Scaling Properties Near Local Bifurcations

In this chapter, we examine the dynamical progression of orbits as they approach a fixed point, focusing both on the bifurcation and on its neighborhood. At the bifurcation, we demonstrate that a generalized homogeneous function captures the decay toward the fixed point, involving three critical exponents: \(\tilde{\alpha }\) , \(\tilde{\beta }\) , and \(\tilde{z}\) . Slightly away from the bifurcation, the relaxation becomes exponential, with a characteristic time that scales as a power law with respect to the distance from the bifurcation point in parameter space. The associated exponent is denoted by \(\tilde{\delta }\) . Collectively, these four exponents ( \(\tilde{\alpha }\) , \(\tilde{\beta }\) , \(\tilde{z}\) , and \(\tilde{\delta }\) ) define the universality class of the bifurcation. The transition from exponential decay to a homogeneous generalized function is referred to as critical slowing down.