Bifurcations in 2D, Part 4: Global Bifurcations, Limit Cycle Creation | Homoclinic Bifurcation

Описание: In two-dimensional systems, there are four common ways in which limit cycles are created or destroyed. The Hopf bifurcation is the best known and occurs 'locally' in phase space, in the small neighborhood of a fixed point. The other three involve large regions of the phase plane and are therefore called 'global' bifurcations.

Chapters
0:00 Introduction
0:51 saddle-node bifurcation of cycles, or fold bifurcation
9:11 saddle-node infinite-period bifurcation on a cycle, or SNIPER
18:07 connection with heartbeat or nerve firing time-series
22:00 homoclinic bifurcation, or saddle-loop
26:11 universal behavior of bifurcations of cycles, a summary of the amplitude and period of the resulting limit cycles as a function of the distance of the parameter from the critical value

► Next, quasi-periodicity, phase-locking, and dynamics on the torus
   • Action-Angle Variables in Hamiltonian Syst...  

► From 'Nonlinear Dynamics and Chaos' (online course).
Playlist https://is.gd/NonlinearDynamics

► Bifurcations in 2D
Zero eigenvalue bifurcations    • Bifurcations in 2D, Part 1: Introduction, ...  
Hopf bifurcation theory    • Bifurcations in 2D, Part 2: Hopf Bifurcati...  
Hopf physical examples    • Bifurcations in 2D, Part 3: Hopf Bifurcati...  
Bifurcations of limit cycles    • Bifurcations in 2D, Part 4: Global Bifurca...  

► Bifurcations in 1D (the zero eigenvalue bifurcations)
Saddle-node    • Bifurcations Part 1, Saddle-Node Bifurcation  
Trans-critical    • Bifurcations Part 2- Transcritical Bifurca...  
Pitchfork    • Bifurcations Part 3- Pitchfork Bifurcation  
Robustness    • Bifurcations Part 4- Robustness of Bifurca...  

► Additional background on 2D dynamical systems
Phase plane introduction    • Phase Portrait Introduction- Pendulum Example  
Classifying 2D fixed points    • Classifying Fixed Points of 2D Systems - L...  
Gradient systems    • Gradient Systems - Nonlinear Differential ...  
Index theory    • Index Theory for Dynamical Systems, Part 1...  
Limit cycles    • Limit Cycles, Part 1: Introduction & Examples  
Averaging theory    • Averaging Theory for Weakly Nonlinear Osci...  

► Advanced lecture on Hopf bifurcations
   • Hopf Bifurcation Example- Normal Forms for...  

► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
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► Course lecture notes (PDF)
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► 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
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📚Lagrangian and 3D Rigid Body Dynamics
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📚Center Manifolds, Normal Forms, and Bifurcations
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References:
Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 8: Bifurcations Revisited

Arnaldo Rodriguez-Gonzalez, Strogatz's example of an infinite-period bifurcation,
   • Strogatz's example of an infinite-period b...  

Arnaldo Rodriguez-Gonzalez, Strogatz's example of a homoclinic bifurcation,
   • Strogatz's example of a homoclinic bifurca...  


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