Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation
An extensive treatment of the parametrized differential-delay equation \((1)\quad \dot x(t)=-\lambda x(t)+\lambda f(x(t-1)),\lambda >0\) is presented with particular emphasis on the bifurcation behaviour as \(\lambda\) is varied from 0 to \(\infty\). In the first section topological fixed point index theory is used to show that the equation (1) has a global continuum of ''slowly oscillating'' periodic solutions (for \(\lambda\in [0,\infty))\). Such solutions are characterized by the fact that the period of the solution is greater than twice the delay. In section 2 similar results are obtained for rapidly oscillating periodic solutions. The period of oscillation of solutions as \(\lambda\to \infty\) is discussed in section 3 and, in particular, it is shown that, under certain conditions, we have \(1<q<1+C/\lambda\), \(2<\bar q<2(1+\bar C/\lambda)\) for some constants \(C,\bar C\) where \(q,\bar q\) are the first two positive zeros of the solution. In the final section it is shown that as \(\lambda\to \infty\) the periodic solutions converge uniformly to a square wave away from the points 0,1,2,.... At these points the nature of the solution is derived by considering the ''transition layer equations'' \[ \dot y(t)=y(t)-f(z(t-r)),\quad \dot z(t)=z(t)-f(y(t-r)), \] and it is shown that, under certain circumstances, one can obtain finite overshoot in any neighbourhood of 0 or 1 as \(\lambda\to \infty\).
parametrized differential-delay equation
scientific article; zbMATH DE number 4000467
scientific article