In this work we consider a three-dimensional autonomous system of nonlinear ordinary differential equations, which may be thought of as a generalization of the well-known Hopf–Langford system introduced about forty years ago. This dynamical system turned out to be equivalent to the nonlinear force-free Duffing oscillator. In three special cases, it is found to be completely integrable. To the best of our knowledge, these facts have not been noticed so far in the rich literature on the subject. In the aforementioned three special cases, the general solutions of the respective systems are expressed in explicit analytical form by means of elementary and Jacobi elliptic functions depending on the values of the system parameters. This allowed us to characterize in details the dynamics of the regarded systems.

在这项工作中，我们考虑了非线性常微分方程的三维自治系统，该系统可以看作是大约40年前引入的著名Hopf-Langford系统的推广。事实证明，该动力学系统等效于非线性无力Duffing振荡器。在三种特殊情况下，它是完全可集成的。据我们所知，这些事实到目前为止尚未在有关该主题的丰富文献中被注意到。在上述三个特殊情况下，根据系统参数的值，通过基本和Jacobi椭圆函数，以明确的解析形式表示各个系统的一般解。这使我们能够详细描述所考虑系统的动力学。