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January  2012, 32(1): 81-100. doi: 10.3934/dcds.2012.32.81

Lefschetz sequences and detecting periodic points

Anna Gierzkiewicz 1, and  Klaudiusz Wójcik 2,

1. 

Department of Applied Mathematics, University of Agriculture in Krakow, Balicka 253c, 30-198 Kraków, Poland
2. 

Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland

Received  July 2010 Revised  December 2010 Published  September 2011

We introduce a dual sequence condition (DSC) for a discrete dynamical system given by a continuous map $f:X\to X$ of some metric space $X$. It is defined in terms of the Lefschetz sequence and its dual sequence of the endomorphism of a graded vector space of finite type associated to the dynamical system $f$. We prove the arithmetical properties of the dual Lefschetz sequence and we show some of its dynamical consequences, mainly concerning the topological methods for detecting chaotic dynamics.
Keywords: discrete Conley index., periodic solution, Lefschetz number, isolated invariant set, periodic isolating segment, Boundary value problems, Poincaré map, fixed point index, isolating block.
Mathematics Subject Classification: Primary: 37B30, 37B10; Secondary: 37B4.
Citation: Anna Gierzkiewicz, Klaudiusz Wójcik. Lefschetz sequences and detecting periodic points. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 81-100. doi: 10.3934/dcds.2012.32.81
References:
[1]

M. Aigner and G. M. Ziegler, "Proofs from the Book,", Third Edition, (2003).   Google Scholar

[2]

I. K. Babienko and S. A. Bogatyi, Behaviour of the index of periodic points under iterations of mapping,, Izv. Akad. Nauk SSSR Ser. Math., 55 (1991), 3.   Google Scholar

[3]

B. Banhelyi, T. Csendes and B. M. Garay, Optimization and the Miranda approach in detecting horseshoe-type chaos by computer,, Inter. J. Bifurcation and Chaos, 17 (2007), 735.  doi: 10.1142/S0218127407017549.  Google Scholar

[4]

F. Battelli and M. Feĉkan, Chaos arising near a topologically transversal homoclinic set,, Topol. Meth. Nonl. Anal., 20 (2002), 195.   Google Scholar

[5]

A. Capietto, W. Dambrosio and D. Papini, Superlinear indefinite equations on the real line and chaotic dynamics,, J. Diff. Eq., 181 (2002), 419.   Google Scholar

[6]

S. N. Chow, J. Mallet-Paret and J. A. Yorke, A periodic orbit index which is a bifurcation invariant,, Lect. Notes in Math., 1007 (1983), 109.  doi: 10.1007/BFb0061414.  Google Scholar

[7]

M. Feckan, "Topological Degree Approach to Bifurcation Problems,", Series: Topological Fixed Point Theory and Its Applications, (2008).   Google Scholar

[8]

M. Gidea and P. Zgliczyński, Covering relations for multidimensional dynamical systems,, J. Differential Equations, 202 (2004), 32.   Google Scholar

[9]

J. Jezierski and W. Marzantowicz, Homotopy methods in topological fixed and periodic points theory,, Series: Topological Fixed Point Theory and Its Applications, 3 (2005).   Google Scholar

[10]

W. Marzantowicz and K. Wójcik, Periodic segment implies infinitely many periodic solutions,, Proc. American Math Society, 135 (2007), 2637.  doi: 10.1090/S0002-9939-07-08750-3.  Google Scholar

[11]

K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: A computer-assisted proof,, Bull. American Math Society, 32 (1995), 66.  doi: 10.1090/S0273-0979-1995-00558-6.  Google Scholar

[12]

K. Mischaikow and M. Mrozek, Isolating neighborhoods and chaos,, Japan J. Indust. and Appl. Math., 12 (1995), 205.   Google Scholar

[13]

M. Mrozek, The method of topological sections in the rigorous numerics of dynamical systems,, Canadian Applied Mathemathics Quartely, 14 (2006), 209.   Google Scholar

[14]

M. Mrozek and K. Wójcik, Disrete version of a geometric method for detecting chaotic dynamics,, Topology Appl., 152 (2005), 70.  doi: 10.1016/j.topol.2004.08.015.  Google Scholar

[15]

P. Oprocha and P. Wilczyński, Distributional chaos via semiconjugacy,, Nonlinearity, 20 (2007), 2661.  doi: 10.1088/0951-7715/20/11/010.  Google Scholar

[16]

P. Oprocha and P. Wilczyński, Distributional chaos via isolating segments,, Discrete and Continuous Dynamical Systems Series B, 8 (2007), 347.  doi: 10.3934/dcdsb.2007.8.347.  Google Scholar

[17]

D. Papini and F. Zanolin, Some results on periodic points and chaotic dynamics arising from the study of the nonlinear Hill equation,, Rend. Semin. Mat. Univ. Politec. Torino, 65 (2007), 115.   Google Scholar

[18]

D. Papini and F. Zanolin, Fixed points, periodic points and coin-tossing sequences for mappings defined on two-dimensional cells,, Fixed Point Theory Appl., 2 (2004), 113.  doi: 10.1155/S1687182004401028.  Google Scholar

[19]

L. Pieniążek and K. Wójcik, Complicated dynamics in nonautonomous ODEs,, Univ. Iagel. Acta Math., XLI (2003), 163.   Google Scholar

[20]

M. Pireddu and F. Zanolin, Cutting surfaces and applications to periodic points and chaotic-like dynamics,, Topol. Methods Nonlinear Anal., 30 (2007), 279.   Google Scholar

[21]

J. C. Sprott, "Elegant Chaos. Algebraically Simple Chaotic Flows,", World Scientific Publishing, (2010).   Google Scholar

[22]

M. Shub and P. Sullivan, A remark on the Lefschetz fixed point formula for differentiable maps,, Topology, 13 (1974), 189.  doi: 10.1016/0040-9383(74)90009-3.  Google Scholar

[23]

R. Srzednicki, Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations,, Nonlinear Anal. TMA, 22 (1994), 707.  doi: 10.1016/0362-546X(94)90223-2.  Google Scholar

[24]

R. Srzednicki, Ważewski method and the Conley index,, Handbook of Dfferential Equations vol. \textbf{1} (2004), 1 (2004), 591.   Google Scholar

[25]

R. Srzednicki, On solutions of two-point boundary value problems inside isolating segments,, Topol. Methods Nonlinear Anal., 13 (1999), 73.   Google Scholar

[26]

Zhi-Wei Sun and R. Tauraso, Congruences for sums of binomial coefficients,, Journal of Number Theory, 126 (2007), 287.  doi: 10.1016/j.jnt.2007.01.002.  Google Scholar

[27]

R. Srzednicki and K. Wójcik, A geometric method for detecting chaotic dynamics,, J. Differential Equations, 135 (1997), 66.   Google Scholar

[28]

R. Srzednicki, K. Wójcik and P. Zgliczyński, Fixed point results based on the Ważewski method,, Handbook of topological fixed point theory, (2005), 905.   Google Scholar

[29]

K. Wójcik, On detecting periodic solutions and chaos in the time periodically forced ODEs,, Nonlinear Anal. TMA, 45 (2001), 19.  doi: 10.1016/S0362-546X(99)00327-2.  Google Scholar

[30]

K. Wójcik and P. Zgliczyński, Isolating segments, fixed point index, and symbolic dynamics,, J. Differential Equations, 161 (2000), 245.   Google Scholar

[31]

P. Zgliczyński, Computer assisted proof of chaos in the Rössler equations and in the Hénon map,, Nonlinearity, 10 (1997), 243.  doi: 10.1088/0951-7715/10/1/016.  Google Scholar

show all references

References:
[1]

M. Aigner and G. M. Ziegler, "Proofs from the Book,", Third Edition, (2003).   Google Scholar

[2]

I. K. Babienko and S. A. Bogatyi, Behaviour of the index of periodic points under iterations of mapping,, Izv. Akad. Nauk SSSR Ser. Math., 55 (1991), 3.   Google Scholar

[3]

B. Banhelyi, T. Csendes and B. M. Garay, Optimization and the Miranda approach in detecting horseshoe-type chaos by computer,, Inter. J. Bifurcation and Chaos, 17 (2007), 735.  doi: 10.1142/S0218127407017549.  Google Scholar

[4]

F. Battelli and M. Feĉkan, Chaos arising near a topologically transversal homoclinic set,, Topol. Meth. Nonl. Anal., 20 (2002), 195.   Google Scholar

[5]

A. Capietto, W. Dambrosio and D. Papini, Superlinear indefinite equations on the real line and chaotic dynamics,, J. Diff. Eq., 181 (2002), 419.   Google Scholar

[6]

S. N. Chow, J. Mallet-Paret and J. A. Yorke, A periodic orbit index which is a bifurcation invariant,, Lect. Notes in Math., 1007 (1983), 109.  doi: 10.1007/BFb0061414.  Google Scholar

[7]

M. Feckan, "Topological Degree Approach to Bifurcation Problems,", Series: Topological Fixed Point Theory and Its Applications, (2008).   Google Scholar

[8]

M. Gidea and P. Zgliczyński, Covering relations for multidimensional dynamical systems,, J. Differential Equations, 202 (2004), 32.   Google Scholar

[9]

J. Jezierski and W. Marzantowicz, Homotopy methods in topological fixed and periodic points theory,, Series: Topological Fixed Point Theory and Its Applications, 3 (2005).   Google Scholar

[10]

W. Marzantowicz and K. Wójcik, Periodic segment implies infinitely many periodic solutions,, Proc. American Math Society, 135 (2007), 2637.  doi: 10.1090/S0002-9939-07-08750-3.  Google Scholar

[11]

K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: A computer-assisted proof,, Bull. American Math Society, 32 (1995), 66.  doi: 10.1090/S0273-0979-1995-00558-6.  Google Scholar

[12]

K. Mischaikow and M. Mrozek, Isolating neighborhoods and chaos,, Japan J. Indust. and Appl. Math., 12 (1995), 205.   Google Scholar

[13]

M. Mrozek, The method of topological sections in the rigorous numerics of dynamical systems,, Canadian Applied Mathemathics Quartely, 14 (2006), 209.   Google Scholar

[14]

M. Mrozek and K. Wójcik, Disrete version of a geometric method for detecting chaotic dynamics,, Topology Appl., 152 (2005), 70.  doi: 10.1016/j.topol.2004.08.015.  Google Scholar

[15]

P. Oprocha and P. Wilczyński, Distributional chaos via semiconjugacy,, Nonlinearity, 20 (2007), 2661.  doi: 10.1088/0951-7715/20/11/010.  Google Scholar

[16]

P. Oprocha and P. Wilczyński, Distributional chaos via isolating segments,, Discrete and Continuous Dynamical Systems Series B, 8 (2007), 347.  doi: 10.3934/dcdsb.2007.8.347.  Google Scholar

[17]

D. Papini and F. Zanolin, Some results on periodic points and chaotic dynamics arising from the study of the nonlinear Hill equation,, Rend. Semin. Mat. Univ. Politec. Torino, 65 (2007), 115.   Google Scholar

[18]

D. Papini and F. Zanolin, Fixed points, periodic points and coin-tossing sequences for mappings defined on two-dimensional cells,, Fixed Point Theory Appl., 2 (2004), 113.  doi: 10.1155/S1687182004401028.  Google Scholar

[19]

L. Pieniążek and K. Wójcik, Complicated dynamics in nonautonomous ODEs,, Univ. Iagel. Acta Math., XLI (2003), 163.   Google Scholar

[20]

M. Pireddu and F. Zanolin, Cutting surfaces and applications to periodic points and chaotic-like dynamics,, Topol. Methods Nonlinear Anal., 30 (2007), 279.   Google Scholar

[21]

J. C. Sprott, "Elegant Chaos. Algebraically Simple Chaotic Flows,", World Scientific Publishing, (2010).   Google Scholar

[22]

M. Shub and P. Sullivan, A remark on the Lefschetz fixed point formula for differentiable maps,, Topology, 13 (1974), 189.  doi: 10.1016/0040-9383(74)90009-3.  Google Scholar

[23]

R. Srzednicki, Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations,, Nonlinear Anal. TMA, 22 (1994), 707.  doi: 10.1016/0362-546X(94)90223-2.  Google Scholar

[24]

R. Srzednicki, Ważewski method and the Conley index,, Handbook of Dfferential Equations vol. \textbf{1} (2004), 1 (2004), 591.   Google Scholar

[25]

R. Srzednicki, On solutions of two-point boundary value problems inside isolating segments,, Topol. Methods Nonlinear Anal., 13 (1999), 73.   Google Scholar

[26]

Zhi-Wei Sun and R. Tauraso, Congruences for sums of binomial coefficients,, Journal of Number Theory, 126 (2007), 287.  doi: 10.1016/j.jnt.2007.01.002.  Google Scholar

[27]

R. Srzednicki and K. Wójcik, A geometric method for detecting chaotic dynamics,, J. Differential Equations, 135 (1997), 66.   Google Scholar

[28]

R. Srzednicki, K. Wójcik and P. Zgliczyński, Fixed point results based on the Ważewski method,, Handbook of topological fixed point theory, (2005), 905.   Google Scholar

[29]

K. Wójcik, On detecting periodic solutions and chaos in the time periodically forced ODEs,, Nonlinear Anal. TMA, 45 (2001), 19.  doi: 10.1016/S0362-546X(99)00327-2.  Google Scholar

[30]

K. Wójcik and P. Zgliczyński, Isolating segments, fixed point index, and symbolic dynamics,, J. Differential Equations, 161 (2000), 245.   Google Scholar

[31]

P. Zgliczyński, Computer assisted proof of chaos in the Rössler equations and in the Hénon map,, Nonlinearity, 10 (1997), 243.  doi: 10.1088/0951-7715/10/1/016.  Google Scholar

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