Asymptotic expansions using blow-up
This paper considers the singularly perturbed problem \[ \varepsilon \dot x=f(x,y,\varepsilon), \quad \dot y= g(x,y,\varepsilon), \] where \(x\), \(y\in\mathbb{R}\), \(0<\varepsilon\leq1\), \( f(x,y,\varepsilon) =-y+x^2+axy+bx^3+\mathcal{O}(y^2,\varepsilon x,\varepsilon y, x^2y,x^4)\), and \(g(x,y,\varepsilon)=-1+cx+\mathcal{O}(x^2,y,\varepsilon x)\). With such assumptions, the reduced problem is defined on a manifold which has a fold. This fold is a jump point for the orbits of the reduced problem. It separates the attractive manifold \(S_a\) and the repelling one \(S_r\). It is standard that these manifolds perturb smoothly to locally invariant manifolds \(S_{a,\varepsilon}\) and \(S_{r,\varepsilon}\) of the perturbed problem. The aim of the paper is to derive asymptotic expansion of these slow manifolds continued beyond the fold point. More precisely, the authors consider suitable sections \(\Delta^{\text{in}}\) and \(\Delta^{\text{out}}\) transversal to the flow just before and just after the jump point and define the transition map \(\Pi:\Delta^{\text{in}}\to\Delta^{\text{out}}\). They write \(\Pi(S_{a,\varepsilon}\cap \Delta^{\text{in}})=(\rho,h(\rho,\varepsilon))\) and compute an asymptotic expansion of the form \[ h(\rho,\varepsilon)= \rho^2 \sum_{j=0}^N\sum_{l=0}^{\pi(j)} c_{jl}(\rho)(\frac{\varepsilon}{\rho^3})^\frac{2+j}{3} \biggl(\ln\frac{\varepsilon}{\rho^3}\biggr)^l \rho^k + \mathcal{O} \biggl(\frac{\varepsilon}{\rho^3}\biggr)^\frac{N+3}{3}. \] This is a rather elaborate computation obtained using blow-up techniques successively in three different regions near the jump point. Such an approach establishes a connection between matched asymptotic expansions and geometric singular perturbation theory. It explains the unexpected structure of the expansion.
scientific article; zbMATH DE number 2172367