A Fast Method for Detecting Interdependence between Time Series and Its Directionality

We propose a fast nonlinear method for assessing quantitatively both the existence and directionality of linear and nonlinear couplings between a pair of time series. We test this method, called Boolean Slope Coherence (BSC), on bivariate time series generated by various models, and compare our results with those obtained from different well-known methods. A similar approach is employed to test the BSC’s capability to determine the prevalent coupling directionality. Our results show that the BSC method is successful for both quantifying the coupling level between a pair of signals and determining their directionality. Moreover, the BSC method also works for noisy as well as chaotic signals and, as an example of its application to real data, we tested it by analyzing neurophysiological recordings from visual cortices.

Ground States of a 𝐾-Component Critical System with Linear and Nonlinear Couplings: The Attractive Case

Consider the system \left\{\begin{aligned} \displaystyle-\Delta u_{i}+\mu_{i}u_{i}&\displaystyle=% \nu_{i}u_{i}^{2^{*}-1}+\beta\mathop{\sum_{j=1,j\neq i}^{k}}u_{j}^{\frac{2^{*}}% {2}}u_{i}^{\frac{2^{*}}{2}-1}+\lambda\mathop{\sum_{j=1,j\neq i}^{k}}u_{j}&&% \displaystyle\phantom{}\text{in}\ \Omega,\\ \displaystyle u_{i}&\displaystyle>0&&\displaystyle\phantom{}\text{in}\ \Omega,% \\ \displaystyle u_{i}&\displaystyle=0&&\displaystyle\phantom{}\text{on}\ % \partial\Omega,\quad i=1,2,\ldots,k,\end{aligned}\right. where {k\geq 2} , {\Omega\subset\mathbb{R}^{N}} ( {N\geq 3} ) is a bounded domain, {2^{*}=\frac{2N}{N-2}} , {\mu_{i}\in\mathbb{R}} and {\nu_{i}>0} are constants, and {\beta,\lambda>0} are parameters. By showing a unique result of the limit system, we prove existence and nonexistence results of ground states to this system by variational methods, which generalize the results in [7, 18]. Concentration behaviors of ground states for {\beta,\lambda} are also established.