Dr. Shane Ross, Virginia Tech. Lecture 8 of a short course on 'Center manifolds, normal forms, and bifurcations'. Normal forms that depend on parameters. For example, the Hopf bifurcation on the birth of a periodic orbit (a limit cycle) from an equilibrium point as a parameter is varied.
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► Previous lecture: Hopf bifurcation example - Normal forms for vector fields depending on parameters
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► Simpler introduction to Hopf bifurcations
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► Online course playlist
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► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
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0:00 Normal forms for vector fields that depend on parameters.
23:54 Hopf bifurcation example: We consider the case where a system which depends on parameters has a complex conjugate pair of eigenvalues that cross the imaginary axis as a parameter is varied. The normal form calculation leads us to the Hopf bifurcation, wherein a periodic orbit is birthed from an equilibrium point.
► See the entire playlist for the short course:
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► entire class lecture notes in PDF form here
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► in OneNote form here:
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► Main textbook:
Stephen Wiggins [2003] Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer.
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Lecture 2020-06-25, Summer 2020
► Make your own phase portrait
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► Related Courses and Series Playlists by Dr. Ross
📚Nonlinear Dynamics and Chaos
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📚Hamiltonian Dynamics
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📚3-Body Problem Orbital Dynamics Course
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📚Space Manifolds
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📚Space Vehicle Dynamics
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📚Lagrangian and 3D Rigid Body Dynamics
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📚Center Manifolds, Normal Forms, and Bifurcations
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► Author's website:
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Hopf bifurcation simulation and visualization from Jonathan Mitchell, "When Math Drops the Bass - Hopf Bifurcation": [ Ссылка ]
#HopfBifurcation #NonlinearDynamics #DynamicalSystems #LimitCycle #mathematics #CenterManifold #RossDynamicsLab #appliedmathematics
