Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor Page: 24
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a particle cannot be reliably calculated after a sufficiently long time because of the uncertainty in
the initial position. The coefficient of time in the exponential separation of adjacent trajectories is
called the global Lyapunov exponent and is used as a criterion of chaos. Strictly speaking, the
global Lyapunov exponent is defined only in the limit as the initial separation distance between the
two trajectories goes to zero and time goes to infinity. Since mathematical limits cannot be at-
tained by numerical methods, other definitions have been proposed for a local Lyapunov exponent
to characterize chaos [30, 31, 47, 48, 49, 50].
In this chapter we define a local Lyapunov exponent that approaches a global Lyapunov in
the limit as the separation distance approaches zero and time approaches infinity. Using our
m-file programs in Matlab , we solve the classical kicked rotor in both chaotic and nonchaotic
regimes. In the chaotic regime the local Lyapunov exponent is calculated as a function of separation
distance and time. Our computations show that for sufficiently long times two adjacent trajectories
in the chaotic regime depart polynomially, instead of exponentially. This behavior is confirmed
analytically by showing that an upper bound to the departure of two adjacent trajectories is also a
polynomial. The corresponding bound on the local Lyapunov exponent decreases asymptotically
as O(N-1 InN). However, in the beginning stage of time development exponential departure of two
adjacent trajectories does occur and extends for a longer time as the initial separation distance
Chaotic regime is characterized by strong sensitivity to the initial conditions. If the two tra-
jectories with initial very small distance depart exponentially, then the trajectory description of the
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Zheng, Yindong. Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor, dissertation, August 2005; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc4824/m1/41/: accessed November 14, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .