We propose a new mathematical model of the TNC oscillator and study its impact on the dynamical properties of the oscillator subjected to an exponential nonlinearity. We establish the existence of hyperchaotic behavior in the system through theoretical analysis and by exploiting electronic circuit experiments. The obtained numerical results are found to be in good agreement with experimental observations. Moreover, the new technique on adaptive control theory is used on our model and it is rigorously proven that the adaptive synchronization can be achieved for hyperchaotic systems with uncertain parameters. 1. Introduction Over the last four decades, an increasing interest has been shown on chaotic systems with higher dimensional attractors known as hyperchaotic. The interest for hyperchaotic dynamics is justified by the rapid development of new techniques in various areas of physics such as nonlinear circuits [1–6], complex system studies [7, 8], laser dynamics [9–11], secure communication [12, 13], and synchronization [14–21]. Hyperchaotic systems are usually classified as chaotic systems with more than one positive Lyapunov exponent, indicating that the chaotic dynamics of the systems are expanded in some directions but rapidly shrink in other directions, which significantly increase the system’s orbital degree or disorder and randomness. Those systems are suitable for engineering application such as secure communications [12–14, 19, 20, 22]. In fact, in 1995, Pérez and Cerdeira [23] have shown that by masking signals with simple chaos with only one Lyapunov exponent does not provide high level of security. Hence, the use of more complex hyperchaotic signals is a straightforward way to overcome this limitation because of their increased randomness and higher unpredictability [24]. Therefore the dynamical behavior of several hyperchaotic electronic circuits has been studied [1, 2, 4–6, 10, 11, 25–27]. Within these, TNC oscillator [2] retains our attention. Indeed, it is extremely simple and provides a powerful tool for understanding the inherent architectures and dynamical behavior of hyperchaotic oscillator as well as their use as chaotic carriers in practical secure communication. Its synchronization and dynamics have been studied using a piecewise linear (PWL) model [4, 28]. Moreover, Peng et al. in [14], address the synchronization of hyperchaotic oscillators using a scalar transmitted signal. Furthermore, this scalar synchronization does not require either the computation of the Lyapunov exponent or initial conditions belonging to the same basin
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