Dynamics of a class of nonlinear discrete-time neural networks
A second-order ordinary differential equation (ODE)-system with piecewise constant terms is transformed into a discrete nonlinear recurrence (\(\mathbb{R}^2\)-point map). Its properties are studied numerically, leading to ``interesting results'', which have turned out to be more complicated than expected. -- The unawoved reason for this cautious statement can be attributed to Birkhoff's proof that the space of functions defined by recurrences is larger than that defined by ODE-s, which extends Poincaré's proof that most ODE systems are non-integrable: In contemporary terminology this means that the corresponding dynamics are complex (`chaotic', although deterministic). Although the ODE-recurrence transformation is singular, the `observed' synchronized periodic solutions correspond to stable cycles. No unstable cycles are reported, although they are well-known to exist in nonlinear recurrences, their accumulations leading to homoclinic and heteroclinic points.
scientific article; zbMATH DE number 2142447