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So, 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, …. Although f( n ) is only defined for the single orbit starting value, the same thing can be done for each point in the 3D space. Then the function “f” becomes a function taking 3D space to itself. The total derivative of f, df, is a linear transformation from 3D space to 3D space (recall that this linear transformation is detrrmined by the Jacobian matrix of partial derivatives. A function “f” is “hyperbolic” if all its eigenvalues are non-zero. For the 3D tangent space at each point, their is a so-called basis of eigenvalues, a so-called “moving frame”.
If the eigenvalue is bigger than 1 then the discretized flow f( n ) is expanding in the unstable direction of the associated eigenvector. If the eigenvalue is less than 1, then the discretized flow is contracting in the stable direction of the associated eigenvector. The space and orbits nearby the point expand pretty much in the same direction. To get the unstable Lyapunov exponent, you multiply all these unstable Lyapunov exponents and then take the 1/n power. Chaos is caused by a positive Lyapunov exponent in the expanding direction. Which causes exponential divergence of orbits. See eight minute video below:
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Thus “sensitive dependence on initial conditions is caused by a positive unstable Lyapunov exponent, representing exponential divergence of orbits.
To be continued: Next time we discuss the archtype of chaotic hyperbolic dynamical systems, the Smale Horseshoe. Smale discovered the horseshoe on the beaches of Rio de Janeiro, while he was visiting the mathematical institute IMPA. See link thirteen minute video below:
