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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, 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, Springer, 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), 3-31. 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), 735-747. 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), 195-215.  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), 419-438.  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), 109-131. doi: 10.1007/BFb0061414.  Google Scholar

[7]

M. Feckan, "Topological Degree Approach to Bifurcation Problems," Series: Topological Fixed Point Theory and Its Applications, Springer Science and Business Media, 2008.  Google Scholar

[8]

M. Gidea and P. Zgliczyński, Covering relations for multidimensional dynamical systems, J. Differential Equations, 202 (2004), 32-58.  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, vol. 3, Springer, (2005).  Google Scholar

[10]

W. Marzantowicz and K. Wójcik, Periodic segment implies infinitely many periodic solutions, Proc. American Math Society, 135 (2007), 2637-2647. 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), 66-72. 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), 205-236.  Google Scholar

[13]

M. Mrozek, The method of topological sections in the rigorous numerics of dynamical systems, Canadian Applied Mathemathics Quartely, 14 (2006), 209-222.  Google Scholar

[14]

M. Mrozek and K. Wójcik, Disrete version of a geometric method for detecting chaotic dynamics, Topology Appl., 152 (2005), 70-82. doi: 10.1016/j.topol.2004.08.015.  Google Scholar

[15]

P. Oprocha and P. Wilczyński, Distributional chaos via semiconjugacy, Nonlinearity, 20 (2007), 2661-2679. 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), 347-356. 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), 115-157.  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), 113-134. doi: 10.1155/S1687182004401028.  Google Scholar

[19]

L. Pieniążek and K. Wójcik, Complicated dynamics in nonautonomous ODEs, Univ. Iagel. Acta Math., XLI (2003), 163-179.  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), 279-319.  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), 189-191. 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), 707-737. doi: 10.1016/0362-546X(94)90223-2.  Google Scholar

[24]

R. Srzednicki, Ważewski method and the Conley index, Handbook of Dfferential Equations vol. 1 (2004), Edited by A. Canada, P. Drabek, A. Fonda, 591-684.  Google Scholar

[25]

R. Srzednicki, On solutions of two-point boundary value problems inside isolating segments, Topol. Methods Nonlinear Anal., 13 (1999), 73-89.  Google Scholar

[26]

Zhi-Wei Sun and R. Tauraso, Congruences for sums of binomial coefficients, Journal of Number Theory, 126 (2007), 287-296. 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), 66-82.  Google Scholar

[28]

R. Srzednicki, K. Wójcik and P. Zgliczyński, Fixed point results based on the Ważewski method, Handbook of topological fixed point theory, Ed: R. Brown, M. Furi, L. Górniewicz, B. Jiang, (2005), 905-943.  Google Scholar

[29]

K. Wójcik, On detecting periodic solutions and chaos in the time periodically forced ODEs, Nonlinear Anal. TMA, 45 (2001), 19-27. 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), 245-288.  Google Scholar

[31]

P. Zgliczyński, Computer assisted proof of chaos in the Rössler equations and in the Hénon map, Nonlinearity, 10 (1997), 243-252. 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, Springer, 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), 3-31. 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), 735-747. 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), 195-215.  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), 419-438.  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), 109-131. doi: 10.1007/BFb0061414.  Google Scholar

[7]

M. Feckan, "Topological Degree Approach to Bifurcation Problems," Series: Topological Fixed Point Theory and Its Applications, Springer Science and Business Media, 2008.  Google Scholar

[8]

M. Gidea and P. Zgliczyński, Covering relations for multidimensional dynamical systems, J. Differential Equations, 202 (2004), 32-58.  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, vol. 3, Springer, (2005).  Google Scholar

[10]

W. Marzantowicz and K. Wójcik, Periodic segment implies infinitely many periodic solutions, Proc. American Math Society, 135 (2007), 2637-2647. 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), 66-72. 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), 205-236.  Google Scholar

[13]

M. Mrozek, The method of topological sections in the rigorous numerics of dynamical systems, Canadian Applied Mathemathics Quartely, 14 (2006), 209-222.  Google Scholar

[14]

M. Mrozek and K. Wójcik, Disrete version of a geometric method for detecting chaotic dynamics, Topology Appl., 152 (2005), 70-82. doi: 10.1016/j.topol.2004.08.015.  Google Scholar

[15]

P. Oprocha and P. Wilczyński, Distributional chaos via semiconjugacy, Nonlinearity, 20 (2007), 2661-2679. 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), 347-356. 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), 115-157.  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), 113-134. doi: 10.1155/S1687182004401028.  Google Scholar

[19]

L. Pieniążek and K. Wójcik, Complicated dynamics in nonautonomous ODEs, Univ. Iagel. Acta Math., XLI (2003), 163-179.  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), 279-319.  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), 189-191. 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), 707-737. doi: 10.1016/0362-546X(94)90223-2.  Google Scholar

[24]

R. Srzednicki, Ważewski method and the Conley index, Handbook of Dfferential Equations vol. 1 (2004), Edited by A. Canada, P. Drabek, A. Fonda, 591-684.  Google Scholar

[25]

R. Srzednicki, On solutions of two-point boundary value problems inside isolating segments, Topol. Methods Nonlinear Anal., 13 (1999), 73-89.  Google Scholar

[26]

Zhi-Wei Sun and R. Tauraso, Congruences for sums of binomial coefficients, Journal of Number Theory, 126 (2007), 287-296. 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), 66-82.  Google Scholar

[28]

R. Srzednicki, K. Wójcik and P. Zgliczyński, Fixed point results based on the Ważewski method, Handbook of topological fixed point theory, Ed: R. Brown, M. Furi, L. Górniewicz, B. Jiang, (2005), 905-943.  Google Scholar

[29]

K. Wójcik, On detecting periodic solutions and chaos in the time periodically forced ODEs, Nonlinear Anal. TMA, 45 (2001), 19-27. 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), 245-288.  Google Scholar

[31]

P. Zgliczyński, Computer assisted proof of chaos in the Rössler equations and in the Hénon map, Nonlinearity, 10 (1997), 243-252. doi: 10.1088/0951-7715/10/1/016.  Google Scholar

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