Lorenz equations strange attractor

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

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. 

Hysteresis between a fixed point and a strange attractor) Consider the Lorenz equations with <7 = 10 and b = 8/3. Suppose that we slowly turn the r knob up and down. Specifically, let r = 24.4 -h sin or, where ft) is small compared to typical orbital frequencies on the attractor. Numerically integrate the equations, and plot the solutions in whatever way seems most revealing. You should see a striking hysteresis effect between an equilibrium and a chaotic state. 

Back in Chapter 9, we found that the solutions of the Lorenz equations settle down to a complicated set in phase space. This set is the strange attractor. As Lorenz (1963) realized, the geometry of this set must be very peculiar, something like an infinite complex of surfaces. In this chapter we develop the ideas needed to describe such strange sets more precisely. The tools come from fractal geometry. 

In theculinary spirit ofthe pastry map and the baker s map, Otto Rossler (1976) found inspiration in a taffy-pulling machine. By pondering its action, he was led to a system of three differential equations with a simpler strange attractor than Lorenz s. The Rossler system has only one quadratic nonlinearity xz ... 

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. 

This attractor is called the Lorenz attractor. The Lorenz attractor falls into a class of so-called strange attractors, corresponding to the quasi--stochastic behaviour of a system. Dynamics of this type has been found to occur in the systems of chemical kinetics equations (see Section 6.3.3). 

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

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. 

As tools for analyzing differential equations. We have already encountered maps in this role. For instance, Poincare maps allowed us to prove the existence of a periodic solution for the driven pendulum and Josephson junction (Section 8.5), and to analyze the stability of periodic solutions in general (Section 8.7). The Lorenz map (Section 9.4) provided strong evidence that the Lorenz attractor is truly strange, and is not just a long-period limit cycle. 

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