Question

2. Consider the logistic equation where a e (0,4) and o E (0,1) are parameters. Interesting things happen in the regime a e (3,4) Plot a trajectory xn as a function of n for some suitable a and xo Make a bifurcation plot: As a function of a (along the r-axis) show an ensemble (for randomly chosen xo) of n (along the y-axis), where n is a reasonably large number of steps For values of a close to, but not equal, four make a histogram of I values. Does the rest depend on ro?

It is better if this question is worked with matlab.

Answer 1

xn+1 = rxn(1 − xn)

is called the ”logistic map”, f(r,x) = rx(1 − x). This simple equation yields many of the properties of non-linear systems. The allowed values for r are 0 ≤ r ≤ 4. If r is restricted to this range, then the xn will lie between zero and one: 0 ≤ xn ≤ 1.

Difference equations appear as a natural way of evolution phenomena because most measurements of time evolving variables are discrete and these equations are important in mathematical models. More importantly, difference equations also appear in the study of discretization methods for differential equations.

The logistic difference equation was popularized in a seminal 1976 paper by the biologist Robert May [2], as a discrete-time demographic model analogous to the logistic equation first created by Pierre Franois Verhulst [1]. The discrete version of the logistic model is written as

xn+1 = axn(1 − xn), n = 0, 1, ....

We mention that if a < 1, the model describes extinction of population. In this talk we consider the fuzzy difference equations xn+1 = βxn(1 − xn),

and xn+1 = βxn βxn 2 , n = 0, 1, ...,

where xn is a sequence of positive fuzzy numbers, β and initial value x0 are positive fuzzy numbers and denotes the Hukuhara difference (H-difference) of two fuzzy numbers.

Proposition 1. Consider Eq. (2) where xn is a sequence of positive fuzzy numbers and x0, β ∈ R + F . If x0,α, βα < 1, ∀α ∈ (0, 1], then for every positive fuzzy number x0, there exists a unique positive solution xn of (2) with initial condition x0.

Proposition 2. Consider Eq. (3) where xn is a sequence of positive fuzzy numbers and β, x0 ∈ R + F such that βα < 1, x0,α ≤ 1 2 , ∀α ∈ (0, 1]. Then for every positive fuzzy number x0, there exists a unique positive solution xn of (3)