The stability criteria for the dynamical system of a homogeneous and isotropic cosmological model are investigated with the interaction of a scalar field in the presence of a perfect fluid. In this paper, we depict the dynamical system perspective to study qualitatively the scalar field cosmology under two special cases, with and without potential. In the absence of potential, we get a two-dimensional dynamical system, and we study the analytical as well as geometrical behavior. For the dynamical system with potential, we analyze different potential forms: simple exponential potential form (), double exponential potential form , and inverse power law potential form (). We generate an autonomous system of ordinary differential equations (ASODE) for each case by introducing new dimensionless variables and obtain respective fixed points. We also analyze the type, nature, and stability of the fixed points and how their behavior reflects towards the cosmological scenarios. Throughout the whole work, the investigation of this model has shown us the deep connection between these theories and cosmic acceleration phenomena. The phase plots of the system at different conditions and different values of have been analyzed in detail, and their geometrical interpretations have been studied. The perturbation plots of the dynamical system have been analyzed with emphasis on our analytical findings. We have evaluated the total energy density () at the fixed points and also found out the suitable range of and for a stable model.

中文翻译：

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最小标量场耦合的宇宙学模型的动力学系统视角

在存在完美流体的情况下，利用标量场的相互作用研究了均质各向同性宇宙学模型动力学系统的稳定性准则。在本文中，我们描绘了动力学系统的观点，以定性研究在两种特殊情况下（无论有无潜力）的标量场宇宙学。在没有潜力的情况下，我们得到了一个二维动力学系统，我们研究了解析以及几何行为。对于具有势能的动力学系统，我们分析了不同的势能形式：简单的指数势能形式（），双指数势形式，与逆幂势形式（）。通过引入新的无量纲变量，我们为每种情况生成了一个常微分方程（ASODE）自治系统，并获得了相应的固定点。我们还将分析固定点的类型，性质和稳定性，以及它们的行为如何反映宇宙场景。在整个工作中，对该模型的研究向我们展示了这些理论与宇宙加速现象之间的深层联系。详细分析了系统在不同条件和不同值下的相图，并对它们的几何解释进行了研究。分析了动力系统的扰动图，重点是我们的分析结果。我们已经评估了总能量密度（）在固定点，并且还发现了的合适的范围，并为一个稳定的模型。