The entry-exit function and geometric singular perturbation theory

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• 作者：Peter De Maesschalck^a ; ^{peter.demaesschalck@uhasselt.be" class="auth_mail" title="E-mail the corresponding author} ; Stephen Schecter^b ; ^{schecter@math.ncsu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Entry&ndash ; exit function ; Geometric singular perturbation theory ; Bifurcation delay ; Blow-up ; Turning point
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 April 2016
• 年：2016
• 卷：260
• 期：8
• 页码：6697-6715
• 全文大小：382 K

文摘

For small e08f2accbedec6ccba198c5" title="Click to view the MathML source">ε>0$\epsilon >0$, the system $\dot{x}=\epsilon$, $\dot{z}=h(x,z,\epsilon )z$, with 14e8f1137eff7f82263b" title="Click to view the MathML source">h(x,0,0)<0$h(x,0,0)<0$ for x<0$x<0$ and h(x,0,0)>0$h(x,0,0)>0$ for x>0$x>0$, admits solutions that approach the x  -axis while x<0$x<0$ and are repelled from it when x>0$x>0$. The limiting attraction and repulsion points are given by the well-known entry–exit function. For h(x,z,ε)z$h(x,z,\epsilon )z$ replaced by h(x,z,ε)z²$h(x,z,\epsilon )z^2$, 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$z=z_0$, z₀>0$z_0>0$, in the limit ε→0$\epsilon \to 0$.