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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 3-D example in which M is time-independent. 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