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

Roman Srzednicki

Annales Mathematicae Silesianae, Tome 29 (2015), p. 7-17 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2015-01-01
EUDML-ID : urn:eudml:doc:276852

Bibliographie

[1] Appel K., Haken W., Every planar map is four colorable. Part I: Discharging, Illinois J. Math. 21 (1977), 429–490. | Zbl 0387.05009

[2] Appel K., Haken W., Koch J., Every planar map is four colorable. Part II: Reducibility, Illinois J. Math. 21 (1977), 491–567. | Zbl 0387.05010

[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.[Crossref] | Zbl 1160.70010

[4] CAPA,

[5] CAPD,

[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.[Crossref] | Zbl 1264.37008

[7] CHomP,

[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. | Zbl 0528.58033

[9] Galias Z., Computer assisted proof of chaos in the Muthuswamy-Chua memristor circuit, Nonlinear Theory Appl. IEICE 5 (2014), 309–319.

[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. | Zbl 1275.37023

[11] Gidea M., Zgliczyński P., Covering relations for multidimensional dynamical systems, J. Differential Equations 202 (2004), 33–58. | Zbl 1061.37013

[12] Hales T.C., A proof of the Kepler conjecture, Ann. of Math. (2) 162 (2005), 1065–1185. | Zbl 1096.52010

[13] Hales T.C. et al., A formal proof of the Kepler conjecture, preprint (2015),

[14] Hass J., Schlafly R., Double bubbles minimize, Ann. of Math. (2) 151 (2000), 459–515. | Zbl 0970.53008

[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. | Zbl 0799.34046

[16] Hastings S.P., Troy W.C., A shooting approach to the Lorenz equations, Bull. Amer. Math. Soc. 27 (1992), 128–131.[Crossref] | Zbl 0764.58023

[17] Hickey T., Ju Q., van Emden M.H., Interval arithmetic: From principles to implementation, J. ACM 48 (2001), 1038–1068.[Crossref] | Zbl 1323.65047

[18] Hutchings M., Morgan F., Ritoré M., Ros A., Proof of the double bubble conjecture, Ann. of Math. (2) 155 (2002), 459–489. | Zbl 1009.53007

[19] Kapela T., Simó C., Computer assisted proofs for non-symmetric planar choreographies and for stability of the Eight, Nonlinearity 20 (2007), 1241–1255.[Crossref] | Zbl 1115.70008

[20] Kapela T., Zgliczyński P., The existence of simple choreographies for the N-body problem – a computer assisted proof, Nonlinearity 16 (2003), 1899–1918.[Crossref] | Zbl 1060.70023

[21] Lam C.W.H., The search for a finite projective plane of order 10, Amer. Math. Monthly 98 (1991), 305–318.[Crossref] | Zbl 0744.51011

[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. | Zbl 0691.51003

[23] Lanford O.E., III, A computer-assisted proof of the Feigenbaum conjecture, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 427–434.[Crossref] | Zbl 0487.58017

[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.

[25] Lorenz E.N., Deterministic nonperiodic flow, J. Atmos. Sci. 20 (1963), 130–141.[Crossref]

[26] Mann A.L., A complete proof of the Robbins conjecture, preprint (2003).

[27] McCune W., Solution of the Robbins problem, J. Autom. Reasoning 19 (1997), 263–276.[Crossref] | Zbl 0883.06011

[28] Mischaikow K., Mrozek M., Chaos in the Lorenz equations: A computer assisted proof, Bull. Amer. Math. Soc. (N.S.) 32 (1995), 66–72.[Crossref] | Zbl 0820.58042

[29] Mischaikow K., Mrozek M., Chaos in the Lorenz equations: A computer assisted proof. II: Details, Math. Comp. 67 (1998), 1023–1046.[Crossref] | Zbl 0913.58038

[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. | Zbl 0979.37015

[31] Mischaikow K., Zgliczyński P., Rigorous numerics for partial differential equations: the Kuramoto-Sivashinsky equation, Found. Comput. Math. 1 (2001), 255–288.[Crossref] | Zbl 0984.65101

[32] Mizar project,

[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.[Crossref] | Zbl 1079.70011

[34] Mrozek M., Leray functor and cohomological Conley index for discrete dynamical systems, Trans. Amer. Math. Soc. 318 (1990),149–178.[Crossref] | Zbl 0686.58034

[35] Mrozek M., From the theorem of Ważewski to computer assisted proofs in dynamics, Banach Center Publ. 34 (1995), 105–120. | Zbl 0848.34032

[36] Mrozek M., Topological invariants, multivalued maps and computer assisted proofs in dynamics, Comput. Math. Appl. 32 (1996), 83–104.[Crossref] | Zbl 0861.58027

[37] Mrozek M., Index pairs algorithms, Found. Comput. Math. 6 (2006), 457–493.[Crossref] | Zbl 1111.37006

[38] Mrozek M., Srzednicki R., Topological approach to rigorous numerics of chaotic dynamical systems with strong expansion, Found. Comput. Math. 10 (2010), 191–220.[Crossref] | Zbl 1192.65152

[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.[Crossref] | Zbl 1325.65175

[40] Mrozek M., Żelawski M., Heteroclinic connections in the Kuramoto-Sivashinsky equations, Reliab. Comput. 3 (1997), 277–285.[Crossref] | Zbl 0888.65091

[41] Robertson N., Sanders D., Seymour P., Thomas R., The four-colour theorem, J. Combin. Theory Ser. B 70 (1997), 2–44. | Zbl 0883.05056

[42] Tucker W., The Lorenz attractor exists, C.R. Math. Acad. Sci. Paris 328 (1999), 1197–1202. | Zbl 0935.34050

[43] Tucker W., A rigorous ODE solver and Smale’s 14th problem, Found. Comput. Math. 2 (2002), 53–117.[Crossref] | Zbl 1047.37012

[44] Wikipedia, Pentium FDIV bug,

[45] Wilczak D., Chaos in the Kuramoto-Sivashinsky equations – a computer assisted proof, J. Differential Equations 194 (2003), 433–459. | Zbl 1050.37017

[46] Wilczak D., The existence of Shilnikov homoclinic orbits in the Michelson system: a computer assisted proof, Found. Comput. Math. 6 (2006), 495–535.[Crossref] | Zbl 1130.37415

[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.[Crossref] | Zbl 1055.70005

[48] Wilczak D., Zgliczyński P., Period doubling in the Rössler system – a computer assisted proof, Found. Comput. Math. 9 (2009), 611–649.[Crossref] | Zbl 1177.37083

[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.[Crossref] | Zbl 1187.37115

[50] Zgliczyński P., Computer assisted proof of chaos in the Rössler equations and in the Hénon map, Nonlinearity 10 (1997), 243–252.[Crossref] | Zbl 0907.58048

[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.[Crossref] | Zbl 1066.65105