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Unformatted text preview: 15 Lyapunov exponents Whereas fractals quantify the geometry of strange attractors, Lyaponov ex ponents quantify the sensitivity to initial conditions that is, in effect, their most salient feature. In this lecture we point broadly sketch some of the mathematical issues con cerning Lyaponov exponents. We also briey describe how they are com puted. We then conclude with a description of a simple model that shows how both fractals and Lyaponov exponents manifest themselves in a simple model. 15.1 Diverging trajectories Lyapunov exponents measure the rate of divergence of trajectories on an attractor. Consider a ow ( t ) in phase space, given by d = F ( ) d t If instead of initiating the ow at (0), it is initiated at (0)+ (0), sensitivity to initial conditions would produce a divergent trajectory: (0) ( (0) ( t) t) Here grows with time. To the first order,   d( + ) F ( ) + M ( t ) d t 141 where F i M ij ( t ) = . j ( t ) We thus find that d = M ( t ) . (31) d t Consider the example of the Lorenz model. The Jacobian M is given by P P Z ( t ) + r M ( t ) = . 1 X ( t ) X ( t ) b Y ( t ) We cannot solve for because of the unknown time dependence of M ( t ). However one may numerically solve for ( t ), and thus ( t ), to obtain (for mally) ( t ) = L ( t ) (0) . 15.2 Example 1: M independent of time Consider a simple 3D example in which M is timeindependent. Assume additionally that the phase space coordinates correspond to M s eigenvectors....
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This note was uploaded on 02/24/2012 for the course MECHANICAL 12.006J taught by Professor Danielrothman during the Fall '06 term at MIT.
 Fall '06
 DanielRothman

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