Chaos Control by Thresholding

We report the experimental verification of thresholding as a versatile tool for efficient and flexible chaos control. The strategy here simply involves monitoring a single state variable and resetting it when it exceeds a threshold. We demonstrate the success of the technique in rapidly controlling different chaotic electrical circuits, including a hyperchaotic circuit, onto stable fixed points and limit cycles of different periods, by thresholding just one variable. The simplicity of this controller entailing no run-time computation, and the ease and rapidity of switching between different targets it offers, suggests a potent tool for chaos based applications. In this paper we will experimentally demonstrate an alternate control strategy: the simple and easily implementable threshold mechanism [1, 2]. This strategy does not involve adjusting any parameter in the system, but only invloves the occasional resetting of one state variable. We will first introduce the general formalism and then focus on experimental implementation on a range of systems, including the challenging task of controlling a hyperchaotic system [3]. Threshold Formalism for Multi-Dimensional Systems: Consider a generalN dimensional dynamical system, described by the evolution equation: ẋ = F (x; t) where x ≡ (x1, x2, . . . xN ) are the state variables, and variable xi is chosen to be monitored and threshold controlled. The prescription for threshold control in this system is as follows: control will be triggered whenever the value of the monitored variable exceeds a critical threshold x∗ (i.e. when xi > x ∗) and the variable xi will then be re-set to x ∗ [4, 5]. The dynamics continues till the next occurence of xi exceeding the threshold, when control resets its value to x∗ again. No run-time knowledge of F (x) is involved, and no computation is needed to obtain the necessary control. The method only involves monitoring a single variable and no parameters are perturbed in the original system. The theoretical basis of the method does not involve stabilising unstable periodic orbits, but [email protected] 2 [email protected]