Defining universality classes for three different local bifurcations

The convergence to the fixed point at a bifurcation and near it is
characterized via scaling formalism for three different types of local
bifurcations of fixed points in differential equations, namely: (i) saddle-node;
(ii) transcritical; and (iii) supercritical pitchfork. At the bifurcation, the
convergence is described by a homogeneous function with three critical exponents
$\alpha$, $\beta$ and $z$. A scaling law is derived hence relating the three
exponents. Near the bifurcation the evolution towards the fixed point is given
by an exponential function whose relaxation time is marked by a power law of
the distance of the bifurcation point with an exponent $\delta$. The four
exponents $\alpha$, $\beta$, $z$ and $\delta$ can be used to defined classes of
universality for the local bifurcations of fixed points in differential
equations. The formalism is proved to be valid and can be used in either mappings and nonlinear differential equations. In a family of logistic-like mapping of the type $x_{n+1}=Rx_n(1-x_n^{\gamma})$, the exponent $\alpha$ is independent on the nonlinearity of the mapping while both $\beta$ and $z$ do indeed depend on $\gamma$ for both transcritical and saddle-node bifurcation. The critical exponents for the period doubling bifurcation however do not depend on $\gamma$ and seem to be universal. Because of the so called normal forms, the three main bifurcations above mentioned can be observed in a set of three distinct differential equations. The present approach can be an alternative to define classes of universality in local bifurcations both in mappings and in differential equations.