The Tricomi problem for the Shimizu-Morioka dynamical system (Q1945195)

Consider the system \[ \begin{aligned} {dx\over dt} &= y,\\ {dy\over dt} &= x-ay- xz,\\ {dz\over dt} &= -bz+ x^2,\end{aligned}\tag{\(*\)} \] where \(a\) and \(b\) are real parameters. The author proves that, for a given \(b> 0\), there exists a number \(a(b)\geq 0\) such that \((*)\) has a homoclinic orbit to the origin. At the same time, the underlying scheme of the proof, the so-called Fisher principle, is justified.