Analytical investigation of the fractional Klein-Gordon equation along with analysis of bifurcation, sensitivity and chaotic behaviors

The fractional Klein-Gordon equation (fKGE) holds a crucial position in various fields of theoretical and applied physics, with wide applications covering different areas such as nonlinear optics, condensed matter physics, and quantum mechanics. In this paper, we carry out analytical investigation to the fKGE with beta fractional derivative by using the Bernoulli (G '/G)-expansion method and improved tan(phi/2)-expansion method. In order to better comprehend the physical structure of the obtained solutions, three-dimensional visualizations, contour diagrams, and line graphs of the exponent function solutions are depicted with the aid of Matlab. Moreover, the phase portraits and bifurcation behaviors of the fKGE under transformation are studied. Sensitivity and chaotic behaviors are analyzed in specific conditions. The phase plots and time series map are exhibited through sensitivity analysis and perturbation factors. These studies enhance our understanding of practical phenomena governed by the model of fKGE, and are crucial for examining the dynamic behaviors and phase portraits of the fKGE system. The strategies utilized here are more direct and effective, which can be effortlessly utilized to various fractional-order differential equations arising in nonlinear optics and quantum mechanics.