Critical aspects of bifurcations in a geometrical approach

Bifurcation is a qualitative change that occurs in the behavior of a dynamic system when the value
of a (control) parameter undergoes a small and smooth change.
In complex systems bifurcations can be associated with phase transitions
in many different contexts: from ecosystems to models of information traffic
on the Internet, and from epidemic to social models.

But if a complex system undergoes a phase transition when the control
parameter reaches some critical value, which are the critical aspects
(in the sense of thermodynamics / statistical mechanics)
of such a transition? Trying to answer this question, it is presented an exploratory
study, based on Thermodynamic Geometry (TG),
of the critical aspects of some common bifurcations (e.~g. transcritical, saddle-node, etc.).

Thermodynamic Geometry (TG), also called
Information Geometry, is an approach based on Riemannian geometry for
the study of thermodynamical systems in equilibrium.
The main idea is that the space of equilibrium states of a system is described
by a metric, which is proportional to the Hessian matrix of entropy (or
another thermodynamic potential) of the system with respect to the thermodynamic
parameters. In this context, the notion of ``distance'' between two
states is associated with probability of fluctuation between them:
the less likely the fluctuation between the states, more distant
they are. From the metric one can obtain the corresponding geodesic equations and
curvature scalar $R$, which in turn is proportional to the correlation volume $ \xi^d$
and therefore closely related to phase transitions.

In order to study the critical aspects of bifurcations in such a geometrical
approach, the normal form (described by a nonlinear first-order differential equation) of
each bifurcation is regarded (after some appropriate transformations) as a geodesic equation of some
hypothetical model. With this we obtain the corresponding metric, curvature scalar
and also the exponent $\alpha$, analogous to the critical exponent
of specific heat in a typical thermodynamical system.