Controlling Chaos! Dylan Thomas and Alex Yang. Why control chaos? One may want a system to be used...
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Transcript of Controlling Chaos! Dylan Thomas and Alex Yang. Why control chaos? One may want a system to be used...
![Page 1: Controlling Chaos! Dylan Thomas and Alex Yang. Why control chaos? One may want a system to be used for different purposes at different times Chaos offers.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ed95503460f94be812c/html5/thumbnails/1.jpg)
Controlling Chaos!
Dylan Thomas and Alex Yang
![Page 2: Controlling Chaos! Dylan Thomas and Alex Yang. Why control chaos? One may want a system to be used for different purposes at different times Chaos offers.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ed95503460f94be812c/html5/thumbnails/2.jpg)
Why control chaos?
One may want a system to be used for different purposes at different times
Chaos offers flexibility (ability to switch between behaviors as circumstances change)
Small changes produce large effects
![Page 3: Controlling Chaos! Dylan Thomas and Alex Yang. Why control chaos? One may want a system to be used for different purposes at different times Chaos offers.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ed95503460f94be812c/html5/thumbnails/3.jpg)
How is it done?
Chaotic systems can be controlled by using the underlying non-linear deterministic structure.
Exploit extreme sensitivity to initial conditions
Use small, appropriately timed changes to bring the system onto the stable manifold of an unstable orbit
![Page 4: Controlling Chaos! Dylan Thomas and Alex Yang. Why control chaos? One may want a system to be used for different purposes at different times Chaos offers.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ed95503460f94be812c/html5/thumbnails/4.jpg)
Famous examples
Chaotic ribbon
Lorentz equations
![Page 5: Controlling Chaos! Dylan Thomas and Alex Yang. Why control chaos? One may want a system to be used for different purposes at different times Chaos offers.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ed95503460f94be812c/html5/thumbnails/5.jpg)
ISEE-3/ICE and the n body problem
![Page 6: Controlling Chaos! Dylan Thomas and Alex Yang. Why control chaos? One may want a system to be used for different purposes at different times Chaos offers.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ed95503460f94be812c/html5/thumbnails/6.jpg)
Two methods Ott, Grebogi, Yorke: modify parameters of the system to move the
stable manifold to the current system state
Garfinkel et. al. (Proportional perturbation feedback): force the system onto the stable manifold by a small perturbation
![Page 7: Controlling Chaos! Dylan Thomas and Alex Yang. Why control chaos? One may want a system to be used for different purposes at different times Chaos offers.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ed95503460f94be812c/html5/thumbnails/7.jpg)
The logistic map
![Page 8: Controlling Chaos! Dylan Thomas and Alex Yang. Why control chaos? One may want a system to be used for different purposes at different times Chaos offers.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ed95503460f94be812c/html5/thumbnails/8.jpg)
The Hénon map
![Page 9: Controlling Chaos! Dylan Thomas and Alex Yang. Why control chaos? One may want a system to be used for different purposes at different times Chaos offers.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ed95503460f94be812c/html5/thumbnails/9.jpg)
Variation of a parameter in the Hénon map
-0.95 -0.9 -0.85 -0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.5 -0.45-0.95
-0.9
-0.85
-0.8
-0.75
-0.7
-0.65
-0.6
-0.55
-0.5
-0.45
a=0a=0.01
a=0.02a=0.03
a=0.04a=0.05
a=0.06a=0.07
a=0.08a=0.09
a=0.1a=0.11a=0.12
a=0.13a=0.14a=0.15a=0.16a=0.17a=0.18a=0.19a=0.2
Legend:Green =stable manifoldRed = unstable manifold
![Page 10: Controlling Chaos! Dylan Thomas and Alex Yang. Why control chaos? One may want a system to be used for different purposes at different times Chaos offers.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ed95503460f94be812c/html5/thumbnails/10.jpg)
Matlab experimental results
0 200 400 600 800 1000 1200-1.4
-1.38
-1.36
-1.34
-1.32
-1.3
-1.28
-1.26
0 200 400 600 800 1000 1200-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
![Page 11: Controlling Chaos! Dylan Thomas and Alex Yang. Why control chaos? One may want a system to be used for different purposes at different times Chaos offers.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ed95503460f94be812c/html5/thumbnails/11.jpg)
Controlling chaos when the equations determining the system are not known
Let Z1, Z2,…,Zn be a trajectory, or a series of piercing of a Poincare surface-of-section
If two successive Zs are close, then there will be a period one orbit Z* nearby
Find other such close successive pairs of points, which will exist because orbits on a strange attractor are ergodic.
Perform a regression to estimate A, an approximation of the Jacobian matrix, and C, a constant vector.
For period 2 points, proceed the same way, for pairs (Zn, Zn+2)
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Altering the dynamics of arrythmia
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Cardiac tissue
![Page 14: Controlling Chaos! Dylan Thomas and Alex Yang. Why control chaos? One may want a system to be used for different purposes at different times Chaos offers.](https://reader030.fdocuments.in/reader030/viewer/2022032607/56649ed95503460f94be812c/html5/thumbnails/14.jpg)
Neurons
Schiff et al. removed and sectioned the hippocampus of rats (where sensory inputs and distributed to the forebrain) and perfused it with artificial cerebrospinal fluid.