Generic bifurcation of refracted systems
Let \(f\) be a smooth function on \(\mathbb{R}^3\), with \(f(0)=0\), and assume that \(0\) is a regular value. Define the sets \(\Sigma=f^{-1}(0)\), \(\Sigma_{+}=\{q|f(q) \geq 0\}\), \(\Sigma_{-}=\{q|f(q) \leq 0\}\). Consider the space of all vector fields \(Z(x,y,z)=X(x,y,z)\) if \((x,y,z)\epsilon \Sigma_{+}\), and \(Z(x,y,z)=Y(x,y,z)\) if \((x,y,z)\epsilon \Sigma_{-}.\) These systems are called refracted systems. The authors consider the space of all fold-fold elliptic systems. These are refracted systems satisfying \(X f(0)=0\), \(X^{2}f(0)<0\) and \(Y^{2}f(0) >0\); the notation \(X f(p)=(\nabla f(p),X(p))\) is used. The authors prove that the set of all structurally stable systems is an open dense set. Moreover, the normal forms of structurally stable systems are given. The authors give a classification of co-dimension one singularities. The main tools are results on embedding diffeomorphism in flows and properties of reversible mappings.
scientific article; zbMATH DE number 6135762