Scholar

[6]

L. Sella and P. Collins, "Discrete Dynamics of Two-Dimensional Nonlinear Hybrid Automata,", Hybrid Systems: Computation and Control, (2008). Google Scholar

[7]

P. Collins, Symbolic dynamics from homoclinic tangles,, HInternational Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 12 (2002), 605. doi: 10.1142/S0218127402004565. Google Scholar

[8]

T. Kaczynski, K. Mischaikow and M. Mrozek, "Computational Homology,", Applied Mathematical Sciences, (). Google Scholar

[9]

S. Day, O. Junge and M. Konstantin, Towards automated chaos verification,, EQUADIFF, (2003), 157. Google Scholar

[10]

Z. Galias and P. Zgliczyński, Abundance of homoclinic and heteroclinic orbits and rigorous bounds for the topological entropy for the Hénon map,, Nonlinearity, 14 (2001), 909. doi: 10.1088/0951-7715/14/5/301. Google Scholar

[11]

A. Szymczak, The Conley index for decompositions of isolated invariant sets,, Fundamenta Mathematicae, 148 (1995), 71. Google Scholar

[12]

P. Collins, Dynamics of surface diffeomorphisms relative to homoclinic and heteroclinic orbits,, Dynamical Systems, 19 (2004), 1. doi: 10.1080/14689360310001623421. Google Scholar

[13]

M. Misiurewicz, Strange attractors for the Lozi mappings,, Nonlinear Dynamics (Internat. Conf., (1979), 348. Google Scholar

[14]

A. Hatcher, "Algebraic Topology,", Cambridge University Press, (2002). Google Scholar

[15]

J. Munkres, "Elements of Algebraic Topology,", Addison-Wesley Publishing Company, (2002). Google Scholar

[16]

R. Gilmore and M. Lefranc, "The Topology of Chaos," Alice in Stretch and Squeezeland,, Wiley-Interscience [John Wiley & Sons], (1984). Google Scholar

[17]

D. Sand, Numerical computations on Lozi maps,, \url{http://topo.math.u-psud.fr/ sands/Programs/Lozi/index.html}., (). Google Scholar

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