# Historical study of dynamical system

I am currently doing a historical study on my school project 'study of ODE' which slowly shift to the study of dynamical system as I am interested in pursuing my study of ode from linear system, phase portrait to dynamical system.

I had made my own study and research about the development of dynamical systems. Based on my research, the study of dynamical system started with DE. DE started since Newton and Leibniz introduced the DEs system. Then in 19th century, most of the techniques for solving elementary level of differential equations had been identified. Then till 20th century, with the works of Poincare in between 1880 - 1886, dynamical system started. My questions are :

• When is the idea of equilibrium point discussed? As the study linear system phase portrait, we often started with simplex autonomous linear system which has equilibrium point at origin.

• When do we have a general theory of ODE?

• When do we see the study of local behaviour for a general ODE, for example 2-D linear system.

• Who is the one that discover the phase portrait of linear autonomous system.

• When do dynamical system view of ODE became popular?

I had found few of the contributors to dynamical system which are Henri Poincare, Stephen Smale, George David Birkhoff, Ivar Otto Bendixson and etc? Can someone help me to classify them based on their contribution according to the year?

• That's a very wide question. Poincare is of the central importance here. Everything about "qualitative analysis of dynamical system" started in his works. He knew a lot of facts, which he even did not bother to dwell on. The first time "dynamical system" term appeared in works by Birhoff. Only in the middle of 20th century it was realized that there are many more dynamical systems apart from ODE. The works by Lyapunov on the stability of movement cannot be ignored. And finally, the Russian school by Andronov and his colleagues put a lot of work into the field to make the exposition clear. – Artem Mar 29 '14 at 15:32
• scholarpedia.org/article/History_of_dynamical_systems – Andrés E. Caicedo Mar 29 '14 at 17:40