Question

Discuss the presence and stability of

4-cycle & 3-cycle of the discrete logistic map _Xn+1=mu x_n (1-x_n)

Hints: Use Schorkovski Theorem for 3-cycle. For 4-cycle you may use Newtons method with mu=3.5

Answer 1

Steady state and stability

The concept of steady state (or equilibrium) relates to the absence of changes in a system. In the context of difference equations, the steady state Xss is defined by Xn+1 = Xn = Xss

For the logistic equation, the steady state is then Xss = rXss(1 ? Xss) (6) rX2 ss ? Xss(r ? 1) = 0

Two steady states are possible: Xss1 = 0 and Xss2 = 1 ? 1/r By definition, a stable steady state is a state that can be reached from the neighbour states, whereas an unstable steady state is a state that the system will leave as soon as a small perturbation will move the system out of this state. The notion of stability is schematized here: The stability is a local property. It can be calculated by applying a small

We are interested in the task of computing fixed points and N-cycles for the well known and much studied discrete logistic equation xn+1 = kxn (1 ? xn) (1)

where k is a constant satisfying 0 ? k ? 4. We are particularly interested in values of k between 3 and 4. Background and a detailed discussion of may be found in

A delightful elementary introduction may be found in ; and a more detailed mathematical treatment may be found . Finding N-cycles entails finding the fixed points of FN (x),

the Nth composition of the function F(x) = kx(1 ? x) with itself.

We thus must find the solutions of FN (x) = x.

We must then determine the N-cycles determined by the fixed points.

We are interested primarily in