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CH01_DYNSYS

# CH01_DYNSYS - Chapter 1 Dynamical Systems 1.1 1.1.1...

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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 ϕ 0 ∈ M . Then we write ϕ ( t ) = g t ϕ 0 , which also is in M . The set braceleftbig g t ϕ 0 vextendsingle vextendsingle t R , ϕ 0 ∈ M bracerightbig is called a phase curve . A graph of the motion ϕ ( t ) in the product space R × M is called an integral curve . 1.1.2 Vector fields The velocity vector V ( ϕ ) is given by the derivative V ( ϕ ) = d dt vextendsingle vextendsingle vextendsingle vextendsingle t =0 g t ϕ . (1.2) The velocity V ( ϕ ) is an element of the tangent space to M at ϕ , abbreviated T M ϕ . If M is N -dimensional, then so is each T M ϕ (for all p ). However, M and T M ϕ 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

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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 ϕ = ( ϕ 1 ,...,ϕ N ) to be an N -tuple, i.e. a point in R N . The equation of motion is then d dt ϕ ( t ) = V ( ϕ ( 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 ˙ ϕ = V ( ϕ ). To see this, define ϕ k = d k 1 x/dt k 1 , with k = 1 ,...,N . Thus, for j < N we have ˙ ϕ j = ϕ j +1 , and ˙ ϕ N = f . In other words, ˙ ϕ bracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright d dt ϕ 1 . . . ϕ N 1 ϕ N = V ( ϕ ) bracehtipdownleft bracehtipuprightbracehtipupleft bracehtipdownright ϕ 2 . . . ϕ N F ( ϕ 1 ,...,ϕ N ) . (1.5) 1.1.3 Existence / uniqueness / extension theorems Theorem : Given ˙ ϕ = V ( ϕ ) and ϕ (0), if each V ( ϕ ) is a smooth vector field over some open set D ∈ M , then for ϕ (0) ∈ D the initial value problem has a solution on some finite time interval ( τ, + τ ) and the solution is unique. Furthermore, the solution has a unique extension forward or backward in time, either indefinitely or until ϕ ( t ) reaches the boundary of D . Corollary : Different trajectories never intersect!
1.1. INTRODUCTION 3 1.1.4 Linear differential equations A homogeneous linear N th order ODE, d N x dt N + c N 1 d N 1 x dt N 1 + ... + c 1 dx dt + c 0 x = 0 (1.6) may be written in matrix form, as d dt ϕ 1 ϕ 2 .

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