Math 2.0

Dynamical systems and ergodic theory Smooth and hyperbolic dynamics Problem

Smale's fourteenth problem

Builds on Lorenz systemStrange attractorAttractor

Problem

Consider the Lorenz system in ℝ3\R^3, x˙=σ(y−x),y˙=x(ρ−z)−y,z˙=xy−βz,\dot x = \sigma (y - x), \qquad \dot y = x(\rho - z) - y, \qquad \dot z = xy - \beta z, with the classical parameters σ=10\sigma = 10, ρ=28\rho = 28 and β=8/3\beta = 8/3. Does the flow of this system have a strange attractor, with the dynamics of the geometric Lorenz attractor?

Remarks

The geometric Lorenz attractor is a model flow, built in the 1970s from the features seen in numerical simulations. For the model, the existence of a robust attractor with sensitive dependence on initial conditions was proved. The question was whether the actual Lorenz equations behave like the model. Smale included it as the fourteenth problem of his 1998 list of problems for the next century. Warwick Tucker gave a positive answer with a computer-assisted proof, published in 2002. His proof combines rigorous numerics with interval arithmetic, far from the origin, with normal form theory near the equilibrium at the origin.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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Attractor, concept, no evidence levelAttractorconceptDifferential 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 bifurcationconceptStrange attractor, concept, no evidence levelStrange attractorconceptOrdinary 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 systemexampleSmale's fourteenth problem, problem, no evidence levelSmale's fourteenth pr…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Attractor concept
  2. Differential equation concept
  3. Function concept
  4. Hopf bifurcation concept
  5. Interval concept
  6. Pitchfork bifurcation concept
  7. Strange attractor concept
  8. Ordinary differential equation definition
  9. Solution of an ordinary differential equation definition
  10. Autonomous system definition
  11. Equilibrium point definition
  12. Lorenz system example
  13. Smale's fourteenth problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Lorenz_system
  2. Web https://en.wikipedia.org/wiki/Smale%27s_problems
About this page
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MSC37D45, 37C10, 37M10
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AboutLorenz system
UsesLorenz system, Strange attractor, Attractor

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