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In this work, new dual-mode nonlinear Schrödinger’s equations (NLSEs) are studied with cubic and quadratic cubic nonlinearities. The new models introduce three different physical parameters such as nonlinearity, phase velocity, and dissipative factor. These models interpret the propagation of two-directional waves simultaneously. The necessary conditions on nonlinearity and dissipative factors for the solitons to exist for dual-mode are also calculated. Lastly, to analyze the effect of phase velocity during the promulgation of dual-waves is highlighted graphically. Moreover, the governing model is converted into the planer dynamical system with the help of Galilean transformation. Every possible form of phase portraits is plotted for pertinent parameters viz. k,β,d1,d2,d3. The Runge–Kutta fourth-order technique to extract the nonlinear periodic solutions of the considered problem and outcomes are presented graphically. Furthermore, quasiperiodic and chaotic behavior of p-FLM is analyzed for different values of the parameters upon the application of an after deploying an external periodic force. While, quasiperiodic–chaotic nature is observed for selected values of parameters k,β,d1,d2,d3by keeping the force and frequency of the perturbed dynamical system fixed.