Bifurcation, Chaotic Behavior, Sensitivity Analysis, and Traveling Wave Solutions of the Nonlinear Telecommunication Model Through Transmission Lines

This paper explores a nonlinear telecommunication transmission line model with the incorporation of the M-truncated fractional derivative, providing memory-dependent effects that go beyond the classical nonlinear transmission line models. The Hamilton–Jacobi and Jacobian dynamical frameworks are, for the first time, utilized in this fractional nonlinear transmission line context to methodically derive fractional travelling wave solutions and define the system’s nonlinear stability. Fractional system dynamics are investigated by bifurcation analysis, Lyapunov exponents, time series behavior, initial condition sensitivity, identifying periodic, quasiperiodic, and chaotic regimes. A rich variety of wave structures like kink, antikink, bell-shaped, singular, M-shaped, and W-shaped solutions are derived and visualized through 2D profiles, 3D surfaces, and contour plots. Stability is investigated by Hamiltonian criteria and linearized perturbation analysis, and it confirms long-time dynamic robustness. The results show that nonlinear wave dynamics and stability characteristics are greatly improved in telecommunication transmission lines by fractional order processes, and thus a new fractional order nonlinear transmission line theory is formed. In this paper, it is shown how the memory effect leads to the change of the stability region and the faster emergence of chaos within the framework of fractional order systems.