Pulsating string solution and stability in two parameter \(\chi\)-deformed background

In this paper, we investigate pulsating string solutions within the context of a two-parameter \(\chi\)-deformed \(\mathcal{R} \times S^2\) background. We derive the energy and oscillation number relation for the deformed \(\mathcal{R} \times S^2\) in a short string limit. Additionally, we derive the first-order perturbation equation for a pulsating string in this specific background. As a special case of the two-parameter deformed background, we examine the stability of the pulsating string solution in one-parameter deformed \(\mathcal{R} \times S^2\) and \(AdS_3\) background by setting one of the deformation parameters to zero.