The onset of instability in autonomous dynamical systems (ADS) of ordinary differential equations is investigated. Binary, ternary and quaternary ADS are taken into account. The stability frontier of the spectrum is analyzed. Conditions necessary and sufficient for the occurring of Hopf, Hopf–Steady, Double-Hopf and unsteady aperiodic bifurcations—in closed form—and conditions guaranteeing the absence of unsteady bifurcations via symmetrizability, are obtained. The continuous triopoly Cournot game of mathematical economy is taken into account and it is shown that the ternary ADS governing the Nash equilibrium stability, is symmetrizable. The onset of Hopf bifurcations in rotatory thermal hydrodynamics is studied and the Hopf bifurcation number (threshold that the Taylor number crosses at the onset of Hopf bifurcations) is obtained.

中文翻译：

---

动力系统中的Hopf分支

研究了常微分方程自治动力系统（ADS）中的不稳定性。二元，三元和四元ADS被考虑在内。分析频谱的稳定性边界。获得了发生Hopf，Hopf-Steady，Double-Hopf和非稳态非周期性分叉（闭合形式）的必要条件和充分条件，以及通过对称性保证不存在非稳态分叉的条件。考虑了数学经济学上连续的三重古诺（Gournot）博弈，证明了控制纳什均衡稳定性的三元ADS是对称的。研究了旋转热流体动力学中霍普夫分叉的发生，并给出了霍普夫分叉数 （阈值是泰勒数在Hopf分叉的起点处交叉）。