# Lyapunov stability ## Microsim (three.js) <div class="microsim-player"> <iframe src="https://wikitube-3d-microsims.netlify.app/Lyapunov_stability.html" width="100%" height="620" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin"></iframe> </div> *A COSMOS-hub companion to celestial and atmospheric dynamics: from the Coriolis-stabilized Trojan points along Keplerian orbits to the runaway error growth that caps [[Weather_forecasting|weather forecasting]], the same question recurs — nudge an equilibrium, and does it hold?* > An equilibrium is **Lyapunov stable** if a system pushed slightly away from it stays nearby ever after, and **asymptotically stable** if it settles back to rest. This microsim makes that physical: a ball sits at a balance point on a shaped landscape, and you shove it to see what follows. Roll it in a **bowl** and it circles back; perch it on a **hilltop** and the faintest push sends it away; set it on a **saddle** and it holds one way but escapes the other. Friction decides whether "stays near" sharpens into "returns exactly." ## About this microsim Pick the equilibrium type with the **Well — stable bowl**, **Saddle — stable one way**, or **Hilltop — unstable equilibrium** choices, then press **Perturb ball** and watch. Three sliders shape the run: **Damping (friction)** (0–0.6, default 0.15) sets how strongly motion is dissipated; **Perturbation size** (0.05–2.2, default 0.60) sets how hard the shove is; and **Steepness of landscape** (0.3–2.5, default 1.00) scales the surface's curvature. **Reset to rest** returns the ball to the fixed point, **Pause** freezes the integration, and **Trails: On** draws the trajectory. These are the only controls. ## Related microsims - Kepler's laws of planetary motion — orbital equilibria and Lagrange points, where stability first mattered to astronomy - Coriolis effect — the rotating-frame force that stabilizes the L4/L5 Trojan points - [[Weather_forecasting]] — Lyapunov exponents and the growth of forecast error - Virial theorem — time-averaged energy balance governing the stability of bound systems - Solar sail — station-keeping at unstable, actively controlled equilibrium points ## Links (Wikipedia order) <!-- injected from _registry/childlinks/Lyapunov_stability.json (2026-07-30T02:09:12Z) --> `Absolute_value` · [[Aerospace_engineering]] · `Aleksandr_Lyapunov` · `American_Mathematical_Society` · `Apsis` · `Argument_of_periapsis` · `Asymptotic_analysis` · [[Attractor]] · `Augustin-Louis_Cauchy` · `Autonomous_system_(mathematics)` · `BIBO_stability` · `Bi-elliptic_transfer` · `Carathéodory's_existence_theorem` · `Celestial_mechanics` · [[Chaos_theory]] · `Circular_orbit` · `Continuous_function` · [[Control_engineering]] · [[Control_theory]] · `Convergence_proof_techniques` · `Crank–Nicolson_method` · `Delay_differential_equation` · `Dependent_and_independent_variables` · `Differential-algebraic_system_of_equations` · [[Differential_equation]] · `Differential_operator` · `Dirac_delta_function` · `Dynamical_friction` · [[Dynamical_system]] · `Elliptic_orbit` · [[Energy]] · `Ernst_Leonard_Lindelöf` · `Escape_velocity` · `Euler_method` · `Exact_differential_equation` · `Exponential_stability` · `Finite_difference_method` · `Finite_element_method` · `Finite_volume_method` · `Galerkin_method` · `Gerald_Teschl` · `Gottfried_Wilhelm_Leibniz` · `Gravity_assist` · `Guidance_system` · `Halo_orbit` · `Hartman–Grobman_theorem` · `Hill_sphere` · `Hohmann_transfer_orbit` · `Holonomic_function` · `Homoclinic_orbit` · `Homogeneous_differential_equation` · `Hyperbolic_trajectory` · `Input-to-state_stability` · `Integral` · [[Integral_transform]] · `Integrating_factor` · `Integration_by_substitution` · `Integro-differential_equation` · [[Isaac_Newton]] · `Jacob_Bernoulli` · `Jacobian_matrix_and_determinant` · `Jet_bundle` · `John_Crank` · `Joint_spectral_radius` · `Joseph-Louis_Lagrange` · `Joseph_P._LaSalle` · `Józef_Maria_Hoene-Wroński` · `Kepler's_equation` · `Kepler's_laws_of_planetary_motion` · `LaSalle's_invariance_principle` · `Lagrange_point` · `Lemma_(mathematics)` · `Leonhard_Euler` · `Libration_point_orbit` · `Linear_differential_equation` · `Linear_system` · `Lissajous_orbit` · `List_of_linear_ordinary_differential_equations` · `List_of_named_differential_equations` · `List_of_nonlinear_ordinary_differential_equations` · `List_of_nonlinear_partial_differential_equations` · `Lyapunov_exponent` · `Lyapunov_function` · `Lyapunov–Malkin_theorem` · `Markus–Yamabe_conjecture` · `Martin_Kutta` · `Mean_anomaly` · `Method_of_undetermined_coefficients` · `Metric_space` · `N-body_problem` · `Nikolay_Gur'yevich_Chetaev` · `Non-autonomous_system_(mathematics)` · [[Nonlinear_system]] · `Notation_for_differentiation` · [[Numerical_integration]] · `Oberth_effect` · `Orbit_insertion` · `Orbital_decay` · `Orbital_eccentricity` · `Orbital_elements` · `Orbital_inclination` · `Orbital_maneuver` · `Orbital_mechanics` · `Orbital_node` · `Orbital_period` · `Orbital_speed` · [[Ordinary_differential_equation]] · `Parabolic_trajectory` · [[Partial_differential_equation]] · `Payload_fraction` · `Peano_existence_theorem` · `Perturbation_(astronomy)` · `Perturbation_theory` · `Phase_portrait` · [[Phase_space]] · `Phyllis_Nicolson` · `Picard–Lindelöf_theorem` · `PlanetMath` · `Positive-definite_function` · `Power_series_solution_of_differential_equations` · `Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America` · `Propellant_mass_fraction` · `Providence,_Rhode_Island` · `Radial_trajectory` · `Rate_of_convergence` · `Rudolf_E._Kálmán` · `Rudolf_Lipschitz` · `Runge–Kutta_methods` · `Semi-major_and_semi-minor_axes` · `Separation_of_variables` · `Sofya_Kovalevskaya` · `Solomon_Lefschetz` · `Specific_orbital_energy` · `Sphere_of_influence_(astrodynamics)` · `Stability_theory` · `Stable_manifold` · `Stochastic_differential_equation` · `Stochastic_partial_differential_equation` · `Structural_stability` · `Surface_gravity` · `Transfer_orbit` · `True_anomaly` · `Tsiolkovsky_rocket_equation` · `Two-body_problem` · `VN_Karazin_Kharkiv_National_University` · `Van_der_Pol_oscillator` · `Variation_of_parameters` · `Vis-viva_equation` · `Wolfgang_Hahn` · `Wronskian` · `Émile_Picard` ## Overview Lyapunov stability, named for the Russian mathematician Aleksandr Lyapunov (1857–1918) and introduced in his 1892 thesis *The General Problem of the Stability of Motion*, is the standard framework for asking whether an equilibrium is robust to small disturbances. It underpins control theory, celestial mechanics, and the study of chaos, giving precise meaning to the difference between a marble in a bowl and a pencil balanced on its tip. ## The science For a system $\dot{x}=f(x)$ with equilibrium $x_e$, the point is **Lyapunov stable** if for every $\varepsilon>0$ there is a $\delta>0$ such that $\lVert x(0)-x_e\rVert<\delta$ implies $\lVert x(t)-x_e\rVert<\varepsilon$ for all $t\ge 0$ — start close, stay close. It is **asymptotically stable** if additionally $x(t)\to x_e$. Lyapunov's *direct method* proves this without solving the equations. For the microsim's ball of mass $m$ in a potential $U(x)$ with damping $c$, $m\ddot{x} = -U'(x) - c\dot{x},$ take the total mechanical energy as a candidate **Lyapunov function** $V=\tfrac12 m\dot{x}^2 + U(x)$, with $V=0$ only at rest at the minimum. Along trajectories, $\dot{V} = \dot{x}\big(-U'(x)-c\dot{x}\big) + U'(x)\dot{x} = -c\dot{x}^2 \le 0.$ Energy never increases, so a well is Lyapunov stable; with $c>0$ it strictly decreases whenever the ball moves, and by LaSalle's invariance principle the well bottom is **asymptotically stable**. The equilibrium type follows from the curvature (the Hessian of $U$): | Landscape | Shape of $U$ | Curvature | Stability | |---|---|---|---| | Well | local minimum | both gt;0$ | Lyapunov stable; asymptotic with damping | | Saddle | saddle point | mixed sign | unstable (stable along one axis only) | | Hilltop | local maximum | both lt;0$ | unstable | ## Controls -> what each maps to | Control | Maps to | Range / values | Meaning | |---|---|---|---| | Damping (friction) | dissipation $c$ | 0–0.6 (0.15) | Sets $\dot V=-c\dot x^2$; larger = faster return, marginal → asymptotic | | Perturbation size | displacement $\delta$ | 0.05–2.2 (0.60) | The $\delta$ of the $\varepsilon$–$\delta$ test | | Steepness of landscape | curvature scale of $U$ | 0.3–2.5 (1.00) | Scales Hessian eigenvalues; steeper restores/diverges faster | | Perturb ball | apply a kick | button | Displaces the ball to test the equilibrium | | Reset to rest | reset state | button | Ball back at the fixed point, zero velocity | | Pause | halt integration | button | Freezes time-stepping to inspect state | | Trails | trajectory history | On / Off | Draws the path to reveal convergence or escape | | Landscape | equilibrium type | Well / Saddle / Hilltop | Selects minimum, saddle, or maximum of $U$ | ## Learning objective See how a perturbed equilibrium can stay bounded, return exactly, or run away — and how damping converts marginal (Lyapunov) stability into asymptotic stability. ## Limits and connections The model is one ball in a fixed, low-dimensional landscape; real systems are high-dimensional and often time-varying. Yet the same test governs the Lagrange points of orbital mechanics — L4 and L5 are potential maxima held stable by the Coriolis force, while L1–L3 are saddles — and the **Lyapunov exponent**, which measures how fast nearby trajectories separate, sets the predictability horizon behind [[Weather_forecasting|weather forecasting]]. ## Poster & source <div class="microsim-fallback"> <!-- poster image pending backfill --> <p><em>Live microsim · <a href="https://wikitube-3d-microsims.netlify.app/Lyapunov_stability.html">open full</a> · source: Microsims for Dissemination/COSMOS_microsims/Lyapunov_stability.html</em></p> </div> <!-- CRAFT-LINK:START g12 --> *Built to the [[WT!Three_js_Microsim_Master_Class|three.js Master Class]].* <!-- CRAFT-LINK:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Lyapunov_stability) : [Wikitube](https://en.wikitube.io/wiki/Lyapunov_stability) ## Previous hub tags Tree parents: [[Control_theory]] · [[Dynamical_system]] · [[Phase_space]]. Legacy hubs: `COSMOS`. --- *Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*