Analysis of a logistic sequential process

In equation (3.83) x is a state variable whereas b is a control parametr. As will turn out, the iterative process (3.83) is virtually independent of the initial value xx. First, let us examine fixed points of the process. Substitution of the stationary state requirement x +1 = x = x into (3.83) yields the equation for fixed points 

Properties of the process (3.83) for small x values are easy to find. For 

Since the sequence xn must be bounded, the parameter b has to satisfy the condition 0 4b 1. Hence, when be[0, 1/4], the sequence xn approaches zero, i.e. the attracting fixed point x (1) irrespective of the xt value (xt must be different from x 2)). 

When a continuous change in the control parametr b results in exceeding the value b0 = 1/4, we have a loss of stability by the fixed point x (1). The new stable fixed point, x (2) = 1 — (1/4b), close to x,(1) for b close to b0, appears. Such a catastrophe is called bifurcation. Catastrophic behaviour of the process (3.83) for b b0 is revealed in the fact that the solution (3.87) for b = b0 + e diverges to infinity for an arbitrarily small positive e. 

To show that for b b0 the fixed point x (2) acquires stability, equation (3.83) will be linearized in the vicinity of this point (approximation of equation (3.83) by (3.86) is linearization in the neighbourhood of the point x (1) = 0). Hence, let us represent x in the form 

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