Lecture 6 of a short course on 'Center manifolds, normal forms, and bifurcations'. We discuss center manifold theory for discrete dynamical systems, i.e., maps or mappings, with fixed points, x=F(x). The Taylor series approximation of the center manifold about the fixed point is discussed, as well as the map restricted to the center manifold, which reveals whether the fixed point in the full nonlinear system is stable or unstable. A 2D example is worked out in detail. The dependence of the center manifold on parameters is discussed and another 2D example discussed where a bifurcation occurs as the parameter is varied, leading to a period-2 orbit.
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https://youtu.be/Wv_b3u1wfKE
► Previous lecture: Center manifolds depending on parameters | related to bifurcations| Lorenz system bifurcation part 1
https://youtu.be/twsO1e3Hqws
► Course playlist
https://is.gd/CenterManifolds
► *Teacher Bio* Dr. Shane Ross is an Aerospace Engineering Professor at Virginia Tech. He has a Caltech PhD, worked at NASA/JPL and Boeing on interplanetary trajectories, and is a world renowned expert in the 3-body problem. He has written a book on the subject (link above).
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► *Website* http://chaotician.com
► *Introduction to discrete time maps*
https://youtu.be/-vV5A4HullY
► Course lecture notes in PDF form here
https://drive.google.com/drive/folders/1tE15obG5EJjlqGyU5h6RjAlb8tcyhoK8?usp=sharing
► in OneNote form here:
https://1drv.ms/u/s!ApKh50Sn6rEDiUIr4Ji8MUkTw7Da?e=YZ6eaZ
► *CHAPTERS*
0:00 Introduction to discrete maps, x_{n+1} = F(x_n)
1:15 The 'orbit' of a state x under a map
3:48 An example 2D map approximation of the restricted 3-body problem, Keplerian map
6:45 Fixed point of a map
9:30 Periodic orbits of a map
15:25 Eigenvalue spectrum of map linearization at a fixed point
20:00 Consider fixed points with only stable and center manifolds
23:08 Reduced dynamics on the center manifold
26:45 Estimating center manifold
29:00 Fundamental equation for center manifold
33:21 Worked example 2D map with 1D center manifold
50:35 Map restricted to the center manifold
56:52 Another 2D example with a parameter
1:06:41 Bifurcation leading to period-2 orbit along center manifold
1:11:22 Bifurcation diagram for a discrete map (pitchfork-like)
► *Related Courses and Series Playlists by Dr. Ross*
📚3-Body Problem Orbital Dynamics
https://is.gd/3BodyProblem
📚Space Manifolds
https://is.gd/SpaceManifolds
📚Space Vehicle Dynamics
https://is.gd/SpaceVehicleDynamics
📚Lagrangian & 3D Rigid Body Dynamics
https://is.gd/AnalyticalDynamics
📚Nonlinear Dynamics & Chaos
https://is.gd/NonlinearDynamics
📚Hamiltonian Dynamics
https://is.gd/AdvancedDynamics
📚Center Manifolds, Normal Forms, & Bifurcations
https://is.gd/CenterManifolds
Lecture 2020-06-18, Summer 2020
period-doubling bifurcation pitchfork bifurcation for maps period doubling route to chaos period doubling cascade
#NonlinearDynamics #DynamicalSystem #CenterManifold #Bifurcations #PeriodDoubling #PitchforkBifurcation #mathematics #manifold #chaos #unstable #PeriodicOrbit #LyapunovOrbit