Differential Equations Decoded 78 Bifurcations When Stability Changes

Free to reuse. Free to remix. No attribution required. Make your own at https://www.patreon.com/cw/MadSciHub QUICK SUMMARY A bifurcation is what happens when a parameter creeps past a critical value and the long-term behavior of a dynamical system flips qualitatively. Four canonical shapes — saddle-node, transcritical, pitchfork, and Hopf — cover almost every bifurcation a student will ever see, including the one that killed a Wisconsin lake and the one that lets a Tufts lab grow two-headed flatworms. KEY CONCEPTS 1. State vs Parameter - state evolves with time, parameter is a knob you turn between experiments 2. Bifurcation Diagram - parameter on horizontal axis, equilibria on vertical, solid stable, dashed unstable 3. The Four Animals - saddle-node, transcritical, pitchfork, Hopf cover the entire alphabet 4. Sub vs Supercritical - sign of the highest odd nonlinear term decides gentle vs catastrophic DEFINITIONS - Bifurcation: qualitative change in long-term behavior as a parameter crosses a critical value - State Variable: the quantity the equation evolves over time, usually x - Parameter: a constant you slowly vary between experiments, usually r - Saddle-Node: stable and unstable equilibria collide and annihilate - Transcritical: two equilibria pass through each other and swap stability - Pitchfork: one stable equilibrium becomes unstable while two new stable ones sprout symmetrically - Hopf: an equilibrium destabilizes and a limit cycle is born around it - Hysteresis: the system retains the new state even after the parameter is reset - Anatomical Compiler: steering bioelectric parameters to push tissue into a chosen attractor HOW IT WORKS 1. Find equilibria by setting f equal to zero and solving for x in terms of r 2. Find stability by computing the partial derivative of f with respect to x at each equilibrium 3. Find the bifurcation point where f and the partial derivative are both zero 4. Plot equilibrium curves, solid for stable and dashed for unstable 5. Match the shape to one of the four canonical diagrams to classify 6. Check the sign of the highest odd nonlinear term to decide sub vs super 7. In two dimensions, watch for a complex conjugate pair of eigenvalues crossing zero 8. Recognize hysteresis when reversing the parameter does not retrace the forward path KEY ARGUMENTS 1. Quantitative change moves an equilibrium, qualitative change makes it cease to exist 2. Arnold's normal form theorem reduces almost every smooth one-dim bifurcation to one of three shapes 3. The landscape picture treats x as a marble on a hillside whose slope is minus f 4. The Wisconsin lake is a saddle-node, clear-water equilibrium and unstable threshold collide 5. Logistic growth is a transcritical, extinction and carrying capacity swap stability at r equals zero 6. The pitchfork is spontaneous symmetry breaking, the same square root that buckles columns and aligns iron 7. Subcritical pitchfork has plus on the cubic and is catastrophic, supercritical has minus and is gentle 8. Hopf is the only bifurcation that births a limit cycle, requires complex eigenvalues crossing imaginary axis 9. Two-headed planaria are a living bifurcation diagram, identical DNA but two stable bioelectric attractors KEY TAKEAWAYS - Parameters are not state variables, you turn the knob between experiments - The bifurcation diagram is the exam, everything else is scaffolding - Saddle-node kills, transcritical swaps, pitchfork splits, Hopf oscillates - The sign of the cubic separates a gentle bend from a catastrophic snap - Hysteresis is why removing the cause does not restore the original state - Anatomy is a solution to a dynamics problem, bifurcation theory writes it down MEMORY HOOKS - The bifurcation zoo has only four animals, an alphabet not a menagerie - The lake holds a grudge, removing the fertilizer does not bring it back - Two-headed planaria are a living bifurcation diagram SOURCE https://www.biorxiv.org/content/10.1101/124149v1.full #DifferentialEquations #Bifurcations #NonlinearDynamics #SaddleNode #Pitchfork #HopfBifurcation #PhaseTransition #Hysteresis #Bioelectricity #AnatomicalCompiler #MathEducation #ExamPrep #Decoded #madscilecture #decoded #differentialequations #math #science