I am trying to come up with the strategy to write my Master's thesis in mathematics. At the moment it is as follows:
- Finding a (preferably) discrete dynamical system that possesses at least 3 distinct attractors.
- Taking many different initial conditions (a grid of them, in 2D case) and coloring them according to the attractor that the system eventually converges to.
- Looking at the boundary of these basins of attraction in terms of fractal dimension, Wada property, etc.
I noticed that the generalized Newton's method $$ z_{n+1}=z_n- a \frac{p(z_n)}{p'(z_n)} $$ gives somewhat interesting results even for a relatively straighforward polynomial $p(z) = z^3-1$.
Below you can see basins of attraction for a few different $a \in \mathbb{C}$ values.
However I received criticism that the model is maybe too simple. The discussion shifted to suggestions to look at some other systems namely Henon-Hailes system (which is not discrete at all) and forced Duffing oscillator (which also has to be integrated numerically).
I am not satisfied with these suggestions. Do you know any dynamical systems that could provide a rich variety of basin boundaries?
Thank you in advance!
There's nothing wrong with "simple" models, especially those that produce interesting results. There's no reason to prefer a more complicated model just because it's more complicated.
Since "reference-request" was one of your tags: you might look at Milnor's book "Dynamics in One Complex Variable".