The following result of Fenichel (see ) 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  extended "Geometric Singular Perturbation Theory
(see Fenichel [6, 7], O'Malley , Jones , and Kaper )" to canards problems in [R.sup.3] and provided a "standard version" of Benolt's theorem .