Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor Page: 40
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To investigate the stability of the oscillator mode we make a small deviation 8 from mi for odd m,
so our new condition is Ksin9 = m + 8. Iterating for some kicks with P0o = 0 shows no instability.
This mode still oscillates with a small deviation from mi for m odd. From trajectory simulation we
also have shown the stability of this oscillator mode . The stability of these modes explains the
finite width of the region around min. An empirical formula for the width of the regions with an odd
multiples m of ic that fits the numerical simulation very well for seven regions is
[(m )2 - 2211/2 < K < [(mc)2 +1]1/2. (4.13)
However, we have not been able to derive this formula from first principles.
To visualize the motion, trajectory simulations are presented here on both accelerator and oscil-
lator modes. Fig. 4.7 (a) shows the trajectory simulation of the two adjacent trajectories at K= 4,
which is at the accelerator mode. Without perform modulo 2n, the trajectories glow 4R at each
kick but the two trajectories stay together. Fig. 4.7 (b) takes K= 9.2643, which is (3ix)2- 3
within the region of 3n stability region. The motion is oscillating with two adjacent trajectories stay
together. Fig. 4.7 (c) takes exactly K= 3. We can see that the two trajectories stay together and
the trajectories are oscillating. Fig. 4.7 (d) presents the trajectory simulation at the chaotic region
with K= 9.6. We can see that the two trajectories are eventually separating substantially.
The classical kicked rotor is found to have chaos for K= 3.5 to 4 in agreement with Ref. . For
even multiples of ic the classical kicked rotor has stable accelerator mode islands. For odd multiples
of ic we find stable oscillator modes. The details of the stable regions for the control parameter near
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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/m1/57/: accessed April 26, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .