Analyzing the Occurrence of Bifurcation and Chaotic Behaviors in Multi-Fractional-Order Stochastic Ginzburg–landau Equations

Abstract

The main novelty of this paper lies in five aspects. (1) Our study on the fractional-order derivative stochastic Ginzburg–Landau equation (FODSGLE) has resulted in numerous exact solutions using Jacobian elliptic functions (JEFs). These solutions offer valuable insights into complex physical phenomena and have practical implications across various fields. (2) By examining the effect of noise on FODSGLE solutions, we found consistent behavior regardless of the form of fractional derivatives. As noise intensity increases, the solutions deteriorate and tend toward zero. This study contributes to the existing literature by providing new insights and surpassing previous efforts in understanding the dynamics of FODSGLE. (3) Our research investigates how fractional order affects noise in solutions of the FODSGLE. Surprisingly, we found that changes in the fractional order ([Formula: see text]) have minimal influence on the system when the noise intensity ([Formula: see text]) is fixed. This indicates an aspect that has not been addressed extensively in existing literatures. (4) We compare different types of fractional derivatives within the context of the FODSGLE, presenting a novel contribution, while keeping the noise intensity or fractional order fixed. Our results demonstrate that the CFOD closely aligns with the MFOD, while displaying more differences with the solutions obtained using the BFOD. This comparison sheds light on an aspect that has not been thoroughly investigated in previous literatures. (5) This paper delves into the phase portraits of the FODSGLE and investigates the associated sensitivity and chaotic behaviors. Notably, previous studies on the stochastic Ginzburg–Landau equation (SGLE) have not extensively examined this aspect. By analyzing the phase portraits, we gain valuable insights into the dynamics and stability of FODSGLE solutions, uncovering intricate behaviors within the system. Our exploration of sensitivity and chaotic behaviors adds another dimension to understanding the (FODSGLE), laying a foundation for further research in this domain.