# Advanced Mathematical Methods 1: Numerical Investigation of Chaos in Nonlinear Hamiltonian Systems

*20 credits / 30 lectures*

Nonlinear Hamiltonian dynamics is used to study the behavior of systems coming from a wide variety of scientific fields, the most important of them being classical mechanics, astronomy, optics, electromagnetism, solid state physics, quantum mechanics, and statistical mechanics. An important phenomenon appearing in nonlinear systems is chaos, which is attributed to the sensitive dependence of a system’s dynamical evolution on its initial conditions. In this course we will implement several modern numerical techniques to investigate and quantify the chaotic behavior of low-dimensional Hamiltonian systems and area preserving symplectic maps. In particular we will discuss the following topics:

- Chaos
- Autonomous Hamiltonian systems and symplectic mappings
- Numerical integration of Hamilton equations of motion
- Poincaré surface of section
- Integrals of motion
- Symplectic integrators
- Variational equations
- Tangent Map Method
- Maximum Lyapunov exponent
- Spectrum of Lyapunov exponents
- Chaos indicators

Prerequisites:

The undergraduate Applied Mathematics courses- MAM1043H
- MAM1044H
- MAM2046W
- MAM3040W