Bifurcation_theory - Wikiparency

# Bifurcation theory  ## Microsims — three.js ### Bifurcation theory (three.js) <div class="microsim-player"> <iframe src="https://wikitube-3d-microsims.netlify.app/Bifurcation_theory.html" width="100%" height="620" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Bifurcation theory — three.js microsim"></iframe> </div> **Open it full-screen:** [Bifurcation_theory.html](https://wikitube-3d-microsims.netlify.app/Bifurcation_theory.html) · library `threejs` · route `microsim/threejs/` ## Microsims — p5.js ### Bifurcation theory (p5.js) · `param → split` <div class="microsim-player"> <iframe src="https://editor.p5js.org/sciencenibber/full/DvoQSoX2z" width="100%" height="480" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Bifurcation theory — p5.js microsim"></iframe> </div> *How a system's behavior changes qualitatively as a parameter crosses a threshold — the road to chaos.* **Open in the editor:** [▶ fork this sketch](https://editor.p5js.org/sciencenibber/sketches/DvoQSoX2z) · movement *IX · Foundations & the rest of the toolbox* · library `p5js` ### Related microsims Live sims on neighbouring articles — 2 of them inside this article's own Wikipedia link tree: - [[Attractor]] *(in tree)* - [[Catastrophe_theory]] *(in tree)* - [[Dissipative_system]] - [[Hierarchy]] - [[Scale-free_network]] - [[Synchronization]] *Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4). Placed by `g08_place_microsims.py`.*  ## MicroSim spec - **Recommended sim type:** chaos / bifurcation - **Microsimmability score:** 90/100 - **Layout:** drawing region (canvas) on top; control region (sliders/buttons) below. ### Parameters (tunable controls) - `Control parameter` - `Initial value` - `Resolution` ### What animates A bifurcation diagram is swept, showing fixed points split and cascade into chaos. ### Learning objective Show how a [[System|system]]'s long-term states change qualitatively as a parameter varies. ## Links (Wikipedia order)  `Addison-Wesley` · `Aleksandr_Lyapunov` · `Amie_Wilkinson` · `Anosov_diffeomorphism` · `Arnold's_cat_map` · `Arnold_tongue` · [[Attractor]] · `Audrey_Terras` · `Axiom_A` · `Baker's_map` · `Benoit_Mandelbrot` · `Bifurcation_diagram` · `Bifurcation_memory` · `Blue_sky_catastrophe` · `Brosl_Hasslacher` · `Bryna_Kra` · `Butterfly_effect` · `Caroline_Series` · [[Catastrophe_theory]] · `Celso_Grebogi` · [[Chaos_theory]] · `Chen_Guanrong` · `Chua's_circuit` · `Codimension` · `Complex_quadratic_polynomial` · [[Complexity]] · `Conservative_system` · `Control_of_chaos` · `Correlation_dimension` · [[Coupled_map_lattice]] · `Crisis_(dynamical_systems)` · `David_Ruelle` · `Delay_differential_equation` · [[Differential_equation]] · `Double_pendulum` · `Duffing_equation` · `Duffing_map` · `Dyadic_transformation` · `Dynamical_billiards` · [[Dynamical_system]] · `Edge_of_chaos` · [[Edward_Norton_Lorenz]] · `Edward_Ott` · `Eigenvalues_and_eigenvectors` · `Elastic_pendulum` · `Ergodic_theory` · `Ergodicity` · `Exponential_map_(discrete_dynamical_systems)` · `Family_of_curves` · `Feigenbaum_constants` · `Fermi–Pasta–Ulam–Tsingou_problem` · `Fixed_point_(mathematics)` · `Floris_Takens` · [[Fractal]] · `Gauss_iterated_map` · `Geomagnetic_reversal` · `Gingerbreadman_map` · `Hausdorff_dimension` · `Hee_Oh` · `Henri_Poincaré` · `Heteroclinic_cycle` · `Homoclinic_orbit` · `Hopf_bifurcation` · `Horseshoe_map` · `Hénon_map` · `Hénon–Heiles_system` · `Ikeda_map` · `Integral_curve` · `Interval_exchange_transformation` · `Invariant_measure` · `Irrational_rotation` · `Itamar_Procaccia` · `Jacobian_matrix_and_determinant` · `James_A._Yorke` · `Kaplan–Yorke_map` · `Kicked_rotator` · `Krylov–Bogolyubov_theorem` · `Lai-Sang_Young` · `Langton's_ant` · `Leon_O._Chua` · `Limit_(mathematics)` · `Limit_cycle` · `Limit_set` · `Liouville's_theorem_(Hamiltonian)` · `List_of_chaotic_maps` · `Logistic_map` · `Lorenz_system` · `Lotka–Volterra_equations` · `Lyapunov_exponent` · [[Lyapunov_stability]] · `Mackey–Glass_equations` · `Marcelo_Viana` · `Martin_Gutzwiller` · [[Mary_Cartwright]] · `Mary_Rees` · `Mary_Tsingou` · `Mathematics` · `Measure-preserving_dynamical_system` · `Michael_Berry_(physicist)` · `Michel_Hénon` · `Mitchell_Feigenbaum` · `Mixing_(mathematics)` · `Multiscroll_attractor` · `Nina_Snaith` · `Normal_form_(dynamical_systems)` · `Orbit_(dynamics)` · [[Ordinary_differential_equation]] · `Otto_Rössler` · `Outer_billiards` · [[Partial_differential_equation]] · `Period-doubling_bifurcation` · `Periodic_point` · `Peter_Grassberger` · `Phase_portrait` · [[Phase_space]] · `Pitchfork_bifurcation` · `Poincaré_recurrence_theorem` · `Poincaré–Bendixson_theorem` · [[Population_dynamics]] · [[Predictability]] · `Quantum_chaos` · `Rabinovich–Fabrikant_equations` · `Rayleigh–Bénard_convection` · `Recurrence_plot` · `Robert_L._Devaney` · `Rufus_Bowen` · `Rössler_attractor` · `Saddle-node_bifurcation` · `Saddle_point` · `Santa_Fe_Institute` · `Stability_theory` · `Stable_manifold` · `Stable_manifold_theorem` · `Standard_map` · `Steven_Strogatz` · `Svetlana_Jitomirskaya` · `Swinging_Atwood's_machine` · `Synchronization_of_chaos` · `Takens's_theorem` · `Tennis_racket_theorem` · `Tent_map` · `Three-body_problem` · `Tilt-A-Whirl` · `Tinkerbell_map` · `Topological_conjugacy` · `Transcritical_bifurcation` · `Van_der_Pol_oscillator` · `Vector_field` · `Vladimir_Arnold` · [[Weather_forecasting]] · `Yakov_Sinai` · `Zaslavskii_map` ## From the vault media library !Bifurcation theory thumb.png *Bifurcation Theory — from the vault's own media holdings, placed 2026-07-09. MTN / Wikitube.io original · CC BY-SA 4.0.*  > p5.js MicroSim stub · part of Systems Science · [Wikipedia source](https://en.wikipedia.org/wiki/Bifurcation_theory) > Relation: subfield of systems science. ## Concept summary Bifurcation theory is the mathematical study of changes in the qualitative or topological [[Structure|structure]] of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. ## Build checklist - [ ] Claim it: set `status: in-progress` + `lease` + `leased_at` - [ ] Write the child page explaining the concept (tie it back to systems science) - [ ] Finalize parameter ranges and defaults - [ ] Implement the p5.js sketch (drawing + control regions) - [ ] Add caption + the learning objective on the page - [ ] Set `status: done` ## p5.js sketch ```javascript // MicroSim: Bifurcation theory let controls = {}; function setup() { // createCanvas(...); create sliders for the parameters above. } function draw() { // background(...); read controls; render chaos / bifurcation; respond live. } ``` --- Back to Systems Science · Wikipedia: [Bifurcation theory](https://en.wikipedia.org/wiki/Bifurcation_theory) ---  ## Semiotic profile > *The semiotic universals this article invokes, machine-derived from the crossref — **unverified** (born so). Populated 2026-07-06 for the Systems room.* **Universals (8):** 🟢 bifurcation (19) · 🟡 system (14) · 🟢 attractor chaos (8) · 🟡 science (5) · 🟡 structure (4) · 🟢 field (2) · 🟢 topology (2) · 🟢 equilibrium (1) **Enter by sign:** Systems Semiotic Gateway · Alphabetum · Icon Registry · ← Systems Portal    *Built to the [[WT!P5_js_Microsim_Master_Class|p5.js Master Class]].*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Bifurcation_theory) : [Wikitube](https://en.wikitube.io/wiki/Bifurcation_theory) ## Previous hub tags Tree parents: [[Complex_system]] · [[Cybernetics]] · [[Dynamical_system]] · [[Emergence]] · [[Feedback]] · [[Self-organization]] · [[Systems_science]] · [[Systems_theory]]. Legacy hubs: none. --- *Sources: 2 legacy notes. Minted wave 1, 2026-07-30 (v1.6 order).*