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Computing slow manifolds of saddle type. (English) Zbl 1176.37026
Slow-fast vector fields have the form
\[ \varepsilon \dot{x}=f(x,y,\varepsilon),\qquad \dot{y}=g(x,y,\varepsilon), \tag{1} \]
with \(x\in \mathbb{R}^m\) the vector of fast variables, \(y\in \mathbb{R}^n\) the vector of slow variables, and \(\varepsilon\) a small parameter that represents the ratio of time scales. The authors propose an algorithm for computing trajectories on slow manifolds that are normally hyperbolic with both stable and unstable fast manifolds and give an estimate of the accuracy order of the algorithm, augmented by the analysis of a linear system for which there are explicit solutions of both the solutions of the differential equations and the boundary value solver.
Numerical investigations of three examples are presented:
1.
a three-dimensional version of the Morris-Lecar model for bursting neurons that was used by D. Terman [SIAM J. Appl. Math. 51, No. 5, 1418–1450 (1991; Zbl 0754.58026)];
2.
a three-dimensional system whose homoclinic orbits yield travelling-wave profiles for the FitzHugh-Nagumo model [J. Guckenheimer and C. Kuehn, Discrete and Continuous Pynamical Systems, Ser. S (in print)];
3.
a four-dimensional model of two coupled neurons studied by J. Guckenheimer, K. Hoffman and W. Weckesser [Int. J. Bifurcation Chaos Appl. Sci. Eng. 10, No. 12, 2669–2687 (2000; Zbl 0978.34038)].

MSC:
37M20 Computational methods for bifurcation problems in dynamical systems
34E17 Canard solutions to ordinary differential equations
34C26 Relaxation oscillations for ordinary differential equations
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