Relaxation self-oscillations in neuron systems. II
Consider the diffusively coupled system of delay-differential equations modelling the interaction of two neurons \[ \begin{aligned} {du_1\over dt} &= \lambda[-1+\alpha f(u_1(t- 1))-\beta g(u_1)]\,u_1+ d(u_2- u_1),\\ {du_2\over dt} &= \lambda[-1+\alpha f(u_2(t- 1))-\beta g(u_2)]\,u_2+ d(u_1- u_2)\end{aligned}\tag{\(*\)} \] under the following assumptions (i) \(\alpha\), \(\beta\), \(d\), \(\lambda\) are positive parameters satisfying \(\alpha>1+\beta\), \(\alpha<2(1+\beta)\), \(\lambda\gg 1\). (ii) \(f\), \(g\) are twice continuously differentiable functions with the properties \[ f(0)= g(0)= 1,\, 0<\beta g(u)+ 1<\alpha\text{ for }u\geq 0, \] \[ f(u),\;g(u),\;uf'(u),\;ug'(u),\;u^2f''(u),\;u^2g''(u)= O\Biggl({1\over u}\Biggr)\text{ as }u\to+\infty. \] Under these conditions, \((*)\) has the so-called homogeneous cycle \(u_1(t,\lambda)= u_2(t,\lambda)= u_*(t,\lambda)\) obeying the equation \[ {du\over dt}= \lambda[-1+\lambda f(u(t-1))-\beta g(u)]\,u \] (see the review of Part I [ibid. 47, No. 7, 927--941 (2011); translation from Differ. Uravn. 47, No. 7, 919--932 (2011; Zbl 1276.34068)]). The authors prove that the homogeneous cycle is an exponentially orbitally stable solution of \((*)\); additionally they show that, for an appropriate choice of the parameters \(\alpha\), \(\beta\) and \(d\), system \((*)\) has at least two stable inhomogeneous periodic solutions \((u^i_1(t,\lambda),u^i_2(t,\lambda))\), \(i= 1,2\), where one solution can be obtained from the other one by the permutation \(u_1\to u_2\), \(u_2\to u_1\).
scientific article; zbMATH DE number 6032728