Canard solutions of two-dimensional singularly perturbed systems
Consider the planar singularly perturbed system \[ \varepsilon{dy\over dt}= y- f(x)+\varepsilon h(x,\lambda,\varepsilon), \qquad {dx\over dt}= g(x,y,\lambda,\varepsilon) \] depending on two real parameters \(\varepsilon\) and \(\lambda\), where \(0<\varepsilon\ll 1\). \(f\), \(g\), \(h\) are sufficiently smooth functions satisfying (i) \(f(0)= f'(0)=\cdots= f^{(m-1)}(0)= 0\), \(f^{(m)}(0)> 0\), (ii) \(g(0,0,0,0)= 0\), \({\partial g\over\partial x}(0, 0,0,0)\neq 0\), \({\partial g\over\partial\lambda}(0,0, 0,0)= -\gamma< 0\). By means of the method of asymptotic analysis (in the sense of W. Eckhaus), the authors prove the existence of canard cycles for \(\lambda= \lambda_c(\varepsilon)+ \sigma\varepsilon^{(2-{1\over m})} e^{-{k\over \varepsilon}^2}\), \(k> 0\), \(\sigma\in\mathbb{R}\). In case \(\sigma{\partial g\over\partial\lambda} (0,0,0,0)< 0\) the canard cycles have a head, in case of alternative sign the canard cycles have no head. The paper contains very sophisticated computations, but a very clear logical structure.
scientific article; zbMATH DE number 2158550