\(C^ p\) singularity theory and heteroclinic bifurcation with a distinguished parameter
A planar dynamical system \(\dot z=h(z,\lambda)\) with one distinguished real parameter \(\lambda\) having for \(\lambda=0\) a hyperbolic saddle \(p_ 0\) and a semihyperbolic equilibrium \(q_ 0\) (a fold) with one negative and one zero eigenvalue of \(Dh(q_ 0)\) is considered. The bifurcation being investigated is the existence of a heteroclinic orbit \(\Gamma\) connecting \(p_ 0\) and \(q_ 0\) tangential to the eigendirection of the negative eigenvalue of \(q_ 0\). Associated with this situation is a map \(g(x,\lambda)=(g_ 1(x,\lambda),g_ 2(x,\lambda))\) where \(x\) is a coordinate along the center subspace of \(q_ 0\), and where zeros of \(g_ 2\) correspond to equilibria near \(q_ 0\) while common zeros of \(g_ 1\) and \(g_ 2\) correspond to heteroclinic bifurcations. The aim are normal forms \(f=(f_ 1,f_ 2)\), recognition criteria and universal unfoldings of \(g=(g_ 1,g_ 2)\). Since even for \(h\in C^ \infty\), \(g\) is only \(C^ p\) with finite \(p\), a new \(C^ p\) singularity theory is stated which enables the author to present several normal forms and universal unfoldings for low codimensional heteroclinic bifurcations together with bifurcation diagrams and phase portraits. It is indicated that this kind of heteroclinic bifurcation being considered is of relevance for shock solutions for a system of two conservation laws in one space dimension.
system of two conservation laws in one space dimension
scientific article; zbMATH DE number 93319