I'm an engineering student. My college courses didn't include advanced mathematical topics, e.g., abstract algebra and topology. I only had mathematics enough to understand linear dynamical systems and stability analysis using lyapunov method. Three months ago, I started reading a reference in non-linear dynamics . I was startled and couldn't understand most of the words used in the proofs. I then had to take a basic course in real analysis which made me become very fascinated with mathematics. I then finished basic courses in complex analysis , discrete mathematics and now I am currently studying abstract algebra.
I enjoy studying mathematics but I want first to focus on mathematics that can be most useful to me in control theory . I came to the conclusion that it might be differential topology as deals with differentiable functions and manifolds and contains theorems such as Poincare-Hopf theorem, Morse Theory and Brouwer's fixed-point theorem which I think can be used in the study of dynamical systems.
Can someone help me by giving me a step by step list of mathematical topics that I need to study with recommendation for the references.
My current knowledge in real analysis is till Taylor series ,derivative and mean value theorem. In complex analysis, I know till residue theorem . no real study in algebraic topology just heard about some basic concepts such as homotopy groups , CW complex and euler characteristic.