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.

All the time, differential dynamical models with delay has witness a tremendous application value in characterizing the internal law among diverse biological populations in biology. In the current article, on the basis of the previous publications, we formulate a new Lotka–Volterra commensal symbiosis system accompanying delay. Utilizing fixed point theorem, inequality tactics and an appropriate function, we gain the sufficient criteria on existence and uniqueness, non-negativeness and boundedness of the solution to the formulated delayed Lotka–Volterra commensal symbiosis system. Making use of stability and bifurcation theory of delayed differential equation, we focus on the emergence of bifurcation behavior and stability nature of the formulated delayed Lotka–Volterra commensal symbiosis system. A new delay-independent stability and bifurcation conditions on the model are presented. By constructing a positive definite function, we explore the global stability. By constructing two diverse hybrid delayed feedback controllers, we can adjusted the domain of stability and time of appearance of Hopf bifurcation of the delayed Lotka–Volterra commensal symbiosis system. The effect of time delay on the domain of stability and time of appearance of Hopf bifurcation of the model is given. Matlab experiment diagrams are provided to sustain the acquired key outcomes.

Establishing dynamical models to characterize the relation of different chemical compositions is an important topic in chemistry and mathematics. However, a lot of dynamical models are merely concerned with the integer-order dynamical models. The report on fractional-order chemical dynamical systems is quite few. In this current article, based on the earlier publications, we establish a new fractional-order coupled Oregonator model incorporating time delay. A set of sufficient conditions which ensure the stability and the onset of Hopf bifurcation of fractional-order coupled Oregonator model incorporating time delay are derived by regarding the time delay as bifurcation parameter. The exploration manifests that time delay has a vital influence on stabilizing system and controlling bifurcation of the investigated fractional-order coupled Oregonator model. At last, Matlab simulation results are adequately displayed to corroborate the derived theoretical achievements.

During the past decades, delayed differential dynamical equation has displayed a great application in portraying the interplay of diverse chemical substances in chemistry. In the research, relying the earlier publications, we propose a novel nutrient-microorganism system accompanying time delay. Exploiting fixed point theorem, inequality skills and a proper function, we acquire the parameter criteria on well-posedness (for example, existence and uniqueness, non-negativeness and boundedness) of the solution of the established delayed nutrient-microorganism system. Utilizing Hopf bifurcation theory and stability criterion of delayed dynamical system, we deal with the onset of Hopf bifurcation phenomenon and stability trait of the established delayed nutrient-microorganism system. A set of novel delay-independent parameter conditions on bifurcation phenomenon and stability of the system are provided. Designing both different hybrid delayed feedback controllers, we have successfully controlled the time of occurrence of bifurcation phenomenon and stability domain of the system. The role of time delay in controlling bifurcation and stabilizing system is analyzed. Numerical experiment figures are given to support the gained primary outcomes. The acquired assertions of this research are completely innovative and can give some important instructions in regulating and balancing the densities of microorganism and nutrient.