Stability in a hebbian network of kuramoto oscillators with second-order couplings for binary pattern retrieve

We study the stability of an oscillatory associative memory network consisting of N coupled Kuramoto oscillators with applications in binary pattern retrieve. In this model, the coupling function consists of a Hebbian term and a second-order Fourier term with nonnegative strength \varepsilon . In [Phys. D, 197 (2004), pp. 134-148] Nishikawa, Hoppensteadt, and Lai studied the stability using the approach of linearization; the criteria for stability/instability is given by the spectrum of linearization which is a matrix of order N. In recent literature [SIAM J. Appl. Dyn. Syst., 14 (2015), pp. 188-201], H\"olzel and Krischer considered the model with \varepsilon = 0 and introduced the orthogonality of binary patterns so that the eigenvalues of linearization can be calculated. In this paper, we will present conditions for stability/instability based on the gradient formulation. First, we use the potential estimate to derive a criteria for stability/instability by the spectrum of a matrix of order N 1. This potential estimate also gives convergence rate under some conditions. Second, we focus on the special case with mutually orthogonal memorized patterns. We find a sufficient and necessary condition for a binary pattern to be stable for any \varepsilon > 0. For any other binary pattern we prove that there exists a critical value of \varepsilon below which it is unstable. A lower bound for this critical strength is provided. A significant advantage of the results in this case is that the conditions for stability/instability are easy to verify and the lower bound of critical strength is easy to compute. Third, when the memorized patterns are not mutually orthogonal, we suggest a framework to transform it into the case of orthogonal memorized patterns. Simulations are presented to illustrate our results.