In this study, the propagation of ultrashort optical pulses in nonlinear dispersive media with third-order dispersion, self-steepening, and stimulated Raman scattering is modelled using the Sasa-Satsuma higher-order nonlinear Schrödinger equation. Exact travelling wave solutions, including bright solitons, dark solitons, singular solutions, and periodic wave solutions, are obtained by applying a travelling wave transformation and applying the $\exp (-\mathrm{\Phi }(\zeta ))$ function method on the resulting ordinary differential equation. A systematic bifurcation analysis of the associated planar Hamiltonian dynamical system validates these solutions, identifying equilibrium points, constructing phase portraits across all parameter regimes, and explicitly linking orbit families to solution types. Chaotic dynamics are analyzed using time series and phase diagrams. Using Poincaré sections, Lyapunov exponent spectra, time series, and bifurcation diagrams, we map the transition from regular to chaotic regimes and demonstrate how sensitivity to initial conditions governs predictability. The sensitivity analysis shows us the point at which the system becomes very sensitive to changes in the parameter values, which measures how uncertainties in initial conditions and parameters propagate and grow over time in the system.