Lecture notes for Introduction to Dynamical Systems: CM131A Based on notes by A. Annibale, R. K¨uhnand H.C. Rae 2018-2019 Lecturer: G. Watts 1 Contents 1 Overview of the course 4 1.1 Revision Exercises . .9 I Differential Equations 10 2 First order Differential Equations 11 2.1 Basic Ideas . 11 2.2 First order differential equations . 12 2.3 General solution of specific equations . 13 2.3.1 First order, explicit . 13 2.3.2 First order, variables separable . 14 2.3.3 First order, linear . 16 2.3.4 First order, homogeneous . 18 2.4 Initial value problems . 20 2.5 Existence and Uniqueness of Solutions | Picard's theorem . 23 2.5.1 Picard iterates . 24 2.6 Exercises . 28 3 Second order Differential Equations 31 3.1 Second order differential equations . 31 3.1.1 Second order linear, with constant coefficients . 32 3.2 Existence and Uniqueness of Solutions | Picard's theorem . 38 3.3 Exercises . 41 II Dynamical Systems 44 4 Introduction to Dynamical Systems 45 Version of Mar 14, 2019 2 5 First Order Autonomous Systems 50 5.1 Trajectories, orbits and phase portraits . 50 5.2 Termination of Motion . 55 5.3 Estimating times of Motion . 60 5.4 Stability | A More General Discussion . 66 5.4.1 Stability of Fixed Points . 66 5.4.2 Structural Stability . 68 5.4.3 Stability of Motion . 71 5.5 Asymptotic Analysis . 72 5.5.1 Asymptotic Analysis and Dynamical Systems . 74 5.6 Exercises . 80 6 Second Order Autonomous Systems 85 6.1 Phase Space and Phase Portraits . 85 6.2 Separable Systems . 95 6.3 The structure of orbits and Phase Space . 98 6.4 Limit cycles . 99 6.5 Fixed points of second-order autonomous systems . 101 6.6 Linear Stability Analysis . 101 6.6.1 Step 1 | Taylor Expansion of Velocity Functions . 102 6.6.2 Step 2 | Finding the Jordan canonical form of the Jacobian . 104 6.6.3 Step 4 | Exploring the Consequences for Dynamics . 105 6.7 Beyond linear stability analysis . 119 6.8 Exercises . 120 III Application to Classical Mechanics 132 7 Elements of Newtonian Mechanics 133 7.1 Motion of a particle . 133 7.2 Newton's Laws of motion . 138 7.2.1 Newton's First Law (N1) . 138 7.2.2 Newton's Second Law (N2) . 138 7.2.3 Newton's Third Law (N3) . 138 7.3 Newton's Law of Gravitation . 140 7.4 Motion in a Straight Line; the Energy Equation . 143 Version of Mar 14, 2019 3 7.5 Equilibrium and Stability . 147 7.6 Exercises . 153 8 Hamiltonian Systems 165 8.1 Hamilton's equations for motion in a potential . 166 8.2 Stability problems . 173 8.3 Summary: how to analyse motion in a potential . 176 8.4 Exercises . 181 A Functions of two variables 187 A.1 The partial derivative . 187 A.2 Continuity of a function of two variables . 189 B Taylor's Theorem 190 B.1 Taylor Expansion for Functions of One Variable . 190 B.2 Taylor Expansion for Functions of two variables . 191 B.3 Vector functions . 192 C Basic Linear Algebra 195 C.1 Fundamental ideas . 195 C.2 Invariance of Eigenvalues Under Similarity Transformations . 197 C.3 Jordan Forms . 197 C.4 Basis Transformation . 199 C.5 Rotations . 200 C.6 Area Preserving Transformations . 200 C.7 Examples and Exercises . 201 D Jacobians and Change of variables 205 D.1 Change of variables . 205 Version of Mar 14, 2019 Chapter 1: Overview of the course 4 Chapter 1 Overview of the course This course is about the study of quantities which vary in time. The quantities are meant to represent some \system" and the variation in time is called the \dynamics", hence the name. These ideas can be very general - the quantities could take real values, such as the coordinates of bodies in a classical mechanics problems, they could take integer values such as the number of individuals in a population - and the variation in time can be continuous (as the position of a body is defined for all values of t) or discrete (maybe we measure the population only once each day). The study of variations of systems with time obviously has gone on for a long time, but the development of the branch of mathematics known as \Dynamical Systems" really started in the 1890s with Poincar´e,inspired by the problems of the motion of planets. It is partly pure mathematics and partly applied and that is reflected in the way this course is organised: there are some theorems, some study of properties of differential equations and some applications. To give a flavour of what's to come we'll look at three models of population size and one physical system which are all typical examples. The simplest model of a population is that of Thomas Malthus who proposed (in 1798) that if there are no constraints on the resources available then the rate of growth of a population will be proportional to its size - the number of births in any time period will be proportional to the number of people. If the size of the population is x and the birth rate is r per unit time then this can be modelled by the differential equations dx = r x : (1.1) dt The solutions of this equation are r t x(t) = x0 e : (1.2) This exponential growth of course leads to problems, as Malthus knew, and so is not a good model for long term population sizes. A better model is the logistic equation (devised by Pierre-Fran¸coisVerhulst in 1844) in which limits of resources mean there is a maximum size c for a stable population: dx x = f(x) ; f(x) = r x 1 − ; (1.3) dt c Version of Mar 14, 2019 Chapter 1: Overview of the course 5 for which we can also find the general solution as the function x(t) c c x = −r t ; where A = 1 − (1.4) 1 − Ae x0 The exponential solution (1.2) is one we are familiar with, the solutions (1.4) may be harder to visualise. One of the aims of the course is to see how to find out properties of the dynamics of a model { exact or approximate { using graphical methods. The simplest method is if one can find the exact solutions and then plot these, so for the exponential growth model we can plot x vs. t or t vs. x, as in figure 1.1 x 1.0 t 4 0.5 2 -2 2 4 t -1.0 -0.5 0.5 1.0 x -0.5 -2 -1.0 Figure 1.1: Sketches of the solutions (1.2) for r = 0:1 together with the phase portrait showing the fixed point at x = 0. Also included is a useful guide to the way the system behaves, known as the \phase portrait" which shows whether x increases or decreases with t and also that x = 0 is a “fixed point". For the more complicated Verhulst model, the corresponding plots are in figure 1.2 x t 2.0 4 1.5 2 1.0 0.5 -1.0 -0.5 0.5 1.0 1.5 2.0 x -4 -2 2 4 t -2 -0.5 -1.0 -4 Figure 1.2: Sketches of the solutions (1.4) for r = 0:1 and c = 1 together with the phase portrait showing the fixed points at x = 0 and x = 1. In both these cases we can solve the equations exactly but the behaviour of the solutions is captured in the phase portrait showing the way x changes with t and the fixed points. Version of Mar 14, 2019 Chapter 1: Overview of the course 6 These models can be generalised to a system with two species, a predator and a prey, where simple models for their interactions lead to two coupled equations dx dy = x(a − by) ; = y(cx − d) : (1.5) dt dt This is the Lotka-Volterra model, also called the hare-lynx or rabbit-fox model for two situ- ations it describes. This is a case where we cannot find an exact solution in general; We can only find exact solutions if x = 0 (no predators), if y = 0 (no prey) or if the populations take the constant values x = d=c; y = a=b. However, we can study other aspects of the system - we can sketch the solutions (technically, sketch the phase curves) and see that there are going to be solutions that stay close to the fixed point at (d=c; a=b), that these will have periodic oscillations and we can find the approximate period of these oscillations. There is another fixed point at (0; 0) which is of a different nature; it is not stable, solutions close to this point do not have to stay close and we can again analyse motion near here. 2.0 1.5 1.0 Predator 0.5 0.0 0 1 2 3 4 5 6 Prey Figure 1.3: Sketch of the time-evolution of prey (x-axis) and predator (y- axis) populations in the case a = 1; b = 2; c = 1; d = 2. The fixed point at (2; 1=2) is the isolated dot, and typical time evolution is a periodic orbit around the fixed point. The first two examples are completely solved using the theory of differential equations; this third example takes us into the study of dynamical systems proper where we pay attention to structural properties of the solutions without necessarily solving them.