Research Papers

Going With the Flow: Enhancing Stochastic Switching Rates in Multigyre Systems

[+] Author and Article Information
Christoffer R. Heckman

Nonlinear Dynamical Systems Section,
Plasma Physics Division,
U.S. Naval Research Laboratory,
Code 6792,
Washington, DC 20375
e-mail: christoffer.heckman.ctr@nrl.navy.mil

M. Ani Hsieh

Scalable Autonomous Systems Laboratory,
Mechanical Engineering & Mechanics,
Drexel University,
Philadelphia, PA 19104
e-mail: mhsieh1@drexel.edu

Ira B. Schwartz

Nonlinear Dynamical Systems Section,
Plasma Physics Division,
U.S. Naval Research Laboratory,
Code 6792,
Washington, DC 20375
e-mail: ira.schwartz@nrl.navy.mil

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 23, 2014; final manuscript received June 5, 2014; published online October 21, 2014. Assoc. Editor: Jongeun Choi.

J. Dyn. Sys., Meas., Control 137(3), 031006 (Oct 21, 2014) (6 pages) Paper No: DS-14-1026; doi: 10.1115/1.4027828 History: Received January 23, 2014; Revised June 05, 2014

A control strategy is employed that modifies the stochastic escape times from one basin of attraction to another in a model of a double-gyre flow. The system studied captures the behavior of a large class of fluid flows that circulate and have multiple almost invariant sets. In the presence of noise, a particle in one gyre may randomly switch to an adjacent gyre due to a rare large fluctuation. We show that large fluctuation theory may be applied for controlling autonomous agents in a stochastic environment, in fact leveraging the stochasticity to the advantage of switching between regions of interest and concluding that patterns may be broken or held over time as the result of noise. We demonstrate that a controller can effectively manipulate the probability of a large fluctuation; this demonstrates the potential of optimal control strategies that work in combination with the endemic stochastic environment. To demonstrate this, stochastic simulations and numerical continuation are employed to tie together experimental findings with predictions.

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Grahic Jump Location
Fig. 1

Phase portrait of the two-gyre vector field with arrows denoting the flow direction and positions of equilibria marked by +. The parameters for this simulated flow are A = 1, s = 1, and μ = 1.

Grahic Jump Location
Fig. 5

Comparison of the MFPT as predicted by large fluctuation theory (dashed lines) and computed from an average over many stochastic trials (points). Three regimes were examined: black represents no control, red represents c ≈ −0.31 and blue represents c ≈ 0.1.

Grahic Jump Location
Fig. 2

Optimal switching path for c = 0 overlaid on a pdf of paths that have been stochastically integrated until transitioning out of the basin of attraction. The colormap represents an exponential scale of 3000 sample paths with D = 1/30.

Grahic Jump Location
Fig. 3

Optimal switching path for c ≈ 0.1 overlaid on a pdf of paths that have been stochastically integrated until transitioning out of the basin of attraction. Paths were generated by the same method as in Fig. 2.

Grahic Jump Location
Fig. 4

The conjugate momenta along the optimal path as generated by Auto for c = 0. The arrows indicate the direction in which the optimal noise is acting at the given point along the optimal path, and the color and length of the arrows indicate the magnitude of the noise.



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