MASTER-SLAVE GLOBAL STOCHASTIC SYNCHRONIZATION OF CHAOTIC OSCILLATORS MAURIZIO PORFIRI AND ROBERTA PIGLIACAMPO * Abstract. We study synchronization of two chaotic oscillators in a master-slave configuration. The two dynamic systems are coupled via a directed feedback that randomly switches among a finite set of given constant functions at a prescribed time rate. We use stochastic Lyapunov stability theory and partial averaging techniques to show that global synchronization is possible if the switching period is sufficiently small and if the two systems globally exponentially synchronize under an average feedback coupling. The approach is applied to the synchronization of two Chua’s circuits. Key words. master-slave synchronization, global synchronization, stochastic synchronization, chaos, chua’s circuit, exponential stability AMS subject classifications. 34C15, 34C29, 34D23, 74H65, 93C10, 93E15 1. Introduction. Chaos synchronization is a topic of great interest, due to its observation in a huge variety of phenomena of different nature. In many biological sys- tems, synchronization plays an important role in self-organization of organisms’ groups [10]. Examples of synchronization include communication among fireflies [9, 36], lo- comotion of animals [13], molecular and cellular activity [24] and cardiac stimulation [21, 27, 42]. The study of neural activity [44, 54, 63] and brain disorders [3, 53] is a correlated issue as well. Other examples and applications can also be found in eco- logical systems [5], meteorology [16], chemistry [24, 37], gas-liquid bubbling dynamics [57, 61] and optics [55, 62]. Many reviews on chaos synchronization are currently available (see for example [2, 6, 12, 23, 43, 48, 50]). In the literature, different paradigms have been studied to describe synchroniza- tion of two or more chaotic oscillators. We mention, among the others, peer-to-peer coupling [22, 52, 56, 59], back-stepping [7], generalized synchronization [64, 65], phase synchronization [6] and master-slave synchronization [11, 33, 40, 60]. In this work, we focus on master-slave synchronization. In this case, one system acts as a “master” by driving the other system that behave consequently as a “slave”. Most of the research efforts on master-slave synchronization focus on time invari- ant coupling (see for example [11, 26, 33, 40, 45, 46, 47]). Nevertheless, experimental and numerical evidences show that synchronization can also be achieved using in- termittent, time-varying master-slave coupling [20, 32, 65, 66]. In [20], experimental results on synchronization of two periodically coupled chaotic circuits are presented. In [32, 65], the slave system is driven by a sequence of samples of the master’s state (impulsive synchronization ). In [32, 66], the signal transmission from the master to the slave system is adaptively controlled. That is, the driving signal is transmitted only when it is expected to reduce the synchronization error (selective synchroniza- tion ). The main goal of this work is to establish sufficient conditions for global synchro- nization of master-slave coupled chaotic systems with time-varying coupling. We fo- cus on the general case where the intermittent coupling changes randomly over time. Intermittent coupling is made possible through a switching function, that changes randomly over time, assuming values among a finite set of constant functions. The * The authors are with the Mechanical, Aerospace and Manufacturing Engineering Department, Polytechnic University, Brooklyn, NY 11201, USA (email: mporfiri@poly.edu) 1