Math 2.0

Dynamical systems and ergodic theory Bifurcation theory Theorem

Shilnikov theorem

Builds on Homoclinic orbitHyperbolic equilibrium pointEigenvaluePeriodic orbitSmale horseshoeHomoclinic bifurcation

Statement

Let x˙=F(x)\dot x = F(x) be a smooth vector field on ℝ3\R^3. Let x0x_0 be a hyperbolic equilibrium whose Jacobian matrix DF(x0)DF(x_0) has eigenvalues λ>0\lambda > 0 and −ρ±iω-\rho \pm i \omega with ρ>0\rho > 0 and ω≠0\omega \ne 0. Let Γ\Gamma be a homoclinic orbit to x0x_0. Assume the Shilnikov condition ρ<λ\rho < \lambda. Then every neighbourhood of Γ∪{x0}\Gamma \cup \set{x_0} contains infinitely many periodic orbits of saddle type.

Remarks

Such an equilibrium is called a saddle-focus: orbits near it spiral in along a plane and leave along a line. The proof studies the return map to a small section transverse to Γ\Gamma. Near x0x_0 the flow is close to linear, and the passage time near the equilibrium makes the returning image a spiral that winds infinitely often. When ρ<λ\rho < \lambda this spiral is stretched rather than contracted, and it crosses the section so as to form infinitely many Smale horseshoes. Each horseshoe carries infinitely many periodic orbits, which gives the conclusion. The condition ρ<λ\rho < \lambda is needed: if ρ>λ\rho > \lambda, a sufficiently small neighbourhood of Γ\Gamma contains at most one periodic orbit. The theorem is due to Leonid Shilnikov.

Going deeper

For a student

The theorem is a main source of chaos in three-dimensional flows, because it reduces chaos to the existence of a single homoclinic orbit plus an inequality on eigenvalues. A homoclinic bifurcation to a saddle-focus that satisfies the Shilnikov condition is therefore a route to chaos in families of flows. In the plane, by contrast, the breaking of a homoclinic loop creates at most one limit cycle.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

Show the path as a graph
Cantor set, example, no evidence levelCantor setexampleEigenvalue, concept, no evidence levelEigenvalueconceptHomoclinic bifurcation, concept, no evidence levelHomoclinic bifurcationconceptHomoclinic orbit, concept, no evidence levelHomoclinic orbitconceptHyperbolic equilibrium point, concept, no evidence levelHyperbolic equilibriu…conceptPeriodic orbit, concept, no evidence levelPeriodic orbitconceptTopological conjugacy, concept, no evidence levelTopological conjugacyconceptSmale horseshoe, example, no evidence levelSmale horseshoeexampleShilnikov theorem, statement, no evidence levelShilnikov theoremstatement
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. Cantor set example
  2. Eigenvalue concept
  3. Homoclinic bifurcation concept
  4. Homoclinic orbit concept
  5. Hyperbolic equilibrium point concept
  6. Periodic orbit concept
  7. Topological conjugacy concept
  8. Smale horseshoe example
  9. Shilnikov theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Homoclinic_orbit
About this page
TagQ134D9EMAZ
URIm2://Q134D9EMAZ
Kindstatement, theorem
WritingW0 stub
MSC37G20, 34C37, 37C29
StatusThis copy of the site has no build data. Run m2 status Q134D9EMAZ.
AboutHomoclinic orbit, Homoclinic bifurcation
UsesHomoclinic orbit, Hyperbolic equilibrium point, Eigenvalue, Periodic orbit, Smale horseshoe, Homoclinic bifurcation

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