Stability Boundary Characterization of Nonlinear Autonomous Dynamical Systems in the Presence of Saddle-Node Equilibrium Points

Abstract

A dynamical characterization of the stability boundary for a fairly large class of nonlinear autonomous dynamical systems is developed in this paper. This characterization generalizes the existing results by allowing the existence of saddlenode equilibrium points on the stability boundary. The stability boundary of an asymptotically stable equilibrium point is shown to consist of the stable manifolds of the hyperbolic equilibrium points on the stability boundary and the stable, stable center and center manifolds of the saddle-node equilibrium points on the stability boundary.

Characteristics of Stability Boundary of Nonlinear Continuous Dynamical Systems

“…x  is a vector of state variables. It is natural to assume that the function : nn f  satisfies some sufficient conditions for the existence and uniqueness of solution to (1). We now give some necessary definitions before exploring the main results in Section 3.…”

“…It is well-known that many nonlinear physical and engineering systems are designed to be operated at an equilibrium state [1][2][3][4]. In other words, the authors are constructed to be operated at an equilibrium point and are described by a nonlinear dynamical system.…”

“…Lemma 5 ([2]). Let i x and j x be hyperbolic critical elements of the nonlinear dynamical system (1). Assume that the intersection of the stable and unstable manifolds of , ij xx satisfies the transversality condition and { ( ) } { ( ) } .…”

“…M Theorem 9 ([8]). Let () s Ax be the stability region of a stable equilibrium point s x of nonlinear dynamical system (1) B or converge to (1,2). Thus, assumption (A3) holds.…”

“…The stability region of the stable equilibrium point (0,0) is the one to the left and below the bold red line. On the other hand, the bold red line and the bold blue line are the stable and unstable manifolds of the equilibrium point of type-one (1,2), respectively. It is easy to see that there is a part of unstable manifold of (1,2) that lies in the stability region of (0,0) and the stable manifold of (1,2) is the stability boundary of (0, 0).…”