In this course, we discuss qualitative analysis of differential and discrete dynamical systems.

First unit is about the scalar autonomous systems. The concepts such as equilibrium points and their stability results are discussed in details. Various types of bifurcations are studied with ample number of examples. The open source software "Winplot" is used to draw solution trajectories, slope fields and bifurcation diagrams.

Second and third unit is about the linear systems of two and higher order. The concepts of linear algebra are used to discuss the qualitative behavior of such systems. An elegant way to solve such systems- exponential of a matrix- is introduced. Different properties of matrix exponential are discussed.

Discrete dynamical systems (DDS) are discussed in fourth unit. DDS are very useful in analyzing natural systems (e.g. population model). Fixed points of these systems and their stability results are proved using the results in analysis (e.g. fixed point theorem). The amazing concept of chaos is introduced to students in this unit. The chaotic phase portraits of such systems are elaborated using mathematical software such as Mathematica.

The last unit is based on the problems based on above four units.

- Teacher: Sachin Bhalekar Maths