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

Abstract

We provide a rigorous numerical computation method to validate periodic, homoclinic and heteroclinic orbits as the continuation of singular limit orbits for the fast-slow system x = f (x, y, ), y = g(x, y, ) with one-dimensional slow variable y. Our validation procedure is based on topological tools called isolating blocks, cone condition 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 > 0 but all in a given half-open interval (0, 0]. Several sample verification examples are shown as a demonstration of applicability. Kaname MatsueRigorous Numerics for Fast-Slow Systems and the dynamics of the reduced problem 0 = f (x, y, 0), y = g(x, y, 0), (1.4) which are the limiting problems for = 0 on the fast and the slow time scale, respectively. Notice that (1.4) makes sense only on f (x, y, 0) = 0, while (1.3) makes sense in whole R n+1 as the yparameter family of x-systems. The meaning of the " → 0-limit" is thus different between (1.1) and (1.2). This is why (1.1) or (1.2) is a kind of singular perturbation problems. In particular, (1.1) or (1.2) is known as fast-slow systems (or slow-fast systems), where x dominates the behavior in the fast time scale and y dominates the behavior in the slow time scale. When we study the dynamical system of the form (1.1), we often consider limit systems (1.3) and (1.4) independently at first. Then ones try to match them in an appropriate way to obtain trajectories for the full system (1.1). One of major methods for completely solving singularly perturbed systems like (1.1) is the geometric singular perturbation theory formulated by Fenichel [10], Jones-Kopell [20], Szmolyan [32] and many researchers. A series of theories are established so that formally constructed singular limit orbits of (1.3) and (1.4) can perturb to true orbits of (1.1) for sufficiently small > 0. In geometric singular perturbation theory, there are mainly two key points to consider. One is the description of slow dynamics for sufficiently small near the nullcline {(x, y) | f (x, y, 0) = 0}. The other is the matching of fast and slow dynamics. As for the former, Fenichel [10] provided the Invariant Manifold Theorem for describing the dynamics on and around locally invariant manifolds, called slow manifolds, for sufficiently small > 0. Such manifolds can be realized as the perturbation of normally hyperbolic invariant manifolds at = 0, which are often given by submanifolds of nullcline in (1.3). As for the latter, Jones and Kopell [20] originally formulated the geometric answer for the matching problem deriving Exchange Lemma. This lemma infor...

Numerical validation of blow-up solutions of ordinary differential equations

“…A series of facts concerning compactifications leads to the translation of blow-up solutions (and grow-up solutions) in terms of dynamical systems as global solutions that are asymptotic to equilibria of (2.7) on ∂D. This translation presents the possibility of validating these solutions with computer assistance, as in previous studies such as [21,25,30]. However, note that the above results do not yet distinguish blow-up solutions from grow-up solutions.…”

“…Step 2 can be performed using standard numerical methods, such as the (interval) Newton or the Krawczyk methods, or topological methods such as isolating blocks (see, e.g., [21,22]). The validity of Step 3 stems from the dynamical properties of (2.7) on D. Proposition 2.6 indicates that a solution of (1.1) that diverges in specific direction x * corresponds to the solution of (2.7) that tends to the critical point at infinity x * .…”

“…In the standard theory of dynamical systems, such solutions are considered to be branches of connecting orbits. There are many existing results for validating connecting orbits with computer assistance (see, e.g., [21,25,30]), and hence the statement in Step 3 can be reasonably executed.…”

“…Thus, we can consider a divergent solution as a global solution that is asymptotic to a critical point at infinity. This global solution can be validated using standard approaches in the theory of dynamical systems (see, e.g., [21,25,30]). The second instrument we employ is the validation of Lyapunov functions.…”

“…Interval arithmetic enables us to compute enclosures in which mathematically correct objects are contained. In dynamical systems theory, there are many applications for interval arithmetic, including validations of global trajectories, determining stability of invariant sets, and determining parameter ranges where dynamical bifurcations occur (see, e.g., [3,6,10,21,22,23,25,30,32]). In our case, affine arithmetic [17,18], an enhanced version of interval arithmetic, is applied to validate explicit enclosures of blow-up solutions and their blow-up times.…”