Analysis of Fractional Ordered Incommensurate Quadratic Jerk System

Fractional ordered dynamical systems (FODS) are being studied in the present due to their innate qualitative and quantitative properties and their applications in various fields. The Jerk system, which is a system involving three differential equations with quadratic complexity, arises naturally in wide ranging fields, and hence a qualitative study of solutions of jerk system and its various parameters under different conditions is important. In this article, we have studied phenomena of the Hopf bifurcation and chaos occurring in fractional ordered commensurate and incommensurate quadratic jerk system. The equilibrium points of the system are obtained and are found to be $(\pm\epsilon,0,0)$, where $\epsilon$ denotes the system parameter versus which bifurcation is analyzed. We have presented the criteria for commensurate and incommensurate quadratic jerk system to undergo a Hopf bifurcation. The value of $\epsilon$ at which system undergoes Hopf bifurcation $\epsilon_{H}$ is obtained for both commensurate as well as incommensurate system. It is known that supercritical Hopf bifurcation leads to chaos. The obtained results are verified through numerical simulations versus the fractional order $\alpha$ and parameter $\epsilon$ and the explicit range in which the system exhibits chaos is found. A number of phase portraits, bifurcation diagrams, and Lyapunov exponent diagrams are presented to affirm the obtained chaotic range of parameters.