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I have a 3d dynamical system that I am investigating that undergoes a series of bifurcations as two parameters ($I_1$ and $I_2$) are varied. The parameters represent input entering the system along different directions. Now what I have observed is that when either input is 1 and the other one is 0, the Jacobian has a conjugate eigenvalue pair with real parts larger than 0 and a purely real eigenvalue that is usually smaller than the real part of the conjugate eigenvalue pair. The total input to the system is normalised, so as you crank up $I_1$ for example, $I_2$ is toned down. The system then undergoes a Hopf-bifurcation, i.e. the conjugate eigenvalue pair crosses the stability line, turning unstable spirals into stable ones. Now when $I_1$ is cranked up a little further, then the real part of the conjugate eigenvalue pair actually becomes smaller than the real eigenvalue, i.e. trajectories now approach the fixed point along a single line rather than as spirals.

My question is: what is this bifurcation called (does it even have a name?)? And what, if anything, is its characteristic form?

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  • $\begingroup$ Bifurcations have very little to do with the dimension of the phase space (because of the existence of center manifold). What is important is the codimension, which, at a first approximation, can be taken as the number of free parameters. All codim 2 generic bifurcations of equilibria are well studied. You can find the full list here. If your system has no internal symmetries the chances are you should be able to identify your bifurcation. $\endgroup$ Commented Jul 7 at 13:48
  • $\begingroup$ @Artem. Well, no. The bifurcation (if it even is one) is co-dimension 1, since the total norm of the input vector is fixed. Say the norm is 1 then you can represent the input as $ \vec{(\cos(\phi), \sin(\phi))}$, meaning there is only one real free parameter, namely $\phi$. What I'm looking for is not really a bifurcation in the classical sense either. The fixed points never change stability. The only thing that changes is the way in which the fixed point is approached. $\endgroup$ Commented Jul 7 at 15:16

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Based on this description, I would say this sounds like some degenerate Hopf bifurcation, and there are several ways how this can unfold in a two-parameter family (see the link below for an example).

That being said, it is a little hard for me to understand the precise assumptions you impose on the bifurcation scenario from your description alone. Could you clarify it a little bit, possibly with a sketch explaining how the parameter space looks like or a precise statement for how the eigenvalues change with the parameters?

http://www.scholarpedia.org/article/Hopf-Hopf_bifurcation

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