Emden-Fowler型非线性振子周期轨道的存在性
Existence of periodic orbits in nonlinear oscillators of Emden-Fowler form
摘要：
Gadella和Lara最近处理的非线性伪振子映射到Emden-Fowler（EF）方程，该方程被写成自治的二维常微分方程系统，我们提供相空间分析和参数解。通过一个不变变换，我们找到了通过可积条件的某类EF方程的周期解。我们证明了这个条件是存在周期解的必要条件，并通过常微分方程分析，我们也找到了周期轨道的充分条件。不通过可积性条件的EF方程可以通过不变变换变得可积，这也允许我们构造它们的周期解。在同一背景下讨论了另两个非线性方程，零频率Ermakov方程和正幂次Emden-Fowler方程
英文摘要：
The nonlinear pseudo-oscillator recently tackled by Gadella and Lara is mapped to an Emden-Fowler (EF) equation that is written as an autonomous two-dimensional ODE system for which we provide the phase-space analysis and the parametric solution. Through an invariant transformation we find periodic solutions to a certain class of EF equations that pass an integrability condition. We show that this condition is necessary to have periodic solutions and via the ODE analysis we also find the sufficient condition for periodic orbits. EF equations that do not pass integrability conditions can be made integrable via an invariant transformation which also allows us to construct periodic solutions to them. Two other nonlinear equations, a zero-frequency Ermakov equation and a positive power Emden-Fowler equation are discussed in the same context
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