Dynamical systems methods are used to investigate a cosmological model with non-minimally coupled scalar field and asymptotically quadratic potential function. We found that for values of the non-minimal coupling constant parameter $\frac{3}{16}<\xi <\frac{1}{4}$ there exists an unstable asymptotic de Sitter state, free from a parallelly propagated singularity for $\frac{5}{24}\le \xi <\frac{1}{4}$, giving rise to non-singular beginning of the universe. The energy density associated with this state depends on value of the non-minimal coupling constant and can be much smaller than the Planck energy density. For $\xi =\frac{1}{4}$ we found that the initial state is in form of the static Einstein universe. Proposed evolutional model, contrary to the seminal Starobinsky's model, do not depend on the specific choice of initial conditions in phase space, moreover, a small change in the model parameters do not change the evolution thus the model is generic and structurally stable. The values of the non-minimal coupling constant can indicate for a new fundamental symmetry in the gravitational theory. We show that Jordan frame and Einstein frame formulation of the theory are physically nonequivalent.

动力学系统方法用于研究具有非最小耦合标量场和渐近二次势函数的宇宙学模型。我们发现对于非最小耦合常数参数的值$\frac{3}{16}<\xi <\frac{1}{4}$ 存在一个不稳定的渐近德西特态，没有平行传播的奇点 $\frac{5}{24}\le \xi <\frac{1}{4}$，导致宇宙的非奇异开始。与此状态相关的能量密度取决于非最小耦合常数的值，并且可能比普朗克能量密度小得多。为了$\xi =\frac{1}{4}$我们发现初始状态是静态爱因斯坦宇宙的形式。提出的演化模型与开创性的 Starobinsky 模型相反，不依赖于相空间初始条件的具体选择，而且模型参数的微小变化不会改变演化，因此该模型是通用的且结构稳定。非最小耦合常数的值可以表明引力理论中新的基本对称性。我们表明该理论的 Jordan 框架和爱因斯坦框架公式在物理上是不等价的。