Stability, bifurcation and characteristics of chaos in a new commensurate and incommensurate fractional-order ecological system

In this paper, a new incommensurate fractional-order ecological system describing the interaction between permafrost melting, vegetation degradation and temperature is introduced to explore its dynamical behavior. At first, the existence and uniqueness of the new system is proved utilizing Picard’s operator and Banach fixed-point theorem. Next, the influence of simultaneous changing system parameter and fractional orders on the stability of the incommensurate system is discussed, and the degree of influence of each fractional order is visually compared and analyzed. In addition, it is also indicated that parameter changes can cause the static bifurcation. Subsequently, the sufficient conditions and an analytical expression for the critical value of Hopf bifurcation caused by system parameter in the incommensurate fractional-order ecological system are provided for the first time, and the bifurcation diagrams are utilized to verify the result. Furthermore, by observing the bifurcation diagram of incommensurate and corresponding commensurate system, it can be deduced that changes in the fractional orders of incommensurate system can cause Hopf bifurcation to be postponed or advanced. Then, the chaotic behaviors of commensurate and incommensurate system are explored by utilizing multiple numerical indicators. It is worth noting that there exists a path from quasi-periodic motion to chaos in the system. Finally, the chaotic domain is proposed to investigate the influence of fractional orders and parameter on chaotic behaviors in the incommensurate fractional-order system.