12.006J F2022 Lecture 19: Introduction to Strange Attractors

Lecture notes for 12.006J/18.353J/2.050J, Nonlinear Dynamics: Chaos D. H. Rothman, MIT October 31, 2022 Contents 1 Introduction to strange attractors 1.1 Dissipation and attraction . . . . 1.2 Attractors with d = 2 . . . . . . . 1.3 Aperiodic attractors . . . . . . . 1.4 Example: Rössler attractor . . . . 1.5 Conclusion . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 3 5 7 11 Introduction to strange attractors References: Bergé et al. [1], Strogatz [2], Abraham and Shaw [3] Thus far, we have studied classical attractors: fixed points and limit cycles. In this lecture we begin our study of strange attractors. We emphasize their generic features. 1.1 Dissipation and attraction Our studies of oscillators have revealed explicitly how forced systems can reach a stationary (yet dynamic) state characterized by an energy balance: average energy supplied = average energy dissipated An example is a limit cycle: 1 θ θ Initital conditions inside or outside the limit cycle always evolve to the limit cycle. Limit cycles are a specific way in which dissipation ⇒ attraction. More generally, we have an n-dimensional flow d ~x(t) = F~ [~x(t)], dt ~x ∈ Rn (1) Assume that the flow ~x(t) is dissipative, with attractor A. Properties of the attractor A: • A is invariant with flow (i.e., it does not change with time). • A is contained within B, the basin of attraction. B is that part of phase space from which all initial conditions lead to A as t → ∞: A B • A has dimension d < n. Consider, for example, the case of a limit cycle: 2 θ Γ θ The surface Γ is reduced by the flow to a line segment on the limit cycle (the attractor). Here d = attractor dimension = 1 n = phase-space dimension = 2. This phenomenon is called reduction of dimensionality. Consequence: loss of information on initial conditions. We have already quantified volume contraction. Given an initial volume V evolving according to the flow (1), the Lie derivative tells us that V changes as n X ∂ ẋi 1 dV = ∇ · ~x˙ = V dt ∂xi i As we showed earlier, dissipation yields volume contraction; i.e., dV < 0. dt Consequently, the attractor cannot have n-dimensional volumes, so d < n. What, then, is the dimension of the attractor? We proceed by example, by considering the case d = 2. 1.2 Attractors with d = 2 What happens when d (the dimension of the attractor) is 2? 3 Assume a quasiperiodic attractor on a torus T 2 : ω2 ω1 C Cut the torus on a small circle C and open it: B A B’ A’ Finally, cut the long way, from A to A0 , and open it again: ω2 t 2π A A’ B B’ ω1 t 2π Note the parallel trajectories. As usual, the quasiperiodic flows are characterized by two cases: ω1 /ω2 rational or irrational. • Rational. Consider, e.g., ω1 /ω2 = 1/3: ω2 t 2π ’ 2π 4 ω1 t The trajectory repeats itself exactly every three times around the 2-axis, or each time around the 1-axis. • Irrational. ω2 t 2π ’ ω1 t 2π The trajectories densely fill the plane. Determinism forbids non-parallel trajectories, because they would cross: ω2 t 2π ’ ω1 t 2π Thus a torus T 2 can only be a periodic or quasiperiodic attractor. The attractor cannot be aperiodic if d = 2. 1.3 Aperiodic attractors We have already shown that the power spectrum of an aperiodic signal x(t) is continuous: xk 2 k 5 And the autocorrelation Ψm = hxj xj+m i has finite width: Ψm m The finite width of Ψm implies that knowledge of no finite interval of x(t) allows prediction of all future x(t). This “unpredictability” is associated with what we call “chaos.” We seek, however, a more precise definition of chaos. On an aperiodic attractor, small differences in initial conditions on the attractor lead at later times to large differences, still on the attractor. In phase space, trajectories on an aperiodic attractor can diverge, e.g., We shall see that the divergence of trajectories is exponential in time. This phenomenon is called sensitivity to initial conditions (SIC). It definitively identifies chaos, i.e., a chaotic attractor. Note that, despite the precision of this definition, we are left with an apparent conundrum: simultaneously we have • attraction, such that trajectories converge. • sensitivity to initial conditions, such that trajectories diverge. The conundrum is solved by noting that trajectories converge to the attractor, 6 but diverge on the attractor. Note further that divergence on the attractor implies that the attractor dimension d > 2, since phase tractories cannot diverge in two dimensions. Thus we conclude that an aperiodic (chaotic) attractor must have phase space dimension n ≥ 3. Assume n = 3. How may trajectories converge, but still remain bounded on an attractor? The trajectories are successively stretched (by SIC) and folded (thus remaining bounded). 1.4 Example: Rössler attractor We use drawings [3] of the Rössler attractor to illustrate how this works: Abraham & Shaw [3] See image credit on Page 12. • Trajectories diverge in plane by spiralling out (stretching). • Trajectories leave plane. • Trajectories return to plane (folding), back to center of spiral. 7 At the same time, we must have volume contraction. One dimension can expand while another contracts: Abraham & Shaw [3] See image credit on Page 12. Let’s consider the stretching and folding in more detail. The Rössler attractor reads ẋ = −y − z ẏ = x + ay ż = b + z(x − c) where we assume a > 0. Assume z and ż are small. Then in the x, y plane the system is approximated by ẋ = −y ẏ = x + ay. Then ẍ = −ẏ = −x + aẋ yielding the negatively damped oscillator ẍ − aẋ + x = 0. Consequently the trajectories spiral out of the origin. How is the spreading confined? From the equation for ż, we see that, for small b, x < c ⇒ ż < 0 x > c ⇒ ż > 0 Thus we expect trajectories to behave as follows: 8 • Divergence from the origin creates x > c. • x > c ⇒ z increases ⇒ x decreases. • Eventually x decreases such that x < c. • Then x < c ⇒ z decreases ⇒ back in the x, y plane. • The process repeats. Thus we have • stretching, from the outward spiral; and • folding, from the feedback of z into x. A sequence of figures shows how endless divergence occurs in bounded space. First, let’s zoom in on the stretching: Abraham & Shaw [3] See image credit on Page 12. Then the attractor folds: Abraham & Shaw [3] See image credit on Page 12. Trajectories never close exactly as a surface, but more like filo dough: 9 Abraham & Shaw [3] See image credit on Page 12. Another iteration of stretching and folding looks like this: Abraham & Shaw [3] See image credit on Page 12. And again: Abraham & Shaw [3] 10 See image credit on Page 12. Each time we stretch and fold, we create layers of layers. And when we slice it with a Poincar/’e section, we would see gaps within gaps, and thus a fractal object. Abraham & Shaw [3] See image credit on Page 12. We’ll say more about this later, after we describe its earliest observation in the Lorenz attractor. 1.5 Conclusion We arrive at the following conclusions: • Aperiodic attractors must have d > 2. • Since dissipation contracts volumes, d < n, where n is the dimension of the phase space. • Suppose n = 3. Then a chaotic attractor must have 2 < d < 3. 11 How can 2 < d < 3? The attractor has a fractional, or fractal dimension. We shall look more closely at this later. For now, we conclude that chaotic attractors have three properties: • Attraction • Sensitivity to initial conditions • Non-integer fractal dimension The combination of these three properties defines a strange attractor. The “strangeness” arises not so much from each individual property but their combined presence. Next we study the most celebrated strange attractor—the Lorenz attractor. References 1. Bergé, P., Pomeau, Y. & Vidal, C. Order within Chaos: Towards a Deterministic Approach to Turbulence (John Wiley and Sons, New York, 1984). 2. Strogatz, S. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (CRC Press, 2018). 3. Abraham, R. H. & Shaw, C. D. Dynamics-The Geometry of Behavior: Part 2: Chaotic Behavior (Aerial Press, Incorporated, 1984). Image Credit Images on Pages 7–11 from Abraham, R. H. & Shaw, C. D. Dynamics-The Geometry of Behavior: Part 2: Chaotic Behavior (Aerial Press, Incorporated, 1984). © Aerial Press. All rights reserved. 12