In this paper, the nonlinear Nurshuak–Tolkunay–Myrkakulov (NTM-III) equation is analyzed using the methods of dynamical systems theory and exact solution techniques. The NTM-III model is reduced to a second-order nonlinear ordinary differential equation by using appropriate reductions, then written as a planar Hamiltonian dynamical system. Analytical study of the equilibrium points, bifurcation structures, and stability properties is conducted using Jacobian matrices and eigenvalue theory. Different parameter combinations are shown in various phase portraits, with the saddle and center equilibrium states present and their stability indicated. An external periodic perturbation is added to the system to study complex nonlinear dynamics. Using phase portraits, time-series analysis, return maps, Lyapunov exponents, sensitivity analysis, and multistability diagnostics, the resulting forced dynamical model is investigated. Chaotic behavior, strong dependence on initial conditions, and multiple coexisting attractors for the same set of parameters are illustrated through numerical simulation. In addition, a set of exact analytical solutions, including trigonometric, hyperbolic, and exponential wave structures, is obtained using the Multivariate Generalized Exponential Rational Integral Function (MGERIF) method. The solutions obtained display interesting nonlinear wave interactions, multi-peakon formations, and localized propagation patterns. The findings show the complex relationship among bifurcation, chaos, multistability, and nonlinear wave propagation in the NTM-III equation and provide new insight into the equation’s mathematical and physical properties.
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