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

DOI: http://dx.doi.org/10.12775/TMNA.2016.072

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#### 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 PDEs-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 PDEs III. An effective algorithm for rigorous integration of dissipative PDEs, 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.

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