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So Lorenz has his “toy weather” model of a 3D flow determined by a 3D system of partial differential equations.
Definition: An attractor in a dynamical system is a set towards which all the other points in the space approach in future time.
Lorenz takes an initial starting point, a three-tuple of coordinates, and ignores the first several thousand points in the orbit in order to get the orbit close to the Lorenz Attractor. Then he plots the future points and n the orbit and find it traces out a fractal-looking object. See computer simulation and graphical display in video below:
Now what about Chaos?
Well, chaotic behavior in dynamical systems is caused by hyperbolic exponential divergence. Recall your second year calculus, in particular the differential calculus of multiple variables. For concreteness we’ll let our space have three dimensions. Then we have a flow in this 3D space (e.g., determined as the solutions of a system of an ordinary differential equations). This flow is parametrized by a time parameter. Thus the moving point ( x , y , z ) follows an orbit f(t) = ( x (t), y(t), z(t) ). The differential vector d( f(t) )/dt = ( dx/dt , dy/dt , dz/dt ) Is pointing in the direction of the flow and its magnitude is determined by its differential coordinates. Suppose that we only track the integer times. Then instead of a curve being the future orbit of a point, the orbit is a sequence f(n). n = 1, 2, 3, ….
