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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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##### References:
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