Minimal instability and unstable set of a phase-locked periodic orbit in a delayed neural network. (English) Zbl 0942.34062

The authors consider the following delay differential system \[ \dot{U}(t)=-\mu U(t)+f(V(t-\tau)),\qquad \dot{V}(t)=-\mu V(t)+f(U(t-\tau)), \] where \(f:\mathbb{R}\to \mathbb{R}\) is a \(C^1\)-smooth function with = \(f(0)=0\). Such a system arises in modelling of artificial neural networks with electronic circuit implementation (Hopfield’s model). It is proved the existence of a nontrivial phase-locked periodic orbit. The instability and the stable manifold of this orbit and the spectrum of the related monodromy operator are discussed. The existence of a smooth solid spindle is proved, contained in the global attractor, separated by a disk bordered by a phase-locked orbit.


34K13 Periodic solutions to functional-differential equations
34K20 Stability theory of functional-differential equations
37C75 Stability theory for smooth dynamical systems
34K60 Qualitative investigation and simulation of models involving functional-differential equations
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
93B35 Sensitivity (robustness)
92B20 Neural networks for/in biological studies, artificial life and related topics
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