Hopf bifurcation analysis for congestion control with heterogeneous delays

In this paper, a congestion control algorithm with heterogeneous delays is considered. Local stability of the equilibrium solution of this algorithm is investigated based on analyzing the corresponding transcendental characteristic equation. Especially, using one of the system parameters of congestion control algorithm as the bifurcation parameter, when the system parameter exceeds a critical value, the congestion control algorithm undergoes a supercritical Hopf bifurcation, and the explicit formulae determining the stability and the direction of periodic solutions bifurcating from the equilibrium are obtained by applying Hassard et al.’s approaches. Finally, some numerical simulations are performed to verify the theoretical results.     

Navigation and control of the motion of a riderless bicycle by using a simplified dynamic model

This work deals with the guidance and control of a riderless bicycle (see Figure 1). Given two points P₁ and p₂ in the horizontal plane and a finite time interval [0, t_f]. Denote by (x₁,y₁,z₁) the coordinates of the center of the bicycle's rear wheel. Based on a simplified dynamical model of the bicycle, and by using the concept of path controllability, control laws are derived for the bicycle's pedalling moment and directional moment such that (x₁,y₁) will move from p₁ to P₂ during the time interval [0, t_f].     

Delay induced Hopf bifurcation in a dual model of Internet congestion control algorithm

This paper focuses on the delay induced Hopf bifurcation in a dual model of Internet congestion control algorithms which can be modeled as a time-delay system described by a one-order delay differential equation (DDE). By choosing communication delay as the bifurcation parameter, we demonstrate that the system loses its stability and a Hopf bifurcation occurs when communication delay passes through a critical value. Moreover, the bifurcating periodic solution of the system is calculated by means of the perturbation method. Discussion of stability of the periodic solutions involves the computation of Floquet exponents by considering the corresponding Poincaré–Lindstedt series expansion. Finally, numerical simulations for verifying the theoretical analysis are provided.     

Bifurcation analysis in a simplified tri-neuron BAM network model with multiple delays

In this paper, we consider a three-dimensional delayed differential equation representing a bidirectional associate memory (BAM) neural network with three neurons and two discrete delays. By analyzing the number and stability of equilibria, the pitchfork bifurcation curve of the system is obtained. Furthermore, on the pitchfork bifurcation curve, by using the sum of two delays as the bifurcation parameter, we find that the system can undergo a Hopf bifurcation at the origin and the three-dimensional ordinary differential equation describing the flow on the center manifold is given.     

Nonlinear dynamics of internet congestion control: A frequency-domain approach

In this paper a fluid-flow model for TCP congestion avoidance combined with different AQM schemes is analyzed. The conditions for the appearance of Hopf bifurcations are stated analytically using frequency-domain techniques. The proposed methodology allows the characterization of the emerging periodic orbits, providing approximations of their amplitude and frequency. In addition, multiple oscillations and limit cycle bifurcations are found via numerical tools.     

Dynamics of a congestion control model in a wireless access network

In this paper, a congestion control algorithm with heterogeneous delays in a wireless access network is considered. We regard the communication time delay as a bifurcating parameter to study the dynamical behaviors, i.e., local asymptotical stability, Hopf bifurcation and resonant codimension-two bifurcation. By analyzing the associated characteristic equation, the Hopf bifurcation occurs when the delay passes through a sequence of critical value. Furthermore, the direction and stability of the bifurcating periodic solutions are derived by applying the normal form theory and the center manifold theorem. In the meantime, the resonant codimension-two bifurcation is also found in this model. Some numerical examples are finally performed to verify the theoretical results.     

Quasiperiodic solutions of variational problems of motion in a central force field

A method is proposed for computing nearly optimal trajectories of dynamic systems with a small parameter by splitting the original variational problem into two separate problems for "fast" and "slow" variables. The problem for "fast" variables is solved by improving the zeroth approximation — the extremals of the linearized problem — by the Ritz method. The solution of the problem for "slow" variables is constructed by passing from a discrete argument — the number of revolutions around the attracting center— to a continuous argument. The proposed method does not require numerical integration of systems of differential equations and produces a highly accurate approximate solution of the problem.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 113–118, 1989.     

SVEIRS epidemic model with delays and partial immunization for internet worms

A delayed SVEIRS model for the transmission of worms in internet with partial immunization is proposed. The impact of the possible combination of the two delays on the model is investigated. By analyzing the corresponding characteristic equations and regarding the possible combination of the two delays as the bifurcation parameter, local stability of the endemic equilibrium and existence of local Hopf bifurcation at the viral equilibrium are addressed, respectively. Further, explicit formulas that determine direction and stability of the Hopf bifurcation are derived with the help of the normal form theory and the center manifold theorem. Finally, some numerical simulations are carried out to verify the obtained theoretical findings.     

Global existence of periodic solutions on a simplified BAM neural network model with delays

A simplified n-dimensional BAM neural network model with delays is considered. Some results of Hopf bifurcations occurring at the zero equilibrium as the delay increases are exhibited. Global existence of periodic solutions are established using a global Hopf bifurcation result of Wu [Wu J. Symmetric functional-differential equations and neural networks with memory. Trans Am Math Soc 1998;350:4799–838], and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney [Li MY, Muldowney J. On Bendixson’s criterion. J Differ Equations 1994;106:27–39]. Finally, computer simulations are performed to illustrate the analytical results found.     

Delayed feedback-based bifurcation control in an Internet congestion model

In this paper, the control of Hopf bifurcation in an Internet congestion model with a single link accessed by a single source is presented. By choosing the gain parameter as a bifurcation parameter, it is found that the system without control cannot guarantee a stationary sending rate. Furthermore, Hopf bifurcation occurs when the positive gain parameter of the system exceeds a critical value. For Internet congestion model, a control model based on delayed feedback is proposed and analyzed for delaying the onset of undesirable Hopf bifurcation. Numerical simulations are given to justify the validity of delayed feedback controller in bifurcation control.     

Hopf bifurcation analysis in a fluid flow model of Internet congestion control algorithm

This paper focuses on the Hopf bifurcation analysis of some classes of nonlinear time-delay models, namely fluid flow models, for the Internet congestion control algorithm of TCP/AQM networks. Using tools from control and bifurcation theory, it is proved that there exists a critical value of communication delay for the stability of the network. When the delay passes through the critical value, the system loses its stability and a Hopf bifurcation occurs. Furthermore, the stability of the bifurcation and direction 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.     

Hopf bifurcation and stability crossing curves in a cobweb model with heterogeneous producers and time delays

We study a continuous time cobweb model with discrete time delays where heterogeneous producers behave as adapters in the market. Specifically, they partially adjust production (which is subject to some gestation lags) towards the profit-maximising quantity under static expectations. The dynamics of the economy is described by a two-dimensional system of delay differential equations. We characterise stability properties of the stationary state of the system and show the emergence of Hopf bifurcations. We also apply some recent mathematical techniques (stability crossing curves) to show how heterogeneous time delays affect the stability of the economy.     

Stability and Hopf bifurcation analysis in a novel congestion control model with communication delay

In this paper, we investigated Hopf bifurcation by analyzing the distributed ranges of eigenvalues of characteristic linearized equation. Using communication delay as the bifurcation parameter, linear stability criteria dependent on communication delay have also been derived, and, furthermore, the direction of Hopf bifurcation as well as stability of periodic solution for the exponential RED algorithm with communication delay is studied. We find that the Hopf bifurcation occurs when the communication delay passes a sequence of critical values. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, a numerical simulation is presented to verify the theoretical results.     

Optimal feedback control for a thermoviscoelastic model of Voigt fluid motion

A system of equations describing the motion of viscous fluids with polymer additives is considered. The system involves a temperature-dependent viscosity and a velocity-dependent external force, which is used as a control. The existence of an optimal feedback control for the system is proved.     

Reinsurance control in a model with liabilities of the fractional Brownian motion type

We propose a model for reinsurance control for an insurance firm in the case where the liabilities are driven by fractional Brownian motion, a stochastic process exhibiting long-range dependence. The problem is transformed to a nonlinear programming problem, the solution of which provides the optimal reinsurance policy. The effect of various parameters of the model, such as the safety loading of the reinsurer and the insurer, the Hurst parameter, etc. on the optimal reinsurance program is studied in some detail. Copyright © 2007 John Wiley & Sons, Ltd.     

Analysis and approximation of optimal control problems for a simplified Ginzburg-Landau model of superconductivity

Summary. This paper is concerned with optimal control problems for a Ginzburg-Landau model of superconductivity that is valid for high values of the Ginzburg-Landau parameter and high external fields. The control is of Neumann type. We first show that optimal solutions exist. We then show that Lagrange multipliers may be used to enforce the constraints and derive an optimality system from which optimal states and controls may be deduced. Then we define finite element approximations of solutions for the optimality system and derive error estimates for the approximations. Finally, we report on some numerical results. Received May 3, 1994 / Revised version received November 28, 1995     

Optimal control in a model of the motion of a viscoelastic medium with objective derivative

In this paper we consider the Jeffreys model of the motion of a viscoelastic incompressible medium with the Yaumann derivative. Within this model we study the optimal control problem for the right-hand sides of the initial boundary value problem. We prove the existence of the optimal strong solution.     

Compressible flow induced by the transient motion of a wavemaker

The effect of fluid compressibility on the evolution of the pressure distribution and free surface elevation, following the initiation of a horizontal motion of a vertical wavemaker, is analysed. This effect is significant even in a liquid (like water) when the time scale of the motion is very short (e.g. impulsive motions).In the leading order the present problem is analogous to that of supersonic flow about a thin wing, thus the solution is represented by means of an appropriate supersonic source distribution. Closed-form results are obtained for the case of impulsive motion (i.e. a step function velocity). The pressure field corresponds to systems of double rarefaction and double compression waves traversing the fluid domain intermittently. Following the passage of a rarefaction (compression) wave, the free surface becomes locally concave (convex). The resulting free surface profile consists of an elongating wavetrain in front of a jet riding up the vertical wall.On the compressible time-scale the pressure and velocity fields approach a steady long-time limit. This limit corresponds to the short-time incompressible flow prevailing after the attenuation of the pressure waves. The spatial nonuniformity of the asymptotic expansion in the neighbourhood of the waterline is briefly discussed.