Adaptive fuzzy H ∞ tracking design of SISO uncertain nonlinear fractional order time-delay systems

In this paper, a fuzzy logic controller equipped with training algorithms is developed such that the H ^?? tracking performance should be satisfied for a model-free nonlinear fractional order time delay system which is infinite dimensional in nature and time delay is a source of instability. In order to deal with the linguistic uncertainties caused from delay terms, the adaptive time delay fuzzy logic system is constructed to approximate the unknown time delay system functions. By incorporating Lyapunov stability criterion with H ^?? tracking design technique, the free parameters of the adaptive fuzzy controller can be tuned on line by output feedback control law and adaptive law. Moreover, the tracking error and external disturbance can be attenuated to arbitrary desired level. The numerical results show the effectiveness of the proposed adaptive H ^?? tracking scheme.     

Dynamics of three coupled limit cycle oscillators with?vastly?different frequencies

A system of three coupled limit cycle oscillators with vastly different frequencies is studied. The three oscillators, when uncoupled, have the frequencies ?? ₁=O(1), ?? ₂=O(1/??) and ?? ₃=O(1/?? ²), respectively, where ???1. The method of direct partition of motion (DPM) is extended to study the leading order dynamics of the considered autonomous system. It is shown that the limit cycles of oscillators 1 and 2, to leading order, take the form of a Jacobi elliptic function whose amplitude and frequency are modulated as the strength of coupling is varied. The dynamics of the fastest oscillator, to leading order, is unaffected by the coupling to the slower oscillator. It is also found that when the coupling strength between two of the oscillators is larger than a critical bifurcation value, the limit cycle of the slower oscillator disappears. The obtained analytical results are formal and are checked by comparison to solutions from numerical integration of the system.     

Stability and bifurcation analysis in a FAST TCP model with feedback delay

This paper focuses on the local stability of a FAST TCP model of Internet congestion control algorithms. Necessary and sufficient stability conditions in terms of key system parameters are given, which can provide exact guideline for setting system parameters. In addition, the complex dynamics of the system is also addressed. We demonstrate that Hopf bifurcation would occur when the gain parameter ?? is less than a critical value. Furthermore, the direction and the stability of the bifurcating periodic solutions are determined by applying the normal form theory and the center manifold theorem. Finally, some numerical examples are given to verify the theoretical analysis.     

Pattern formation of an epidemic model with diffusion

One subject of spatial epidemiology is spatial variation in disease risk or incidence. The spread of epidemics can result in strong spatial patterns of such risk or incidence: for example, pathogen dispersal might be highly localized, vectors or reservoirs for pathogens might be spatially restricted, or susceptible hosts might be clumped. Here, spatial pattern of an epidemic model with nonlinear incidence rates is investigated. The conditions for Hopf bifurcation and Turing bifurcation are gained and, in particular, exact Turing domain is found in the two parameters space. Furthermore, numerical results show that force of infection, namely???, plays an important role in the spatial pattern. More specifically, different patterns emerge as ?? increases. The mathematical analysis and numerical results well extend the finding of pattern formation in the epidemic models and may well explain the field observed in some areas.     

A pair of van der Pol oscillators coupled by fractional derivatives

We consider the stability of the in-phase and out-of-phase modes of a pair of fractionally-coupled van der Pol oscillators: 1 2 where D ^?? x is the order ?? derivative of x(t), and 0<??<1. We use a two-variable perturbation method on the system??s corresponding variational equations to derive expressions for the transition curves separating regions of stability from instability in the ??, ?? parameter plane. The perturbation results are validated with numerics and through direct comparison with known results in the limiting cases of ??=0 and ??=1, where the fractional coupling reduces to position coupling and velocity coupling, respectively.     

Adaptive boundary control of the unforced generalized Korteweg–de Vries–Burgers equation

This paper deals with the adaptive control problem of the unforced generalized Korteweg?Cde Vries?CBurgers (GKdVB) equation when the spatial domain is [0,1]. Three adaptive control laws are designed for the GKdVB equation when either the kinematic viscosity ?? or the dynamic viscosity ?? is unknown, or when both viscosities ?? and ?? are unknowns. Using the Lyapunov theory, the L ²-global exponential stability of the solutions of this equation is shown for each of the proposed control laws. Also, numerical simulations based on the Finite Element method (FEM) are given to illustrate the analytical results.     

Fractional-order PD control at Hopf bifurcations in a fractional-order congestion control system

In this paper, we address the problem of the bifurcation control of a delayed fractional-order dual model of congestion control algorithms. A fractional-order proportional–derivative (PD) feedback controller is designed to control the bifurcation generated by the delayed fractional-order congestion control model. By choosing the communication delay as the bifurcation parameter, the issues of the stability and bifurcations for the controlled fractional-order model are studied. Applying the stability theorem of fractional-order systems, we obtain some conditions for the stability of the equilibrium and the Hopf bifurcation. Additionally, the critical value of time delay is figured out, where a Hopf bifurcation occurs and a family of oscillations bifurcate from the equilibrium. It is also shown that the onset of the bifurcation can be postponed or advanced by selecting proper control parameters in the fractional-order PD controller. Finally, numerical simulations are given to validate the main results and the effectiveness of the control strategy.

     

Stability switches and global Hopf bifurcation in a nutrient-plankton model

This paper is devoted to the analysis of a nutrient-plankton model with delayed nutrient cycling. Firstly, stability and Hopf bifurcation of the positive equilibrium are given, and the direction and stability of Hopf bifurcation are also studied. We show that delay, which is considered in the decomposition of dead phytoplankton, can induce stability switches, such that the positive equilibrium switches from stability to instability, to stability again and so on. One can observe that the influence of delay on the system dynamics is essential. Then, we prove that there exists at least one positive periodic solution as the time delay varies in some regions using the global Hopf bifurcation result of Wu (1998, Trans Am Math Soc 350:4799–4838) for functional differential equations. Furthermore, the impact of input rate of nutrient is discussed along with numerical results, and the role of delay in the nutrient cycling is interpreted ecologically. Finally, several groups of illustrations are performed to justify analytical findings.     

Multiple stability switches and Hopf bifurcation in a damped harmonic oscillator with delayed feedback

This paper takes into consideration a damped harmonic oscillator model with delayed feedback. After transforming the model into a system of first-order delayed differential equations with a single discrete delay, the single stability switch and multiple stability switches phenomena as well as the existence of Hopf bifurcation of the zero equilibrium of the system are explored by taking the delay as the bifurcation parameter and analyzing in detail the associated characteristic equation. Particularly, in view of the normal form method and the center manifold reduction for retarded functional differential equations, the explicit formula determining the properties of Hopf bifurcation including the direction of the bifurcation and the stability of the bifurcating periodic solutions are given. In order to check the rationality of our theoretical results, numerical simulations for some specific examples are also carried out by means of the MATLAB software package.

     

Design and analysis of a first order time-delayed chaotic system

The present paper reports the design and analysis of a new time-delayed chaotic system and its electronic circuit implementation. The system is described by a first-order nonlinear retarded type delay differential equation with a closed form mathematical function describing the nonlinearity. We carry out stability and bifurcation analysis to show that with the suitable delay and system parameters the system shows sustained oscillation through supercritical Hopf bifurcation. It is shown through numerical simulations that the system depicts bifurcation and chaos for a certain range of the system parameters. The complexity and predictability of the system are characterized by Lyapunov exponents and Kaplan?CYork dimension. It is shown that, for some suitably chosen system parameters, the system shows hyperchaos even for a small or moderate delay. Finally, we set up an experiment to implement the proposed system in electronic circuit using off-the-shelf circuit elements, and it is shown that the behavior of the time delay chaotic electronic circuit agrees well with our analytical and numerical results.     

Multiple bifurcation analysis in a ring of delay coupled oscillators with neutral feedback

Spatiotemporal periodic patterns, including phase-locked oscillations, mirror-reflecting waves, standing waves, in-phase or anti-phase oscillations are investigated in a ring of bidirectionally coupled oscillators with neutral delay feedback. It is confirmed that neutral feedback makes Hopf bifurcation occur in a larger domain of parameters. We calculate the normal forms near Hopf bifurcation, D _N equivariant Hopf bifurcation and double-Hopf bifurcation in this neutral equation by using the method of multiple scales. Theoretically, the appearance of the in-phase, anti-phase and phase-locked oscillations that we observed in the simulation about a ring of delay coupled Hindmarsh–Rose neurons with neutral feedback is explained.     

Neural network based robust hybrid control for robotic system: an H �� approach

A novel robust hybrid tracking control for robotic system is proposed. This hybrid control scheme combines computed torque control (CTC) with neural network, variable structure control (VSC) and nonlinear H _?? control methods. It is assumed that the nominal system of robotic system is completely known, which is controlled by using CTC method. Neural network is designed to approximate parameter uncertainties, VSC is used to eliminate the effect of approximation error, and H _?? control is employed to achieve a desired robust tracking performance. Based on Lyapunov stability theorem, it can be guaranteed that all signals in closed loop are bounded and a specified H _?? tracking performance is achieved by employing the proposed robust hybrid control. The validity of the control scheme is shown by computer simulation of a two-link robotic manipulator.     

Bifurcation analysis of a tri-neuron neural network model in the frequency domain

In this paper, a class of neural network models with three neurons is considered. By applying the frequency domain approach and analyzing the associated characteristic equation, the existence of the bifurcation parameter point is determined. If the coefficient μ is chosen as a bifurcation parameter, it is found that Hopf bifurcation occurs when the parameter μ passes through a critical value. The direction and the stability of Hopf bifurcation periodic solutions are determined by the Nyquist criterion and the graphical Hopf bifurcation theorem. Some numerical simulations for justifying the theoretical analysis are also provided.     

Stability and Neimark-Sacker bifurcation analysis for a discrete single genetic negative feedback autoregulatory system with delay

In this paper, we deal with a discrete single genetic negative feedback autoregulatory system with delay by using Euler method. Choosing the delay $\tau $ as the bifurcation parameter and analyzing the associated characteristic equation corresponding to the unique positive fixed point, it is found that the stability of the positive equilibrium and Neimark-Sacker bifurcation may occur when $\tau $ crosses some critical values. Then the explicit formula which determines the stability, direction, and other properties of bifurcating periodic solution is derived by using the center manifold theorem and normal form theory. Finally, in order to illustrate our theoretical analysis, numerical simulations are also included in the end.     

Using delay to quench undesirable vibrations

A van der Pol type system with delayed feedback is explored by employing the two variable expansion perturbation method. The perturbation scheme is based on choosing a critical value for the delay corresponding to a Hopf bifurcation in the unperturbed ε=0 system. The resulting amplitude–delay relation predicts two Hopf bifurcation curves, such that in the region between these two curves oscillations will be quenched. The perturbation results are verified by comparison with numerical integration.     

Bifurcations and stability of a discrete singular bioeconomic system

The paper analyzes the stability and bifurcations of a discrete singular bioeconomic system in the closed first quadrant $R_{+}^{3}$ . First, applying the Poincaré scheme to a differential-algebraic predator–prey system where the economic interest of harvesting is taken into account, a discrete singular bioeconomic system is proposed. Then, local stability and the existing conditions of the flip bifurcation and Neimark–Sacker bifurcation around the interior equilibria of the proposed model are discussed by using the normal form of the discrete singular bioeconomic system, the center manifold theorem and the bifurcation theory, when choosing the step size δ as the parameter of the bifurcation. Finally, the results are illustrated and the complex dynamical behaviors are exhibited by computer numerical simulations.     

Mixed Convection Boundary-Layer Flow Along a Vertical Cylinder Embedded in a Porous Medium Filled by a Nanofluid

The steady mixed convection boundary-layer flow on a vertical circular cylinder embedded in a porous medium filled by a nanofluid is studied for both cases of a heated and a cooled cylinder. The governing system of partial differential equations is reduced to ordinary differential equations by assuming that the surface temperature of the cylinder and the velocity of the external (inviscid) flow vary linearly with the axial distance x measured from the leading edge. Solutions of the resulting ordinary differential equations for the flow and heat transfer characteristics are evaluated numerically for various values of the governing parameters, namely the nanoparticle volume fraction ${\phi}$ , the mixed convection or buoyancy parameter ?? and the curvature parameter ??. Results are presented for the specific case of copper nanoparticles. A critical value ?? _c of ?? with ?? _c <?0 is found, with the values of | ?? _c| increasing as the curvature parameter ?? or nanoparticle volume fraction ${\phi}$ is increased. Dual solutions are seen for all values of ?? >??? _c for both aiding, ?? >?0 and opposing, ?? <?0, flows. Asymptotic solutions are also determined for both the free convection limit ${(\lambda \gg 1)}$ and for large curvature parameter ${(\gamma \gg 1)}$ .     

Simulating the electric activity of FitzHugh–Nagumo neuron by using Josephson junction model

The resistive-capacitive-inductance Josephson junction (RCLSJ) model can simulate the electric activities of neurons. In this paper, the RCLSJ system is controlled to reproduce the dynamical properties of the FitzHugh?CNagumo system neuron by using the improved adaptive synchronization scheme. Improved Lyapunov functions with two controllable gain coefficients (??,??) is constructed, and the controller is approached analytically to realize linear generalized synchronization defined as $x=k\hat{x}+C$ . The summation of error function during the process of synchronization and the power consumption of controller are calculated in the dimensionless model to measure the effect of the two gain coefficients (??,??) by selecting different constants (k,C) to represent different kinds of generalized synchronization. The results are approached as follows: (1) the power consumption of the controller is independent of the selection of the two gain coefficients (??,??); (2) the synchronization region is marked in the phase space of the two gain coefficients; (3) the power consumption of controller is dependent on the selection of constants (k,C), smaller power consumption of the controller is required with larger k at fixed C; larger power consumption costs with larger C at fixed k. The specific case for C=0,k=1 is also discussed to understand the case for complete synchronization.     

Chaos control of a φ 6-Van der Pol oscillator driven by external excitation

This paper studies the dynamics of a ?? ⁶-Van der Pol oscillator subjected to an external excitation. Numerical analysis is presented to observe its periodic and chaotic motions, and a method called Multiple-prediction Delayed Feedback Control is proposed to control chaos effectively via periodic feedback gain. The controller is designed based on plural Poincaré maps which are defined to regard the nonautonomous system as a T-periodic discrete time system, therefore, the stability of the closed-loop system can be evaluated from the theory of monodromy matrix. Numerical simulations are provided to illustrate the validity of the proposed control strategy.     

van der Pol-Duffing时滞系统的稳定性和Hopf分岔

研究了具有三次项的ｖａｎ ｄｅｒ Ｐｏｌ－Ｄｕｆｆｉｎｇ非线性时滞系统的稳定性和Ｈｏｐｆ分岔,分析了当线性化特征方程随两参数（时滞量和增益系数）变化时特征根的分布;证明了Ｈｏｐｆ分岔的存在性,通过构造中心流形并且使用范式方法给出的Ｈｏｐｆ分岔的方向以及周期解的稳定性,讨论时滞量对该系统的Ｈｏｐｆ分岔的影响。