Comprehensive analysis of noise behavior influenced by random effects in stochastic differential equations

Stochastic differential equations are practical tools for modeling systems in which stochastic effects prevail, distinguishing it from deterministic models. Qualitative and quantitative analyses of a specific observed model are possible with the help of a thorough discrimination framework of such systems. The effectiveness of the method is supported by exact results derived from the model using necessary constraint conditions. This study looks at how model parameters influence solution behavior with two and three dimensions. In addition, numerical studies are conducted to validate the theoretical findings and determine the stability of the system under different circumstances. Therefore, when the model is reformulated as a dynamical system, we get the Hamiltonian and topological characteristics, bifurcation theory, Lyapunov coefficients, quasiperiodic, and chaos. The analysis of the sustained chaotic behavior by outer forms of control offers greater insight into the dynamics of the proposed model. The findings further indicate possible uses of this model in areas such as climatology where stochastic disturbances play a major role in system behavior. Hence, the current study shares enough methodological improvement for analytical problems in engineering, physics, and mathematics, especially the non-linearities solved with stochastic models.