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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 brieﬂy 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 ﬂow θ ψ ( t ) in phase space, given by d θ = F ψ ( θ ψ ) d t If instead of initiating the ﬂow at θ ψ (0), it is initiated at θ ψ (0)+ π (0), sensitivity to initial conditions would produce a divergent trajectory: φ(0) φ( ε(0) ε( t) t) Here ψ π grows with time. To the first order, | | d( θ ψ + ψ π ) F ψ ( θ ψ ) + M ( t ) ψ π d t ◦ 141 where ωF i M ij ( t ) = . ωθ j π τ ( t ) We thus find that d ψ π = M ( t ) ψ π. (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 ψ π because of the unknown time dependence of M ( t ). However one may numerically solve for θ ψ ( t ), and thus ψ π ( t ), to obtain (for- mally) ψ π ( t ) = L ( t ) ψ π (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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- Fall '06
- Exponential Function, Fractal, Smale, Lyapunov exponent, Lyaponov