Abstract
This paper focuses on the dynamical analysis of the advection-diffusion-reaction equation under various conditions that highlight the system's sensitivity and potential for chaotic behavior. Traveling wave solutions for the underlying equation are derived using a novel modified [Formula: see text] expansion method based on the traveling wave transformation. A broad spectrum of exact traveling wave solutions, including solitons, kinks, periodic solutions, and rational solutions, is obtained. These solutions are recognized as having significant potential applications in fields such as engineering and plasma physics. The proposed method is demonstrated to successfully generate various exponential solutions, such as bright, dark, single, rational, and periodic solitary wave solutions. MATLAB simulations were carried out to visualize the results, producing 3D, 2D, and contour graphs that emphasize the impact of the advection-diffusion-reaction equation. Furthermore, the Galilean transformation is applied to derive the corresponding planar dynamical system, enabling deeper insights into its dynamical behavior. Sensitivity analysis is performed to evaluate the system's response to different initial conditions, with symmetrical properties and equilibrium points being represented through phase portraits. The chaotic behavior of the planar dynamical system under the influence of an external force is also examined. It is revealed that the system exhibits periodic, quasi-periodic, and chaotic processes, with significant increases in intensity and frequency being observed. Additionally, we apply Poincaré maps and Lyapunov exponent to analyze the behavior of the dynamical system by different initial conditions.