Are one-dimensional bifurcations dissipative or conservative?

Question

I have elementary questions about one dimensional bifurcations.

It should be remarked that I have also searched and read previous posts of Physics stack exchange. However, these did not help me with the questions to be presented here. Owing to the fact that there are few references on the subject, I think that these questions, and their possible answers, may be of great interest to those that subscribe to physicsstacksexchange.

With that said, let me first recall the definition of bifurcation theory from the standpoint of differential equations. The main goal of the bifurcation theory is to express the sudden qualitative changes in the phase portrait of dynamical systems nearby their local solution branches when control parameters are changed continuously and smoothly. From the standpoint of differential equations, the following dynamical systems

$\displaystyle\frac{dx}{dt}=m-x^{2}$, (Saddle-node bifurcation)

$\displaystyle\frac{dx}{dt}=mx-x^{2}$, (Transcritical bifurcation)

$\displaystyle\frac{dx}{dt}=mx-x^{3}$, (Pitchfork bifurcation)

may show qualitative changes in their phase portraits as a result of the variation of the control parameter ($m$). Based on the above, I ask:

1. Are the above systems conservative? From the framework of two-dimensional systems, it is clear how I may determine whether a dynamical system is conservative from Liouville's theorem. However, how may one determine if these systems are or are not conservative?

2. Based on the answer to the first question, May you suggest to me references in which those systems are described as conservative/dissipative?

Answer 1

In general, a dynamical system $\dot{x} = f(x)$ is conservative if and only if $∇·f = \mathrm{div} f = 0$ (in the entire phase-space volume of interest). This works irrespective of the dimension of the system.

This follows almost directly from Liouville’s theorem which (in the variant I know) states that a given phase-space volume stays constant in volume over time in a Hamiltonian system. Since the divergence quantifies the change of phase-space volume, it needs to be zero for conservative systems.

With this, it is straightforward to compute that neither of your systems is conservative.

Note that most bifurcations are changes of the stability of fixed points, limit cycles, etc. Since there are no stable dynamics (attractors) in Hamiltonian systems, there is no good analogue in Hamiltonian systems.

There is such a thing as bifurcations in Hamiltonian systems. For example, consider the dynamics of a ball rolling frictionless on the one-dimensional geography described by $h(x) = x^4+mx^2$. When $m$ becomes negative, we go from one to three fixed points (all marginal, before and after) in something vaguely resembling a pitchfork bifurcation.