Lorenz equations chaos

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. 

Sparrow, C, The Lorenz Equations Bifurcation, Chaos, and Strange Attractions (Appl. Math. Sci. 41). Springer-Verlag, New York, 1982. 

The Lorenz equations may produce deterministic chaos because we know how it will instantaneously change. However, for high enough Rayleigh numbers, the system becomes chaotic. Small changes in the initial conditions can lead to very different behavior after long time interval, since the small differences grow nonlinearly with feedback over time (known as the Butterfly effect). These equations are fairly well behaved and the overall patterns repeat in a quasi-periodic fashion. 

We begin our study of chaos with the Lorenz equations... 

Show numerically that the Lorenz equations can exhibit transient chaos when r = 21 (with <7 = 10 and b = as usual). 

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. 

Smale, S. (1967) Differentiable dynamical systems. Bull. Am. Math. Soc. 73, 747. Sparrow, C. (1982) The Lorenz Equations Bifurcations, Chaos, and Strange Attrac-... 

It has been known since 1975 that chaotic behavior is possible in some lasers under specific conditions, due to the similarity between the Lorenz equations (which predict chaos in fluids) and the semiclassical Maxwell-Bloch equations describing single mode lasers including the ef-... 

Sparrow, C. 1982. The Lorenz Equations Bifurcations, Chaos and Strange Attractors. Springer-Verlag New York. 

In the Lorenz model, the saddle value is positive for the parameter values corresponding to the homoclinic butterfly. Therefore, upon splitting the two symmetric homoclinic loops outward, a saddle periodic orbit is born from each loop. Furthermore, the stable manifold of one of the periodic orbits intersects transversely the unstable manifold of the other one, and vice versa. The occurrence of such an intersection leads, in turn, to the existence of a hyperbolic limit set containing transverse homoclinic orbits, infinitely many saddle periodic orbits and so on [1]. In the case of a homoclinic butterfly without symmetry there is also a region in the parameter space for which such a rough limit set exists [1, 141, 149]. However, since this limit set is unstable, it cannot be directly associated with the strange attractor — a mathematical image of dynamical chaos in the Lorenz equation. 

It was assumed that a description of evolution of deterministic systems required a solution of the equations of motion, starting from some initial conditions. Although Poincare [1] knew that it was not always true, this opinion was common. Since the work of Lorenz [2] in 1963, unpredictability of deterministic systems described by differential nonlinear equations has been discovered in many cases. It has been established that given infinitesimally different initial conditions, the outcomes can be wildly different, even with the simplest equations of motion. This feature means the occurrence of deterministic chaos. The literature devoted to this multidisciplinary and rapidly developing discipline of science is huge. There are many excellent textbooks, monographs, and collections of main papers, and we mention only a few [3-8]. 

A major development reported in 1964 was the first numerical solution of the laser equations by Buley and Cummings [15]. They predicted the possibility of undamped chaotic oscillations far above a gain threshold in lasers. Precisely, they numerically found almost random spikes in systems of equations adopted to a model of a single-mode laser with a bad cavity. Thus optical chaos became a subject soon after the appearance Lorenz paper [2]. 

A Poincare map is established by cutting across the trajectories in a certain region in the phase space, say with dimension n, with a surface that is one dimension less than the dimension of the phase space, n — 1. One such cut is also shown in Fig. 1. The equation that produces the return to the crossing the next time is a discrete evolution equation and is called the Poincare map. The dynamics of the continuous system that creates the Poincare map can be analyzed by the discrete equation. Therefore, chaotic behavior of the Poincare map can be used to identify chaos in the continuous system. For example, for certain parameters, the Henon discrete evolution equation is the Poincare map for the Lorenz systems. 

The phenomenon of deterministic chaos was established by the work of Lorenz who showed for the first time that chaos is not unnecessarily an unpredictable phenomenon. A certain class can be governed by mathematical equations suggesting it s predictability. This type of chaos is called deterministic chaos. 

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