Nonlinear Sciences – Chaotic Dynamics
Scientific paper
Jan 1995
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1995phdt........18b&link_type=abstract
Thesis (PH.D.)--UNIVERSITY OF COLORADO AT BOULDER, 1995.Source: Dissertation Abstracts International, Volume: 56-09, Section: B,
Nonlinear Sciences
Chaotic Dynamics
1
Scientific paper
The sensitivity that defines chaotic dynamics makes accessible a wide range of behaviors using arbitrarily small control signals. "Controlling chaos" attempts to cause large changes in the dynamics using only small perturbations. In targeting, one attempts to find a fast path from an initial condition {bf a} to a target point {bf b} by exploiting the fact that transport times for a chaotic system are highly sensitive to initial conditions and parameter values. The main difficulty is finding the switching points, the times and places to apply judiciously chosen perturbations. I present a new technique to find rough orbits (epsilon chains) that rapidly achieve a desired transport. The strategy is to build the epsilon chain from segments of a long orbit. In two-dimensional maps, long orbits have recurrences in neighborhoods where faster orbits must also pass. Long orbits of higher dimensional maps are likely to have recurrences, albeit less frequently. The recurrences are used as switching points between segments. If a local hyperbolicity condition is satisfied, then a nearby shadow orbit might be constructed. In one example, I show that transport times for the standard map can typically be reduced by a factor of 10^4. In another example, I apply the technique to the restricted three-body problem from which I find a low energy Earth-Moon transfer orbit which requires 38% less characteristic velocity than a comparable Hohmann transfer orbit. In yet another example, a symbol dynamics model has a closed-form expression for the optimal transporting orbit from near {bf a} to near {bf b}. I compare the optimal orbit to the targeted orbit resulting from removing recurrences, which also takes a particularly simple form in symbol dynamics. The techniques developed here do not require a closed-form representation of the map. Using the standard map as an example, I demonstrate that predictions from a time series may be sufficient for targeting. Finally, as a contribution to the understanding of barriers in high-dimensional Hamiltonian maps, I present a technique to investigate the breakup of invariant tori with fixed frequency of a four-dimensional generalization of the complex, semi-standard map.
No associations
LandOfFree
Controlling Chaos, Targeting, and Transport. does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Controlling Chaos, Targeting, and Transport., we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Controlling Chaos, Targeting, and Transport. will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-837691