Two-Delay System Including the Logistic Map: Disappearance of Attractors Owing to Anomalous Bifurcation Process

Abstract

We study bifurcations of fixed points and oscillations generated by them for a two-delay differential equation in which the logistic maps xi+1 = μf(xi) with f(x) = x(1 − x) are embedded; in particular, we consider the behavior where two fixed points xo (= 0) and xc interchange their properties through a transcritical bifurcation. We focus on the case that the system has a long delay time t2 with positive strength α2 (> 0) and a short delay time t1 with negative strength α1 (< 0) such that 0 < t1 < t2 and −α2 < α1 < 0. In the range μf′(xc) < 0, oscillation modes appearing at the first bifurcations of the fixed point xc change with t1/t2 and obey the mode selection rule of the boosted bifurcation process (BBP). However, in the range μf′(xc) > 0, if |α1| is not extremely small, there exists a non-attractor region of μ in the neighborhood of the transcritical bifurcation point. This is different from the case of a two-delay system with Gaussian maps and one fixed point, in which attractors appear by an Anomalous Bifurcation Process (ABP). When t1/t2 is a rational number, n/m, the two-delay system can be expressed as an m-dimensional map in the nondispersive limit, i.e., a singular perturbation limit. Analyses of the 2D map at t1/t2 = 1/2 and the 3D maps at t1/t2 = 1/3 and 2/3 show that the non-attractor region is attributable to the subcritical bifurcations of xo and xc induced by ABP that occur at the lower and upper ends of the region, respectively. Neither xo nor xc are stable in the non-attractor region. Numerical calculations for the dispersive system show that non-attractor regions are observed for almost the whole range of the parameters, t1 and α1, and become wider as α1 approaches −α2.