A simple Hopf bifurcation
A simple dynamical system exhibiting a Hopf bifurcation is the system:
having two degrees of freedom (x,y). In polar coordinates (r, θ) with x = r cos θ and y = r sin θ, these equations can be written as:
For μ < μc, there is only one steady state r = 0 (or x = y = 0). For μ > μc, however, there are two solutions, r = 0 and r = √(μ - μc). The non-trivial solution is a periodic orbit with an angular frequency ω and period 2π/ω. Hence at the critical point (μ = μc) periodic behavior with a frequency ω is spontaneously generated through an instability of the trivial solution x = y = 0. This is known as a Hopf bifurcation.
Addition of noise
The simple Hopf bifurcation described above can be modified to include stochastic noise:
where λ is the amplitude of the additive noise and Wt is a Wiener process with increment dWt. The expectation value E[Rt], where R²t = X²t + Y²t, resulting from the stochastic integration of this system is shown in figure 1 for several values of λ; the deterministic case is shown for λ = 0. Clearly, there is a response for values μ < μc which increases with increasing noise level λ.
Figure 1: Response near a stochastic Hopf bifurcation at μ = μc. In the deterministic case (λ = 0), r = 0 for μ < μc. When noise is included, the expectation value E[Rt], where R²t = X²t + Y²t, increases with increasing λ for any value of μ - μc (from ).
 Dijkstra et al. (2008), Phil. Trans. Royal Soc. A, vol. 366, pages 2545-2560.