Discretization strategies for computing Conley indices and Morse decompositions of flows

Previous Article
Towards a formal tie between combinatorial and classical vector field dynamics
JCD Home
This Issue
Next Article

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
Related Papers

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, 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.

[1]	Todd Young. A result in global bifurcation theory using the Conley index. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 387-396. doi: 10.3934/dcds.1996.2.387
[2]	M. C. Carbinatto, K. Mischaikow. Horseshoes and the Conley index spectrum - II: the theorem is sharp. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 599-616. doi: 10.3934/dcds.1999.5.599
[3]	Mauro Patrão, Luiz A. B. San Martin. Morse decomposition of semiflows on fiber bundles. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 561-587. doi: 10.3934/dcds.2007.17.561
[4]	Marian Gidea. Leray functor and orbital Conley index for non-invariant sets. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 617-630. doi: 10.3934/dcds.1999.5.617
[5]	Jintao Wang, Desheng Li, Jinqiao Duan. On the shape Conley index theory of semiflows on complete metric spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1629-1647. doi: 10.3934/dcds.2016.36.1629
[6]	Anna Go??biewska, S?awomir Rybicki. Equivariant Conley index versus degree for equivariant gradient maps. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985
[7]	Ketty A. De Rezende, Mariana G. Villapouca. Discrete conley index theory for zero dimensional basic sets. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1359-1387. doi: 10.3934/dcds.2017056
[8]	Anna Belova. Rigorous enclosures of rotation numbers by interval methods. Journal of Computational Dynamics, 2016, 3 (1) : 81-91. doi: 10.3934/jcd.2016004
[9]	Tomás Caraballo, Juan C. Jara, José A. Langa, José Valero. Morse decomposition of global attractors with infinite components. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2845-2861. doi: 10.3934/dcds.2015.35.2845
[10]	Jan Bouwe van den Berg, Ray Sheombarsing. Rigorous numerics for ODEs using Chebyshev series and domain decomposition. Journal of Computational Dynamics, 2021, 8 (3) : 353-401. doi: 10.3934/jcd.2021015
[11]	Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure and Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003
[12]	Zongming Guo, Zhongyuan Liu, Juncheng Wei, Feng Zhou. Bifurcations of some elliptic problems with a singular nonlinearity via Morse index. Communications on Pure and Applied Analysis, 2011, 10 (2) : 507-525. doi: 10.3934/cpaa.2011.10.507
[13]	Belgacem Rahal, Cherif Zaidi. On finite Morse index solutions of higher order fractional elliptic equations. Evolution Equations and Control Theory, 2021, 10 (3) : 575-597. doi: 10.3934/eect.2020081
[14]	Kelei Wang. Recent progress on stable and finite Morse index solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29 (6) : 3805-3816. doi: 10.3934/era.2021062
[15]	M. D. Todorov, C. I. Christov. Conservative numerical scheme in complex arithmetic for coupled nonlinear Schrödinger equations. Conference Publications, 2007, 2007 (Special) : 982-992. doi: 10.3934/proc.2007.2007.982
[16]	Radoslaw Pytlak. Numerical procedure for optimal control of higher index DAEs. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 647-670. doi: 10.3934/dcds.2011.29.647
[17]	Bao Qing Hu, Song Wang. A novel approach in uncertain programming part I: new arithmetic and order relation for interval numbers. Journal of Industrial and Management Optimization, 2006, 2 (4) : 351-371. doi: 10.3934/jimo.2006.2.351
[18]	Pooja Bansal. Sequential Malmquist-Luenberger productivity index for interval data envelopment analysis. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022058
[19]	Jiaquan Liu, Yuxia Guo, Pingan Zeng. Relationship of the morse index and the $L^\infty$ bound of solutions for a strongly indefinite differential superlinear system. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 107-119. doi: 10.3934/dcds.2006.16.107
[20]	Yejuan Wang, Tomás Caraballo. Morse decomposition for gradient-like multi-valued autonomous and nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2303-2326. doi: 10.3934/dcdss.2020092

Impact Factor:

Tools

Metrics

Other articles
by authors

on AIMS
Konstantin Mischaikow Marian Mrozek Frank Weilandt
on Google Scholar
Konstantin Mischaikow Marian Mrozek Frank Weilandt

[Back to Top]