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In this paper, we study the synchronization problem in complex dynamic networks of Piece Wise Linear (PWL) systems. PWL systems exhibit multi-scrolls and belong to a special class of Unstable Dissipative Systems (UDS). We consider strongly connected digraphs and linear diffusive couplings. The synchronization regions are computed using the concept of disagreement vectors, generalized algebraic connectivity of the network topology, and Lyapunov functions, which provide lower bounds on the coupling gain of the network. Then, different combinations of linear diffusive coupling are explored by changing the observed and measured variables to illustrate the contribution of our results. The theoretical results are validated by numerical simulations.

Complex systems pervade nature and form the core of many technological applications. An exciting feature of these systems is that they exhibit a wide range of temporal behaviors, ranging from collective motion, synchronization, pattern formation, and chaos, among others. This has not only caught the attention of scientists, but also the interest of a wider audience. Consequently, our goal in this work is to provide a simple but descriptive explanation of some concepts related to complex systems. Specifically, the reader embarks on a journey that begins in the 17th century with the discovery of synchronization by Dutch scientist Christiaan Huygens and ends in the chaotic world explored by meteorologist Edward Lorenz around 1963. The journey is filled with examples, including synchronized clocks and metronomes, electronic fireflies that flash harmoniously, and even a chaotic dress.

This study presents the effectiveness of dynamic coupling as a synchronization strategy for fractional chaotic systems. Using an auxiliary system as a link between the oscillators, we investigate the onset of synchronization in the coupled systems and we analytically determine the regions where both systems achieve complete synchronization. In the analysis, the integration order is considered as a key parameter affecting the onset of full synchronization, considering the stability conditions for fractional systems. The local stability of the synchronous solution is studied using the linearized error dynamics. Moreover, some statistical metrics such as the average synchronization error and Pearson’s correlation are used to numerically identify the synchronous behavior. Two particular examples are considered, namely, the fractional-order Rössler and Chua systems. By using bifurcation diagrams, it is also shown that the integration order has a strong influence not only on the onset of full synchronization but also on the individual dynamic behavior of the uncoupled systems.

This work presents a multiscroll generator system, which addresses the issue by the implementation of 9-level saturated nonlinear function, SNLF, being modified with a new control parameter that acts as a bifurcation parameter. By means of the modification of the newly introduced parameter, it is possible to control the number of scrolls to generate. The proposed system has richer dynamics than the original, not only presenting the generation of a global attractor; it is capable of generating monostable and bistable multiscrolls. The study of the basin of attraction for the natural attractor generation (9-scroll SNLF) shows the restrictions in the initial conditions space where the system is capable of presenting dynamical responses, limiting its possible electronic implementations.