Robust Chaos

It has been proposed to make practical use of chaos in communication [1], in enhancing mixing in chemical processes [2] and in spreading the spectrum of switch-mode power suppies to avoid electromagnetic interference [3, 4]. In such applications it will be necessary to obtain reliable operation in the chaotic mode. It is known that for most smooth chaotic systems (take the logistic map [5] for example), there is a dense set of periodic windows for any range of parameter values. Therefore in practical systems working in chaotic mode, slight inadvertent fluctuation of a parameter may take the system out of chaos. The question is, how to guarantee that there is no periodic window for a given range of parameter values and the maximal Lyapunov exponent remains positive throughout the range? In this Letter, we address this problem. We say a chaotic attractor is robust if, for its parameter values there exists a neighborhood in the parameter space with no periodic attractor and the chaotic attractor is unique in that neighborhood. It is known that robust chaos cannot occur in smooth systems. In this Letter we show that such situations can occur in piecewise smooth maps and obtain the conditions of existence of robust chaos. We first give a practical example from electrical engineering to demonstrate robust chaos. The circuit shown in Fig.1 is known as the boost converter. It consists of a controlled switch S, an uncontrolled switch D, an inductor L, a capacitor C and a load resistor R. When the controlled switch is turned on, the current in the inductor increases and energy is stored in it. When the controlled switch is turned off, the stored energy in the inductor drops and the polarity of the inductor voltage changes so that it adds to the input voltage. The voltage across the inductor and the input voltage together “boosts” the output voltage to a value higher than the input voltage. Such circuits are widely used in regulated dc switch-mode power supplies. Regulation of the output current is achieved by controlling the switching by current feedback — known as Figure 1: The current mode controlled boost converter