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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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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).
|
[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. |
[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. |
[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. |
[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. |
[14] |
T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology,, Applied Mathematical Sciences Vol. 157, (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.
doi: 10.1007/s10208-004-0163-9. |
[16] |
W. Massey, Homology and Cohomology Theory,, Marcel Dekker, (1978).
|
[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. |
[18] |
M. Mrozek, The Conley index on compact ANR's is of finite type,, Results Math., 18 (1990), 306.
doi: 10.1007/BF03323175. |
[19] |
M. Mrozek, Index pairs algorithms,, Found. Comput. Math., 6 (2006), 457.
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.
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).
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.
|
[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. |
[24] |
K. P. Rybakowski, The Homotopy Index and Partial Differential Equations,, Universitext, (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.
doi: 10.1017/S0308210500026901. |
[26] |
G. Teschl, Ordinary Differential Equations and Dynamical Systems,, Graduate Studies in Mathematics, 140 (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.
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.
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.
doi: 10.1115/1.2338651. |
[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. |
[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. |
[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. |
[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).
|
[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. |
[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. |
[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. |
[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. |
[14] |
T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology,, Applied Mathematical Sciences Vol. 157, (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.
doi: 10.1007/s10208-004-0163-9. |
[16] |
W. Massey, Homology and Cohomology Theory,, Marcel Dekker, (1978).
|
[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. |
[18] |
M. Mrozek, The Conley index on compact ANR's is of finite type,, Results Math., 18 (1990), 306.
doi: 10.1007/BF03323175. |
[19] |
M. Mrozek, Index pairs algorithms,, Found. Comput. Math., 6 (2006), 457.
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.
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).
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.
|
[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. |
[24] |
K. P. Rybakowski, The Homotopy Index and Partial Differential Equations,, Universitext, (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.
doi: 10.1017/S0308210500026901. |
[26] |
G. Teschl, Ordinary Differential Equations and Dynamical Systems,, Graduate Studies in Mathematics, 140 (2012).
doi: 10.1090/gsm/140. |
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