Study on the dynamical behaviors of a two-dimensional discrete system

The generalized Ushiki map is investigated. It is theoretically proven that there are transcritical and flip bifurcations and there exists a chaotic phenomenon in the sense of Marotto. And numerical simulations not only show the consistency with the theoretical analysis but also exhibit the complex dynamical behaviors including the period-6, 5, 3 windows, intermittency mechanics, attractor merging crisis and boundary crisis. The computation of Lyapunov exponents conforms the dynamical behaviors.     

Effects of system parameters on the optimal policy structure in a class of queueing control problems

This paper studies a class of queueing control problems involving commonly used control mechanisms such as admission control and pricing. It is well established that in a number of these problems, there is an optimal policy that can be described by a few parameters. From a design point of view, it is useful to understand how such an optimal policy varies with changes in system parameters. We present a general framework to investigate the policy implications of the changes in system parameters by using event-based dynamic programming. In this framework, the control model is represented by a number of common operators, and the effect of system parameters on the structured optimal policy is analyzed for each individual operator. Whenever a queueing control problem can be modeled by these operators, the effects of system parameters on the optimal policy follow from this analysis.      

Bifurcation and chaos in a discrete time predator-prey system of Leslie type with generalized Holling type III functional response

This paper is devoted to study a discrete time predator-prey system of Leslie type with generalized Holling type III functional response obtained using the forward Euler scheme. Taking the integration step size as the bifurcation parameter and using the center manifold theory and bifurcation theory, it is shown that by varying the parameter the system undergoes flip bifurcation and Neimark-Sacker bifurcation in the interior of $\mathbb{R}_+^2$. Numerical simulations are implemented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascade of period-doubling bifurcation in period-$2$, $4$, $8$, quasi-periodic orbits and the chaotic sets. These results shows much richer dynamics of the discrete model compared with the continuous model. The maximum Lyapunov exponent is numerically computed to confirm the complexity of the dynamical behaviors. Moreover, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

On the stability of inner and outer approximations of a convex compact set by a ball

The finite-dimensional problems of outer and inner estimation of a convex compact set by a ball of some norm (circumscribed and inscribed ball problems) are considered. The stability of the solution with respect to the error in the specification of the estimated compact set is generally characterized. A new solution criterion for the outer estimation problem is obtained that relates the latter to the inner estimation problem for the lower Lebesgue set of the distance function to the most distant point of the estimated compact set. A quantitative estimate for the stability of the center of an inscribed ball is given under the additional assumption that the compact set is strongly convex. Assuming that the used norm is strongly quasi-convex, a quantitative stability estimate is obtained for the center of a circumscribed ball.     

Control of a novel fractional hyperchaotic system using a located control method

Fractional order dynamics and chaotics systems have been recently combined, yielding interesting behaviours. In this paper, a novel integer order hyperchaotic system is considered. Then, a fractional order hyperchaotic representation of said system is proposed using a natural fractionalization. Two different linear control methodologies to deal with the complexity which introduce such systems are proposed. Those methods are able to modify the hyperchaotic behaviour of the system and force it to move towards a fixed point; i.e. steady state. These approaches give a general framework for taming such complex systems using simple linear controllers. The main tools for analysing the controlled system are Matignon stability criterion and RouthHurwitz test. Using a reliable numerical simulation, the designed system is simulated to verify the theoretical analysis.     

Movement of hot spots on the exterior domain of a ball under the Neumann boundary condition

We consider the Cauchy-Neumann problem of the heat equation in the exterior domain of a ball in R^N, and study the movement of hot spots of the solution as t→∞.     

Analytical solutions to the heat conduction problems for a cylinder and a ball based on determining the temperature perturbation front

Analytical solutions to the heat conduction problems for a cylinder and a ball are obtained by the integral method of heat balance. To improve the accuracy of the solutions, the temperature function is approximated by polynomials of high degrees. Their coefficients are determined via introducing additional boundary conditions, which are found from the governing differential equation and the basic boundary conditions, including those specified at the temperature perturbation front. It is shown that the additional boundary conditions, even in the second approximation, lead to a considerable improvement in the solution accuracy.     

Unique determination of a sound-soft ball by the modulus of a single far field datum

In this paper, we prove uniqueness in determining a sound-soft ball in inverse acoustic scattering by the modulus of the far field patterns. Since the modulus of the far field pattern is invariant under translation (Kress and Rundell, 1997 [6], Kwon and Seo, 2000 [7]), we can only reconstruct the shape of the obstacle but not the location. Assume that the ball is centered at the origin. Under the condition that the ball is small, it is uniquely determined by the modulus of a single far field datum measured at a fixed spot corresponding to a single incident plane wave.     

Spread of a disease and its effect on population dynamics in an eco-epidemiological system

In this paper, an eco-epidemiological model with simple law of mass action and modified Holling type II functional response has been proposed and analyzed to understand how a disease may spread among natural populations. The proposed model is a modification of the model presented by Upadhyay et al. (2008) [1]. Existence of the equilibria and their stability analysis (linear and nonlinear) has been studied. The dynamical transitions in the model have been studied by identifying the existence of backward Hopf-bifurcations and demonstrated the period-doubling route to chaos when the death rate of predator (μ₁) and the growth rate of susceptible prey population (r) are treated as bifurcation parameters. Our studies show that the system exhibits deterministic chaos when some control parameters attain their critical values. Chaotic dynamics is depicted using the 2D parameter scans and bifurcation analysis. Possible implications of the results for disease eradication or its control are discussed.     

Control of AMIRA’s ball and beam system via improved fuzzy feedback linearization approach

This paper first studies the tracking and almost disturbance decoupling problem of nonlinear AMIRA’s ball and beam system based on the feedback linearization approach and fuzzy logic control. The main contribution of this study is to construct a controller, under appropriate conditions, such that the resulting closed-loop system is valid for any initial condition and bounded tracking signal with the following characteristics: input-to-state stability with respect to disturbance inputs and almost disturbance decoupling, i.e., the influence of disturbances on the L₂ norm of the output tracking error can be arbitrarily attenuated by changing some adjustable parameters. One example, which cannot be solved by the first paper on the almost disturbance decoupling problem, is proposed in this paper to exploit the fact that the tracking and the almost disturbance decoupling performances are easily achieved by our proposed approach. The simulation results show that our proposed approach has achieved the almost disturbance decoupling performance perfectly.     

Continuous dependence of the controllability set of a nonlinear control system on a parameter

In this note, we consider the mapping which associates a controllability set to each value of the parameter of a system. Sufficient conditions for the existence of breaks of this mapping are obtained. They show that nearly all such mappings have a finite number of breaks.This note was written during the author's stay at the University of Hamburg. He expresses his gratitude to the Institut für Angewandte Mathematik for its warm hospitality.     

基于主动滑模控制的混沌系统函数投影同步

研究了一类混沌系统的函数投影同步问题.基于Lyapunov稳定性理论和主动滑模控制方法,设计了主动滑模控制器,实现混沌系统的函数投影同步.数值仿真验证了该控制器的有效性和正确性.     

On the optimal control of a single-server queueing system: Comment

In a recent paper by Scott and Jefferson, the optimal control of the service rate for a single-server queue with limited waiting space is treated by the maximum principle. We show that their control policies are necessarily suboptimal. Characterizations for optimal control are derived and used to obtain corresponding optimal trajectories in both nonsingular and singular regions.     

On the optimal control of a single-server queueing system: Reply

This comment replies to a criticism due to Klein and Gruver (Ref. 1) of our earlier paper (Ref. 2) on the application of control theory to a queueing system. The criticism concerns the state-space diagram and the table which we inadvertently gave for the terminal-reward problem, albeit incorrectly labeled, rather than for the free-endpoint problem considered in our paper. We show that the solution given by Klein and Gruver is itself incorrect and nonoptimal.     

On the optimal control of a single-server queueing system: Further remarks

This comment is in response to a reply by Scott and Jefferson (Ref. 3) concerning the application of control theory to a queueing problem.     

Asymptotic analysis of certain classes of singularly perturbed problems on the semiaxis

We propose a new method for asymptotic integration of certain classes of singularly perturbed Cauchy problems on the semiaxis for nonhomogeneous systems of linear ordinary differential equations; this method is a further development of the ideas of the regularization method. This method enables us to prove the existence of a unique bounded (as ε→+0) solution of such problems and leads to a simpler and more constructive algorithm for obtaining the asymptotic expansion of the solution and singling out all of its singularities in closed analytic form (including the critical case in which the spectral points of the limit operator may touch the imaginary axis). The proposed method supplements and sharpens earlier results. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 111–117, July, 1997. Translated by N. K. Kulman     

On the 0-local controllability of a linear control system

The problem of the 0-local controllability of a linear control system when the origin does not belong to the constraint set is considered. Sufficient conditions for the 0-local controllability are presented here, extending a previous theorem by Saperstone and connecting this property with the distance of the constraint set from the origin.