Integral manifolds, canards and black swans. (English) Zbl 0985.34039
Consider the singularly perturbed system of nonautonomous differential equations
\[
dx/dt =f(x,y,t,\varepsilon), \quad \varepsilon dy/dt =2ty+a+g(x,y,t,a,\varepsilon), \tag \(*\)
\]
with \(x \in \mathbb{R}^n\), \(y \in \mathbb{R}\), \(0<\varepsilon \ll 1\). Assume that for \(\varepsilon =0\), \(a =0\), system \((*)\) has the invariant manifold \(y=0\). The authors derive conditions on \(f\) and \(g\) ensuring the existence of a function \(a(x,\varepsilon)\) such that \((*)\) has an integral manifold \({\mathcal{M}}_\varepsilon : = \{(x,y,t,\varepsilon)\in \mathbb{R}^{n+3}: y=h(x,t,\varepsilon)\}\) that is attracting for \(t<0\) and repelling for \(t>0\); moreover, \(a\) and \(h\) tend to zero as \(\varepsilon\) tends to zero. The authors call the integral manifold \({\mathcal{M}}_\varepsilon\) a black swan (higher-dimensional canard solution).
Reviewer: Klaus R.Schneider (Berlin)
MSC:
| 34C45 | Invariant manifolds for ordinary differential equations |
| 34E15 | Singular perturbations for ordinary differential equations |
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