Dynamical Systems An LTCC course Wolfram Just and Franco Vivaldi School of Mathematical Sciences Queen Mary University of London c The University of London, 2018 Last updated: January 16, 2019 These lecture notes provide a brief introduction to some topics in dynamical systems theory, suitable for a short course. 1) Dynamical systems: continuous and discrete time. 2) Linear stability analysis and beyond: hyperbolicity, structural stability. 3) Bifurcations: what happens when a state becomes unstable. 4) Symbolic dynamics: turning orbits into strings of symbols. 5) Chaos: when the future of a deterministic system cannot be determined. The presentation is informal and hands-on. The accompanying supplementary notes by F Vivaldi provide some background material in analysis and linear algebra, and also deal with one-dimensional flows, which are not part of this course. Suggestions for further reading will be found in the last section. 1 DYNAMICAL SYSTEMS 1 1 Dynamical systems Informally, a dynamical system is a system whose states are represented by the points of a set; these states evolve with time according to a deterministic law, specified by a differential or difference equation. The system is not subjected to the effects of noise1. For instance, let us consider a single particle in one-dimension, subjected to a force. The state of this system is determined by the particle’s position and velocity; the time- evolution is governed by Newton’s law F = ma, which can be written as a pair of first- order ordinary differential equations:x ˙ = y,y˙ = F(x)/m. This law determines the state at time t provided the state at time 0 is given. This is a dynamical system on the set R2; the time is a real number. Let M be any set and f : M M be any function. Choose x0 M, and, using the → ∈ first-order difference equation xt+1 = f (xt), t > 0, define recursively an infinite sequence (x0,x1,x2,...) of elements of M. This is a dynamical system on M; the time is an integer. Definition 1.1 A dynamical system consists of a set M, called the phase space, and a one-parameter family of transformations Φt : M M such that, for all x M, → ∈ i) Φ0(x)= x; ii) Φt(Φs(x)) = Φt+s(x), for all s,t > 0. If t R (respectively, t Z), then we speak of a continuous time (respectively, discrete time) ∈ ∈ dynamical system. The family Φt is a semi-group of transformations. If ii) holds also for negative t,s, then { } we have a group, and the system is said to be invertible. The set Γ(x0)= Φt(x0),t > 0 { } is called the (forward) orbit of the point x0, called the initial condition. For discrete time dynamical systems, the set M can be quite arbitrary. For continuous time, or for systems of physical origin, M is a manifold2. For continuous time, the family of functions Φt is called a flow. 1In presence of noise, we speak of a stochastic dynamical system. 2A set that, locally, looks like Rn for some n, see page 16. 1 DYNAMICAL SYSTEMS 2 1.1 Flows We begin with some examples of continuous-time dynamical systems. Example 1.1 The harmonic oscillator. This is one of the simplest dynamical system of mathematical physics, yet a very significant one. The phase space is M = R2; the basic information is displayed below3. equations initial condition solution y x˙ = y x(0) = x x(t) = x cos(ωt)+ 0 sin(ωt) 0 0 ω 2 y˙ = ω x y(0) = y0 y(t) = ωx0 sin(ωt)+ y0 cos(ωt) − − with Φt(x0,y0)=(x(t,x0,y0),y(t,x0,y0)). The phase space space foliates into invariant ellipses. The figure on the right illustrates the geometrical meaning of the associative property ii) in definition 1.1. Φ 0 s y (x ,y ) 0 0 s Φ x t Φ s+t (x(t),y(t)) s+t The definition of a dynamical systems requires that, for any initial condition in phase space, the orbit must be unique and defined for all future times. This is not guaranteed for the solutions of a differential equationx ˙ = f (x), and so there are differential equations which cannot be regarded as dynamical systems. 3The dot denotes differentiation with respect to time. 1 DYNAMICAL SYSTEMS 3 Example 1.2 The solutions are not defined for all times. The system 2 x˙ = x x(0)= x0 (1) has solution x0 x(t,x0)= . 1 x0t − For x0 = 0, this function is not defined over the whole R. It is defined in the interval 6 ( ∞,1/x0) for x0 > 0, and (1/x0,∞) for x0 < 0. This is because the real function f (x)= x˙ − grows too rapidly near infinity, so that it is possible to reach infinity, or come back from infinity, in a finite time. Example 1.3 Lack of uniqueness. The differential equation x˙ = 2 x , with x(0)= 0 | | has two distinct solutions for t > 0,p namely x(t)= 0 and x(t)= t2. This is because the function f (x)= x˙ is not Lipschitz continuous at the equilibrium point x = 0 (the constant solution), and hence it does not vanish rapidly enough near it. So it is possible to reach the equilibrium point, or move away from it, in a finite time. To ensure uniqueness of solutions, it suffices to require that f be of class C1. Example 1.4 Forced van der Pol oscillator. x˙ = x σy x(x2 + y2) − − (2) y˙ = σx + y y(x2 + y2) γ. − − 1.5 For any value of the real parameters σ,γ, this equation defines a vector field v(z)= z˙ = 1.0 2 (x˙,y˙) in R . Plotting this vector field (here 0.5 for σ = 0.2 and γ = 0.3) gives already an 0.0 impression of the behaviour of the solutions. Dynamical systems theory is about under- -0.5 standing how the flow relates to the vector -1.0 field. -1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 1 DYNAMICAL SYSTEMS 4 For special parameter values we are able to compute the flow analytically. If γ = 0, then using polar coordinates x = r cos(ϕ), y = r sin(ϕ), the equations of motion read r˙ = r r3, ϕ˙ = σ . (3) − The special solution r(t)= 0 yields the constant solution z(t)=(x(t),y(t))=(0,0) (an equilibrium point, or fixed point), while r(t)= 1, ϕ(t)= σt +ϕ(0) results in the harmonic solution z(t)=(cos(σt + ϕ(0)),sin(σt + ϕ(0)). Let T = 2π/σ. Then this solution sat- isfies z(t)= z(t + T) for all t, and we call it a periodic orbit (or a cycle), with period T. A periodic orbit of a flow is a closed non-self intersecting curve in phase space. In the present example, if σ = 0, then this periodic orbit is a so-called limit cycle, since it is 6 approached by nearby orbits —see below. For γ = 0, it is even possible to write down the general solution. The long-time behaviour of the orbits of one- and two-dimensional flows is well- understood. In one dimension, the solutions are either constant, or strictly monotonic. Indeed letx ˙ = f (x) be a smooth differential equation over R, and let x = x(t) be a non- constant solution (i.e.,x ˙(t)= f (x(t)) and f (x0) = 0). Then separation of variables yields x(t) 6 x(0) dx/ f (x)= t. Since the integral on the left-hand side is bounded (and smooth in t), we have f (x) = 0 for x [x(0),x(t)]. Because t is arbitrary, f (x(t)) does not change sign. R 6 ∈ Therefore the solution is monotonic, and hence it is either unbounded or tends to a finite limit x(t) x . It then follows by continuity that x(t)= x is a constant solution of the → ∗ ∗ differential equation. In higher dimensions, limit sets of flows can be much more complicated than fixed points, and we need a machinery to characterise this form of convergence. Let Γ(z0)= Φt(z0) : t R be the orbit of z0. { ∈ } A point w is an ω-limit point of the orbit Γ(z0) if there is a sequence (tk) of times such that lim tk = ∞ and lim Φt (z0)= w. (4) j ∞ j ∞ k → → In other words, w is an ω-limit point if for any neighbourhood U of w there is a (smallest) time t = t (U) such that Φt (z0) U. ∗ ∗ ∗ ∈ The set of all ω-limit points of Γ(z0) is called the ω-limit set of Γ(z0), denoted by ω(z0) [or ω(Γ), since this set is a property of the orbit]. By reversing the direction of time, and replacing ∞ by ∞, we obtain the analogous concept of α-limit set. − 1 DYNAMICAL SYSTEMS 5 The simplest situation occurs when limt ∞ Φt(z0)= z , where z is a fixed point. → ∗ ∗ Then, by letting tk = k and w = z in (4), we find that z ω(z0). Since there cannot be ∗ ∗ ∈ any other ω-limit point (why?), we have ω(z0)= z . { ∗} Another possibility is that ω(z0)= α(z0)= z , with z0 = z , that is, the orbit of z0 ∗ 6 ∗ approaches z in both time directions. We then say that z0 is a homoclinic point, and that ∗ Γ(z0) is a homoclinic orbit. The point z0 is heteroclinic if ω(z0)= z , and α(z0)= w ∗ ∗ where w is an equilibrium distinct from z .