Lecture notes #9 Nonlinear Dynamics YFX1520 Lecture 9: Attractor and strange attractor, chaos, analysis of Lorenz attractor, transient and intermit- tent chaos, Lyapunov exponents, Kolmogorov entropy, predictability horizon, examples of chaos, cobweb dia- gram and recurrence map or recurrence relation (1-D map) Contents 1 Lorenz attractor 2 2 Exponential divergence of neighbouring trajectories 3 2.1 Lyapunov exponents . .3 2.2 Kolmogorov entropy . .4 2.3 Predictability horizon . .4 3 Conceptual definition of chaos and strange attractor 6 3.1 Identifying attractors . .7 3.2 Examples of strange attractors . .8 3.3 Examples of chaotic systems . .9 4 Introduction to 1-D maps (short) 12 5 Dynamics of the Lorenz attractor in chaotic regime 13 Material teaching aid: Double pendulum, Möbius strip D. Kartofelev 1/16 K As of March 8, 2021 Lecture notes #9 Nonlinear Dynamics YFX1520 1 Lorenz attractor The Lorenz system was presented in the previous lecture. The following slides present an overview of selected properties of the system. The bifurcation analysis is shown for varied r while keeping the other parameters constant. Slides: 3–8 Lorenz attractor Properties of Lorenz attractor Lorenz attractor has the form There exists a symmetric pair of solutions. If (x, y) ( x, y) x˙ = d (y x), → − − − the system stays the same. If solution (x(t), y(t), z(t)) exists y˙ = rx y xz, (1) − − then solution ( x(t), y(t), z(t)) is also a solution. z˙ = xy bz, − − − Lorenz system is dissipative1: volumes in phase space contract where r, d, and b are the parameters. For r = 28, d = 10, and under the flow. As t , V 0 b = 8/3, the Lorenz system has the chaotic solution. Other → ∞ → ~x˙ = (d + 1 + b) < 0 = const., (2) parameter values may generate other types of solutions. ∇ · − V˙ = (d + 1 + b) V. (3) Lorenz equations also arise in simplified models for lasers, dynamos, − thermosyphons, brushless DC motors, electric circuits, chemical reactions and forward osmosis. 1See Mathematica .nb file uploaded to the course webpage. D. Kartofelev YFX1520 3 / 40 D. Kartofelev YFX1520 4 / 40 Bifurcation analysis of Lorenz attractor Bifurcation analysis of Lorenz attractor For r < 1 (d = 10, b = 8/3) there is only one stable fixed point located at the origin. This point corresponds to no convection. All orbits converge to the origin – a global attractor. As we decrease r from rHopf, the unstable limit cycles expand and A supercritical pitchfork bifurcation occurs at r = 1, and for pass precariously close to the saddle point at the origin. At r > 1 two additional fixed points appear at r = 13.926 the cycles touch the saddle point and become homoclinic orbits; hence we have a homoclinic bifurcation which (x∗, y∗, z∗) = C± = β(r 1), β(r 1), r 1 . (4) ± − ± − − is referred to as the first homoclinic explosion. Below r = 13.926 p p there are no limit cycles. These correspond to steady convection. Fixed points are stable only if The region 13.926 < r < 24.06 is referred to as transient chaos2 d + b + 3 + r < rHopf, rHopf = d = 24.74, (5) region. Here, the chaotic trajectories eventually settle at C or C . d b 1 − − − which can hold only for positive r and d > b + 1. At a critical value r = rHopf, both stable fixed points lose stability through a subcritical Hopf bifurcation. 2See Mathematica .nb file uploaded to the course webpage. D. Kartofelev YFX1520 5 / 40 D. Kartofelev YFX1520 6 / 40 Bifurcation analysis of Lorenz attractor Bifurcation analysis of Lorenz attractor For r > 24.06 and r > rHopf = 28 (immediate vicinity): no stable limit-cycles exist; trajectories do not escape to infinity (dissipation); do not approach an invariant torus (quasi-periodicity). Almost all initial conditions (I.C.s) will tend to an invariant set – the Lorenz attractor – a strange attractor and a fractal. Note: No quasi-periodic solutions for r > rHopf are possible because of the dissipative property of the flow. For r rHopf different types of chaotic dynamics exist e.g. noisy periodicity, transient and intermittent chaos. One can even find transient chaos settling to periodic orbits3. 3See Mathematica .nb file uploaded to the course webpage. D. Kartofelev YFX1520 7 / 40 D. Kartofelev YFX1520 8 / 40 Slide 7 shows the behaviour for small values of r while keeping other parameters constant where x∗ is the distance from the origin. The origin is globally stable node for r < 1. At r = 1 the origin loses stability by a supercritical pitchfork bifurcation, and a symmetric pair of attracting fixed points (stable spirals) is born, in the above schematic, only one of the pair is shown. At rHopf = 24:74 the fixed points lose stability by absorbing an unstable limit-cycle in a subcritical Hopf bifurcation. D. Kartofelev 2/16 K As of March 8, 2021 Lecture notes #9 Nonlinear Dynamics YFX1520 As we decrease r from rHopf, the unstable limit-cycles expand and pass precariously close to the sad- dle point at the origin. At r = 13:926 the cycles touch the saddle point and become homoclinic orbits; hence we have a homoclinic bifurcation of unstable limit-cycles. Below r = 13:926 there are no limit-cycles. Viewed in the other direction, we could say that a pair of unstable limit-cycles are created as r increases through r = 13:926. Numerics: nb#1 Dissipative chaotic flow (Lorenz attractor) visualisation via particle tracking. Calculation of divergence of 3-D and higher dimensional flows. Examples of transient, intermittent and periodic dynamics in Lorenz system for r rHopf (d = 10, b = 8=3) are shown in Sec. 3 where these phenomena are defined and in numerical files nb#3. 2 Exponential divergence of neighbouring trajectories 2.1 Lyapunov exponents From numerical integration of the Lorenz system we know that the motion/flow on the attractor exhibits sensitive dependence on initial conditions. This means that two trajectories starting very close to each other will rapidly diverge, and thereafter have totally different futures, see Fig. 1. xTtI j.gg in o Figure 1: Rapidly diverging trajectories where the initial separation ~δ0 = δ0 0. en181 j j ≈ It can be demonstrated that after then.EE.ietransient behaviour has elapsed and a trajectory has settled on the attracting set (Lorenz attractor) the norm (magnitude) of the separation vector F xTtI ~ λt f δ(t) δ0e ; (1) j j ∼ where in the case of Lorenz system λ 0:9. Hence,t neighbouring trajectoriesin diverge exponentially fast. ≈ j.gg Equivalently, if we plot ln ~δ vs. t, a logarithmic plot, we find a curve that is close to a straight line with a j j o positive slope of λ, see Fig. 2. sa en181 to n.EE.ie f F et t Figure 2: Norm of separation vector ~δ as a function of t. Slope λ is shown with the straight red line. j j The exponential divergence mustU stop when the separation is comparable to the diametersa of the attractor (attracting set, think Lorenz butterfly)to — the trajectories obviously can’t get any farther apart than that. This explains the levelling off or saturation of the curve ln ~δ(t) shown with the dashed line in Fig. 2. j j The exponent λ is often called the Lyapunov exponent, although this is a sloppy use of the term, for two reasons: • First, there are actually n different Lyapunov exponentset for each degree of freedom of an n-dimensional system (λx, λy, λz;:::). D. KartofelevU 3/16 K As of March 8, 2021 Lecture notes #9 Nonlinear Dynamics YFX1520 • Second, λ depends slightly on which trajectory we study. We should average over many trajectories (initial conditions) and over many different points on the same trajectory to get the true value of λ. Lyapunov exponent λ of a system is usually close to the biggest of the exponents related to the degrees of freedom (λx, λy, λz;:::). The biggest exponent governs the resulting slope shown in Figs. 2 and 3. Note: λ > 0 is a signature of chaos. Numerics: nb#2 Lyapunov exponent of 3-D chaotic systems (visual demonstration). Plotted below is the quantitatively accurate ln ~δ vs. t graphs shown in Fig. 2 for the Lorenz attractor j j with the same parameters as used in Sec. 1. 5 0 )| t ( 5 δ - | ln -10 -15 0 10 20 30 40 t Figure 3: Exponential divergence of trajectories in the Lorenz system shown with the blue graph. ln(δ eλt) λ = 0:9 Graph of 0 where the Lyapunov exponent is indicated with the red line. 2.2 Kolmogorov entropy Also known as the metric entropy, Kolmogorov-Sinai entropy, or KS entropy.xTtI The Kolmogorov entropy is zero for non-chaotic motion and positive for chaotic motion. For λi > 0 one can find the Kolmogorov entropy using the following formula: j.gg in K = λi; o (2) Xi where i spans the number of degrees of freedom of a given chaotic system. Kolmogorov entropy is a measure of the growth of uncertaintyen due181 to the expansion rate given by the positive Lyapunov exponents λi. n.EE.ie 2.3 Predictability horizon F The predictability horizon is also calledf Lyapunov time. When a system has a positive Lyapunov exponent λ, there is a time horizon beyond which prediction breaks downt , as shown schematically in Fig. 4. sa to Figure 4: Deviation of two trajectories with close-by initial conditions, where a is the allowed tolerance representing noticeable deviation from the true trajectory. After a time period t the discrepancy, shown in Fig.et 4, grows to ~δ(t) = δ(t) δ eλt: (3) j j ≈ 0 Let a be a measure of our tolerance, i.e.,U if prediction is within a of the true/nominal state, we consider it acceptable.