内耦合系统的能量分布：弹簧摆模型
Energy Distribution in Intrinsically Coupled Systems: The Spring Pendulum Paradigm
摘要：
内在非线性耦合系统呈现出不同的振荡分量，它们之间交换能量。我们提出了一种新的方法来处理这种能量交换，并探讨它如何依赖于系统的控制参数。该方法包括在写系统的总能量，并正确地识别每个组件的能量项，特别是，他们的耦合。为了说明所提出的方法，我们与二维弹簧摆，这是一个范例，研究非线性耦合系统，并被用作几个系统的模型。对于弹簧摆，我们确定了三个能量分量，类似于弹簧和钟摆一样的运动，以及它们之间的耦合。有了这些解析表达式，我们分析了单个轨迹的能量交换，我们还获得了弹簧摆能量分布的全局特性，通过计算大量的轨迹（周期，准周期和混沌）在整个相空间的空间和时间平均能量分量。考虑到由于非线性耦合的能量项，我们确定的区域中的参数空间，对应于强和弱耦合。所提出的方法可以应用于非线性耦合系统，以揭示如何耦合介导的内能交换，以及如何根据系统参数的能量分布的变化。
英文摘要：
Intrinsically nonlinear coupled systems present different oscillating components that exchange energy among themselves. We present a new approach to deal with such energy exchanges and to investigate how it depends on the system control parameters. The method consists in writing the total energy of the system, and properly identifying the energy terms for each component and, especially, their coupling. To illustrate the proposed approach, we work with the bi-dimensional spring pendulum, which is a paradigm to study nonlinear coupled systems, and is used as a model for several systems. For the spring pendulum, we identify three energy components, resembling the spring and pendulum like motions, and the coupling between them. With these analytical expressions, we analyze the energy exchange for individual trajectories, and we also obtain global characteristics of the spring pendulum energy distribution by calculating spatial and time average energy components for a great number of trajectories (periodic, quasi-periodic and chaotic) throughout the phase space. Considering an energy term due to the nonlinear coupling, we identify regions in the parameter space that correspond to strong and weak coupling. The presented procedure can be applied to nonlinear coupled systems to reveal how the coupling mediates internal energy exchanges, and how the energy distribution varies according to the system parameters.
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