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Stabilizing The Lorenz Flows Using A Closed Loop Quotient Controller, James P. Braselton, Yan Wu

Department of Mathematical Sciences Faculty Publications

In this study, we introduce a closed loop quotient controller into the three-dimensional Lorenz system. We then compute the equilibrium points and analyze their local stability. We use several examples to illustrate how cross-sections of the basins of attraction for the equilibrium points look for various parameter values. We then provided numerical evidence that with the controller, the controlled Lorenz system cannot exhibit chaos if the equilibrium points are locally stable.

Applying Linear Controls To Chaotic Continuous Dynamical Systems, James P. Braselton, Yan Wu

Department of Mathematical Sciences Faculty Publications

In this case-study, we examine the effects of linear control on continuous dynamical systems that exhibit chaotic behavior using the symbolic computer algebra system Mathematica. Stabilizing (or controlling) higher-dimensional chaotic dynamical systems is generally a difficult problem, Musielak and Musielak, [1]. We numerically illustrate that sometimes elementary approaches can yield the desired numerical results with two different continuous higher order dynamical systems that exhibit chaotic behavior, the Lorenz equations and the Rössler attractor.

On The Stability Of Cycles By Delayed Feedback Control, Dmitriy Dmitrishin, Paul Hagelstein, Anna Khamitova, Alexander M. Stokolos

Department of Mathematical Sciences Faculty Publications

We present a delayed feedback control (DFC) mechanism for stabilizing cycles of one dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizingT-cycles of a differentiable functionf:R→Rof the form

x(k+1)=f(x(k))+u(k)

where

u(k)=(a₁−1)f(x(k))+a₂f(x(k−T))+...+a_Nf(x(k−(N−1)T)),

with a₁+...+a_N=1. Following an approach of Morgül, we construct a map F:R^T+1→R^T+1 whose fixed points correspond to T-cycles of f. We then analyze the local stability of the above DFC mechanism by evaluating the stability of the corresponding equilibrum points of F. We associate to each periodic orbit of f an …