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Bifurcation and stability analysis of nonlinear waves in $\Bbb {D}_n$ symmetric delay differential systems. (English) Zbl 1118.34067
The paper is concermed with the array of identical coupled oscillators with delay $$ \dot x_j(t) = -x_j(t) + \alpha f(x_j(t-\tau)) + \beta \left[ g(x_{j-1}(t-\tau))+g(x_{j+1}(t-\tau)) \right],$$ where $j$ is considered mod $n$, $\tau>0$, $f(0)=g(0)=0$, and $f'(0)=g'(0)=1$. The system is ${\Bbb D}_n$-equivariant and has an equilibrium at the origin. The author studies stability of the origin and properties of the bifurcating solutions. In particular, the analysis of the characteristic equation reveals multiple Hopf (or pitchfork) bifurcations, which give rise to a variety of symmetric periodic solutions (or equilibria) as the delay or another bifurcation parameter varies.

34K18Bifurcation theory of functional differential equations
92B20General theory of neural networks (mathematical biology)
34C14Symmetries, invariants (ODE)
34K13Periodic solutions of functional differential equations
Full Text: DOI
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