Topological Methods in Nonlinear Analysis

Rigorous numerics for fast-slow systems with one-dimensional slow variable: topological shadowing approach

Kaname Matsue

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We provide a rigorous numerical computational method to validate periodic, homoclinic and heteroclinic orbits as the continuation of singular limit orbits for the fast-slow system \begin{equation*} \begin{cases} x' = f(x,y,\varepsilon), & \\ y' =\varepsilon g(x,y,\varepsilon) & \end{cases} \end{equation*} with one-dimensional slow variable $y$. Our validation procedure is based on topological tools called isolating blocks, cone conditions and covering relations. Such tools provide us with existence theorems of global orbits which shadow singular orbits in terms of a new concept, the covering-exchange. Additional techniques called slow shadowing and $m$-cones are also developed. These techniques give us not only generalized topological verification theorems, but also easy implementations for validating trajectories near slow manifolds in a wide range, via rigorous numerics. Our procedure is available to validate global orbits not only for sufficiently small $\varepsilon > 0$ but all $\varepsilon$ in a given half-open interval $(0,\varepsilon_0]$. Several sample verification examples are shown as a demonstration of applicability.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 50, Number 2 (2017), 357-468.

Dates
First available in Project Euclid: 27 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1506477635

Digital Object Identifier
doi:10.12775/TMNA.2016.072

Mathematical Reviews number (MathSciNet)
MR3747023

Zentralblatt MATH identifier
1383.34079

Citation

Matsue, Kaname. Rigorous numerics for fast-slow systems with one-dimensional slow variable: topological shadowing approach. Topol. Methods Nonlinear Anal. 50 (2017), no. 2, 357--468. doi:10.12775/TMNA.2016.072. https://projecteuclid.org/euclid.tmna/1506477635


Export citation

References

  • G. Arioli and H. Koch, Existence and stability of traveling pulse solutions of the Fitzhugh–Nagumo equation, Nonlinear Anal. 113 (2015), 51–70.
  • E. Benoit, J.L. Callot, F. Diener and M. Diener, Chasse au canards, Collect. Math. 31 (1981), 37–119.
  • CAPD, Computer Assisted Proof of Dynamics software, http://capd.ii.uj.edu.pl
  • G.A. Carpenter, A geometric approach to singular perturbation problems with applications to nerve impulse equations, J. Differential Equations 23 (1977), 335–367.
  • R. Castelli and J.-P. Lessard, Rigorous numerics in Floquet theory: computing stable and unstable bundles of periodic orbits, SIAM J. Appl. Dyn. Syst. 12 (2013), 204–245.
  • S.-N. Chow, W. Liu and Y. Yi, Center manifolds for smooth invariant manifolds, Trans. Amer. Math. Soc. 352 (2000), 5179–5211.
  • C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, vol. 38, Amer. Math. Soc., Providence, R.I., 1978.
  • A. Czechowski and P. Zgliczyński, Existence of periodic solutions of the FitzHugh–Nagumo equations for an explicit range of the small parameter, SIAM J. Appl. Dyn. Sys. 15 (2016), 1615–1655.
  • B. Deng The existence of infinitely many traveling front and back waves in the FitzHugh–Nagumo equations, SIAM J. Math. Anal. 22 (1991), 1631–1650.
  • F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc. 121 (577) (1996), 100 pp., with an appendix by Cheng Zhi Li.
  • N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations 31 (1979), 53–98.
  • M. Gameiro, T. Gedeon, W. Kalies, H. Kokubu, K. Mischaikow and H. Oka, Topological horseshoes of traveling waves for a fast-slow predator-prey system, J. Dynam. Differential Equations 19 (2007), 23–654.
  • R. Gardner and J. Smoller, The existence of periodic travelling waves for singularly perturbed predator-prey equations via the Conley index, J. Differential Equations 47 (1983), 133–161.
  • T. Gedeon, H. Kokubu, K. Mischaikow and H. Oka, The Conley index for fast-slow systems II. Multidimensional slow variable, J. Differential Equations 225 (2006), 242–307.
  • T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka and J.F. Reineck, The Conley index for fast-slow systems I. One-dimensional slow variable, J. Dynam. Differential Equations 11 (1999), 427–470.
  • J. Guckenheimer, T. Johnson and P. Meerkamp, Rigorous enclosures of a slow manifold, SIAM J. Appl. Dyn. Syst. 11 (2012), 831–863.
  • J. Guckenheimer and C. Kuehn, Computing slow manifolds of saddle type, SIAM J. Appl. Dyn. Syst. 8 (2009), 854–879.
  • C.K.R.T. Jones, Stability of the travelling wave solution of the FitzHugh–Nagumo system, Trans. Amer. Math. Soc. 286 (1984), 431–469.
  • C.K.R.T. Jones, Geometric singular perturbation theory, Dynamical Systems (Montecatini Terme, 1994), Lecture Notes in Math., vol. 1609, Springer, Berlin, 1995, 44–118.
  • C.K.R.T. Jones, T.J. Kaper and N. Kopell, Tracking invariant manifolds up to exponentially small errors, SIAM J. Math. Anal. 27 (1996), 558–577.
  • C.K.R.T. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Differential Equations 108 (1994), 64–88.
  • M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points – fold and canard points in two dimensions, SIAM J. Math. Anal. 33 (2001), 286–314.
  • W. Liu, Exchange lemmas for singular perturbation problems with certain turning points, J. Differential Equations 167 (2000), 134–180.
  • K. Matsue, Rigorous numerics for stationary solutions of dissipative \romPDEs-existence and local dynamics, Nonlinear Theory and Its Applications, IEICE 4 (2013), 62–79.
  • C.K. McCord, Mappings and homological properties in the Conley index theory, Ergodic Theory Dynam. Systems 8 (8*) (1988), 175–198.
  • K. Mischaikow, Conley index theory, Dynamical Systems (Montecatini Terme, 1994), Lecture Notes in Math., vol. 1609, Springer, Berlin, 1995, 119–207.
  • C. Robinson, Dynamical Systems, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, second edition, 1999. Stability, symbolic dynamics, and chaos.
  • S. Schecter, Exchange lemmas 1: Deng's lemma, J. Differential Equations 245 (2008), 392–410.
  • S. Schecter, Exchange lemmas 2: General exchange lemma, J. Differential Equations 245 (2008), 411–441.
  • S. Schecter and P. Szmolyan, Composite waves in the Dafermos regularization, J. Dynam. Differential Equations 16 (2004), 847–867.
  • S. Schecter and P. Szmolyan, Persistence of rarefactions under Dafermos regularization: blow-up and an exchange lemma for gain-of-stability turning points, SIAM J. Appl. Dyn. Systems 8 (2009), 822–853.
  • J. Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften, vol. 258, Springer, New York, second edition, 1994.
  • P. Szmolyan, Transversal heteroclinic and homoclinic orbits in singular perturbation problems, J. Differential Equations 92 (1991), 252–281.
  • S.-K. Tin, N. Kopell and C.K.R.T. Jones, Invariant manifolds and singularly perturbed boundary value problems, SIAM J. Numer. Anal. 31 (1994), 1558–1576.
  • J.B. van den Berg, J.D. Mireles-James, J.P. Lessard and K. Mischaikow, Rigorous numerics for symmetric connecting orbits: even homoclinics of the Gray–Scott equation, SIAM J. Math. Anal. 43 (2011), 1557–1594.
  • D. Wilczak, The existence of Shilnikov homoclinic orbits in the Michelson system: a computer assisted proof, Found. Comput. Math. 6 (2006), 495–535.
  • D. Wilczak, Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system, Discrete Contin. Dyn. Syst. Ser. B 11 (2009), 1039–1055.
  • D. Wilczak and P. Zgliczyński, Topological method for symmetric periodic orbits for maps with a reversing symmetry, Discrete Contin. Dyn. Syst. 17 (2007), 629–652 (electronic).
  • P. Zgliczyński, $C^1$ Lohner algorithm, Found. Comput. Math. 2 (2002), 429–465.
  • P. Zgliczyński, Covering relations, cone conditions and the stable manifold theorem, J. Differential Equations 246 (2009), 1774–1819.
  • P. Zgliczyński, Rigorous numerics for dissipative \romPDEs III. An effective algorithm for rigorous integration of dissipative \romPDEs, Topol. Methods Nonlinear Anal. 36 (2010), 197–262.
  • P. Zgliczyński and M. Gidea, Covering relations for multidimensional dynamical systems, J. Differential Equations 202 (2004), 32–58.
  • P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: the Kuramoto–Sivashinsky equation, Found. Comput. Math. 1 (2001), 255–288.