Stability of Periodic Orbits | Floquet Theory | Stable & Unstable Invariant Manifolds | Lecture 21

Lecture 21, course on Hamiltonian and nonlinear dynamics. Stability of periodic orbits and invariant manifolds. The monodromy matrix, M, is the state transition matrix for one period T of a periodic orbit, obtained from integrating the variational equations related to the Jacobian of the vector field. The monodromy matrix is the linearized approximation of the nonlinear Poincare map near a periodic orbit and the eigenvalues are Floquet multipliers (characteristic multipliers). For a Hamiltonian system, the monodromy matrix M is a symplectic matrix, which puts constraints on the eigenspectrum. Eigenvalues outside the unit circle correspond to unstable manifolds.

► Next: Chaos in Hamiltonian Systems | Separatrix Splitting and Turnstile Lobe Dynamics | Homoclinic Tangle

► Previous, Dynamics of Driven Damped Nonlinear Oscillators | From Analytical and Geometrical Points of View

► Simpler introduction to limit cycles (special type of periodic orbit)

► Advanced Dynamics - Hamiltonian Systems and Nonlinear Dynamics

► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)

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► See the entire playlist for the course:
Advanced Dynamics - Hamiltonian Systems and Nonlinear Dynamics

► Continuation of this course on a related topic
Center manifolds, normal forms, and bifurcations

► Course lecture notes (PDF)

► Course lecture notes (OneNote)

► Chapters
0:00 State transition matrix introduction
2:57 State transition matrix for periodic orbit (monodromy matrix)
12:25 Stability of the periodic orbit from monodromy matrix eigenvalues
16:30 Floquet multipliers, characteristic multipliers
29:33 Example scenarios in 3D
47:11 Saddle-type periodic orbit with stable and unstable manifolds
49:04 Periodic orbits in Hamiltonian systems
59:33 Example scenarios for 3 degrees of freedom (6D phase space)
1:04:41 Chaos in Hamiltonian systems, introduction via Duffing system

Lecture 2020-04-30

Part of a graduate level course:
Advanced Dynamics (ESM/AOE 6314)
Spring Semester, 2020

#NonlinearDynamics #DynamicalSystems #PeriodicOrbit #stability #manifolds #Floquet #FloquetTheory #InvariantManifolds #UnstableManifolds #StableManifolds #FloquetMultipliers #LimitCycle #Poincare #PoincareMap #Monodromy #MonodromyMatrix

[terraeri]

chaos dynamics is so fascinating... thank you ... great presentation...

[hamaiziasara]

Hi, how can i calculate the monodromy matrix if i have the jacobian matrix numerically ?

[deeper]

Great lecture Professor Ross. I wanted to ask, could you recommend any text book which teaches Floquet theory in nonlinear dynamical systems in details. I am currently working on piece wise-smooth (PWS) systems (like impact oscillator, Filippov systems which show chattering and sliding motion near a discontinuity boundary) which shows route to chaos via period doubling, period adding cascade etc. I want to understand/study the Floquet spectrum for such PWS systems. If you could recommend any book/paper/s; it would be very helpful.