Eckhaus Instability in Systems with Large Delay

The dynamical behavior of various physical and biological systems under the influence of delayed feedback or coupling can be modeled by including terms with delayed arguments in the equations of motion. In particular, the case of long delay may lead to complicated and high-dimensional dynamics. We investigate the effects of delay in systems that display an oscillatory instability (Hopf bifurcation) in the absence of delay. We show by analytical and numerical methods that the dynamical scenario includes the coexistence of multiple stable periodic solutions and can be described in terms of the Eckhaus instability, which is well known in the context of spatially extended systems.

Figure 1

Linear stability of system (1). (a) Curves of pseudocontinuous spectrum, along which eigenvalues accumulate for large delay, shown for three different parameter values. Solid line: $\alpha =0.8$; dashed line: $\alpha =1$; dotted line: $\alpha =1.2$. Destabilization occurs at $|\alpha |=1$. (b) The interval of unstable frequencies $\omega$ is shown in gray for different values of parameter $\alpha$. These frequencies correspond to the interval of $\mathrm{Im}\lambda$, for which the pseudocontinuous spectrum from (a) is unstable (weak instability). In addition, the strong instability region for $\alpha >0$ is indicated. $\beta =1$ is fixed.

Figure 2

“Space-time” representation of the asymptotic states of system (1) for $\alpha =-0.8$ shows numerically the coexistence of stable solutions with different frequencies. The real part of $z$ is plotted. The horizontal axis represents the spacelike direction ranging from 0 to $\tau$. Solutions in (a) and (b) have a different number of maxima per delay interval. They are obtained by choosing different initial conditions (see details in the text). $\tau =80$, $\beta =1$.

Figure 3

Frequency of the limiting periodic attractor ${\omega }_a$ as a function of the frequency of the initial condition ${\omega }_{\mathrm{in}}$. Initial condition: $z_0(s)=0.01(1+i)\cos ({\omega }_{\mathrm{in}}s)$. $\alpha =-0.2$, $\beta =1$.

Figure 4

Dependence of the frequencies of the asymptotic states ${\omega }_a$ on the bifurcation parameter $\alpha$. Close to the destabilization point at $\alpha =-1$, the system exhibits typical Eckhaus instability. The Eckhaus parabola (E) delineates the stability boundary for the periodic states. For $\alpha >0$, the zero state is strongly unstable with respect to the single mode with frequency $\beta =1$ as in the usual Hopf scenario, where no splitting of frequencies occurs.