P- and D-Bifurcations of the Multiplicatively Perturbed Logistic Map

The multiplicative perturbation chosen is a dichotomic noise process, i.e. a Markov chain assuming only two values a and b with symmetric transition probability matrix where 0 < < 1 is the probability to keep the current state. The transition matrix is primitive and thus the resulting Markov chain is ergodic. For = 0.5 one obtains an i.i.d. process. We will study the dependence of the bifurcation behavior on the two values the noise process assumes and on the transition probabilities which are completely specified by .

The multiplicatively perturbed logistic map is given by

xn+1 = xn (1 - xn)

where is the Markov chain introduced above. Note that zero is a solution of this equation for any noise process.

The links given below lead to P- and D-bifurcation diagrams of the multiplicatively perturbed logistic map. We fix different values of α  and vary the mean of the noise process ξ. We further assume that is symmetrically distributed, i.e. assumes the two states + a and - a, where a is a real constant.

The first list of links leads to P- and D-bifurcation diagrams. The second list of links is devoted to Lyapunov exponents only. There the dependence of the Lyapunov exponent on the two states a,b in [0,4] the Markov chain assumes is depicted. The Lyapunov exponent diagrams associated to the first list are sections parallel to the diagonal of the corresponding figures in the second list of links.

Complete P- and D- bifurcation diagrams.
= 0.5
= µ ± 0.01 		= 0.5
= µ ± 0.1 		= 0.5
= µ ± 0.15 		= 0.5
= µ ± 0.2
= 0.01
= µ ± 0.15 		= 0.25
= µ ± 0.15 		= 0.85
= µ ± 0.15 		= 0.95
= µ ± 0.15


Lyapunov exponent diagrams for a,b in [0,4] as 2d presentations accompanied by interpretations and as 3d plots.
= 0.01 (3d) = 0.25 (3d) = 0.50 (3d) = 0.85 (3d)


In summary, the numerical results lead to the conjectures


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Introduction
Additive noise case
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