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The Critical Amplitude for Bifurcation Points of the Logistic Map

Abstract The phenomenon called critical slowing down is studied for some analytically tractable bifurcation points of the logistic map. The critical amplitude and the critical amplitude ratio are introduced to describe the critical behaviors more precisely when the critical exponents are identically unity for two critical behaviors which are to be compared. Since Hao predicted unity of the critical exponent for 1-d maps assuming exponential damping of the distance from the attractor, the assumption is checked. The result is that the shrinkage of the valid region of the assumption occurs as the adjustable parameter approaches one of the bifurcation points.