Logistic Map Potentials

Prof. Curtright has published a new physics paper with a connection to the logistic map, which has inspired so much work. 

After more than thirty years that this simple equation entered in physics, we do not seem to get enough!

 

We can read Curtright and Veitia's conclusion:

"V. CONCLUSION
 

The point of view supported in this paper is that the logistic map, and other discrete time-stepped dynamical models, may be regarded as continuously evolving Hamiltonian systems sampled at integer times. For this view to be valid, the continuous system must be allowed to undergo a series of switchbacks whereupon the potential affecting the dynamics changes when the evolving particle encounters a turning point. From a perspective of configuration space covering manifolds, in the case of simple one-dimensional motion, the particle moves from one sheet of a Riemann surface to another, to experience a different branch of the underlying analytic potential.
 

The methods of this paper may be used directly to determine such branches of the potential for the logistic map, for any value of s, as well as other one-dimensional maps. A more extensive study of other examples is underway [9]. While peculiar behavior is possible for exceptional maps (say, for special values‡ of s), so far as we are aware, any such behavior can always be analyzed using the potential framework presented here.
 

In total, for all parameter values governing a particular map, the collection of potential sequence families constitute what we may call a “potential fractal” with self-similarities qualitatively evident in the various graphs, as visible in the above. Perhaps such potential fractals have a significant role to play in continuum physics. Applications might involve any of the usual systems exhibiting chaotic behavior [7], including accelerator beams [10], or perhaps cosmological models [11]."