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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.

Background. Depression disorder has been associated with altered oscillatory brain activity. The common methods to quantify oscillatory activity are Fourier and wavelet transforms. Both methods have difficulties distinguishing synchronized oscillatory activity from nonrhythmic and large-amplitude artifacts. Here we proposed a method called self-synchronization index (SSI) to quantify synchronized oscillatory activities in neural data. The method considers temporal characteristics of neural oscillations, amplitude, and cycles, to estimate the synchronization value for a specific frequency band. Method. The recorded electroencephalography (EEG) data of 45 depressed and 55 healthy individuals were used. The SSI method was applied to each EEG electrode filtered in the alpha frequency band (8-13 Hz). The multiple linear regression model was used to predict depression severity (Beck Depression Inventory-II scores) using alpha SSI values. Results. Patients with severe depression showed a lower alpha SSI than those with moderate depression and healthy controls in all brain regions. Moreover, the alpha SSI values negatively correlated with depression severity in all brain regions. The regression model showed a significant performance of depression severity prediction using alpha SSI. Conclusion. The findings support the SSI measure as a powerful tool for quantifying synchronous oscillatory activity. The data examined in this article support the idea that there is a strong link between the synchronization of alpha oscillatory neural activities and the level of depression. These findings yielded an objective and quantitative depression severity prediction.

Obtaining the master stability function is a well-known approach to study the synchronization in networks of chaotic oscillators. This method considers a normalized coupling parameter which allows for a separation of network topology and local dynamics of the nodes. The present study aims to understand how the dynamics of oscillators affect the master stability function. In order to examine the effect of various characteristics of oscillators, a flexible oscillator with adjustable amplitude, Lyapunov exponent, and Kaplan–Yorke dimension is used. Not surprisingly, it is demonstrated that the amplitude of the oscillations has no substantial effect on the master stability function, i.e. the coupling strength needed for the complete synchronization is not changed. However, the flexible oscillators with larger maximal Lyapunov exponent synchronize with larger coupling strength. Interestingly, it is shown that there is no linear connection between the Kaplan–Yorke dimension and coupling strength needed for complete synchronization.