Stability of Periodic Orbits - Floquet Theory - Stable and Unstable Invariant Manifolds

Explore the stability of periodic orbits and invariant manifolds in this comprehensive lecture on Hamiltonian and nonlinear dynamics. Delve into the monodromy matrix, Floquet theory, and the analysis of stable and unstable manifolds. Learn about the state transition matrix for periodic orbits, the relationship between monodromy matrices and Poincaré maps, and the constraints on eigenspectra in Hamiltonian systems. Examine various scenarios in 3D and 6D phase spaces, including saddle-type periodic orbits and their manifolds. Conclude with an introduction to chaos in Hamiltonian systems using the Duffing system as an example.

State transition matrix introduction.
State transition matrix for periodic orbit (monodromy matrix).
Stability of the periodic orbit from monodromy matrix eigenvalues.
Floquet multipliers, characteristic multipliers.
Example scenarios in 3D.
Saddle-type periodic orbit with stable and unstable manifolds.
Periodic orbits in Hamiltonian systems.
Example scenarios for 3 degrees of freedom (6D phase space).
Chaos in Hamiltonian systems, introduction via Duffing system.