Fig. 1. Kaplan-Yorke dimension and Lyapunov exponents versus b
showing the route to chaos.
Fig. 2. Bifurcation diagram (local maximum of x
) and Lyapunov exponents versus b
showing the route to chaos in
Fig. 3. Kaplan-Yorke dimension and Lyapunov exponents versus b
showing the route to chaos. In
the upper plot, the circles with error bars are values of the
Fig. 4. Cross-section of the attractor in yz
-space at x
= 0 for four values of b
. The axes are -20 to 20 for each
Fig. 5. Standard deviation and kurtosis for the excursion of the
trajectory from the origin for the attractors as a function of b
Fig. 6. Plot showing regions of multiple coexisting attractors as a
function of b
Fig. 7. Multiple coexisting attractors.
Fig. 8. Six coexisting strange attractors at b
Fig. 9. Cross-section of the chaotic sea at (x
mod 2pi) = 0 for the
conservative case with b
Fig. 10. Stereogram showing the regions where quasiperiodic
trajectories occur for b
The view is looking down along the x
different colors denote the six directions in which
Fig. 11. Cross-section at (x
mod 2pi) = 0 showing the quasiperiodic orbits for b
= 0 surrounded by a KAM surface.
Fig. 12. Brownian motion of a trajectory in the chaotic sea (black)
along with a quasiperiodic trajectory (red).
Fig. 13. Probability distribution function of x
for 5 × 106
initial conditions near the origin after a time of 4 × 103
curve is a Gaussian distribution with the same standard
deviation and area.
Fig 14. Projection of the trajectory onto the x
-axis showing an example of
intermittency where the trajectory approaches the quasiperiodic region
with initial conditions (0.05, 0.09, 0.05).
Fig. 15. Standard deviation of 1.5 × 106
starting near the origin versus time.
Fig. 16. (a) Range versus time, and (b) autocorrelation function of dx/dt
versus delay for an initial
condition of (0.2, 0, 0).
Fig. 17. Allowable transitions for symbolic sequence.
Fig. 18. Iterated function system representation of symbolic dynamic.