Exact traveling-wave solutions and dynamical behavior of nonlinear low-pass electrical models in the fractional framework

Abstract

The paper is an analytical study of a low-pass electrical model of nonlinear type in a fractional perspective, in which the classical derivative is generalized to the Katugampola fractional operator. Precise traveling-wave solutions are built based on an extended Riccati-Bernoulli sub-ODE scheme together with a Bäcklund transformation. The families of obtained solutions contain bright and dark kink type structures. These solutions have a dynamical behavior that is demonstrated with the help of detailed 3D and 2D visualizations. The 3D plots reveal how sensitive the integer-order parameter is to the waveform whereas the 2D plots show how sensitive the waveform is to the changes in the fractional order ($\alpha$). To deeper examine the qualitative dynamics, a hamiltonian formulation is created and phase-portrait diagrams are plotted. These unveil the local and global organization of the nonlinear flow underlying. Besides, chaotic behavior is also studied by analyzing sensitivity to initial conditions by determining the largest Lyapunov exponent $\lambda _{max}$. The findings validate the occurrence of regular, quasi-periodic and chaotic regimes in the parameter space. The entire process of analytical calculations and visualization is implemented in MATLAB, which provides the numerical accuracy of calculations and high-resolution graphical confirmation of fractions solutions. The results illustrate the presence of significant enrichment of the dynamical behavior of the nonlinear electrical model by the fractional extension. It also offers a practical and efficient model to study intricate waves phenomena in the systems of the fractional-order.