Lambert W functions and exponential type systems
seem to have complex dynamic that is a lot like your
Hopf bifurcations. Look at the fractals at this link:
http://users.forthnet.gr/ath/jgal/math/exponents.html
at the bottom of the page.
It seems to me that you are getting some kind of parabolic
attractor set that is generating the Hopf bifurcation.
I'm sure I may be wrong, but it is at least a lead.
Kostis Chlouverakis wrote:

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