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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), 393-417.
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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