On computer-assisted proving the existence of periodic and bounded orbits
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.
Keywords
Poincaré map; Conley index; interval arithmetic; rigorous numerical algorithm
References
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46. Wilczak D., The existence of Shilnikov homoclinic orbits in the Michelson system: a computer assisted proof, Found. Comput. Math. 6 (2006), 495–535.
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.
48. Wilczak D., Zgliczyński P., Period doubling in the Rössler system – a computer assisted proof, Found. Comput. Math. 9 (2009), 611–649.
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.
50. Zgliczyński P., Computer assisted proof of chaos in the Rössler equations and in the Hénon map, Nonlinearity 10 (1997), 243–252.
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.
2. Appel K., Haken W., Koch J., Every planar map is four colorable. Part II: Reducibility, Illinois J. Math. 21 (1977), 491–567.
3. Bánhelyi B., Csendes T., Garay B.M., Hatvani L., A computer-assisted proof of Σ_3-chaos in the forced damped pendulum equation, SIAM J. Appl. Dyn. Syst. 7 (2008), 843–867.
4. CAPA, http://www2.math.uu.se/~warwick/CAPA
5. CAPD, http://capd.ii.uj.edu.pl
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.
7. CHomP, http://chomp.rutgers.edu
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.
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.
11. Gidea M., Zgliczyński P., Covering relations for multidimensional dynamical systems, J. Differential Equations 202 (2004), 33–58.
12. Hales T.C., A proof of the Kepler conjecture, Ann. of Math. (2) 162 (2005), 1065–1185.
13. Hales T.C. et al., A formal proof of the Kepler conjecture, preprint (2015), http://arxiv.org/pdf/1501.02155.pdf
14. Hass J., Schlafly R., Double bubbles minimize, Ann. of Math. (2) 151 (2000), 459–515.
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.
16. Hastings S.P., Troy W.C., A shooting approach to the Lorenz equations, Bull. Amer. Math. Soc. 27 (1992), 128–131.
17. Hickey T., Ju Q., van Emden M.H., Interval arithmetic: From principles to implementation, J. ACM 48 (2001), 1038–1068.
18. Hutchings M., Morgan F., Ritoré M., Ros A., Proof of the double bubble conjecture, Ann. of Math. (2) 155 (2002), 459–489.
19. Kapela T., Simó C., Computer assisted proofs for non-symmetric planar choreographies and for stability of the Eight, Nonlinearity 20 (2007), 1241–1255.
20. Kapela T., Zgliczyński P., The existence of simple choreographies for the N-body problem – a computer assisted proof, Nonlinearity 16 (2003), 1899–1918.
21. Lam C.W.H., The search for a finite projective plane of order 10, Amer. Math. Monthly 98 (1991), 305–318.
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.
23. Lanford O.E., III, A computer-assisted proof of the Feigenbaum conjecture, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 427–434.
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.
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.
28. Mischaikow K., Mrozek M., Chaos in the Lorenz equations: A computer assisted proof, Bull. Amer. Math. Soc. (N.S.) 32 (1995), 66–72.
29. Mischaikow K., Mrozek M., Chaos in the Lorenz equations: A computer assisted proof. II: Details, Math. Comp. 67 (1998), 1023–1046.
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.
31. Mischaikow K., Zgliczyński P., Rigorous numerics for partial differential equations: the Kuramoto-Sivashinsky equation, Found. Comput. Math. 1 (2001), 255–288.
32. Mizar project, http://mizar.org
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.
34. Mrozek M., Leray functor and cohomological Conley index for discrete dynamical systems, Trans. Amer. Math. Soc. 318 (1990), 149–178.
35. Mrozek M., From the theorem of Ważewski to computer assisted proofs in dynamics, Banach Center Publ. 34 (1995), 105–120.
36. Mrozek M., Topological invariants, multivalued maps and computer assisted proofs in dynamics, Comput. Math. Appl. 32 (1996), 83–104.
37. Mrozek M., Index pairs algorithms, Found. Comput. Math. 6 (2006), 457–493.
38. Mrozek M., Srzednicki R., Topological approach to rigorous numerics of chaotic dynamical systems with strong expansion, Found. Comput. Math. 10 (2010), 191–220.
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.
40. Mrozek M., Żelawski M., Heteroclinic connections in the Kuramoto-Sivashinsky equations, Reliab. Comput. 3 (1997), 277–285.
41. Robertson N., Sanders D., Seymour P., Thomas R., The four-colour theorem, J. Combin. Theory Ser. B 70 (1997), 2–44.
42. Tucker W., The Lorenz attractor exists, C.R. Math. Acad. Sci. Paris 328 (1999), 1197–1202.
43. Tucker W., A rigorous ODE solver and Smale’s 14th problem, Found. Comput. Math. 2 (2002), 53–117.
44. Wikipedia, Pentium FDIV bug, http://en.wikipedia.org/wiki/Pentium_FDIV_bug
45. Wilczak D., Chaos in the Kuramoto-Sivashinsky equations – a computer assisted proof, J. Differential Equations 194 (2003), 433–459.
46. Wilczak D., The existence of Shilnikov homoclinic orbits in the Michelson system: a computer assisted proof, Found. Comput. Math. 6 (2006), 495–535.
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.
48. Wilczak D., Zgliczyński P., Period doubling in the Rössler system – a computer assisted proof, Found. Comput. Math. 9 (2009), 611–649.
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.
50. Zgliczyński P., Computer assisted proof of chaos in the Rössler equations and in the Hénon map, Nonlinearity 10 (1997), 243–252.
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.
SrzednickiR. (2015). On computer-assisted proving the existence of periodic and bounded orbits. Annales Mathematicae Silesianae, 29, 7-17. Retrieved from https://www.journals.us.edu.pl/index.php/AMSIL/article/view/13973
Roman Srzednicki
srzednicki@im.uj.edu.pl
Wydział Matematyki i Informatyki, Uniwersytet Jagielloński Poland
Wydział Matematyki i Informatyki, Uniwersytet Jagielloński Poland
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