Large amplitude oscillations for a class of symmetric polynomial differential systems in R³

Abstract

In this paper we study a class of symmetric polynomial differential systems in R 3 , which has a set of parallel invariant straight lines, forming degenerate heteroclinic cycles, which have their two singular endpoints at infinity. The global study near infinity is performed using the Poincaré compactification. We prove that for all n ∈ N there is ε n > 0 such that for 0 < ε < ε n the system has at least n large amplitude periodic orbits bifurcating from the heteroclinic loop formed by the two invariant straight lines closest to the x-axis, one contained in the half-space y > 0 and the other in y < 0.

On the Darboux integrability of Blasius and Falkner–Skan equation

“…For the derivation of this equation see [1]. For the existence and uniqueness of the solutions [9] and [13]; and for more recent works on the bifurcations in this equation…”

“…Let H = H(x, y, z) be a first integral of system (2), then it must satisfy that (13) y ∂H ∂x + z ∂H ∂y − [xz + λ(1 − y 2 )] ∂H ∂z = 0.…”

On the Pitchfork Bifurcation of the Folded Node and Other Unbounded Time-Reversible Connection Problems in $\mathbb R^3$

“…aim of the following section, is therefore to give a simple proof using the Melnikov approach, in particular Theorem 2.8, and the recipe in Section 2.2, which is based upon -as is more standard in dynamical systems -invariant manifolds. See also [18], for a similar approach in this context. In this reference, however, periodic orbits are constructed through an analysis of a return mapping.…”

“…The Falkner-Skan equation (1.4) are well-known examples of systems (without equilibria) possessing such bifurcations, see e.g. [24], and [18] for other examples. The Falkner-Skan equation (1.3) initially appeared in the study of boundary layers in fluid dynamics, see [5].…”

On the pitchfork bifurcation of the folded node and other unbounded time-reversible connection problems in $\mathbb R^3$

“…are well-known examples of systems (without equilibria) possessing such bifurcations, see e.g. [23], and [17] for other examples. The Falkner-Skan equation (1.3) initially appeared in the study of boundary layers in fluid dynamics, see [6].…”