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Feedback control methods for stabilizing unstable equilibrium points in a new chaotic system. (English) Zbl 1181.34067

Consider the autonomous system \[ \begin{aligned} {dx\over dt} & = 20(y- x),\;{dy\over dt}= 14x+ 10.6y- xz,\\ {dz\over dt} & = x^2- 2.8z,\end{aligned}\tag{\(*\)} \] which has a chaotic attractor and three unstable equilibria. Together with \((*)\), the author considers the control system
\[ \begin{aligned} {dx\over dt} & = 2-(y- x)+ u,\;{dy\over dt}= 14x+ 10.6y- xz+ v,\\ {dz\over dt} & = x^2- 2.8z\end{aligned} \]
and presents the controls \(u= 0\), \(v=-kx\); \(u=- kx\), \(v= -ky\), which stabilize the origin for \(k> k_i\). Applying the control \(u= 0\), \(v= -k\dot x\), another equilibrium point becomes stable.

MSC:

34H05 Control problems involving ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
93D15 Stabilization of systems by feedback
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References:

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