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【6h】A Cross Section of Oscillator Dynamics.

机译振荡器动力学的横截面。

【摘要】The goal of this research is to explore criteria sufficient to produce oscillations, sample some dynamical systems that oscillate, and investigate synchronization. A discussion on linear oscillators attempts to demonstrate why autonomous oscillators are inherently nonlinear in nature. After describing some criteria on second-order dynamics that ensure periodic orbits, we explore the dynamics of two second-order oscillators in both autonomous and periodically driven fashion. Finally, we investigate the phenomena of synchronization with the nonlinear phase-locked loop. Methods of analysis are exemplified as they become relevant including Poincare maps and the Zero-One test for chaos.;The Poincare-Bendixson theorem is used to demonstrate the existence of periodic orbits in R2 under extraordinarily general conditions. Lienard's equation and theorem are introduced, which provide an intuitive parameterization for a class of oscillators. Lienard's equation is a second-order, ordinary differential equation that characterizes an oscillator with respect a state dependent damping function and a restoring force function. Lienard's theorem establishes sufficient criteria under which the Lienard's equation has a unique, stable, limit cycle.;The Duffing equation conforms with the Lienard equation, yet produces limit cycles without satisfying Lienard's theorem. Our Duffing dynamics are explained in the context of a nonlinear spring model. We survey the parameter space, which form both pitchfork and hyperbolic potential wells with respect to the displacement. These two wells characterize the bifurcations between the four fundamental undamped dynamical modes. One interesting result is that chaotic trajectories of the Duffing equation are able to quickly shed light on a multitude of quasi-periodic trajectories at the boundaries of the Poincare map.;Next we introduce an oscillator that is similar to many engineered oscillators. The Van der Pol (VDP) oscillator model is presented in the context of a nonlinear current source in parallel with an inductor, a capacitor, and a resistor. It provides a net negative conductance destabilizing the equilibrium, and is tamed into global stability by increasing damping by the square of the voltage. The VDP oscillator is the opposite of the Duffing equation in that its nonlinearity is in the damping function, with a linear restoring force function. Like the VDP oscillator, many engineered oscillators are self-excited, autonomous systems that produce limit cycles.;Finally, we investigate the process of synchronization with the phase-locked loop (PLL). Synchronization is a nonlinear process in which systems entrain their frequencies to external signals or other systems. Naturally occurring PLLs lie at the foundation of synchronization. We describe the basic topology of the PLL. Interestingly, the phase model introduced conforms with Lienard's equation and is similar to the model used for the Josephson junction and the driven pendulum. Perhaps explaining the prevalence of synchronization, we show that almost any nonlinear functional can serve as a phase detector. We briefly demonstrate a phase-lock of two oscillators with phase-noise analysis. Finally, we report on the nonlinear behavior of the PLL when subjected to a modulated input.

【摘要机译】这项研究的目的是探索足以产生振荡的标准,对一些振荡的动力系统进行采样,并研究同步性。关于线性振荡器的讨论试图证明为什么自主振荡器本质上是非线性的。在描述了确保周期轨道的二阶动力学的一些标准之后,我们以自治和周期驱动的方式探索了两个二阶振荡器的动力学。最后,我们研究了与非线性锁相环同步的现象。举例说明了与之相关的分析方法,包括庞加莱图和零一检验混沌。庞加莱-本迪克森定理用于证明R2在非常普通的条件下是否存在周期轨道。介绍了Lienard方程和定理,它们为一类振荡器提供了直观的参数设置。 Lienard方程是一个二阶常微分方程,它根据状态相关的阻尼函数和恢复力函数来表征振荡器。 Lienard定理建立了足够的标准,在该条件下Lienard方程具有唯一,稳定的极限环。Duffing方程与Lienard方程一致,但产生的极限环不满足Lienard定理。我们的Duffing动力学是在非线性弹簧模型的背景下进行解释的。我们调查了参数空间,该参数空间相对于位移形成了干草叉和双曲线势阱。这两个井表征了四个基本的非阻尼动力学模式之间的分叉。一个有趣的结果是,达芬方程的混沌轨迹能够迅速阐明庞加莱图边界上的许多准周期轨迹。接下来,我们介绍一种与许多工程振荡器相似的振荡器。范德波尔(VDP)振荡器模型是在非线性电流源与电感器,电容器和电阻器并联的情况下提出的。它提供了一个使平衡不稳定的净负电导率,并通过以电压的平方增加阻尼来使全局稳定。 VDP振荡器与Duffing方程相反,其非线性在于阻尼函数,具有线性恢复力函数。像VDP振荡器一样,许多工程振荡器都是产生极限周期的自激式自治系统。最后,我们研究了与锁相环(PLL)同步的过程。同步是一个非线性过程,其中系统将其频率携带到外部信号或其他系统。自然发生的PLL是同步的基础。我们描述了PLL的基本拓扑。有趣的是,引入的相位模型符合Lienard方程,并且类似于用于约瑟夫森结和从动摆的模型。也许可以解释同步的普遍性,我们证明几乎所有非线性函数都可以用作鉴相器。我们用相位噪声分析简要演示了两个振荡器的锁相。最后,我们报告了经过调制输入时PLL的非线性行为。

【作者】De Salvo, Jason A.;

【作者单位】University of Colorado at Boulder.;

【年(卷),期】2010(),

【年度】2010

【页码】65 p.

【总页数】65

【原文格式】PDF

【正文语种】eng

【中图分类】;

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