Modified Kadomtsev Petviashvili (mKP) equation is derived by considering the dynamics of the dust-acoustic waves (DAWs) with kappa-distributed hot and cold ions and Boltzmannian electrons. The bifurcation theory of the planar dynamical system is used to obtain the phase portrait of the DAWs in the framework of the mKP equation. The formation of dust-acoustic solitary waves (DASWs) is analyzed for different plasma parameters in Saturn's magnetosphere (SMS). In the current plasma model, the external periodic force is introduced to study the quasiperiodic and chaotic behavior of the DAWs. It is noted that periodic initial conditions lead to the emergence of many outlandish features in the system by comparison with the solitary initial conditions.