## 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 < μ_{c}, there is only one steady state r = 0 (or x = y = 0). For μ > μ_{c},
however, there are two solutions, r = 0 and r = √(μ - μ_{c}). The non-trivial solution is a periodic orbit with an angular frequency ω and period 2π/ω. Hence at the critical point (μ = μ_{c}) 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 W_{t} is a Wiener process with
increment dW_{t}. The expectation value E[R_{t}], where R²_{t} = X²_{t} + Y²_{t}, 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 < μ_{c} which increases with increasing noise level λ.

Figure 1: Response near a stochastic Hopf bifurcation at μ = μ_{c}. In the deterministic case (λ = 0), r = 0 for μ < μ_{c}. When noise is included, the expectation value E[R_{t}], where R²_{t} = X²_{t} + Y²_{t}, increases with increasing λ for any value of μ - μ_{c} (from [1]).

### References

[1] Dijkstra et al. (2008), *Phil. Trans. Royal Soc. A*, vol. 366, pages 2545-2560.