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140IntroductoryNotes

140IntroductoryNotes - Page 1 1 Basic Theory of Dynamical...

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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 di ff erential equa- tions and partial di ff erential equations. In these notes we shall focus on the case of ordinary di ff erential equations (ODE), and start o ff thinking of these equations in Euclidean space R n as 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. For now, lets assume that f is smooth. One is to find solutions x ( t ) to this equation satisfying some initial conditions, say x ( t 0 ) 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.
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2 1. Basic Theory of Dynamical Systems A Simple Example. Let us start o ff 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.
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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 where the right hand side of equation ( 1.1.2 ) vanish. Finding and analyzing such points is a useful thing to do. Notice that if x 0 is such a point, then the solution curve through that point is simply the constant curve x
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