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

Discretization strategies for computing Conley indices and Morse decompositions of flows

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

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.
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, http://capd.ii.uj.edu.pl/.

[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, http://capd.ii.uj.edu.pl/.

[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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