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Lecture 10: 1-D Maps, Lorenz Map, Logistic Map, Sine Map, Period

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Lecture 10: 1-D Maps, Lorenz Map, Logistic Map, Sine Map, Period Lecture notes #10 Nonlinear Dynamics YFX1520 Lecture 10: 1-D maps, Lorenz map, Logistic map, sine map, period doubling bifurcation, tangent bifurcation, tran- sient and intermittent chaos in maps, orbit diagram (or Feigenbaum diagram), Feigenbaum constants, universals of unimodal maps, universal route to chaos Contents 1 Lorenz map 2 1.1 Cobweb diagram and map iterates . .2 1.2 Comparison of fixed point x∗ in 1-D continuous-time systems and 1-D discrete-time maps .3 1.3 Period-p orbit and stability of period-p points . .4 2 1-D maps, a proper introduction 6 3 Logistic map 6 3.1 Lyapunov exponent . .7 3.2 Bifurcation analysis and period doubling bifurcation . .8 3.3 Orbit diagram . 13 3.4 Tangent bifurcation and odd number period-p points . 14 4 Sine map and universality of period doubling 15 5 Universal route to chaos 17 Handout: Orbit diagram of the Logistic map D. Kartofelev 1/17 K As of August 7, 2021 Lecture notes #10 Nonlinear Dynamics YFX1520 1 Lorenz map In this lecture we continue to study the possibility that the Lorenz attractor might be long-term periodic. As in previous lecture, we use the one-dimensional Lorenz map in the form zn+1 = f(zn); (1) where f was incorrectly assumed to be a continuous function, to gain insight into the continuous time three- dimensional flow of the Lorenz attractor. 1.1 Cobweb diagram and map iterates The cobweb diagram (also called the cobweb plot) introduced in previous lecture is a graphical way of thinking about and determining the iterates of maps including the Lorenz map, see Fig. 1. In order to construct a cobweb plot move vertically to function f and horizontally to the diagonal and repeat (see Lecture 9 for basic examples). Zz it iii it z Figure 1: (Left) Cobweb diagram of the Lorenz map. (Right) Lorenz map iterates corresponding to the cobweb diagram shown on the left. In the previous lecture we showed, using linearisation procedure, that the fixed point z∗ (period-1 point) satisfying I f(z∗) = z∗; I (2) A IFF where f(z) is the function defining the Lorenz map, is unstable. This conclusion can be graphically confirmed with the aid of a cobweb diagram. The following interactive numerical file shows the Lorenz map and its iterates. D. Kartofelev 2/17 K As of August 7, 2021 Lecture notes #10 Nonlinear Dynamics YFX1520 Numerics: nb#1 Cobweb diagram and iterates of the (generalised) Lorenz map. Lyapunov exponent λ(r) of Logistic map. 1.0 1.0 0.5 0.5 1 + n 0.0 0.0 max z z -0.5 -0.5 -1.0 -1.0 -1.0 -0.5 0.0 0.5 1.0 0 2 4 6 8 10 zn Zz n Figure 2: (Left) Cobweb diagram of the normalised Lorenz map showing the first ten iterates for z z 0 ≈ ∗ where the fixed point z∗ is unstable, since f 0(z∗) > 1. The Lorenz map has property f 0(z) > 1, z in the map basin. (Right) Lorenz map iteratesj correspondingj to the cobweb diagram shownj onj the8 left. it iii it 1.2 Comparison of fixed point x∗ in 1-D continuous-time systems and 1-D discrete- time maps z The fixed points of one-dimensional continuous-time systems (ODEs) and of discrete-time one-dimensional maps (with period-1 point) are clearly analogous. Figures 3 and 4 show that analogy. I I A IFF Figure 3: (Top-left) Stable fixed point x∗ of a one-dimensional system given by x_ = f(x) where f 0(x∗) < 0 is shown with the filled bullet. (Top-right) Family of time-series solutions shown for several initial con- ditions positioned close to x∗ corresponding to the phase portrait shown on the left. (Bottom-left) Unstable fixed point x∗ shown with the empty bullet where f 0(x∗) > 0. (Bottom-right) Family of time- series solutions shown for several initial conditions positioned close to x∗ corresponding to the phase portrait shown on the left. D. Kartofelev 3/17 K As of August 7, 2021 Lecture notes #10 Nonlinear Dynamics YFX1520 iti i i T j i i t i iti i i Figure 4: (Top-left) Stable fixed point x∗ (period-1 point) of a one-dimensional map given by xn+1 = f(xn) where f (x ) < 1 is shown with the filled bullet. (Top-right) Three sets of map iterates x shown for initial j 0 ∗ j n conditions x01, x∗ and x02 corresponding to the cobweb diagram shown on the left. (Bottom-left) Unstable fixed point x (period-1 point) shown with the empty bullet where of (x ) > 1. (Bottom-right) Three sets ∗ T j 0 ∗ j of map iterates xn shown for initialj conditions x01, x∗ and x02 corresponding to the cobweb diagram shown on the left. i t i i i 1.3 Period-p orbit and stability of period-p points 1 t PERIOD PERIOD We ended the previous lecture with two open ended questions: Can trajectories in a cobweb diagram of the Lorenz map close onto themselves? And, how does a closed trajectory in the cobweb plot translatei into the three-dimensional continuous-time Lorenz flow? o i i 1 t PERIOD PERIOD Figure 5: (Left) Period-4 orbit, shown with the blue graph, where zn+4 = zn. (Right) Lorenz map iterates zn corresponding to the cobweb diagram shown on the left. Map iterates are repeat every four iterates. So, can trajectories in a cobweb diagram of the Lorenz map close onto themselves? We could imagine a trajectory of the Lorenz map’s cobweb plot closing onto itself in a manner shown in Fig. 5. Visual inspection of the cobweb diagram shown in Fig. 5 reveals that the iterates zn (local maxima of the three-dimensional Lorenz flow) repeat themselves such that z4 = z0 or more generally z = z ; n; (3) n+4 n 8 D. Kartofelev 4/17 K As of August 7, 2021 Lecture notes #10 Nonlinear Dynamics YFX1520 if this dynamics can be shown to be possible, then it would strongly suggest that a limit-cycle might be possible in the three-dimensional Lorenz attractor. This conclusion can be generalised further: z = z ; n; (4) n+p n 8 where p Z+ is the period of a limit-cycle. This type of fixed point is called the period-p point and it represents2 the period-p orbit of the map. Period-p points are a new type of fixed points or in other words a new type of dynamics. A period-1 point coincides with our old friend — fixed point z∗. The period-p points where p > 1 don’t have corresponding analogies in one-dimensional continuous systems in the manner as was discussed and shown in Sec. 1.2. This is because the period-p points represent oscillatory limit-cycle solutions which are not possible in one-dimensional systems (see Lecture 2: Impossibility of oscillations in 1-D systems). It is natural to assume that period-p points or orbits, very much like fixed points z∗ (period-1 point), can be either stable (attracting trajectories) or unstable (repelling trajectories). If we can show that stable period-p orbits are possible in the Lorenz map, then that would strongly suggest a possibility of periodic solutions in the continuous-time Lorenz attractor. Let’s find the analytical expression for the relationship between zn+p and zn in (4). The n + 1 iterate of z0 is z = f(f(f(: : : f(z ) :::))) f n(z ): (5) n+1 0 ≡ 0 n times The subsequent iterates of the closed period-p| {z orbits,} similar to the one shown in Fig. 5 with the blue trajectory, expressed analytically are z = f(z ) f 1(z ) 1 0 ≡ 0 z = f(z ) = f(f(z )) f 2(z ) 2 1 0 ≡ 0 3 z3 = f(z2) = f(f(f(z0))) f (z0) ≡ (6) z = f(z ) = f(f(f(f(z ))) f 4(z ) 4 3 0 ≡ 0 . p zn+p = f (zn); where z is the period-p point in the period-p orbit and f p is the pth iterate map. Definition: z is a period-p point if equation f p(z) = z; (7) where p is minimal, is satisfied. The stability of a period-p point is determined via linearisation. For simplicity we consider stability of period-2 point f 2(z) f(f(z)) = z: (8) ≡ Note that z is the period-2 point for map f but the fixed point (period-1) for map f 2. In the previous lec- ture we showed that the stability of fixed point z∗ depends on the slope of the map at that point. Small perturbations η 1 evolve according to j j η f 0(z∗) η : (9) n+1 ≈ j j n We find chain rule + 2 d 2 3 (f (z))0 = f(f(z)) = orbit points = f 0(f(z )) f 0(z ) = f 0(z ) f 0(z ) > 1; (10) j j dz j j j 0 · 0 j j 1 j · j 0 j 6 z = z ; 7 6 n+2 n 7 z1 6we select z 7 6 07 4 5 | {z } D. Kartofelev 5/17 K As of August 7, 2021 Lecture notes #10 Nonlinear Dynamics YFX1520 because for the Lorenz map f 0(z) > 1; z in the map basin. The period-2 point is thus unstable. The evolution of perturbations definedj j by (9) generalised8 to all period-p points in any closed period-p orbit are the following: p 1 − ηn+p f 0(zn+k) ηn; (11) ≈ k=0 Y here, again, by the Lorenz map property p 1 − f 0(zn+k) > 1: (12) k=0 Y Thus, all period-p points are unstable.
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