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The Falkner-Skan equation. II: Dynamics and the bifurcations of \(P\)- and \(Q\)-orbits. (English) Zbl 1017.34042
Consider the Falkner-Skan equation \[ y'''+y''y+\lambda ({1-y'}^2)=0\eqno(1) \] with the parameter \(\lambda >0\), which is a reversible 3-dimensional system of ordinary differential equations with \(x\) as time variable. Equation (1) has two distinguished straight-line trajectories \(Y_{\pm}:\;y'=\pm 1,y''=y'''=0\), which form a heteroclinic loop between two fixed points \(Q_+\) and \(Q_-\) at infinity. In a previous paper [J. Diff. Equations 119, No. 2, 336-394 (1995; Zbl 0828.34028)], the same authors showed that, as the parameter \(\lambda\) increases through positive integer values greater than 1, large and complicated sets of admissible trajectories (on which \(y'\) is bounded) are created by bifurcation away from \(Y_{\pm}\). They mainly focussed on two particular types of admissible trajectory created in these bifurcations; these were periodic trajectories, which are called \(P-\)orbits, and trajectories bi-asymptotic to \(Y_+\) which are called \(Q-\)orbits. They showed that for \(\lambda>1\) each bifurcation creates infinitely many of these two types of orbits, and for \(\lambda=1\), they obtain a single \(P-\)orbit but infinitely many \(Q-\)orbits.
In the present paper, the authors continue their concentration on these trajectories. They show that all but two of these trajectories are destroyed in another sequence of bifurcation as \(\lambda\to\infty\), and by considering topological invariants and orderings on certain manifolds, they obtain unusually detailed information about the sequence of bifurcations which can occur.

MSC:
34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
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