| Volume 2, Number 2, 2012, Pages 213-240 DOI：10.11948/2012016 |
| Evans functions and bifurcations of standing wave solutions in delayed synaptically coupled neuronal networks |
| Linghai Zhang |
| Keywords:Delayed synaptically coupled neuronal networks |
| Abstract: |
| Consider the following nonlinear singularly perturbed system of integral differential equations &\frac{\partial u}{\partial t}+f(u)+w\\ =&(\alpha-au)\int^{\infty}_0\xi(c)\left[\int_{\mathbb R}K(x-y) H\left(u\left(y,t-\frac1c|x-y|\right)-\theta\right){\rm d}y\right]{\rm d}c\\ &+(\beta-bu)\int^{\infty}_0\eta(\tau)\left[\int_{\mathbb R}W(x-y)H\big(u(y,t-\tau)-\Theta\big){\rm d}y\right]{\rm d}\tau,\\ &\frac{\partial w}{\partial t}=\varepsilon[g(u)-w], and the scalar integral differential equation &\frac{\partial u}{\partial t}+f(u)\\ =&(\alpha-a u)\int^{\infty}_0\xi(c)\left[\int_{\mathbb R}K(x-y) H\left(u\left(y,t-\frac1c|x-y|\right)-\theta\right){\rm d}y\right]{\rm d}c\\ &+(\beta-bu)\int^{\infty}_0\eta(\tau)\left[\int_{\mathbb R}W(x-y)H\big(u(y,t-\tau)-\Theta\big){\rm d}y\right]{\rm d}\tau. There exist standing wave solutions to the nonlinear system. Similarly, there exist standing wave solutions to the scalar equation. The author constructs Evans functions to establish stability of the standing wave solutions of the scalar equation and to establish bifurcations of the standing wave solutions of the nonlinear system. |
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