Application of nonlinear dynamics in nonlinear electrical transmission line networks: Exact solutions, fractional analysis and new discoveries

This study focuses on nonlinear electrical transmission line networks (NETNs), exploring their dynamics via high-order modeling and fractional-order analysis. It first uses the extended tanh method to solve the quartic nonlinear voltage equation of multi-coupled discrete networks, yielding exact solutions like multi-peak and parabolic solitons. Fractional-order operators significantly regulate signal distribution, amplitude, and propagation; dispersive element Cs decisively affects voltage amplitude and compensation (e.g., peak rising linearly as Cs increases from 20 to 60 pF). Differences between fractional-order operators (e.g., Conformable, Riemann–Liouville) in impacting system potential are negligible. Via 3D phase portraits, Lyapunov exponents (first quantifying chaos), and Runge–Kutta–Nyström method (errors $<10^{-9}$), it reveals signal distortion from modulation instability. Compared to prior second-order models, it supplements high-order stability analysis via quartic equations, offering a precision-efficiency adjustable theoretical toolchain for aerospace power anti-interference and high-voltage transmission parameter optimization.