Bifurcation and chaotic behaviors in nonlinear fractional electrical transmission line: Solitons, 3D frameworks, and algorithmic advances

This paper presents some innovations in studying the nonlinear fractional electrical transmission line equation (NFETLE). First, the unified F-expansion method is applied for the first time to derive exact solutions of NFETLE, uncovering double-periodic voltage soliton solutions expressed as Jacobi elliptic functions (e.g., V11(ξ), V21(ξ)), which extend traditional single-periodic solutions and reveal new dual-frequency energy transmission modes. Second, through computational simulations, diverse soliton waveforms (such as spike-like and multi-peak structures) are identified, and their relationships with fractional order (α) and circuit parameters (m, δ, u0) are analyzed, providing insights into the regulation mechanism of voltage propagation. Third, a three-dimensional dynamic framework is constructed to replace conventional two-dimensional analysis, enabling the visualization of chaotic behaviors (e.g., sensitivity to initial conditions) and bifurcation characteristics, which deepens the understanding of the system’s nonlinear dynamics. In addition, the high-order Huta Runge–Kutta algorithm is introduced, achieving numerical errors below 10−13 and demonstrating superior stability for solving fractional-order equations. These findings provide a theoretical basis for optimizing power transmission systems and fault diagnosis.