Center Manifold Theory- Computing Center Manifolds, Lecture 3

Lecture 3 of a short course on 'Center manifolds, normal forms, and bifurcations'. Center manifold theory for continuous dynamical systems (ODEs) with equilibrium points that have only stable and center directions. The Taylor series approximation of the center manifold is discussed, as well as the dynamics (that is, the vector field) restricted to the center manifold, which reveals whether the equilibrium point in the full space is stable or unstable. A 2D example is given and the failure of the center subspace approximation, i.e., the Galerkin method, is illustrated. A 3D example is also given, where a linear coordinate transformation must first be done to put the ordinary differential equation into the standard form. ► Jump to center manifold theory computations: 15:05 ► Next lecture: Center manifolds depending on parameters | related to bifurcations| Lorenz system bifurcation part 1 https://youtu.be/twsO1e3Hqws ► Previous lecture: Hyperbolic vs non-hyperbolic fixed points and computing their invariant manifolds via Taylor series https://youtu.be/5d0UhnBm16g ► Course playlist 'Center manifolds, normal forms, and bifurcations' https://is.gd/CenterManifolds ► Dr. Shane Ross, chaotician, Virginia Tech professor (Caltech PhD) Subscribe https://is.gd/RossLabSubscribe​ Research http://chaotician.com​ ► Follow me on Twitter https://twitter.com/RossDynamicsLab ► Class lecture notes (PDF) https://drive.google.com/drive/folders/1tE15obG5EJjlqGyU5h6RjAlb8tcyhoK8?usp=sharing ► in OneNote form https://1drv.ms/u/s!ApKh50Sn6rEDiUIr4Ji8MUkTw7Da?e=YZ6eaZ ► Are you a beginner? Go to the 'Nonlinear Dynamics and Chaos' online course Course playlist https://is.gd/NonlinearDynamics ► Courses and Playlists by Dr. Ross 📚Attitude Dynamics and Control https://is.gd/SpaceVehicleDynamics 📚Nonlinear Dynamics and Chaos https://is.gd/NonlinearDynamics 📚Hamiltonian Dynamics https://is.gd/AdvancedDynamics 📚Three-Body Problem Orbital Mechanics https://is.gd/SpaceManifolds 📚Lagrangian and 3D Rigid Body Dynamics https://is.gd/AnalyticalDynamics 📚Center Manifolds, Normal Forms, and Bifurcations https://is.gd/CenterManifolds Chapters: 0:00 Center Manifold Theory introduction 1:10 Motivation from linear vector fields with block diagonal matrix D=diag{A,B} where A has only eigenvalues of zero real part and B is a matrix having only eigenvalues of negative real part. We need to focus on exp(A*t) to know the stability of the equilibrium. 7:45 Nonlinear case, expanding about an equilibrium point. Need to know the nonlinear vector field along the center manifold. 15:05 Center manifold theory computation 20:55 Approximate the center manifold locally as a function and do a Taylor series expansion to obtain it 24:45 Vector field on the center manifold 30:03 the tangency condition, main computational 'workhouse' 32:15 2D example: two-dimensional system where stability of the origin is not obvious 50:26 Why not do a tangent space (Galerkin) approximation for center manifold dynamics? 59:03 3D example with 2D center manifold Lecture 2020-06-09, Summer 2020 Reference: Stephen Wiggins [2003] Introduction to Applied Nonlinear Dynamical Systems and Chaos, Second Edition, Springer. https://www.springer.com/gp/book/9780387001777 #CenterManifold #NonlinearDynamics #DynamicalSystems #Bifurcations #Slowmanifold #Wiggins #Verhulst #ZeroEigenvalue #DifferentialEquation #mathematics #DynamicalSystem #Centre #Wiggins #CentreManifold #CenterManifoldTheorem #Nonhyperbolic #Equilibria #Equilibrium #FixedPoint #TaylorSeries #TaylorExpansion #Subspace #InvariantManifold #InvariantSet #Bifurcation #NormalForm #AppliedMath #AppliedMathematics #Math