In this study, we propose a novel fractional-order Jerk system. Experiments show that, under some suitable parameters, the fractional-order Jerk system displays a chaotic phenomenon. In order to suppress the chaotic behavior of the fractional-order Jerk system, we design two control strategies. Firstly, we design an appropriate time delay feedback controller to suppress the chaos of the fractional-order Jerk system. The delay-independent stability and bifurcation conditions are established. Secondly, we design a suitable mixed controller, which includes a time delay feedback controller and a fractional-order PD^σ controller, to eliminate the chaos of the fractional-order Jerk system. The sufficient condition ensuring the stability and the creation of Hopf bifurcation for the fractional-order controlled Jerk system is derived. Finally, computer simulations are executed to verify the feasibility of the designed controllers. The derived results of this study are absolutely new and possess potential application value in controlling chaos in physics. Moreover, the research approach also enriches the chaos control theory of fractional-order dynamical system.

Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays: In this paper, a two-species Lotka-Volterra predator-prey model with two delays is considered. By analyzing the associated characteristic transcendental equation, the linear stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and direction of Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using normal form theory and center manifold theory. Some numerical simulations for supporting the theoretical results are also included.

Formulating suitable dynamical models to describe the interplay of different chemical substance in biology and chemistry is a very interesting topic. In this article, we set up a new plankton population dynamical system accompanying delay. Taking advantage of fixed point theorem and appropriate function, we derive the criteria on existence and uniqueness, boundedness of the solutions of the established plankton population dynamical system. By applying Hopf bifurcation and stability theorem of delayed dynamical system, we analyze the emergence of Hopf bifurcation and stability trait of the established plankton population dynamical system. Novel delay-independent criteria guaranteeing the emergence of Hopf bifurcation and stability of the system are derived. By virtue of two different controllers, we have effectively controlled the time of occurrence of Hopf bifurcation and stability domain of the established system. The influence of time delay on Hopf bifurcation and stability of the system is shown. The study shows that when the delay crosses a critical value, then Hopf bifurcations of the established plankton population dynamical system and its controlled systems arise. Numerical experiment outcomes are included to support the derived main fruits. The derived results of this article are completely new and have important theoretical value in balancing and controlling the concentrations of distinct chemical substances.