On Computer-Assisted Proving The Existence Of Periodic And Bounded Orbits

Roman Srzednicki

doi:10.1515/amsil-2015-0001

.blurhash-client-img { display: none !important; }

On Computer-Assisted Proving The Existence Of Periodic And Bounded Orbits

Annales Mathematicae Silesianae

Volume 29 (2015): Issue 1 (September 2015)

By: Roman Srzednicki

Open Access

|Sep 2015

Abstract

We announce a new result on determining the Conley index of the Poincaré map for a time-periodic non-autonomous ordinary differential equation. The index is computed using some singular cycles related to an index pair of a small-step discretization of the equation. We indicate how the result can be applied to computer-assisted proofs of the existence of bounded and periodic solutions. We provide also some comments on computer-assisted proving in dynamics.

References

[1] Appel K., Haken W., Every planar map is four colorable. Part I: Discharging, Illinois J. Math. 21 (1977), 429–490.
Search in Google Scholar
[2] Appel K., Haken W., Koch J., Every planar map is four colorable. Part II: Reducibility, Illinois J. Math. 21 (1977), 491–567.
Search in Google Scholar
[3] Bánhelyi B., Csendes T., Garay B.M., Hatvani L., A computer-assisted proof of ∑3-chaos in the forced damped pendulum equation, SIAM J. Appl. Dyn. Syst. 7 (2008), 843–867.10.1137/070695599
Search in Google Scholar
[4] CAPA, http://www2.math.uu.se/~warwick/CAPA
Search in Google Scholar
[5] CAPD, http://capd.ii.uj.edu.pl
Search in Google Scholar
[6] Capiński M.J.,Computer assisted existence proofs of Lyapunov orbits at L2 and transversal intersections of invariant manifolds in the Jupiter-Sun PCR3BP, SIAM J. Appl. Dyn. Syst. 11 (2012), 1723–1753.10.1137/110847366
Search in Google Scholar
[7] CHomP, http://chomp.rutgers.edu
Search in Google Scholar
[8] Eckmann J.-P., Koch H., Wittwer P., A computer-assisted proof of universality for area-preserving maps, Mem. Amer. Math. Soc. 47 (1984), 1–121.
Search in Google Scholar
[9] Galias Z., Computer assisted proof of chaos in the Muthuswamy-Chua memristor circuit, Nonlinear Theory Appl. IEICE 5 (2014), 309–319.10.1587/nolta.5.309
Search in Google Scholar
[10] Galias Z., Tucker W., Numerical study of coexisting attractors for the Hénon map, Int. J. Bifurcation Chaos 23 (2013), no. 7, 1330025, 18 pp.10.1142/S0218127413300255
Search in Google Scholar
[11] Gidea M., Zgliczyński P., Covering relations for multidimensional dynamical systems, J. Differential Equations 202 (2004), 33–58.10.1016/j.jde.2004.03.014
Search in Google Scholar
[12] Hales T.C., A proof of the Kepler conjecture, Ann. of Math. (2) 162 (2005), 1065–1185.10.4007/annals.2005.162.1065
Search in Google Scholar
[13] Hales T.C. et al., A formal proof of the Kepler conjecture, preprint (2015), http://arxiv.org/pdf/1501.02155.pdf
Search in Google Scholar
[14] Hass J., Schlafly R., Double bubbles minimize, Ann. of Math. (2) 151 (2000), 459–515.10.2307/121042
Search in Google Scholar
[15] Hassard B., Zhang J., Existence of a homoclinic orbit of the Lorenz system by precise shooting, SIAM J. Math. Anal. 25 (1994), 179–196.10.1137/S0036141092234827
Search in Google Scholar
[16] Hastings S.P., Troy W.C., A shooting approach to the Lorenz equations, Bull. Amer. Math. Soc. 27 (1992), 128–131.10.1090/S0273-0979-1992-00327-0
Search in Google Scholar
[17] Hickey T., Ju Q., van Emden M.H., Interval arithmetic: From principles to implementation, J. ACM 48 (2001), 1038–1068.10.1145/502102.502106
Search in Google Scholar
[18] Hutchings M., Morgan F., Ritoré M., Ros A., Proof of the double bubble conjecture, Ann. of Math. (2) 155 (2002), 459–489.10.2307/3062123
Search in Google Scholar
[19] Kapela T., Simó C., Computer assisted proofs for non-symmetric planar choreographies and for stability of the Eight, Nonlinearity 20 (2007), 1241–1255.10.1088/0951-7715/20/5/010
Search in Google Scholar
[20] Kapela T., Zgliczyński P., The existence of simple choreographies for the N-body problem – a computer assisted proof, Nonlinearity 16 (2003), 1899–1918.10.1088/0951-7715/16/6/302
Search in Google Scholar
[21] Lam C.W.H., The search for a finite projective plane of order 10, Amer. Math. Monthly 98 (1991), 305–318.10.1080/00029890.1991.12000759
Search in Google Scholar
[22] Lam C.W.H., Thiel L., Swiercz S., The non-existence of finite projective planes of order 10, Canad. J. Math. 41 (1989), 1117–1123.10.4153/CJM-1989-049-4
Search in Google Scholar
[23] Lanford O.E., III, A computer-assisted proof of the Feigenbaum conjecture, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 427–434.10.1090/S0273-0979-1982-15008-X
Search in Google Scholar
[24] Lanford O.E., III, Computer-assisted proofs in analysis, in: Proceedings of the International Congress of Mathematicians, Berkeley, California, USA, 1986, pp. 1385–1394.
Search in Google Scholar
[25] Lorenz E.N., Deterministic nonperiodic flow, J. Atmos. Sci. 20 (1963), 130–141.10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
Search in Google Scholar
[26] Mann A.L., A complete proof of the Robbins conjecture, preprint (2003).
Search in Google Scholar
[27] McCune W., Solution of the Robbins problem, J. Autom. Reasoning 19 (1997), 263–276.10.1023/A:1005843212881
Search in Google Scholar
[28] Mischaikow K., Mrozek M., Chaos in the Lorenz equations: A computer assisted proof, Bull. Amer. Math. Soc. (N.S.) 32 (1995), 66–72.10.1090/S0273-0979-1995-00558-6
Search in Google Scholar
[29] Mischaikow K., Mrozek M., Chaos in the Lorenz equations: A computer assisted proof. II: Details, Math. Comp. 67 (1998), 1023–1046.
Search in Google Scholar
[30] Mischaikow K., Mrozek M., Szymczak A., Chaos in the Lorenz equations: A computer assisted proof. III: Classical parameter values, J. Differential Equations 169 (2001), 17–56.10.1006/jdeq.2000.3894
Search in Google Scholar
[31] Mischaikow K., Zgliczyński P., Rigorous numerics for partial differential equations: the Kuramoto-Sivashinsky equation, Found. Comput. Math. 1 (2001), 255–288.10.1007/s002080010010
Search in Google Scholar
[32] Mizar project, http://mizar.org
Search in Google Scholar
[33] Moeckel R., A computer-assisted proof of Saari’s conjecture for the planar three-body problem, Trans. Amer. Math. Soc. 357 (2005), 3105–3117.10.1090/S0002-9947-04-03527-5
Search in Google Scholar
[34] Mrozek M., Leray functor and cohomological Conley index for discrete dynamical systems, Trans. Amer. Math. Soc. 318 (1990),149–178.10.1090/S0002-9947-1990-0968888-1
Search in Google Scholar
[35] Mrozek M., From the theorem of Ważewski to computer assisted proofs in dynamics, Banach Center Publ. 34 (1995), 105–120.10.4064/-34-1-105-120
Search in Google Scholar
[36] Mrozek M., Topological invariants, multivalued maps and computer assisted proofs in dynamics, Comput. Math. Appl. 32 (1996), 83–104.10.1016/0898-1221(96)00127-7
Search in Google Scholar
[37] Mrozek M., Index pairs algorithms, Found. Comput. Math. 6 (2006), 457–493.10.1007/s10208-005-0182-1
Search in Google Scholar
[38] Mrozek M., Srzednicki R., Topological approach to rigorous numerics of chaotic dynamical systems with strong expansion, Found. Comput. Math. 10 (2010), 191–220.10.1007/s10208-009-9053-5
Search in Google Scholar
[39] Mrozek M., Srzednicki R., Weilandt F., A topological approach to the algorithmic computation of the Conley index for Poincaré maps, SIAM J. Appl. Dyn. Syst. 14 (2015), 1348–1386.10.1137/15M100794X
Search in Google Scholar
[40] Mrozek M., Żelawski M., Heteroclinic connections in the Kuramoto-Sivashinsky equations, Reliab. Comput. 3 (1997), 277–285.10.1023/A:1009974824292
Search in Google Scholar
[41] Robertson N., Sanders D., Seymour P., Thomas R., The four-colour theorem, J. Combin. Theory Ser. B 70 (1997), 2–44.10.1006/jctb.1997.1750
Search in Google Scholar
[42] Tucker W., The Lorenz attractor exists, C.R. Math. Acad. Sci. Paris 328 (1999), 1197–1202.10.1016/S0764-4442(99)80439-X
Search in Google Scholar
[43] Tucker W., A rigorous ODE solver and Smale’s 14th problem, Found. Comput. Math. 2 (2002), 53–117.10.1007/s002080010018
Search in Google Scholar
[44] Wikipedia, Pentium FDIV bug, http://en.wikipedia.org/wiki/Pentium_FDIV_bug
Search in Google Scholar
[45] Wilczak D., Chaos in the Kuramoto-Sivashinsky equations – a computer assisted proof, J. Differential Equations 194 (2003), 433–459.10.1016/S0022-0396(03)00104-9
Search in Google Scholar
[46] Wilczak D., The existence of Shilnikov homoclinic orbits in the Michelson system: a computer assisted proof, Found. Comput. Math. 6 (2006), 495–535.10.1007/s10208-005-0201-2
Search in Google Scholar
[47] Wilczak D., Zgliczyński P., Heteroclinic connections between periodic orbits in planar restricted circular three body problem – a computer assisted proof, Comm. Math. Phys. 234 (2003), 37–75.10.1007/s00220-002-0709-0
Search in Google Scholar
[48] Wilczak D., Zgliczyński P., Period doubling in the Rössler system – a computer assisted proof, Found. Comput. Math. 9 (2009), 611–649.10.1007/s10208-009-9040-x
Search in Google Scholar
[49] Wilczak D., Zgliczyński P., Computer assisted proof of the existence of homoclinic tangency for the Hénon map and for the forced-damped pendulum, SIAM J. Appl. Dyn. Syst. 8 (2009), 1632–1663.10.1137/090759975
Search in Google Scholar
[50] Zgliczyński P., Computer assisted proof of chaos in the Rössler equations and in the Hénon map, Nonlinearity 10 (1997), 243–252.10.1088/0951-7715/10/1/016
Search in Google Scholar
[51] Zgliczyński P., Rigorous numerics for dissipative partial differential equations II. Periodic orbit for the Kuramoto-Sivashinsky PDE – a computer assisted proof, Found. Comput. Math. 4 (2004), 157–185.10.1007/s10208-002-0080-8
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.1515/amsil-2015-0001 | Journal eISSN: 2391-4238 | Journal ISSN: 0860-2107

Journal RSS Feed

Language: English

Page range: 7 - 17

Submitted on: Mar 1, 2015

Published on: Sep 30, 2015

Published by: University of Silesia in Katowice, Institute of Mathematics

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

37B30,

37C27,

65L99

Related subjects:

Mathematics,

General mathematics

© 2015 Roman Srzednicki, published by University of Silesia in Katowice, Institute of Mathematics
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 29 (2015): Issue 1 (September 2015)