Lecture 1 of a short course on 'Center manifolds, normal forms, and bifurcations'. Here we discuss the types of dynamical systems to be considered (continuous and discrete time, that is, ODEs and maps, respectively). Course playlist https://is.gd/CenterManifolds
We consider the linearized dynamics near a reference trajectory for a continuous time system, specializing to the case of an equilibrium point. The eigenvalues and eigenvectors of the linearization contain important information about the local phase space structure, the local dynamics. Several examples of 2D and 3D systems are considered.
► Chapters
0:00 Introduction and Definitions
17:20 Linearized dynamics about a reference trajectory
27:50 Eigen-decomposition into stable, unstable, and center subspaces
51:14 Numerical example of complex conjugate eigenvalues & interpretation
55:36 General solution to a linear system
59:57 Numerical examples
► Next lecture: Hyperbolic vs non-hyperbolic fixed points and computing their invariant manifolds via Taylor series
https://youtu.be/5d0UhnBm16g
► Are you a newcomer? Go to the 'Nonlinear Dynamics and Chaos' online course
https://is.gd/NonlinearDynamics
► Dr. Shane Ross, chaotician, Virginia Tech professor (Caltech PhD)
Subscribe https://is.gd/RossLabSubscribe
Research http://chaotician.com
Video introduction https://youtu.be/iAlNnYJinRs
► Follow me on Twitter
https://twitter.com/RossDynamicsLab
► Course playlist 'Center manifolds, normal forms, and bifurcations'
https://is.gd/CenterManifolds
► Class lecture notes (PDF)
https://drive.google.com/drive/folders/1tE15obG5EJjlqGyU5h6RjAlb8tcyhoK8?usp=sharing
► in OneNote form
https://1drv.ms/u/s!ApKh50Sn6rEDiUIr4Ji8MUkTw7Da?e=YZ6eaZ
► Other courses
📚Nonlinear Dynamics and Chaos
https://is.gd/NonlinearDynamics
📚Lagrangian and 3D Rigid Body Dynamics
https://is.gd/AnalyticalDynamics
📚Hamiltonian Dynamics
https://is.gd/AdvancedDynamics
📚Three-Body Problem Orbital Mechanics
https://is.gd/SpaceManifolds
📚Space Vehicle Dynamics
https://is.gd/SpaceVehicleDynamics
📚Center Manifolds, Normal Forms, and Bifurcations
https://is.gd/CenterManifolds
Lecture 2020-06-02, Summer 2020
Nonlinear Dynamics Dynamical System manifolds Bifurcation Normal Forms Wiggins Strogatz Center Manifold Jordan canonical form
#NonlinearDynamics #DynamicalSystem #manifolds #Bifurcation #NormalForms #Wiggins #Strogatz #CenterManifold