The Perturbed Rat
Recently, Blanchard, Devaney, Garijo, and Russell have studied rational maps of the form , where and are complex-valued parameters. This Demonstration considers the special case when and . When both the real and imaginary parts of are equal to zero, we obtain a Julia set called "the rat". When is nonzero, there is a pole of order three at the origin that changes the structure of the Julia set and perturbs the rat.
Contributed by: Elizabeth Russell, Sebastian Marotta, and Jeff Hamrick (March 2011)
Based on work by: Paul Blanchard, Bob Devaney, Antonio Garijo, and Elizabeth Russell
Open content licensed under CC BY-NC-SA
When is sufficiently small, the Julia set has a structure analogous to a Cantor set of simple, closed curves for the map , for . This family of maps ("the rat maps") generalizes a result of Kurt McMullen.
To make higher-quality pictures, download the source notebook, increase the number of iterates, and decrease the step size used to create the mesh of points in the complex plane.
P. Blanchard, R. L. Devaney, A. Garijo, and E. Russell, "A Generalized Version of the McMullen Domain," International Journal of Bifurcation and Chaos, 18(8), 2008 pp. 2309–2318.