Cusp Catastrophe: Sudden Transitions in Nonlinear Systems

Описание: Cusp Catastrophe & Codimension-2 Bifurcations | Nonlinear Dynamics

Large, sudden changes in system behavior can arise from smooth parameter variations.
This lecture introduces the cusp catastrophe, a codimension-2 bifurcation that explains discontinuous transitions in nonlinear systems.

We motivate the theory using the physical example of a bead in an off-center rotating hoop, then connect the mathematics to broader applications such as population outbreaks and structural instability.

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Chapters

0:00 Intro
1:52 Imperfect bifurcation
8:40 Stability diagram
12:40 Surfaces

Next lecture (in sequence): Insect outbreak model
   • Cusp Catastrophe in Ecology: The Spruce Bu...  

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Related videos (recommended order)

Bead in a rotating hoop
• Equations of motion from Newton’s laws
   • Bead on a Rotating Hoop: Deriving the Equa...  
• Nondimensionalization of the governing equation
   • Nondimensionalizing Equations of Motion | ...  
• High-damping limit and pitchfork bifurcation
   • Pitchfork Bifurcation Explained via a Bead...  

Other bifurcations
• Saddle-node bifurcation
   • Bifurcations Part 1, Saddle-Node Bifurcation  
• Transcritical bifurcation
   • Bifurcations Part 2- Transcritical Bifurca...  
• Pitchfork bifurcation
   • Bifurcations Part 3- Pitchfork Bifurcation  
• Robustness of bifurcations
   • Bifurcations Part 4- Robustness of Bifurca...  

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Course playlist

Nonlinear Dynamics & Chaos (full course):
   • Nonlinear Dynamics and Chaos | Online Course  

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Instructor

Dr. Shane Ross
Professor, Virginia Tech (Caltech PhD)
Ross Dynamics Lab: http://chaotician.com

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Course notes

PDF lecture notes:
https://drive.google.com/drive/folder...

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Reference

Steven Strogatz, Nonlinear Dynamics and Chaos, Chapter 3

Animation credit:
“Bifurcation Examples: Cusp Unfolding” – Prof. Tom Ghrist
   • AppDynSys : Bifurcation Examples : Cusp Un...  

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