Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How can I study bifurcations in a dynamical system consisting of $\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt}$? The system depends on two parameters, and I want to find the bifurcation curves and, then, to plot these curves in a $(a,b)$-plane, the plane of the parameters $a$ and $b$.

share|cite|improve this question

Here's how to find the places where the fixed points bifurcate. First find the equilibria of the system by finding for each $(a,b)$ all values of $(x,y,z)$ with $F(x,y,z) = 0$, where the ODE is $\frac {d}{dt} (x,y,z) = F(x,y,z)$. Then find the eigenvalues of the linearisation $DF$ at these equilibria. For each equilibrium these eigenvalues may depend on $a,b$ -- you can find the bifurcation curves by finding the values of $(a,b)$ at which the eigenvalues cross the imaginary axis, which corresponds to the fixed point changing stability from stable to unstable, or vice versa.

There are other sorts of bifurcations that may occur, but these are the simplest.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.