This paper reviews a systematic dynamical analysis on a general form of scalar field as Dark Energy (DE) with dark matter (DM) to sort out the “cosmic coincidence” problem. Here the autonomous system of differential equations is two-dimensional (2D) as well as nonlinear. So we have utilized nonlinear dynamical theory to explain various cosmological implications of this model. Nowadays, we have noted that some works are undertaking this nonlinear systems theory. Although we have seen that most of the works are simplifying the underlying nonlinear dynamical systems similar to a linear one, that can lead to flawed conclusions about the evolution of the universe. Since an important theorem, Poincare–Bendixson theorem asserts linearization of the nonlinear system and does not give “global” stability, unlike the linear one if the dimension is more than two. Anyway, our work is different from others in this regard. Here the dimension of the system is two, and we have obtained some interesting stuffs also. We have applied the above theorem of nonlinear dynamical systems and others to find the “global” stability. This theorem offers completely different stable solutions, contrary to the prediction of linear analysis. As a result, we have obtained two fixed points; one of them is a stable “attractor” (it is attracting “node” actually), and thereafter, we have analyzed the stability. To investigate the dynamical system behavior, we have drawn different figures. These figures include vector field and a new plotting strategy (explained later). These investigations suggest a way out of the coincidence problem (or, precisely speaking, what should be the mathematical form of the term “C”, which indicates interaction between DE and DM to reduce coincidence). In this scenario, if the equation of state (EoS) of DE and DM obeys ωd < ωm, then coincidence problem may be avoided.

中文翻译：

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任意暗能量与重合问题的动力学分析

本文回顾了对一般形式的标量场的系统动力学分析，即暗能量（DE）与暗物质（DM），以理清“宇宙巧合”问题。在这里，微分方程的自治系统是二维（2D）以及非线性的。因此，我们利用非线性动力学理论来解释该模型的各种宇宙学含义。如今，我们注意到一些工作正在研究这种非线性系统理论。尽管我们已经看到大多数工作都在简化类似于线性系统的潜在非线性动力系统，但这可能会导致关于宇宙演化的有缺陷的结论。由于一个重要的定理，Poincare-Bendixson 定理断言非线性系统的线性化并且不给出“全局”稳定性，如果维度大于两个，则与线性维度不同。无论如何，我们的工作在这方面与其他人不同。这里系统的维度是2，我们也得到了一些有趣的东西。我们已经应用上述非线性动力系统定理和其他定理来找到“全局”稳定性。与线性分析的预测相反，该定理提供了完全不同的稳定解。结果，我们得到了两个不动点；其中一个是稳定的“吸引子”（实际上是吸引“节点”），之后我们分析了稳定性。为了研究动力系统的行为，我们绘制了不同的图形。这些数字包括矢量场和新的绘图策略（稍后解释）。这些调查提出了解决巧合问题的方法（或者，准确地说，C”，表示 DE 和 DM 之间的相互作用以减少巧合）。在这种情况下，如果 DE 和 DM 的状态方程 (EoS) 服从ωd < ω米，则可以避免巧合问题。