In this paper, the stability, stabilization, and $L_{2}$ -gain problems are investigated for periodic piecewise systems with time-varying subsystems. Continuous Lyapunov function with time-varying Lyapunov matrix is adopted. A condition guaranteeing the negative definiteness of a matrix polynomial, deriving from the Lyapunov derivative, is first obtained. Based on such a condition, an exponential stability condition is provided. Moreover, a state-feedback controller with time-varying gain is developed to stabilize the unstable periodic piecewise time-varying system. The $L_{2}$ -gain criterion for periodic piecewise time-varying system is also studied. Numerical examples are given to show the validity of the proposed techniques.

Stability and $L_2$ Synthesis of a Class of Periodic Piecewise Time-Varying Systems

Global dynamic behavior of a predator-prey model under ratio-dependent state impulsive control

Dynamic behaviors of the FitzHugh–Nagumo neuron model with state-dependent impulsive effects

In this article, we discuss the dynamics of a Leslie–Gower ratio-dependent predator–prey model incorporating fear in the prey population. Moreover, the Allee effect in the predator growth is added into account from both biological and mathematical points of view. We explore the influence of the Allee and fear effect on the existence of all positive equilibria. Furthermore, the local stability properties and possible bifurcation behaviors of the proposed system about positive equilibria are discussed with the help of trace and determinant values of the Jacobian matrix. With the help of Sotomayor’s theorem, the conditions for existence of saddle-node bifurcation are derived. Also, we show that the proposed system admits limit cycle dynamics, and its stability is discussed with the value of first Lyapunov coefficient. Moreover, the numerical simulations including phase portrait, one- and two-parameter bifurcation diagrams are performed to validate our important findings.

/pdf/the-dynamics-of-a-leslie-type-predator-prey-model-with-fear-1hvb8ybnr3.pdf

The dynamics of a Leslie type predator–prey model with fear and Allee effect

In this paper, a predator–prey model with ratio-dependent and impulsive state feedback control is constructed, where the pest growth rate is related to an Allee effect. Firstly, the existence condition of the homoclinic cycle is obtained by analyzing the control parameter q. The existence, uniqueness and asymptotic stability of the periodic orbit are discussed by using the geometric theory of the differential equations, the method of successor functions and analog of the Poincare criterion. Secondly, we formulate a control optimization with a minimal total cost in pest management, and we obtain an optimal economic threshold. Finally, we verify the main results by numerical simulation.

/pdf/control-optimization-and-homoclinic-bifurcation-of-a-prey-40ftocqjbb.pdf

Control optimization and homoclinic bifurcation of a prey–predator model with ratio-dependent

In this paper, a Leslie predator–prey system with ratio-dependent and state impulsive feedback control is investigated by applying the geometry theory of differential equation. When the economic threshold level is under the positive equilibrium, the existence, uniqueness and orbital asymptotical stability of order-1 periodic solution for the system can be obtained. When the economic threshold level is above the positive equilibrium, and the positive equilibrium is a focus point, sufficient conditions of the existence, uniqueness and orbital asymptotical stability of order-1 periodic solution for the system are also acquired. Furthermore, when the positive equilibrium is an unstable focus point, the existence of order-1 periodic solution of the impulsive system can be obtained within limit cycle of the continuous system. The mathematical results can be verified by numerical simulations.

Periodic solution of a Leslie predator–prey system with ratio-dependent and state impulsive feedback control

In this paper, a predator–prey system with mutual interference and impulsive state feedback control is constructed. Firstly, by using the stability theorem, we obtain the global asymptotically stability of the positive equilibrium for the system without impulse effects. Secondly, by using the geometric theory of differential equations, the method of successor functions and mathematical analysis, we obtain some sufficient conditions for the existence, uniqueness and orbital asymptotically stability of the order-1 periodic solution for the system with impulse effects. Finally, the main conclusions are verified by numerical simulation, and the pest prevention and control strategies on combining light trapping and water-gun spraying are proposed.

Qualitative analysis of a predator–prey system with mutual interference and impulsive state feedback control

In this paper, the system with impulsive state feedback control corresponding to the sulphitation reaction in process of manufacture of sugar is considered. By means of square approximation and a s...

Impulsive state feedback control during the sulphitation reaction in process of manufacture of sugar

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Periodic solution of a Leslie predator–prey system with ratio-dependent and state impulsive feedback control

Qualitative analysis of a predator–prey system with mutual interference and impulsive state feedback control

Impulsive state feedback control during the sulphitation reaction in process of manufacture of sugar