Controlled Positive Dynamic Systems with an Entropy Operator: Fundamentals of the Theory and Applications

Controlled dynamic systems with an entropy operator (DSEO) are considered. Mathematical models of such systems were used to study the dynamic properties in demo-economic systems, the spatiotemporal evolution of traffic flows, recurrent procedures for restoring images from projections, etc. Three problems of the study of DSEO are considered: the existence and uniqueness of singular points and the influence of control on them; stability in “large” of the singular points; and optimization of program control with linear feedback. The theorems of existence, uniqueness, and localization of singular points are proved using the properties of equations with monotone operators and the method of linear majorants of the entropy operator. The theorem on asymptotic stability of the DSEO in “large” is proven using differential inequalities. Methods for the synthesis of quasi-optimal program control and linear feedback control with integral quadratic quality functional, and ensuring the existence of a nonzero equilibrium, were developed. A recursive method for solving the integral equations of the DSEO using the multidimensional functional power series and the multidimensional Laplace transform was developed. The problem of managing regional foreign direct investment is considered, the distribution of flows is modeled by the corresponding DSEO. It is shown that linear feedback control is a more effective tool than program control.

Travelling Wave Solutions of Wu–Zhang System via Dynamic Analysis

In this paper, based on the dynamical system method, we obtain the exact parametric expressions of the travelling wave solutions of the Wu–Zhang system. Our approach is much different from the existing literature studies on the Wu–Zhang system. Moreover, we also study the fractional derivative of the Wu–Zhang system. Finally, by comparison between the integer-order Wu–Zhang system and the fractional-order Wu–Zhang system, we see that the phase portrait, nonzero equilibrium points, and the corresponding exact travelling wave solutions all depend on the derivative order α. Phase portraits and simulations are given to show the validity of the obtained solutions.

Study on Amplitude Modulation Principle of Chaotic System

Exploring the amplitude modulation phenomenon of chaotic signal has become a subject of great concern in recent years. This paper mainly concentrates on the preliminary study on amplitude modulation principle of a chaotic system. First, two 3D chaotic systems with quadratic product terms are introduced for studying the amplitude modulation phenomenon of chaotic signal. It is found that the signal amplitude of the first system can be controlled by partial quadratic coefficient. But for the second system, none of nonlinear coefficient can be employed to control the signal amplitude. Then, the amplitude modulation principle of chaotic system is preliminarily studied by exploring the intrinsic relationship between nonzero equilibrium point and phase space trajectory, and it is further validated by introducing unified parameter to the two 3D chaotic systems. As a necessary condition, the principle provides a feasible and simple method for constructing and analyzing an amplitude modulation chaotic system.

Jacobi Stability Analysis of the Chen System

In this paper, the research of the Jacobi stability of the Chen system is performed by using the KCC-theory. By associating a nonlinear connection and a Berwald connection, five geometrical invariants of the Chen system are obtained. The Jacobi stability of the Chen system at equilibrium points and a periodic orbit is investigated in terms of the eigenvalues of the deviation curvature tensor. The obtained results show that the origin is always Jacobi unstable, while the Jacobi stability of the other two nonzero equilibrium points depends on the values of the parameters. And a periodic orbit of the Chen system is proved to be also Jacobi unstable. Furthermore, Jacobi stability regions of the Chen system and the Lorenz system are compared. Finally, the dynamical behavior of the components of the deviation vector near the equilibrium points is also discussed.

Coexistence of Multiple Attractors in an Active Diode Pair Based Chua’s Circuit

This paper focuses on the coexistence of multiple attractors in an active diode pair based Chua’s circuit with smooth nonlinearity. With dimensionless equations, dynamical properties, including boundness of system orbits and stability distributions of two nonzero equilibrium points, are investigated, and complex coexisting behaviors of multiple kinds of disconnected attractors of stable point attractors, limit cycles and chaotic attractors are numerically revealed. The results show that unlike the classical Chua’s circuit, the proposed circuit has two stable nonzero node-foci for the specified circuit parameters, thereby resulting in the emergence of multistability phenomenon. Based on two general impedance converters, the active diode pair based Chua’s circuit with an adjustable inductor and an adjustable capacitor is made in hardware, from which coexisting multiple attractors are conveniently captured.

Bifurcation Gaps in Asymmetric and High-Dimensional Hypercycles

Hypercycles are catalytic systems with cyclic architecture. These systems have been suggested to play a key role in the maintenance and increase of information in prebiotic replicators. It is known that for a large enough number of hypercycle species ([Formula: see text]) the coexistence of all hypercycle members is governed by a stable periodic orbit. Previous research has characterized saddle-node (s-n) bifurcations involving abrupt transitions from stable hypercycles to extinction of all hypercycle members, or, alternatively, involving the outcompetition of the hypercycle by so-called mutant sequences or parasites. Recently, the presence of a bifurcation gap between a s-n bifurcation of periodic orbits and a s-n of fixed points has been described for symmetric five-member hypercycles. This gap was found between the value of the replication quality factor [Formula: see text] from which the periodic orbit vanishes ([Formula: see text]) and the value where two unstable (nonzero) equilibrium points collide ([Formula: see text]). Here, we explore the persistence of this gap considering asymmetries in replication rates in five-member hypercycles as well as considering symmetric, larger hypercycles. Our results indicate that both the asymmetry in Malthusian replication constants and the increase in hypercycle members enlarge the size of this gap. The implications of this phenomenon are discussed in the context of delayed transitions associated to the so-called saddle remnants.

ASSESSING THE EFFECTS OF TEMPERATURE AND DENGUE VIRUS LOAD ON DENGUE TRANSMISSION

In this study, we propose a model to assess the effect of temperature on the incidence of dengue fever. For this, we take into account the dependence of the entomological and epidemiological parameters of the transmitter vector Aedes aegypti with respect to the temperature. The model consists of an ODE system that describes the transmission between humans and mosquitoes considering the aquatic stage of the vector population. The qualitative analysis of the model is made in terms of the parameters [Formula: see text] and [Formula: see text], which represent the basic offspring of mosquitoes, and the basic reproductive number of the disease, respectively. If [Formula: see text] mosquito population extinguishes while for [Formula: see text] it tends asymptotically to a nonzero equilibrium. Analogously, the disease transmission is eliminated if [Formula: see text], and it approaches an endemic equilibrium for [Formula: see text]. Using entomological data of mosquitoes as well as experimental data of disease transmission we evaluate [Formula: see text] and [Formula: see text] at different temperatures, obtaining that around [Formula: see text]C both parameters attain their maximum. Sensitivity analysis reveals that infection rates and mosquito mortality are the parameters for which [Formula: see text] is more sensitive.

Controlling Chaos in Power System Based on Tridiagonal Structure Matrix Stability Theory

In the case of periodic load disturbance, the chaos oscillation phenomena may caused by power system to threaten the safety operation of the network. In order to solve this problem, this paper presents the strategy on tridiagonal structure matrix stability theory. Using the designed controller, the chaotic system operation state turn into a stable operation state, and system is stabilized an unstable nonzero equilibrium point.