Examples Lorenz attractor

Fig. 1. Examples of w-limit sets, (a) Rest point (b) limit cycle (c) Lorenz attractor [projection on the (Cj, c3) plane, a = 10, r = 30, b = 8/3].

The voltages u,v,w at three different points in the circuit are proportional to Lorenz s x, y,z. Thus the circuit acts like an analog computer for the Lorenz equations. Oscilloscope traces of u t) vs. w t), for example, confirmed that the circuit was following the familiar Lorenz attractor. Then, by hooking up the circuit to a loudspeaker, Cuomo enabled us to hear the chaos—it sounds like static on the radio. 

These same issues confronted scientists in the mid-1970s. At the time, the only known examples of strange attractors were the Lorenz attractor (1963) and some mathematical constructions of Smale (1967). Thus there was a need for other concrete examples, preferably as transparent as possible. These were supplied by Henon (1976) and Rdssler (1976), using the intuitive concepts of stretching and folding. These topics are discussed in Sections 12.1-12.3. The chapter concludes with experimental examples of strange attractors from chemistry and mechanics. In addition to their inherent interest, these examples illustrate the techniques of attractor reconstruction and Poincare sections, two standard methods for analyzing experimental data from chaotic systems. 

Examples of ta-limit sets (A) rest point, (B) limit cycle, (C) Lorenz attractor (projection on the x,y) plane a = 0, p = 30, f = 8/3). Reprinted from Yablonskii, G.S., Bykov, V.i, Gorban, A.N., Elokhin, V.I., 1991. Kinetic models of catalytic reactions. In Compton, R.G. (Ed.), Comprehensive Chemical Kinetics, vol. 32. Elsevier, Amsterdam, Copyright (1991), with permission from Elsevier. 

An example of such a set is the Lorenz attractor which occurs in a variety of models. The wild spiral attractor [153] is another fascinating example. ... 

The significance of higher degeneracies (starting from codimension three) in the linear part is that the effective normal forms become three-dimensional, and may, as a result, exhibit complex dynamics, the so-called instant chaos, even in the normal form itself. Such examples include the normal forms for a bifurcation of an equilibrium state with a triplet of zero characteristic exponents, and a complete or incomplete Jordan block, in which there may be a spiral strange attractor [18], or a Lorenz attractor [129], respectively (the latter case requires an additional symmetry). Since we will focus our considerations only on simple dynamics, we do not include these topics in this book. 

Example 13.1 Lorenz equations The strange attractor The Lorenz equations (published in 1963 by Edward N. Lorenz a meteorologist and mathematician) are derived to model some of the unpredictable behavior of weather. The Lorenz equations represent the convective motion of fluid cell that is warmed from below and cooled from above. Later, the Lorenz equations were used in studies of lasers and batteries. For certain settings and initial conditions, Lorenz found that the trajectories of such a system never settle down to a fixed point, never approach a stable limit cycle, yet never diverge to infinity. Attractors in these systems are well-known strange attractors. 

This textbook is aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. The presentation stresses analytical methods, concrete examples, and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors. 

When the control parameters a, b have fixed values, for example a = 10, b = 8/3, and the parameter r increases from the value r = 0, the nature of phase trajectories of the system (5.14) changes qualitatively. We shall describe properties of the Lorenz system only for the values of parameter r larger than r = 24.74 and smaller than a certain value rx (Lorenz in his work has studied the case a = 10, b = 8/3, r = 28 < rOT). When 24.74 < < r < rx, all three stationary points are unstable. However, trajectories do not escape to infinity — all the trajectories are attracted by a certain region of the phase space (attractor), containing the stationary points and which is approximately a two-dimensional surface. 

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