Introduction to Dynamical Systems - Lecture 1

Explore the fundamentals of dynamical systems in this comprehensive lecture by Soumitro Banerjee from the International Centre for Theoretical Sciences. Begin with discrete-time examples and ordinary differential equations, then progress to equilibrium points, linear ODEs, and eigenvalue calculations. Delve into nonlinear systems, examining attractors, limit cycles, and the Lorenz system. Investigate chaos theory and orbits on a torus before engaging in a Q&A session. Conclude by studying Poincaré sections, one-dimensional maps, fixed point stability, and various bifurcation types, including saddle-node and period-doubling bifurcations.

Start
Example: Discrete-time
ODE:
Equilibrium points
Example
Solution of linear ODEs
Eigenvalues and eigenvectors
Calculation of eigenvalues
Complex eigenvalues
3D systems
On to nonlinear systems
Attractors in nonlinear systems
Limit cycle
The Lorenz system
Chaos
Orbit on a torus
Q&A
The Poincare section
The Poincare map
One-dimensional maps
Graphical iteration
Stability of fixed points
Bifurcation diagram
Saddle-node bifurcation
Period doubling bifurcation