The following result of Fenichel (see [19]) on the preservation of normally hyperbolic invariant manifolds plays a central role in geometric singular perturbation theory.

Fenichel, "Geometric singular perturbation theory for ordinary differential equations," Journal of Differential Equations, vol.

Lozano, "Lyapunov-based controller using

singular perturbation theory: An application on a mini-UAV," in Proceedings of the 20131st American Control Conference, ACC 2013, pp.

In this section, based on the singular perturbation theory, the two-timescale approach is proposed to separate the full-system into the fast and slow subsystems, providing a way for analyzing the interaction of the two-timescale dynamics.

By the proposed two-timescale approach based on singular perturbation theory, the state variables are separated into fast and slow variables.

Based on the

singular perturbation theory, the slow subsystem of the closed-loop system (16) is

Via a blow up we establish a bridge between non-smooth dynamical systems and the geometric singular perturbation theory. This paper deals almost exclusively with the critical (or singular) dynamics, namely the limit r = 0 in a singular perturbation of the form r[??] = a(x, y, r), [??] = b(x, y, r), except for giving regularized vector fields in a form that allows them to be analyzed in the nonsingular limit.

The paper is organized as follows: in Section 2 we give the basic theory about Non-Smooth Vector Fields on the plane, in Section 3 we give the theory about the regularization process, in Section 4 we present a few relevant methods of Geometric Singular Perturbation Theory, in Section 5 we present the singularities treated in Theorem 1, give its topological normal forms and prove Theorem 1, in Section 6 we present the singularities treated in Theorem 2, give its topological normal forms and prove Theorem 2.

Narang-Siddarth and Valasek describe control design techniques that extend

singular perturbation theory to a larger class of systems than previously, specifically systems in which different processes run at different time scales.

Nearly twenty years later, Szmolyan and Wechselberger [5] extended "Geometric

Singular Perturbation Theory (see Fenichel [6, 7], O'Malley [8], Jones [9], and Kaper [10])" to canards problems in [R.sup.3] and provided a "standard version" of Benolt's theorem [2].