Hard Loss of Stability in Painlevé-2 Equation

Abstract

A special asymptotic solution of the Painlevé-2 equation with small parameter is studied. This solution has a critical point t * corresponding to a bifurcation phenomenon. When t < t * the constructed solution varies slowly and when t > t * the solution oscillates very fast. We investigate the transitional layer in detail and obtain a smooth asymptotic solution, using a sequence of scaling and matching procedures.

The capture of a particle into resonance at potential hole with dissipative perturbation

“…Formulas for solutions which are fit for all the regions enable one to investigate the properties of single trajectories. The papers [26,27,28,29] contain such formulas for autoresonance problems, constructed by the method of matching of asymptotic expansions [17].…”

Oscillations near a separatrix in the duffing equation

“…Lemma 3 There exists an asymptotic solution in form (7) as ε → 0, where u n (τ ) has form (8) and |τ | ≪ − 1 4 ln(ε).…”

“…D.C.Diminie and R.Haberman studied a separatrix crossing near a saddle-center bifurcation and a pitchfork bifurcation [7]. Full asymptotic expansions for the problem of the separatrix crossing near the saddle-center bifurcation was obtained in [8,9].…”

An asymptotic structure of the bifurcation boundary of the perturbed Painlevé-2 equation

“…A careful study of the relationship between the pitchfork bifurcation and the Painlevé-2 equation was given in [2]. The asymptotic behaviour of the Painlevé-2 equation due to the hard loss of stability loss was constructed in [3] for the Painlevé-2 equation and in [4] for hard stability loss in the main resonance equation.…”