Analysis of nonlinear dynamics, bifurcation, and chaos in a low-pass electrical transmission line system

Abstract This study presents a comprehensive analytical and dynamical investigation of a nonlinear low-pass electrical transmission line model derived from Kirchhoff’s circuit laws by incorporating nonlinear inductive capacitive effects. The governed model accounts for both quadratic and cubic nonlinearities as well as higher-order spatial dispersion, providing a generalized framework to explore rich nonlinear phenomena including bistability, modulational instability, and solitary wave propagation. To extract exact traveling-wave solutions, the Riccati–Bernoulli sub-ODE method is used, which efficiently reduces the governing nonlinear partial differential equation to an integrable form. Various families of localized structures are obtained, such as dark, bright, hyperbolic, trigonometric, mixed, and hybrid soliton profiles under distinct parametric regimes. The corresponding three-dimensional, contour, and two-dimensional graphical representations signify how nonlinear, dispersive, and higher-order terms modulate the amplitude, polarity, and symmetry of the evolving waveforms. Furthermore, a detailed chaotic assessment of the reduced dynamical system is carried out through phase portraits, power spectra, bifurcation diagrams, return maps, and Lyapunov exponent analysis. The coexistence of positive and negative Lyapunov exponents confirms the system’s chaotic nature and sensitive dependence on initial conditions, while bifurcation transitions highlight parameter-dependent shifts between periodic, quasi-periodic, and chaotic regimes. The interplay between analytical soliton structures and numerical chaos analysis provides a comprehensive physical picture of the nonlinear energy transport in the nonlinear low-pass electrical transmission line framework. The findings demonstrate the versatility of the Riccati–Bernoulli approach in capturing multi-type nonlinear excitations and chaotic transitions, establishing its potential for modeling real-world phenomena such as nonlinear signal propagation in electrical transmission networks, ultrafast pulse modulation in optical fibers, and controlled wave localization in metamaterial systems.