No abstract

Localizing hidden attractors of chaotic systems is practically and theoretically important. Differing from self-excited attractors, hidden ones do not have any equilibria on the boundaries of their basin of attraction. This characteristic makes hidden attractors hard to localize. Some theoretical and numerical methods have been developed to recognize these attractors, yet the problem remains highly uncertain. For this purpose, the theory of connecting curves is utilized in this work. These curves are one-dimensional set-points that describe the structure of chaotic attractors even in the absence of zero-dimensional fixed-points. In this study, a new four-dimensional chaotic system with hidden attractors is presented. Despite the controversial idea of connecting curves that pass through fixed-points, the connecting curves of a system with no equilibria are considered. This analysis confirms that connecting curves provide more critical information about attractors even if they are hidden.

Studying the stability of synchronization of coupled oscillators is one of the prominent topics in network science. However, in most cases, the computational cost of complex network analysis is challenging because they consist of a large number of nodes. This study includes overcoming this obstacle by presenting a method for reducing the dimension of a large-scale network, while keeping the complete region of stable synchronization unchanged. To this aim, the first and last non-zero eigenvalues of the Laplacian matrix of a large network are preserved using the eigen-decomposition method and, Gram-Schmidt orthogonalization. The method is only applicable to undirected networks and the result is a weighted undirected network with smaller size. The reduction method is studied in a large-scale a small-world network of Sprott-B oscillators. The results show that the trend of the synchronization error is well maintained after node reduction for different coupling schemes.