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Cusp Catastrophe Theory

The cusp catastrophe can lead to large discontinuous changes in the state of a system. We analyze a bifurcation that depends on two parameters, motivated by the physical example of fixed points for a bead in an off-center rotating hoop. Other examples abound, for example, an insect outbreak model    • The Science of Insect Outbreaks: Expl...   ► Jump to cusp catastrophe theory 1:56 ► Next: Insect outbreak model    • The Science of Insect Outbreaks: Expl...   ► From 'Nonlinear Dynamics and Chaos' (online course). Other topics posted regularly. Subscribe https://is.gd/RossLabSubscribe​ ► Bead in a rotating hoop, earlier related videos: Equation derived from Newton's Laws    • Bead in a Rotating Hoop, Part 1- Deri...   nondimensionalizing bead in hoop    • Nondimensionalization of Equations, P...   Graphical analysis of high damping limit    • Bead in a Rotating Hoop, Part 2: High...   ► Other bifurcations videos: Saddle-node    • Bifurcations Part 1, Saddle-Node Bifu...   Trans-critical    • Bifurcations Part 2- Transcritical Bi...   Pitchfork    • Bifurcations Part 3- Pitchfork Bifurc...   Robustness of bifurcations    • Bifurcations Part 4- Robustness of Bi...   ► Online course playlist https://is.gd/NonlinearDynamics ► For a more in-depth mathematical discussion of bifurcations    • Bifurcation Theory: Saddle-Node, Hopf...   ► Dr. Shane Ross, chaotician, Virginia Tech professor (Caltech PhD) http://chaotician.com​ ► Course lecture notes (PDF) https://is.gd/NonlinearDynamicsNotes Reference: Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 3: Bifurcations Animation at the end, AppDynSys : Bifurcation Examples : Cusp Unfolding, by Prof Ghrist Math:    • AppDynSys : Bifurcation Examples : Cu...   ► Courses and Playlists by Dr. Ross 📚Three-Body Problem Orbital Mechanics https://is.gd/3BodyProblem 📚Attitude Dynamics and Control https://is.gd/SpaceVehicleDynamics 📚Nonlinear Dynamics and Chaos https://is.gd/NonlinearDynamics 📚Hamiltonian Dynamics https://is.gd/AdvancedDynamics 📚Space Manifold Dynamics https://is.gd/SpaceManifolds 📚Lagrangian and 3D Rigid Body Dynamics https://is.gd/AnalyticalDynamics 📚Center Manifolds, Normal Forms, and Bifurcations https://is.gd/CenterManifolds Cusp catastrophe theory co-dimension codimension 2 bifurcation robustness fragility cusp unfolding perturbations structural stability emergence critical point critical slowing down supercritical bifurcation subcritical bifurcations buckling beam model change of stability nonlinear dynamics dynamical systems differential equations dimensions phase space Poincare Strogatz graphical method Fixed Point Equilibrium Equilibria Stability Stable Point Unstable Point Linear Stability Analysis Vector Field One-Dimensional 1-dimensional Functions #NonlinearDynamics #Bifurcations #DynamicalSystems #CuspCatastrophe #CatastropheTheory #CuspBifurcation #CriticalPoint #SaddleNode #buckling #pitchforkbifurcation #robust #StructuralStability #DifferentialEquations #dynamics #dimensions #PhaseSpace #Poincare #Strogatz #graphicalmethod #FixedPoints #EquilibriumPoints #Stability #StablePoint #UnstablePoint #Stability #LinearStability #LinearStabilityAnalysis #StabilityAnalysis #VectorField #OneDimensional #Functions