The entry–exit function and geometric singular perturbation theory

For small ε>0, the system x˙=ε, z˙=h(x,z,ε)z, with h(x,0,0)<0 for x<0 and h(x,0,0)>0 for x>0, admits solutions that approach the x-axis while x<0 and are repelled from it when x>0. The limiting attraction and repulsion points are given by the well-known entry–exit function. For...

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Bibliographic Details
Published inJournal of Differential Equations Vol. 260; no. 8; pp. 6697 - 6715
Main Authors De Maesschalck, Peter, Schecter, Stephen
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.04.2016
Subjects
Bifurcation delay
Blow-up
Entry–exit function
Geometric singular perturbation theory
Turning point
Bifurcation delay
Entry–exit function
Geometric singular perturbation theory
Turning point
Blow-up
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Summary:For small ε>0, the system x˙=ε, z˙=h(x,z,ε)z, with h(x,0,0)<0 for x<0 and h(x,0,0)>0 for x>0, admits solutions that approach the x-axis while x<0 and are repelled from it when x>0. The limiting attraction and repulsion points are given by the well-known entry–exit function. For h(x,z,ε)z replaced by h(x,z,ε)z2, we explain this phenomenon using geometric singular perturbation theory. We also show that the linear case can be reduced to the quadratic case, and we discuss the smoothness of the return map to the line z=z0, z0>0, in the limit ε→0.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2016.01.008