论文标题
一种用于检测微分方程中正常双曲线不变的托里的机制
A mechanism for detecting normally hyperbolic invariant tori in differential equations
论文作者
论文摘要
确定紧凑型不变流形的存在是微分方程定性理论中的一个核心追求。奇点，周期性解决方案和不变的托里是这种不变流形的例子。平均理论的经典且有用的结果将以特定标准形式给出的非自治周期性微分方程的孤立周期性解决方案的存在与所谓的指导系统的简单奇异性存在，这是第一个非偏差的较高较高级别平均功能。在本文中，我们为存在不变的Tori提供了类似的结果。也就是说，我们表明，以标准形式给出的非自治周期性微分方程在扩展相空间中具有正常双曲线的圆环，前提是指导系统具有双曲线极限循环。我们将此结果应用于一个混蛋差分方程中的正常双曲线不变的托里。
Determining the existence of compact invariant manifolds is a central quest in the qualitative theory of differential equations. Singularities, periodic solutions, and invariant tori are examples of such invariant manifolds. A classical and useful result from the averaging theory relates the existence of isolated periodic solutions of non-autonomous periodic differential equations, given in a specific standard form, with the existence of simple singularities of the so-called guiding system, which is an autonomous differential equation given in terms of the first non-vanishing higher order averaged function. In this paper, we provide an analogous result for the existence of invariant tori. Namely, we show that a non-autonomous periodic differential equation, given in the standard form, has a normally hyperbolic invariant torus in the extended phase space provided that the guiding system has a hyperbolic limit cycle. We apply this result to show the existence of normally hyperbolic invariant tori in a family of jerk differential equations.
