Interdisciplinary research on potential analysis

货车der Pol方程是描述电路中弛豫振荡的方程，对非线性振荡理论和Hopf分岔理论的发展起了重要作用。众所周知，该方程只有一个极限环和两个无界分离面。尽管分界线与极限环密切相关，但人们对分界线的位置知之甚少。在这项研究中，我们估计的位置，使用相平面分析和一些李雅普诺夫函数。讨论了货车del Pol方程的推广Lie^^'nard系统，给出了Lie^^' nard系统具有同宿轨的条件，讨论了各种Euler型微分方程和非线性自伴微分方程的振动性，给出了所有非平凡解振动的充要条件.所得定理推广了前人关于此问题的许多结果。我们还讨论了具有时滞（或衰减系数）的非线性微分方程的所有解是否振动。通过改变变量，我们可以将这些方程改写成李纳德型方程组。因此，通过相平面分析，我们可以详细地研究解的渐近性态，并将上述结果与所谓的“上下解方法”相结合，得到了拟线性椭圆型方程（或Schro^^<..>dinger方程）存在衰减于无穷远处的正解的充分条件.

Van der Pol's equation was formulated to describe relaxation oscillations in electrical circuits, and played an important role in development of the theory of nonlinear oscillations and the theory of Hopf bifurcation. It is well-known that this equation has exactly one limit cycle with two unbounded separatrices. Although the separatrices are closely related to the limit cycle, little is known about the position of separatrices. In this research, we estimate the position by use of phase plane analysis and some Liapunov functions. Also, we consider the Lie^^'nard system which is a generalization of van del Pol's equation and give some conditions under which the Lie^^'nard system has homoclinic orbits.We deal with the oscillation problem for various differential equations of Euler type and nonlinear self-adjoint differential equations, and present necessary and sufficient conditions for all nontrivial solutions to be oscillatory. The obtained theorems extend many previous results on this problem. We also discuss whether all solutions of nonlinear differential equations with time delay (or with decaying coefficients) oscillate or not. Changing variables, we can rewrite those equations into systems of Lie^^'nard type. For this reason, by means of phase plane analysis of the systems, we can examine the asymptotic behaviour of solutions in detail.Combining the above results and the so-called "supersolution-subsolution method", we obtain sufficient conditions for quasilinear elliptic equations (or Schro^^<..>dinger equations) to have a positive solution which decays at infinity.