## 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.

### MSC:

 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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 [1] Bélair, J.; Campbell, S.A.; van den Driessche, P., SIAM J. appl. math., 56, 1, 245, (1996) [2] Bélair, J.; Dufour, S., Canad. appl. math. quarterly, 4, 135, (1996) [3] R.H. Bing, The Geometric Topology of 3-Manifolds, vol. 40, American Mathematical Society, Providence. RI, 1983. · Zbl 0535.57001 [4] Cohen, M.A.; Grossberg, S., IEEE trans. SMC., 13, 815, (1983) [5] Y. Chen, J. Wu, Existence and attraction of a phase-locked oscillation in a delayed network of two neurons, Advances in Differential Equations, in press. · Zbl 1023.34065 [6] Gopalsamy, K.; He, X.Z., Physica D, 76, 344, (1994) [7] Hopfield, J.J., Proc. natl. acad. sci., 79, 2554, (1982) [8] Hopfield, J.J., Proc. natl. acad. sci., 81, 3088, (1984) [9] T. Krisztin, H.-O. Walther, Unique periodic orbits for delayed positive feedback and the global attractor, preprint. · Zbl 1008.34061 [10] T. Krisztin, H.-O. Walther, J. Wu, The Fields Institute Monographs Series, vol. 11, American Mathematical Society, Providence. RI, 1999. [11] Lani-Wayda, B.; Walther, H.-O., Differential and integral equations, 8, 1407, (1995) [12] Mallet-Paret, J.; Sell, G., J. diff. eq., 125, 385, (1996) [13] Mallet-Paret, J.; Sell, G., J. diff. eq., 125, 441, (1996) [14] Marcus, C.M.; Westervelt, R.M., Phys. rev. A, 39, 347, (1989) [15] K. Pakdaman, C. Grotta-Ragazzo, C.P. Malta, O. Arino, J.-F. Vibert, Effect of Delay on the Boundary of the Basin of Attraction in a System of Two Neurons, preprint. · Zbl 0869.68086 [16] Pakdaman, K.; Malta, C.P.; Grotta-Ragazzo, C.; Vibert, J.-F., Neural computation, 9, 319, (1957) [17] Smith, H.L., J. diff. eq., 66, 420, (1987) [18] Wu, J., Trans. amer. math. soc., 350, 12, 4799, (1998)
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