Slowly Varying Jump and Transition Phenomena Associated with Algebraic Bifurcation Problems

Parameter-dependent equilibrium solutions are analyzed as the parameter slowly varies through critical values corresponding to a bifurcation or to a jump phenomena. At these critical times, interior nonlinear transition layers are necessary. Depending on the particular situation, local scaling analysis yields the first and a second Painlevé transcendent among other generic equations. In specific cases the resulting boundary layer solutions either increase algebraically or explode (via a singularity). The algebraic growth corresponds to a smooth transition to a bifurcated equilibrium. When a jump phenomena is expected, an explosion can occur. In this case, the solution of first-order differential equations approaches the equilibrium, describing the slow evolution through such a jump. However, second-order differential equations have finite amplitude oscillations around the new equilibrium.