Properties of ODE systems with small parameters

The analysis of ordinary differential equation (ODE) systems with small parameters e (with 0 C 1) is generally referred to as perturbation analysis or perturbation theory. Perturbation theory has been the subject of many fundamental research contributions (Fenichel 1979, Ladde and Siljak 1983), finding applications in many areas, including linear and nonlinear control systems, fluid mechanics, and reaction engineering (see, e.g., Kokotovic et al. 1986, Kevorkian and Cole 1996, Verhulst 2005). The main concepts of perturbation theory are presented below, following closely the developments in (Kokotovic et al. 1986). 

Equation (2.1) is an ODE system, and, since the values of the variables xi and X2 at t = 0 are provided, it is an initial value problem. By employing a small perturbation parameter 0 e C 1, (2.1) and implicitly its solution are perturbed. The perturbation can occur in different manners. Consider, for example, 

Equation (2.2) is said to be a regular perturbation problem. Notice that in the limiting case, as e — 0, the regular perturbation problem reduces to the original problem (2.1). Intuitively, the solution of the regular perturbation problem should not differ significantly from that of the unperturbed problem. For example, for n = 1, m = 0, the solution of Equation (2.2) is of the form 

The solution (2.3) is known as a regular perturbation expansion, Xio(f) is the solution of the original problem (2.1), and the higher-order terms xi i(t). are determined successively by substituting the regular expansion (2.3) into the original differential equation (2.1) (Haberman 1998). 

Assuming that the fracture is at the same height as the outlet pipe, an equation for the time evolution of the tank level h can easily be written as 

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