This paper is devoted to investigate the tracking control and generalized synchronization of the hyperchaotic Lorenz-Stenflo system using the tracking model and the feedback control scheme. We suppress the chaos to unstable equilibrium via three feedback methods, and we achieve three globally generalized synchronization controls. Novel tracking controllers with corresponding parameter update laws are designed such that the Lorenz-Stenflo systems can be synchronized asymptotically. Moreover, numerical simulations are presented to demonstrate the effectiveness, through the contrast between the orbits before being stabilized and the ones after being stabilized.

This paper investigates the adaptive stabilization problem for a class of stochastic nonholonomic systems with strong drifts. By using input-state-scaling technique, backstepping recursive approach, and a parameter separation technique, we design an adaptive state feedback controller. Based on the switching strategy to eliminate the phenomenon of uncontrollability, the proposed controller can guarantee that the states of closed-loop system are global bounded in probability.

We apply the linear matrix inequality method to consensus and $H_{\infty }$ consensus problems of the single integrator multiagent system with heterogeneous delays in directed networks. To overcome the difficulty caused by heterogeneous time-varying delays, we rewrite the multiagent system into a partially reduced-order system and an integral system. As a result, a particular Lyapunov function is constructed to derive sufficient conditions for consensus of multiagent systems with fixed (switched) topologies. We also apply this method to the $H_{\infty }$ consensus of multiagent systems with disturbances and heterogeneous delays. Numerical examples are given to illustrate the theoretical results.

We analyze a three species predator-prey chain model with stochastic perturbation. First, we show that this system has a unique positive solution and its $p$th moment is bounded. Then, we deduce conditions that the system is persistent in time average. After that, conditions for the system going to be extinction in probability are established. At last, numerical simulations are carried out to support our results.

We investigate the complex dynamics of a SIRS epidemic model incorporating media coverage with random perturbation. We first deal with the boundedness and the stability of the disease—free and endemic equilibria of the deterministic model. And for the corresponding stochastic epidemic model, we prove that the endemic equilibrium of the stochastic model is asymptotically stable in the large. Furthermore, we perform some numerical examples to validate the analytical finding, and find that if the conditions of stochastic stability are not satisfied, the solution for the stochastic model will oscillate strongly around the endemic equilibrium.

We examine the relationships between lower exhausters, quasidifferentiability (in the Demyanov and Rubinov sense), and optimal control for switching systems. Firstly, we get necessary optimality condition for the optimal control problem for switching system in terms of lower exhausters. Then, by using relationships between lower exhausters and quasidifferentiability, we obtain necessary optimality condition in the case that the minimization functional satisfies quasidifferentiability condition.

We investigate a neutral multispecies logarithmic population model with feedback control and impulse. By applying the contraction mapping principle and some inequality techniques, a set of easily applicable criteria for the existence, uniqueness, and global attractivity of positive periodic solution are established. The conditions we obtained are weaker than the previously known ones and can be easily reduced to several special cases. We also give an example to illustrate the applicability of our results.

A schistosomiasis model is proposed including single schistosomes, paired schistosomes, snails, and the latent period of infected snails. A reasonable sex ratio of schistosomes and the minimum mating function are considered. A threshold condition determining the stability of the system is given, and the stability of equilibrium for the model is shown. The impact of the latent period of infected snails on schistosomiasis transmission can be found through numerical simulations. Finally, preferable control strategies are obtained by sensitivity analyses. Killing snails may be the preferred control measure. If we choose chemotherapy, we should use some drugs which are sufficient for reducing egg-associated pathology, since paired schistosomes are mostly harmful to definitive hosts.

A Lotka-Volterra predator-prey model with time-varying delays is investigated. By using the differential inequality theory, some sufficient conditions which ensure the permanence and global asymptotic stability of the system are established. The paper ends with some interesting numerical simulations that illustrate our analytical predictions.

We investigate the Hölder regularity of the local time of the fractional Ornstein-Uhlenbeck process $X^H=\{X_t^H,t\ge 0\}$. As a related problem, we study the collision local time of two independent fractional Ornstein-Uhlenbeck $X^{H_i}=\{X_t^{H_i},t\ge 0\}$, $i=1,2$ with respective indices $H_1,H_2\in (0,1)$.

The analytical method to predict the period-doubling bifurcation of the three-dimensional (3D) system is improved by using the undetermined fundamental frequency method. We compute the stable response of the system subject to the quadratic and cubic nonlinearity by introducing the undetermined fundamental frequency. For the occurrence of the first and second period-doubling bifurcation, the new bifurcation criterion is accomplished. It depends on the stability of the limit cycle on the central manifold. The explicit applications show that the new results coincide with the results of the numerical simulation as compared with the initial methods.

The main goal of this paper is to present a theory of approximation of periodic orbits of vector fields in the plane. From the theory developed here, it is possible to obtain an approximation to the curve of nonhyperbolic periodic orbits in the bifurcation diagram of a family of differential equations that has a transversal Hopf point of codimension two. Applications of the developed theory are made in Liénard-type equations and in Bazykin’s predator-prey system.

This paper mainly modifies and further develops the Reyleigh price model. By modifying the basic Reyleigh model, we can more accurately illustrate the economic phenomena with price varying. First, we research the dynamics of the modified Reyleigh model with time delay. By employing the normal form theory and center manifold theory, we obtain some testable results on these issues. The conclusion confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. Finally, some numerical simulations are given to illustrate the effectiveness of our results.

The optimal control problem with integral boundary condition is considered. The sufficient condition is established for existence and uniqueness of the solution for a class of integral boundary value problems for fixed admissible controls. First-order necessary condition for optimality is obtained in the traditional form of the maximum principle. The second-order variations of the functional are calculated. Using the variations of the controls, various optimality conditions of second order are obtained.

The problem of bounded-input bounded-output (BIBO) stabilization in mean square for a class of discrete-time stochastic control systems with mixed time-varying delays and nonlinear perturbations is investigated. Some novel delay-dependent stability conditions for the previously mentioned system are established by constructing a novel Lyapunov-Krasovskii function. These conditions are expressed in the forms of linear matrix inequalities (LMIs), whose feasibility can be easily checked by using MATLAB LMI Toolbox. Finally, a numerical example is given to illustrate the validity of the obtained results.

We consider the model of a Caputo $q$-fractional boundary value problem involving $p$-Laplacian operator. By using the Banach contraction mapping principle, we prove that, under some conditions, the suggested model of the Caputo $q$-fractional boundary value problem involving $p$-Laplacian operator has a unique solution for both cases of and $p>2$. It is interesting that in both cases solvability conditions obtained here depend on $q$, $p$, and the order of the Caputo $q$-fractional differential equation. Finally, we illustrate our results with some examples.

Many control systems can be written as a first-order differential equation if the state space enlarged. Therefore, general conditions on controllability, stated for the first-order differential equations, are too strong for these systems. For such systems partial controllability concepts, which assume the original state space, are more suitable. In this paper, a sufficient condition for the partial complete controllability of semilinear control system is proved. The result is demonstrated through examples.

We investigate the complex dynamics of an epidemic model with nonlinear incidence rate of saturated mass action which depends on the ratio of the number of infectious individuals to that of susceptible individuals. We first deal with the boundedness, dissipation, persistence, and the stability of the disease-free and endemic points of the deterministic model. And then we prove the existence and uniqueness of the global positive solutions, stochastic boundedness, and permanence for the stochastic epidemic model. Furthermore, we perform some numerical examples to validate the analytical findings. Needless to say, both deterministic and stochastic epidemic models have their important roles.