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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) · Zbl 0840.92003
[2] Bélair, J.; Dufour, S., Canad. Appl. Math. Quarterly, 4, 135 (1996) · Zbl 0880.34075
[3] R.H. Bing, The Geometric Topology of 3-Manifolds, vol. 40, American Mathematical Society, Providence. RI, 1983.; 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) · Zbl 0553.92009
[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.; 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) · Zbl 0815.92001
[7] Hopfield, J. J., Proc. Natl. Acad. Sci., 79, 2554 (1982) · Zbl 1369.92007
[8] Hopfield, J. J., Proc. Natl. Acad. Sci., 81, 3088 (1984) · Zbl 1371.92015
[9] T. Krisztin, H.-O. Walther, Unique periodic orbits for delayed positive feedback and the global attractor, preprint.; 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.; T. Krisztin, H.-O. Walther, J. Wu, The Fields Institute Monographs Series, vol. 11, American Mathematical Society, Providence. RI, 1999. · Zbl 1004.34002
[11] Lani-Wayda, B.; Walther, H.-O., Differential and Integral Equations, 8, 1407 (1995) · Zbl 0827.34059
[12] Mallet-Paret, J.; Sell, G., J. Diff. Eq., 125, 385 (1996) · Zbl 0849.34055
[13] Mallet-Paret, J.; Sell, G., J. Diff. Eq., 125, 441 (1996) · Zbl 0849.34056
[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.; 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) · Zbl 0612.34067
[18] Wu, J., Trans. Amer. Math. Soc., 350, 12, 4799 (1998) · Zbl 0905.34034
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