Abstract
Chapter one [1] 1.1. Background Differential Equation is an equation with unknown function that contains one or more derivatives of the unknown function. The order of the differential equation is the highest derivative in the equation, and the Differential equations can be classified based on the order: I- First order: - just the first derivative appear in the equation. For example II- Higher order: - derivatives higher than the first appear in the equation. For example: ( ) Differential equations can be classified as based on the number of functions that are involved. (1)-A single differential equation is a single unknown function. For example: (2)- A system of differential equations -there is more than one unknown function. For example, together with , Differential equations can be classified as based on the type of unknown function:- (a)-Ordinary - unknown function is a function in a single variable. For exemple: , etc. [2] (b)-Partial - unknown function is a function in more than one variable. For example: An ordinary differential equation ( ) ( ( )) is called non- linear iff the function (t, u) ( ) is non-linear in the second argument. To see that the solutions of the nonlinear system near the origin resemble those of the linearized system. 1.2. Introduction: [11] A nonlinear system refers to a set of nonlinear equations (algebraic, difference, differential, integral, functional, or abstract operator equations, or a combination of some of these) used to describe a physical device or process that otherwise cannot be clearly defined by a set of linear equations of any kind. Dynamical system is used as a synonym for mathematical or physical system when the describing equations represent evolution of a solution with time and, sometimes, with control inputs and/or other varying parameters as well. The theory of nonlinear dynamical systems, or nonlinear control systems if control inputs are involved, has been greatly advanced since the nineteenth century. Today, nonlinear control systems are used to describe a great variety of scientific and engineering phenomena ranging from social, life, and physical sciences to engineering and technology. This theory has been applied to a broad spectrum of problems in physics, chemistry, mathematics, biology, medicine, economics, and various engineering disciplines.