Hebbian Network of Kuramoto Oscillators with Second-Order Couplings for Binary Pattern Retrieve: II. Nonorthogonal Standard Patterns and Structural Stability

We continue the study on the Hebbian network of Kuramoto oscillators with a second-order Fourier term, aiming to apply the system to the binary pattern retrieve task with nonorthogonal standard binary patterns. In [SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 1124-1159] the authors considered the system and studied the stability/instability by introducing the so-called ϵ-independent stability; this theory deals with the scenario where the memorized patterns coincide with the standard ones and the key assumption is the mutual orthogonality in standard patterns (or memorized ones). The key idea was to memorize these mutually orthogonal binary patterns in the network and retrieve one of them for a given defective pattern. However, in practice the orthogonality usually fails. When the orthogonality in the standard patterns fails, we find it is not a proper use of the system memorizing these standard patterns. In this paper we propose a new strategy which converts the problem to a case with orthogonality, and the standard patterns are ϵ-independently asymptotically stable. The structural stability is also considered, and sharp quantitative estimates on the strength of perturbations for the stability of binary patterns are presented. Numerical simulations illustrate the approach and the results.