StuartLandau

Systems / ODEs / Oscillatory systems

The normal form of a supercritical Hopf bifurcation — an exactly circular limit cycle of radius √mu, the reference case for every limit-cycle method.

continuous · ODE2 dimensionslimit cyclenormal form

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= \mu x - \omega y - \left(x^{2} + y^{2}\right) \left(- b y + x\right) \\ \dot{y} &= \mu y + \omega x - \left(x^{2} + y^{2}\right) \left(b x + y\right) \end{aligned} \]

Parameters

Symbol	Default	Role
b	0.5	nonlinear frequency shift (shear); b = 0 is isochronous
mu	1	distance past the Hopf bifurcation (cycle radius √mu)
omega	1	linear angular frequency at the bifurcation

State variables: x, y

Properties

Lyapunov spectrum

$0,\; -2$
analytic: r' = mu r - r^3 gives the transverse exponent -2*mu

Kaplan–Yorke dimension

$D_{KY} = 1$

Divergence ∇·f

$\nabla\!\cdot f = 2 \mu - 4 x^{2} - 4 y^{2}$
state-dependent

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.StuartLandau()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference

Stuart (1960), J. Fluid Mech. 9, 353-370
doi:10.1017/s002211206000116x

BibTeX

@misc{stuartlandau,
  title = {StuartLandau system},
  note = {Stuart (1960), J. Fluid Mech. 9, 353-370},
  doi = {10.1017/s002211206000116x}
}