Lorenz equations fixed points

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. 

Area contraction is the analog of the volume contraction that we found for the Lorenz equations in Section 9.2. As in that case, it yields several conclusions. For instance, the attractor A for the baker s map must have zero area. Also, the baker s map cannot have any repelling fixed points, since such points would expand area elements in their neighborhood. 

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