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. doi: 10.1007/BF03186541. Google Scholar

[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. doi: 10.1137/080734935. Google Scholar

[3]

H. Ban and W. D. Kalies, A computational approach to Conley's decomposition theorem,, J. Comput. Nonlinear Dynam., 1 (2006), 312. doi: 10.1115/1.2338651. Google Scholar

[4]

E. Boczko, W. D. Kalies and K. Mischaikow, Polygonal approximation of flows,, Topology Appl., 154 (2007), 2501. doi: 10.1016/j.topol.2006.04.033. Google Scholar

[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). doi: 10.1063/1.4767672. Google Scholar

[6]

J. B. van den Berg and J. P. Lessard, Rigorous numerics in dynamics,, Notices Amer. Math. Soc., 62 (2015), 1057. doi: 10.1090/noti1276. Google Scholar

[7]

The CAPD Group, Computer assisted proofs in dynamics software library,, , (). Google Scholar

[8]

C. Conley, Isolated Invariant Sets and the Morse Index,, CBMS Regional Conference Series in Mathematics, 38 (1978). Google Scholar

[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. Google Scholar

[10]

M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems,, Chapter 5 in Handbook of dynamical systems, 2 (2002), 221. doi: 10.1016/S1874-575X(02)80026-1. Google Scholar

[11]

J. Franks and D. Richeson, Shift equivalence and the Conley index,, Transactions AMS, 352 (2000), 3305. doi: 10.1090/S0002-9947-00-02488-0. Google Scholar

[12]

M. Gidea and P. Zgliczynski, Covering relations for multidimensional dynamical systems I,, J. Differential Equations, 202 (2004), 32. doi: 10.1016/j.jde.2004.03.013. Google Scholar

[13]

M. Gidea and P. Zgliczynski, Covering relations for multidimensional dynamical systems II,, J. Differential Equations, 202 (2004), 59. doi: 10.1016/j.jde.2004.03.014. Google Scholar

[14]

T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology,, Applied Mathematical Sciences Vol. 157, (2004). doi: 10.1007/b97315. Google Scholar

[15]

W. D. Kalies, K. Mischaikow and R. C. A. M. VanderVorst, An algorithmic approach to chain recurrence,, Found. Comp. Math., 5 (2005), 409. doi: 10.1007/s10208-004-0163-9. Google Scholar

[16]

W. Massey, Homology and Cohomology Theory,, Marcel Dekker, (1978). Google Scholar

[17]

K. Mischaikow and M. Mrozek, Conley index,, Chapter 9 in Handbook of dynamical systems, 2 (2002), 393. doi: 10.1016/S1874-575X(02)80030-3. Google Scholar

[18]

M. Mrozek, The Conley index on compact ANR's is of finite type,, Results Math., 18 (1990), 306. doi: 10.1007/BF03323175. Google Scholar

[19]

M. Mrozek, Index pairs algorithms,, Found. Comput. Math., 6 (2006), 457. doi: 10.1007/s10208-005-0182-1. Google Scholar

[20]

M. Mrozek, Leray functor and cohomological Conley index for discrete dynamical systems,, Trans. Amer. Math. Soc., 318 (1990), 149. doi: 10.1090/S0002-9947-1990-0968888-1. Google Scholar

[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). doi: 10.1088/1751-8113/45/12/125502. Google Scholar

[22]

P. Pilarczyk, Computer assisted method for proving existence of periodic orbits,, Topol. Methods Nonlinear Anal., 13 (1999), 365. Google Scholar

[23]

J. W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index,, Ergodic Theory Dynamical Systems, (1988), 375. doi: 10.1017/S0143385700009494. Google Scholar

[24]

K. P. Rybakowski, The Homotopy Index and Partial Differential Equations,, Universitext, (1987). doi: 10.1007/978-3-642-72833-4. Google Scholar

[25]

A. Szymczak, A combinatorial procedure for finding isolating neighbourhoods and index pairs,, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1075. doi: 10.1017/S0308210500026901. Google Scholar

[26]

G. Teschl, Ordinary Differential Equations and Dynamical Systems,, Graduate Studies in Mathematics, 140 (2012). doi: 10.1090/gsm/140. Google Scholar

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. doi: 10.1007/BF03186541. Google Scholar

[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. doi: 10.1137/080734935. Google Scholar

[3]

H. Ban and W. D. Kalies, A computational approach to Conley's decomposition theorem,, J. Comput. Nonlinear Dynam., 1 (2006), 312. doi: 10.1115/1.2338651. Google Scholar

[4]

E. Boczko, W. D. Kalies and K. Mischaikow, Polygonal approximation of flows,, Topology Appl., 154 (2007), 2501. doi: 10.1016/j.topol.2006.04.033. Google Scholar

[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). doi: 10.1063/1.4767672. Google Scholar

[6]

J. B. van den Berg and J. P. Lessard, Rigorous numerics in dynamics,, Notices Amer. Math. Soc., 62 (2015), 1057. doi: 10.1090/noti1276. Google Scholar

[7]

The CAPD Group, Computer assisted proofs in dynamics software library,, , (). Google Scholar

[8]

C. Conley, Isolated Invariant Sets and the Morse Index,, CBMS Regional Conference Series in Mathematics, 38 (1978). Google Scholar

[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. Google Scholar

[10]

M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems,, Chapter 5 in Handbook of dynamical systems, 2 (2002), 221. doi: 10.1016/S1874-575X(02)80026-1. Google Scholar

[11]

J. Franks and D. Richeson, Shift equivalence and the Conley index,, Transactions AMS, 352 (2000), 3305. doi: 10.1090/S0002-9947-00-02488-0. Google Scholar

[12]

M. Gidea and P. Zgliczynski, Covering relations for multidimensional dynamical systems I,, J. Differential Equations, 202 (2004), 32. doi: 10.1016/j.jde.2004.03.013. Google Scholar

[13]

M. Gidea and P. Zgliczynski, Covering relations for multidimensional dynamical systems II,, J. Differential Equations, 202 (2004), 59. doi: 10.1016/j.jde.2004.03.014. Google Scholar

[14]

T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology,, Applied Mathematical Sciences Vol. 157, (2004). doi: 10.1007/b97315. Google Scholar

[15]

W. D. Kalies, K. Mischaikow and R. C. A. M. VanderVorst, An algorithmic approach to chain recurrence,, Found. Comp. Math., 5 (2005), 409. doi: 10.1007/s10208-004-0163-9. Google Scholar

[16]

W. Massey, Homology and Cohomology Theory,, Marcel Dekker, (1978). Google Scholar

[17]

K. Mischaikow and M. Mrozek, Conley index,, Chapter 9 in Handbook of dynamical systems, 2 (2002), 393. doi: 10.1016/S1874-575X(02)80030-3. Google Scholar

[18]

M. Mrozek, The Conley index on compact ANR's is of finite type,, Results Math., 18 (1990), 306. doi: 10.1007/BF03323175. Google Scholar

[19]

M. Mrozek, Index pairs algorithms,, Found. Comput. Math., 6 (2006), 457. doi: 10.1007/s10208-005-0182-1. Google Scholar

[20]

M. Mrozek, Leray functor and cohomological Conley index for discrete dynamical systems,, Trans. Amer. Math. Soc., 318 (1990), 149. doi: 10.1090/S0002-9947-1990-0968888-1. Google Scholar

[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). doi: 10.1088/1751-8113/45/12/125502. Google Scholar

[22]

P. Pilarczyk, Computer assisted method for proving existence of periodic orbits,, Topol. Methods Nonlinear Anal., 13 (1999), 365. Google Scholar

[23]

J. W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index,, Ergodic Theory Dynamical Systems, (1988), 375. doi: 10.1017/S0143385700009494. Google Scholar

[24]

K. P. Rybakowski, The Homotopy Index and Partial Differential Equations,, Universitext, (1987). doi: 10.1007/978-3-642-72833-4. Google Scholar

[25]

A. Szymczak, A combinatorial procedure for finding isolating neighbourhoods and index pairs,, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1075. doi: 10.1017/S0308210500026901. Google Scholar

[26]

G. Teschl, Ordinary Differential Equations and Dynamical Systems,, Graduate Studies in Mathematics, 140 (2012). doi: 10.1090/gsm/140. Google Scholar

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