Discretization strategies for computing Conley indices and Morse decompositions of flows

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January 2016, 3(1): 1-16. doi: 10.3934/jcd.2016001

Discretization strategies for computing Conley indices and Morse decompositions of flows

Konstantin Mischaikow ^1, , Marian Mrozek ^2, and Frank Weilandt ^2,

1.	Department of Mathematics, Rutgers, The State University of New Jersey, 110 Frelinghuysen Rd, Piscataway, NJ 08854-8019, United States
2.	Division of Computational Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland, Poland

Received April 2015 Revised July 2016 Published August 2016

Abstract
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Conley indices and Morse decompositions of flows can be found by using algorithms which rigorously analyze discrete dynamical systems. This usually involves integrating a time discretization of the flow using interval arithmetic. We compare the old idea of fixing a time step as a parameter to a time step continuously varying in phase space. We present an example where this second strategy necessarily yields better numerical outputs and prove that our outputs yield a valid Morse decomposition of the given flow.

Keywords: Morse decomposition, rigorous numerical algorithm., Conley index, interval arithmetic.

Mathematics Subject Classification: Primary: 37B30, 37B35, 65G20; Secondary: 37-04, 37C1.

Citation: Konstantin Mischaikow, Marian Mrozek, Frank Weilandt. Discretization strategies for computing Conley indices and Morse decompositions of flows. Journal of Computational Dynamics, 2016, 3 (1) : 1-16. doi: 10.3934/jcd.2016001

References:

[1]	Z. Arai, H. Kokubu and P. Pilarczyk, Recent development in rigorous computational methods in dynamical systems, Japan J. of Indust. Appl. Math., 26 (2009), 393-417. doi: 10.1007/BF03186541.
[2]	Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka and P. Pilarczyk, A database schema for the analysis of global dynamics of multiparameter systems, SIAM J. Applied Dyn. Syst., 8 (2009), 757-789. doi: 10.1137/080734935.
[3]	H. Ban and W. D. Kalies, A computational approach to Conley's decomposition theorem, J. Comput. Nonlinear Dynam., 1 (2006), 312-319. doi: 10.1115/1.2338651.
[4]	E. Boczko, W. D. Kalies and K. Mischaikow, Polygonal approximation of flows, Topology Appl., 154 (2007), 2501-2520. doi: 10.1016/j.topol.2006.04.033.
[5]	J. Bush, M. Gameiro, S. Harker, H. Kokubu, K. Mischaikow, I. Obayashi and P. Pilarczyk, Combinatorial-topological framework for the analysis of global dynamics, Chaos, 22 (2012), 047508, 16pp. doi: 10.1063/1.4767672.
[6]	J. B. van den Berg and J. P. Lessard, Rigorous numerics in dynamics, Notices Amer. Math. Soc., 62 (2015), 1057-1061. doi: 10.1090/noti1276.
[7]	The CAPD Group, Computer assisted proofs in dynamics software library,, , ().
[8]	C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38, AMS, 1978.
[9]	G. Chen, K. Mischaikow, R. S. Laramee and E. Zang, Efficient Morse decompositions of vector fields, IEEE Transactions on Visualizations and Computer Graphics, 14 (2008), 848-862.
[10]	M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, Chapter 5 in Handbook of dynamical systems, Elsevier, 2 (2002), 221-264. doi: 10.1016/S1874-575X(02)80026-1.
[11]	J. Franks and D. Richeson, Shift equivalence and the Conley index, Transactions AMS, 352 (2000), 3305-3322. doi: 10.1090/S0002-9947-00-02488-0.
[12]	M. Gidea and P. Zgliczynski, Covering relations for multidimensional dynamical systems I, J. Differential Equations, 202 (2004), 32-58. doi: 10.1016/j.jde.2004.03.013.
[13]	M. Gidea and P. Zgliczynski, Covering relations for multidimensional dynamical systems II, J. Differential Equations, 202 (2004), 59-80. doi: 10.1016/j.jde.2004.03.014.
[14]	T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology, Applied Mathematical Sciences Vol. 157, Springer-Verlag New York, 2004. doi: 10.1007/b97315.
[15]	W. D. Kalies, K. Mischaikow and R. C. A. M. VanderVorst, An algorithmic approach to chain recurrence, Found. Comp. Math., 5 (2005), 409-449. doi: 10.1007/s10208-004-0163-9.
[16]	W. Massey, Homology and Cohomology Theory, Marcel Dekker, New York and Basel, 1978.
[17]	K. Mischaikow and M. Mrozek, Conley index, Chapter 9 in Handbook of dynamical systems, Elsevier, 2 (2002), 393-460. doi: 10.1016/S1874-575X(02)80030-3.
[18]	M. Mrozek, The Conley index on compact ANR's is of finite type, Results Math., 18 (1990), 306-313. doi: 10.1007/BF03323175.
[19]	M. Mrozek, Index pairs algorithms, Found. Comput. Math., 6 (2006), 457-493. doi: 10.1007/s10208-005-0182-1.
[20]	M. Mrozek, Leray functor and cohomological Conley index for discrete dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 149-178. doi: 10.1090/S0002-9947-1990-0968888-1.
[21]	P. Pilarczyk, L. García, B. A. Carreras and I. Llerena, A dynamical model for plasma confinement transitions, J. Phys. A: Math. Theor., 45 (2012), 125502. doi: 10.1088/1751-8113/45/12/125502.
[22]	P. Pilarczyk, Computer assisted method for proving existence of periodic orbits, Topol. Methods Nonlinear Anal., 13 (1999), 365-377.
[23]	J. W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynamical Systems, $\mathbf{8^*}$ (1988), 375-393. doi: 10.1017/S0143385700009494.
[24]	K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Universitext, Springer-Verlag, 1987. doi: 10.1007/978-3-642-72833-4.
[25]	A. Szymczak, A combinatorial procedure for finding isolating neighbourhoods and index pairs, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1075-1088. doi: 10.1017/S0308210500026901.
[26]	G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics, 140, AMS, 2012. doi: 10.1090/gsm/140.

show all references

References:

[1]	Z. Arai, H. Kokubu and P. Pilarczyk, Recent development in rigorous computational methods in dynamical systems, Japan J. of Indust. Appl. Math., 26 (2009), 393-417. doi: 10.1007/BF03186541.
[2]	Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka and P. Pilarczyk, A database schema for the analysis of global dynamics of multiparameter systems, SIAM J. Applied Dyn. Syst., 8 (2009), 757-789. doi: 10.1137/080734935.
[3]	H. Ban and W. D. Kalies, A computational approach to Conley's decomposition theorem, J. Comput. Nonlinear Dynam., 1 (2006), 312-319. doi: 10.1115/1.2338651.
[4]	E. Boczko, W. D. Kalies and K. Mischaikow, Polygonal approximation of flows, Topology Appl., 154 (2007), 2501-2520. doi: 10.1016/j.topol.2006.04.033.
[5]	J. Bush, M. Gameiro, S. Harker, H. Kokubu, K. Mischaikow, I. Obayashi and P. Pilarczyk, Combinatorial-topological framework for the analysis of global dynamics, Chaos, 22 (2012), 047508, 16pp. doi: 10.1063/1.4767672.
[6]	J. B. van den Berg and J. P. Lessard, Rigorous numerics in dynamics, Notices Amer. Math. Soc., 62 (2015), 1057-1061. doi: 10.1090/noti1276.
[7]	The CAPD Group, Computer assisted proofs in dynamics software library,, , ().
[8]	C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38, AMS, 1978.
[9]	G. Chen, K. Mischaikow, R. S. Laramee and E. Zang, Efficient Morse decompositions of vector fields, IEEE Transactions on Visualizations and Computer Graphics, 14 (2008), 848-862.
[10]	M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, Chapter 5 in Handbook of dynamical systems, Elsevier, 2 (2002), 221-264. doi: 10.1016/S1874-575X(02)80026-1.
[11]	J. Franks and D. Richeson, Shift equivalence and the Conley index, Transactions AMS, 352 (2000), 3305-3322. doi: 10.1090/S0002-9947-00-02488-0.
[12]	M. Gidea and P. Zgliczynski, Covering relations for multidimensional dynamical systems I, J. Differential Equations, 202 (2004), 32-58. doi: 10.1016/j.jde.2004.03.013.
[13]	M. Gidea and P. Zgliczynski, Covering relations for multidimensional dynamical systems II, J. Differential Equations, 202 (2004), 59-80. doi: 10.1016/j.jde.2004.03.014.
[14]	T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology, Applied Mathematical Sciences Vol. 157, Springer-Verlag New York, 2004. doi: 10.1007/b97315.
[15]	W. D. Kalies, K. Mischaikow and R. C. A. M. VanderVorst, An algorithmic approach to chain recurrence, Found. Comp. Math., 5 (2005), 409-449. doi: 10.1007/s10208-004-0163-9.
[16]	W. Massey, Homology and Cohomology Theory, Marcel Dekker, New York and Basel, 1978.
[17]	K. Mischaikow and M. Mrozek, Conley index, Chapter 9 in Handbook of dynamical systems, Elsevier, 2 (2002), 393-460. doi: 10.1016/S1874-575X(02)80030-3.
[18]	M. Mrozek, The Conley index on compact ANR's is of finite type, Results Math., 18 (1990), 306-313. doi: 10.1007/BF03323175.
[19]	M. Mrozek, Index pairs algorithms, Found. Comput. Math., 6 (2006), 457-493. doi: 10.1007/s10208-005-0182-1.
[20]	M. Mrozek, Leray functor and cohomological Conley index for discrete dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 149-178. doi: 10.1090/S0002-9947-1990-0968888-1.
[21]	P. Pilarczyk, L. García, B. A. Carreras and I. Llerena, A dynamical model for plasma confinement transitions, J. Phys. A: Math. Theor., 45 (2012), 125502. doi: 10.1088/1751-8113/45/12/125502.
[22]	P. Pilarczyk, Computer assisted method for proving existence of periodic orbits, Topol. Methods Nonlinear Anal., 13 (1999), 365-377.
[23]	J. W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynamical Systems, $\mathbf{8^*}$ (1988), 375-393. doi: 10.1017/S0143385700009494.
[24]	K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Universitext, Springer-Verlag, 1987. doi: 10.1007/978-3-642-72833-4.
[25]	A. Szymczak, A combinatorial procedure for finding isolating neighbourhoods and index pairs, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1075-1088. doi: 10.1017/S0308210500026901.
[26]	G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics, 140, AMS, 2012. doi: 10.1090/gsm/140.

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