The derivation of conservation laws and invariant functions is an essential procedure for the investigation of nonlinear dynamical systems. In this study, we consider a two-field cosmological model with scalar fields defined in the Jordan frame. In particular, we consider a Brans–Dicke scalar field theory and for the second scalar field we consider a quintessence scalar field minimally coupled to gravity. For this cosmological model, we apply for the first time a new technique for the derivation of conservation laws without the application of variational symmetries. The results are applied for the derivation of new exact solutions. The stability properties of the scaling solutions are investigated and criteria for the nature of the second field according to the stability of these solutions are determined.

中文翻译：

---

具有额外标量场的 Brans-Dicke 宇宙学中的新守恒定律和精确宇宙学解

守恒定律和不变函数的推导是研究非线性动力系统的基本步骤。在这项研究中，我们考虑在 Jordan 框架中定义标量场的双场宇宙学模型。特别是，我们考虑了 Brans-Dicke 标量场理论，对于第二个标量场，我们考虑了与重力最小耦合的典型标量场。对于这个宇宙学模型，我们首次应用了一种新技术来推导守恒定律，而无需应用变分对称性。结果用于推导新的精确解。研究了标度解的稳定性属性，并根据这些解的稳定性确定了第二场性质的标准。