I am interested in Dynamical Systems.
Dynamical Systems are models for real-world deterministic processes.
A dynamical system with continuous time is a system of differential equations. For example, our Solar system is governed by Newton's gravity laws, i.e. by the system of ordinary differential equations. (see a famous 3-body problem).
A dynamical system with discrete time is a single function f, and we consider its iterates f(x), f(f(x)) f(f(f(x)))... For example, let us model the number of bugs in the forest: if we have x bugs this year, then we will have f(x) bugs next year. A good model here is a Logistic map f(x) = ax(1-x), a is a parameter. Iterates of f describe what happens in several years.
Dynamical Systems Theory studies the limit behavior of the system after a lot of time has passed. Is the Solar system stable? Will the bugs die out in several years?.
Bifurcation Theory studies bifurcations - abrupt changes in the behavior of the system. Example: the famous
Mandelbrot set is a bifurcation diagram for a simple map z → z2+c in the complex plane.
In Dynamical Systems, I am interested in: