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 → z^{2}+c in the complex plane.

In Dynamical Systems, I am interested in: