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Unformatted text preview: Page 1 1 Basic Theory of Dynamical Systems 1.1 Introduction and Basic Examples Dynamical systems is concerned with both quantitative and qualitative properties of evolution equations, which are often ordinary differential equa tions and partial differential equations. In these notes we shall focus on the case of ordinary differential equations (ODE), and start off dealing with such equations in Euclidean space R n . They are equations of the form ˙ x = f ( x,t ) (1.1.1) where f is a map of an open set in R n × R to R n with some regularity properties to be examined as we develop the theory a bit more in depth. For now, we assume that f is smooth. A goal is to find solutions x ( t ) to this equation satisfying some initial conditions, say x ( t ) = x is given. To further simplify things, let us assume for now that f is autonomous ; that is, f does not depend explicitly on t . Then the equation becomes ˙ x = f ( x ) (1.1.2) However, in many examples, f can depend on parameters. We shall see concrete examples of this as we proceed. If we denote these parameters by μ ∈ R p , then equation ( 1.1.1 ) becomes ˙ x = f ( x,μ ) (1.1.3) and we think of solving this equation for each fixed μ and then consider how things change as μ varies. 2 1. Basic Theory of Dynamical Systems A Simple Example. Let us start off by examining a simple system that is mechanical in nature. We will have much more to say about examples of this sort later on. Basic mechanical examples are often grounded in New ton’s law, F = ma . For now, we can think of a as simply the acceleration, given in R n by a = ¨ x , the second time derivative. 1 Often the forces F are derived from a potential; namely F ( x ) =∇ V ( x ) for some real valued function V , the potential energy . In this case, the equations take the form m ¨ x =∇ V ( x ) . (1.1.4) Here is a simple example in one dimension; choosing V ( x ) = 1 2 x 2 + 1 4 x 4 and m = 1, we get the equation ¨ x = x x 3 Intuitively, think of a particle moving in the potential field given by V , as in Figure 1.1.1 . Figure 1.1.1. Particle at position x on the line, moving in a potential field V . To analyze this system, some basic observations are useful. First of all, we can put the equation in first order form ( 1.1.2 ) by introducing the velocity v as a separate variable, so what we called x before becomes the pair ( x,v ): ˙ x = v ˙ v = x x 3 (1.1.5) Now let us now pause for a basic observation: 1 As we shall see later, the acceleration does not take this simple form in coordinates more general than Euclidean coordinates and also is more subtle when one is considering systems that are in motion, such as a pendulum on a rotating Earth. 1.1 Introduction and Basic Examples 3 First Order Form and Equilibria. Suppose we have an equation of the form ( 1.1.4 ). We first write that equation in first order form as we did in the preceding example: ˙ x = v ˙ v = 1 m (∇ V ( x )) (1.1.6) By definition, equilibrium points of an autonomous system are points...
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This note was uploaded on 01/04/2012 for the course CDS 140A taught by Professor Marsden during the Fall '09 term at Caltech.
 Fall '09
 Marsden

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