Plot of the fifth iterate of the logistic function. Zeros are the fixed points of the map.
The Logistic Map Fixed Points model finds periodic trajectories of the logistic map such that n iterations of the map starting with an initial value x return to that value. To find these fixed points, we need to find values of x such that
f(n)(x)=x
where f(n) is the n-th iteration of the logistic map applied to x. Algorithms for find x are called root-finding algorithms because we are finding the roots of the function g(x)=0 where
g(x) = f(n)(x)-x.
The most common root finding algorithms are bisection and Newton's method. This model uses an Apache Commons Mathematical Library implementation of the Brent-Dekker algorithm which combines root bracketing, bisection, and inverse quadratic interpolation.
The key idea for this model is that for values of the logistic map control parameter r in the chaotic regime, there are periodic but unstable trajectories. In other words, although chaotic orbits are almost always aperiodic, there exist periodic unstable orbits embedded in this aperiodic sea. This property of the chaotic regime means that if we choose the value of the seed x0 to be precisely equal to a point on an unstable trajectory with period n, the subsequent trajectory will have this period. However, if we choose a value of x0 that differs ever so slightly from this special value, the trajectory will not be periodic. This model shows how to find these special values of x0.
The following EJS models are described in Chapter 6 of the EJS adaptation of An Introduction to Computer Simulation Methods.
The Logistic Fixed Point model was developed by Wolfgang Christian using the Easy Java Simulations (EJS) modeling tool version 4.3. You can examine and modify the model for this simulation if you have EJS installed by right-clicking within the program and selecting "Open Ejs Model" from the pop-up menu.
Information about EJS is available at: <http://www.um.es/fem/Ejs/> and in the OSP ComPADRE collection <http://www.compadre.org/OSP/>.