Dynamical System Analysis of Chameleon Mechanism in Brans-Dicke Scalar-Tensor Model

We investigated the stability of the chameleon screening mechanism in the Brans-Dicke scalar-tensor model. We define a constraint on the Brans-Dicke parameter \(\varvec{\omega }_{BD}^*\) identifying two stability groups, \(\varvec{\omega }_{BD}>\varvec{\omega }_{BD}^*\) and \(0<\varvec{\omega }_{BD}<\varvec{\omega }_{BD}^*\). The first group achieves stability with both appropriate eigenvalues and a density profile consistent with dark energy dominance. The second exhibits eigenvalue stability but contradicts conditions for a stable universe. We explore the impact of variations in the scalar field potential and matter coupling by analyzing different parameter sets. Each unique set of parameters results in a distinct \(\varvec{\omega }_{BD}^*\). Dynamic analysis reveals that stability is achieved when the scalar field dominates, highlighting the importance of the kinetic and potential terms while minimizing the influence of matter density. In high matter density regions, the scalar field’s negligible presence aligns with standard gravitational behavior, whereas in low matter density regions, the scalar field grows exponentially, driving dark energy and cosmic acceleration.