  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 $\omega_{BD}^*$ identifying two stability groups,
$\omega_{BD}>\omega_{BD}^*$ and $0<\omega_{BD}<\omega_{BD}^*$. 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
$\omega_{BD}^*$. 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.
