# Dynamical system A dynamical system is a mathematical formalization of state evolution over time, typically via a [[Differential_equation|differential equation]], map, or flow in [[Phase_space|phase space]]. Trajectories, [[Attractor|attractors]], bifurcations, Lyapunov stability, chaos (sensitive dependence), ergodicity, and structural stability characterize continuous, discrete, stochastic, or hybrid systems across [[Physics|physics]], [[Biology|biology]], and [[Engineering|engineering]]. <!-- LEGACYSIM:BEGIN v1.5 — generated by g03_mint_wave.py; three.js first; do not hand-edit inside --> ## Microsims (promoted from legacy — three.js first) ### MicroSim spec - **Recommended sim type:** [[Phase_space|phase space]] / [[Attractor|attractor]] - **Microsimmability score:** 88/100 - **Layout:** drawing region (canvas) on top; control region (sliders/buttons) below. ### Parameters (tunable controls) - `Parameter a` - `Time step` - `Initial condition` ### What animates A state evolves step by step along a trajectory that bends as the parameters change. ### Learning objective Show how an update rule moves a system's state through time. <!-- LEGACYSIM:END --> <!-- GIFPLATE:BEGIN v1.0 g16 — Commons hotlink; do not hand-edit inside --> ## Images <figure class="wt-gifplate"> <img src="https://commons.wikimedia.org/wiki/Special:FilePath/Lorenz_attractor.gif" alt="Attractor Dynamics" loading="lazy" decoding="async"> <figcaption><strong>Attractor Dynamics</strong> — Watch a trajectory trace a butterfly-shaped path. This is the Lorenz attractor—how deterministic rules generate endlessly complex, bounded motion.<br> <span class="wt-credit">Wikimedia Commons &middot; <strong>licence pending verification</strong> (run <code>g17_gif_verify.py</code> on a networked lane) &middot; <a href="https://commons.wikimedia.org/wiki/File:Lorenz_attractor.gif">Details</a></span></figcaption> </figure> *The hub concept of [[PORTAL_Dynamical_system]], and this page has no interactive build yet — the plate carries it.* <!-- GIFPLATE:END --> ## Reveal %%REVEAL:p5%% %%REVEAL:three%% --- *Concept aligned with [Wikipedia](https://en.wikipedia.org/wiki/Dynamical_system); adapted text, where present, is licensed [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/).* ## Overview Continuous flow is written as an [[Ordinary_differential_equation|ordinary differential equation]] or [[Partial_differential_equation|partial differential equation]]; discrete evolution as iterated maps and the [[Cellular_automaton|cellular automaton]]. [[Bifurcation_theory|Bifurcation theory]] tracks qualitative change, [[Chaos_theory|chaos theory]] follows [[Edward_Norton_Lorenz|Edward Norton Lorenz]] into sensitive dependence, and [[Statistical_mechanics|statistical mechanics]] treats ensembles of trajectories. The geometry threads through [[Fractal|fractals]], [[Topology|topology]], and [[Functional_analysis|functional analysis]]. Applications feed [[Control_theory|control theory]], [[Population_dynamics|population dynamics]], [[System_dynamics|system dynamics]], the [[Dissipative_system|dissipative system]] and [[Self-organized_criticality|self-organized criticality]], [[Network_theory|network]] dynamics, and modern [[Machine_learning|machine learning]], making the [[Mathematical_model|mathematical model]] of time-evolution the shared grammar of [[Systems_theory|systems theory]]. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Dynamical_system) : [Wikitube](https://en.wikitube.io/wiki/Dynamical_system) ## Previous hub tags Hubs: `Systems`. Portals: [[PORTAL_Systems]], [[PORTAL_Dynamical_system]].