This work studies the behavior of electrical signals in resonant tunneling diodes through the application of the Lonngren wave equation. Utilizing the method of Lie symmetries, we have identified optimal systems and found symmetry reductions; we have also found soliton wave solutions by applying the tanh technique. The bifurcation and Galilean transformation are found to determine the model implications and convert the system into a planar dynamical system. In this experiment, the equilibrium state, sensitivity, and chaos are investigated and numerical simulations are conducted to show how the frequency and amplitude of alterations affect the system. Furthermore, local conservation rules are demonstrated in more detail to unveil the whole system of movements.

This study presents a dynamical and statistical investigation of the Rabinovich Fabrikant system within the bounded control parameter space $(p,q)\in [-1,1]\times [-1,1]$, using ensembles of initial conditions sampled from $[-0.1,0.1]$. A systematic grid-based exploration of the parameter space revealed clearly distinguishable regions corresponding to stable equilibria, sustained periodic oscillations, and unstable dynamical regimes. A grid-based systematic scan of the parameter space indicated that the space is evidently divided into easily recognizable domains that are characterized by a stable equilibrium, periodic oscillations, and unstable dynamical states. The analysis of time series and power spectral density was used to describe the time dynamics of system responses and to identify a change between steady, oscillatory, and broadband states of the system. In addition to these classical dynamical diagnostics, an ensemble-based statistical analysis was carried out to measure the variability of the system dynamics. The probability distributions of the dynamical states were measured by kernel density estimation, which demonstrated that stable and periodic regimes have convergent and sharp distributions, and transitional regions have wider distributions and slower convergence. Moreover, empirical cumulative distribution functions will gave more information about the structure and variability of the distribution of the system responses in various parameter regimes. Sensitivity analysis also implied that the impact of initial conditions is also highly parameter sensitive and is especially pronounced at regime boundaries. These findings offer a quantitative and reproducible parameter space mapping of the dynamics of RF systems and make ensemble-based statistical diagnostics fundamental instruments to measuring robustness and uncertainty in nonlinear chaotic systems.