Math 2.0

Ordinary differential equations Qualitative theory Example

Lorenz system

Builds on Equilibrium pointHopf bifurcationPitchfork bifurcation

Example

Let σ,ρ,β>0\sigma, \rho, \beta > 0. The Lorenz system is the autonomous system in ℝ3\R^3 x′=σ(y−x),y′=x(ρ−z)−y,z′=xy−βz.x' = \sigma (y - x), \qquad y' = x (\rho - z) - y, \qquad z' = x y - \beta z. The classical parameter values are σ=10\sigma = 10, β=8/3\beta = 8/3 and ρ=28\rho = 28.

The origin is an equilibrium point for all parameters. For ρ>1\rho > 1 there are two more equilibria C±=(±β(ρ−1),±β(ρ−1),ρ−1)C_\pm = \left(\pm\sqrt{\beta(\rho - 1)}, \pm\sqrt{\beta(\rho - 1)}, \rho - 1\right). Check: x=yx = y makes the first equation vanish, then z=ρ−1z = \rho - 1 makes the second vanish, and x2=β(ρ−1)x^2 = \beta(\rho - 1) makes the third vanish.

Remarks

The divergence of the vector field is −(σ+1+β)<0-(\sigma + 1 + \beta) < 0, so the flow shrinks volumes at a constant exponential rate. At ρ=1\rho = 1, the two equilibria C±C_\pm branch off the origin in a pitchfork bifurcation. For σ>β+1\sigma > \beta + 1, the linearisation at C±C_\pm has a pair of purely imaginary eigenvalues at ρ=σ(σ+β+3)/(σ−β−1)\rho = \sigma(\sigma + \beta + 3)/(\sigma - \beta - 1). For σ=10\sigma = 10 and β=8/3\beta = 8/3 this value is 470/19≈24.74470/19 \approx 24.74, and there C±C_\pm lose stability in a subcritical Hopf bifurcation. At the classical values, numerical solutions approach a butterfly-shaped set, the Lorenz attractor, and nearby solutions separate quickly. Warwick Tucker gave a computer-assisted proof in 2002 that this set is a strange attractor.

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Differential equation, concept, no evidence levelDifferential equationconceptFunction, concept, no evidence levelFunctionconceptHopf bifurcation, concept, no evidence levelHopf bifurcationconceptInterval, concept, no evidence levelIntervalconceptPitchfork bifurcation, concept, no evidence levelPitchfork bifurcationconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionAutonomous system, definition, no evidence levelAutonomous systemdefinitionEquilibrium point, definition, no evidence levelEquilibrium pointdefinitionLorenz system, example, no evidence levelLorenz systemexample
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  1. Differential equation concept
  2. Function concept
  3. Hopf bifurcation concept
  4. Interval concept
  5. Pitchfork bifurcation concept
  6. Ordinary differential equation definition
  7. Solution of an ordinary differential equation definition
  8. Autonomous system definition
  9. Equilibrium point definition
  10. Lorenz system example

Further reading

  1. Web https://en.wikipedia.org/wiki/Lorenz_system
About this page
TagWYRC41CQN9
URIm2://WYRC41CQN9
Kindexample
WritingW0 stub
MSC34C28, 37D45
Instance ofAutonomous system
UsesEquilibrium point, Hopf bifurcation, Pitchfork bifurcation
Changed2 November 2026

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