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Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation
Lefschetz sequences and detecting periodic points
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 |
References:
[1] |
M. Aigner and G. M. Ziegler, "Proofs from the Book," Third Edition, Springer, 2003. |
[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. |
[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. |
[4] |
F. Battelli and M. Feĉkan, Chaos arising near a topologically transversal homoclinic set, Topol. Meth. Nonl. Anal., 20 (2002), 195-215. |
[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. |
[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. |
[7] |
M. Feckan, "Topological Degree Approach to Bifurcation Problems," Series: Topological Fixed Point Theory and Its Applications, Springer Science and Business Media, 2008. |
[8] |
M. Gidea and P. Zgliczyński, Covering relations for multidimensional dynamical systems, J. Differential Equations, 202 (2004), 32-58. |
[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). |
[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. |
[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. |
[12] |
K. Mischaikow and M. Mrozek, Isolating neighborhoods and chaos, Japan J. Indust. and Appl. Math., 12 (1995), 205-236. |
[13] |
M. Mrozek, The method of topological sections in the rigorous numerics of dynamical systems, Canadian Applied Mathemathics Quartely, 14 (2006), 209-222. |
[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. |
[15] |
P. Oprocha and P. Wilczyński, Distributional chaos via semiconjugacy, Nonlinearity, 20 (2007), 2661-2679.
doi: 10.1088/0951-7715/20/11/010. |
[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. |
[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. |
[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. |
[19] |
L. Pieniążek and K. Wójcik, Complicated dynamics in nonautonomous ODEs, Univ. Iagel. Acta Math., XLI (2003), 163-179. |
[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. |
[21] |
J. C. Sprott, "Elegant Chaos. Algebraically Simple Chaotic Flows," World Scientific Publishing, 2010. |
[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. |
[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. |
[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. |
[25] |
R. Srzednicki, On solutions of two-point boundary value problems inside isolating segments, Topol. Methods Nonlinear Anal., 13 (1999), 73-89. |
[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. |
[27] |
R. Srzednicki and K. Wójcik, A geometric method for detecting chaotic dynamics, J. Differential Equations, 135 (1997), 66-82. |
[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. |
[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. |
[30] |
K. Wójcik and P. Zgliczyński, Isolating segments, fixed point index, and symbolic dynamics, J. Differential Equations, 161 (2000), 245-288. |
[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. |
show all references
References:
[1] |
M. Aigner and G. M. Ziegler, "Proofs from the Book," Third Edition, Springer, 2003. |
[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. |
[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. |
[4] |
F. Battelli and M. Feĉkan, Chaos arising near a topologically transversal homoclinic set, Topol. Meth. Nonl. Anal., 20 (2002), 195-215. |
[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. |
[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. |
[7] |
M. Feckan, "Topological Degree Approach to Bifurcation Problems," Series: Topological Fixed Point Theory and Its Applications, Springer Science and Business Media, 2008. |
[8] |
M. Gidea and P. Zgliczyński, Covering relations for multidimensional dynamical systems, J. Differential Equations, 202 (2004), 32-58. |
[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). |
[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. |
[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. |
[12] |
K. Mischaikow and M. Mrozek, Isolating neighborhoods and chaos, Japan J. Indust. and Appl. Math., 12 (1995), 205-236. |
[13] |
M. Mrozek, The method of topological sections in the rigorous numerics of dynamical systems, Canadian Applied Mathemathics Quartely, 14 (2006), 209-222. |
[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. |
[15] |
P. Oprocha and P. Wilczyński, Distributional chaos via semiconjugacy, Nonlinearity, 20 (2007), 2661-2679.
doi: 10.1088/0951-7715/20/11/010. |
[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. |
[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. |
[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. |
[19] |
L. Pieniążek and K. Wójcik, Complicated dynamics in nonautonomous ODEs, Univ. Iagel. Acta Math., XLI (2003), 163-179. |
[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. |
[21] |
J. C. Sprott, "Elegant Chaos. Algebraically Simple Chaotic Flows," World Scientific Publishing, 2010. |
[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. |
[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. |
[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. |
[25] |
R. Srzednicki, On solutions of two-point boundary value problems inside isolating segments, Topol. Methods Nonlinear Anal., 13 (1999), 73-89. |
[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. |
[27] |
R. Srzednicki and K. Wójcik, A geometric method for detecting chaotic dynamics, J. Differential Equations, 135 (1997), 66-82. |
[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. |
[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. |
[30] |
K. Wójcik and P. Zgliczyński, Isolating segments, fixed point index, and symbolic dynamics, J. Differential Equations, 161 (2000), 245-288. |
[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. |
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