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Unformatted text preview: Chapter 1 Dynamical Systems 1.1 Introduction 1.1.1 Phase space and phase curves Dynamics is the study of motion through phase space. The phase space of a given dynamical system is described as an N-dimensional manifold, M . A (differentiable) manifold M is a topological space that is locally diffeomorphic to R N . 1 Typically in this course M will R N itself, but other common examples include the circle S 1 , the torus T 2 , the sphere S 2 , etc. Let g t : M M be a one-parameter family of transformations from M to itself, with g t =0 = 1, the identity. We call g t the t-advance mapping. It satisfies the composition rule g t g s = g t + s . (1.1) Let us choose a point M . Then we write ( t ) = g t , which also is in M . The set braceleftbig g t vextendsingle vextendsingle t R , M bracerightbig is called a phase curve . A graph of the motion ( t ) in the product space R M is called an integral curve . 1.1.2 Vector fields The velocity vector V ( ) is given by the derivative V ( ) = d dt vextendsingle vextendsingle vextendsingle vextendsingle t =0 g t . (1.2) The velocity V ( ) is an element of the tangent space to M at , abbreviated T M . If M is N-dimensional, then so is each T M (for all p ). However, M and T M may 1 A diffeomorphism F : M N is a differentiable map with a differentiable inverse. This is a special type of homeomorphism , which is a continuous map with a continuous inverse. 1 2 CHAPTER 1. DYNAMICAL SYSTEMS Figure 1.1: An example of a phase curve. differ topologically. For example, if M = S 1 , the circle, the tangent space at any point is isomorphic to R . For our purposes, we will take = ( 1 ,..., N ) to be an N-tuple, i.e. a point in R N . The equation of motion is then d dt ( t ) = V ( ( t ) ) . (1.3) Note that any N th order ODE, of the general form d N x dt N = F parenleftbigg x, dx dt ,... , d N 1 x dt N 1 parenrightbigg , (1.4) may be represented by the first order system = V ( ). To see this, define k = d k 1 x/dt k 1 , with k = 1 ,... ,N . Thus, for j &lt; N we have j = j +1 , and N = f . In other words, bracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright d dt 1 . . . N 1 N = V ( ) bracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright 2 . . . N F ( 1 ,... , N ) . (1.5) 1.1.3 Existence / uniqueness / extension theorems Theorem : Given = V ( ) and (0), if each V ( ) is a smooth vector field over some open set D M , then for (0) D the initial value problem has a solution on some finite time interval ( , + ) and the solution is unique. Furthermore, the solution has a unique extension forward or backward in time, either indefinitely or until ( t ) reaches the boundary of D ....
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- Fall '11