Controlling bifurcations and chaos in TCP–UDP–RED

We study the bifurcation and chaotic behavior of the Transmission Control Protocol (TCP) and User Datagram Protocol (UDP) network with Random Early Detection (RED) queue management. These bifurcation and chaotic behaviors may cause heavy oscillation of an average queue length and induce network instability. We propose an impulsive control method for controlling bifurcations and chaos in the internet congestion control system. The theoretical analysis and the simulation experiments show that this method can obtain the stable average queue length without sacrificing the other advantages of RED.     

Stochastic nonlinear dynamics of interpersonal and romantic relationships

Current theories from biosocial (e.g., the role of neurotransmitters in behavioral features), ecological (e.g., cultural, political, and institutional conditions), and interpersonal (e.g., attachment) perspectives have grounded interpersonal and romantic relationships in normative social experiences. However, these theories have not been developed to the point of providing a solid theoretical understanding of the dynamics present in interpersonal and romantic relationships, and integrative theories are still lacking. In this paper, mathematical models are used to investigate the dynamics of interpersonal and romantic relationships, via ordinary and stochastic differential equations, in order to provide insight into the behaviors of love. The analysis starts with a deterministic model and progresses to nonlinear stochastic models capturing the stochastic rates and factors (e.g., ecological factors, such as historical, cultural and community conditions) that affect proximal experiences and shape the patterns of relationship. Numerical examples are given to illustrate various dynamics of interpersonal and romantic behaviors with particular emphases placed on sustained oscillations and transitions between locally stable equilibria that are observable in stochastic models (closely related to real interpersonal dynamics), but absent in deterministic models.     

Interpreting supply chain dynamics: A quasi-chaos perspective

Chaotic phenomena, chaos amplification and other interesting nonlinear behaviors have been observed in supply chain systems. Chaos can be defined theoretically if the dynamics under study are produced only by deterministic factors. However, deterministic settings rarely present themselves in reality. In fact, real data are typically unknown. How can the chaos theory and its related methodology be applied in the real world? When the demand is stochastic, the interpretation and distribution of the Lyapunov exponents derived from the effective inventory at different supply chain levels are not similar to those under deterministic demand settings. Are the observed dynamics of the effective inventory random, chaotic, or simply quasi-chaos? In this study, we investigate a situation whereby the chaos analysis is applied to a time series as if its underlying structure, deterministic or stochastic, is unknown. The result shows clear distinction in chaos characterization between the two categories of demand process, deterministic vs. stochastic. It also highlights the complexity of the interplay between stochastic demand processes and nonlinear dynamics. Therefore, caution should be exercised in interpreting system dynamics when applying chaos analysis to a system of unknown underlying structure. By understanding this delicate interplay, decision makers have the better chance to tackle the problem correctly or more effectively at the demand end or the supply end.     

Stability and Analysis of TCP Connections with RED Control and Exogenous Traffic

In this paper we study the stability and performance of a system involving several TCP connections passing through a tandem of RED controlled queues each of which has an incoming exogenous stream. The exogenous stream, representing the superposition of all incoming UDP connections into a queue, has been modeled as an MMPP stream. We consider both the TCP Tahoe and the TCP Reno versions. We start with the analysis of a single TCP connection sharing a RED controlled queue with an exogenous stream. The effect of the exogenous stream (which is almost always present in large networks) is to cause the system to converge to a stationary distribution from any initial conditions. This stability result holds good for any work conserving discipline. We also obtain the performance indices of the system; specifically the goodputs and the mean sojourn times of the various connections. The complexity involved in computation of performance indices for the system is reduced by approximating the evolution of the average queue length process of the RED queue by a deterministic ODE. Then, by using a decomposition approach of two time scales, we reduce the study of the system to that of a simplified one for which the performance measures can be obtained under stationarity. Finally, we extend the above results to the case when multiple TCP connections share a RED controlled queue with an exogenous stream and to the case when a TCP connection passes through several RED controlled tandem queues each of which has an incoming exogenous stream. We also consider an example of multiple TCPs passing through a tandem of queues. A number of simulation results have been provided which support the analysis.     

The efficient solution of the (quietly constrained) noisy, linear regulator problem

In a previous paper we gave a new, natural extension of the calculus of variations/optimal control theory to a (strong) stochastic setting. We now extend the theory of this most fundamental chapter of optimal control in several directions. Most importantly we present a new method of stochastic control, adding Brownian motion which makes the problem “noisy.” Secondly, we show how to obtain efficient solutions: direct stochastic integration for simpler problems and/or efficient and accurate numerical methods with a global a priori error of O(h3/2) for more complex problems. Finally, we include “quiet” constraints, i.e. deterministic relationships between the state and control variables. Our theory and results can be immediately restricted to the non “noisy” (deterministic) case yielding efficient, numerical solution techniques and an a priori error of O(h²). In this event we obtain the most efficient method of solving the (constrained) classical Linear Regulator Problem. Our methods are different from the standard theory of stochastic control. In some cases the solutions coincide or at least are closely related. However, our methods have many advantages including those mentioned above. In addition, our methods more directly follow the motivation and theory of classical (deterministic) optimization which is perhaps the most important area of physical and engineering science. Our results follow from related ideas in the deterministic theory. Thus, our approximation methods follow by guessing at an algorithm, but the proof of global convergence uses stochastic techniques because our trajectories are not differentiable. Along these lines, a general drift term in the trajectory equation is properly viewed as an added constraint and extends ideas given in the deterministic case by the first author.     

Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part I

This paper, together with the accompanying work (Part II, Stochastic Process. Appl. 93 (2001) 205–228) is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial differential equations. We introduce a definition of stochastic viscosity solution in the spirit of its deterministic counterpart, with special consideration given to the stochastic integrals. We show that a stochastic PDE can be converted to a PDE with random coefficients via a Doss–Sussmann-type transformation, so that a stochastic viscosity solution can be defined in a “point-wise” manner. Using the recently developed theory on backward/backward doubly stochastic differential equations, we prove the existence of the stochastic viscosity solution, and further extend the nonlinear Feynman–Kac formula. Some properties of the stochastic viscosity solution will also be studied in this paper. The uniqueness of the stochastic viscosity solution will be addressed separately in Part II where the relation between the stochastic viscosity solution and the ω-wise, “deterministic” viscosity solution to the PDE with random coefficients will be established.     

Analysis of a Kind of Stochastic Dynamics Model with Nonlinear Function

In this paper, we establish stochastic differential equations on the basis of a nonlinear deterministic model and study the global dynamics. For the deterministic model, we show that the basic reproduction number $\Re _0$ determines whether there is an endemic outbreak or not: if $\Re _0< 1$, the disease dies out; while if $\Re _0> 1$, the disease persists. For the stochastic model, we provide analytic results regarding the stochastic boundedness, perturbation, permanence and extinction. Finally, some numerical examples are carried out to confirm the analytical results. One of the most interesting findings is that stochastic fluctuations introduced in our stochastic model can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics.     

Detecting high-dimensional determinism in time series with application to human movement data

We numerically investigate the ability of a statistic to detect determinism in time series generated by high-dimensional continuous chaotic systems. This recently introduced statistic (denoted V_E2) is derived from the averaged false nearest neighbors method for analyzing data. Using surrogate data tests, we show that the proposed statistic is able to discriminate high-dimensional chaotic data from their stochastic counterparts. By analyzing the effect of the length of the available data, we show that the proposed criterion is efficient for relatively short time series. Finally, we apply the method to real-world data from biomechanics, namely postural sway time series. In this case, the results led us to exclude the hypothesis of nonlinear deterministic underlying dynamics for the observed phenomena.     

Nonlinear analysis of RED––a comparative study

Random Early Detection (RED) is an active queue management (AQM) mechanism for routers on the Internet. In this paper, performance of RED and Adaptive RED are compared from the viewpoint of nonlinear dynamics. In particular, we reveal the relationship between the performance of the network and its nonlinear dynamical behavior. We measure the maximal Lyapunov exponent and Hurst parameter of the average queue length of RED and Adaptive RED, as well as the throughput and packet loss rate of the aggregate traffic on the bottleneck link. Our simulation scenarios include FTP flows and Web flows, one-way and two-way traffic. In most situations, Adaptive RED has smaller maximal Lyapunov exponents, lower Hurst parameters, higher throughput and lower packet loss rate than that of RED. This confirms that Adaptive RED has better performance than RED.     

A chaotic attractor in ecology: theory and experimental data

Chaos has now been documented in a laboratory population. In controlled laboratory experiments, cultures of flour beetles (Tribolium castaneum) undergo bifurcations in their dynamics as demographic parameters are manipulated. These bifurcations, including a specific route to chaos, are predicted by a well-validated deterministic model called the “LPA model”. The LPA model is based on the nonlinear interactions among the life cycle stages of the beetle (larva, pupa and adult). A stochastic version of the model accounts for the deviations of data from the deterministic model and provides the means for parameterization and rigorous statistical validation. The chaotic attractor of the deterministic LPA model and the stationary distribution of the stochastic LPA model describe the experimental data in phase space with striking accuracy. In addition, model-predicted temporal patterns on the attractor are observed in the data. This paper gives a brief account of the interdisciplinary effort that obtained these results.     

Stochastic flocking dynamics of the inertial spin model with state-dependent noises

We study stochastic flocking dynamics of the inertial spin (IS) model with state-dependent noises. The IS model was considered to describe the collective behaviors of starling flocks moving with constant speed. Unlike mechanical flocking models extensively studied in the literature, this model incorporates an internal dynamic observable, namely spin (internal angular momentum) in addition to mechanical observables (position and velocity), and it describes how spin interacts with mechanical observables. In previous works, emergent dynamics of the deterministic counterparts for the IS model and its mean-field limit have been investigated under some specific setting in which network topology is multiplicatively separable. In this work, we present sufficient frameworks for stochastic flocking dynamics of the IS model, which state-dependent noises vanish at the equilibria of the deterministic IS model. The proposed frameworks are in terms of coupling strength, friction, and inertial coefficients, and our asymptotic convergence results for sample paths are given in both an almost sure and an expectation sense. We have also conducted several numerical experiments to verify our analytical results and to explore what can be studied further in future work     

Pursuit on a cyclic graph — The symmetric stochastic case

In the cyclic pursuit game the evader, BLUE, and the pursuer, RED, choose one of the vertices of ann point cyclic graph at discrete time 1. If they initially choose the same vertex BLUE receives payoff one. At each subsequent time BLUE may remain where he is or move to an adjacent vertex. RED has the same capability. At no time do RED or BLUE know the other's location. The game ends when RED and BLUE arrive at the same vertex. BLUE then receives a payoff equal to the time of this arrival, i.e. the amount of time for which he eludes RED. In this paper we solve this game under the assumption that both RED and BLUE are restricted to stochastic strategies for which each moves right or left with equal probility.     

Dynamics of the Internet TCP–RED congestion control system

In this paper, the ideal case for the important congestion control algorithms, i.e., the TCP (transmission control protocol) algorithm and the RED (random early detection) algorithm, is analyzed, and the following results are found. First, mathematical analysis proves the existence of two equilibria of this dynamical system (of DDEs—delay differential equations), which has not been established in previous works. Second, reduction of the round-trip delay leads to the optimal design of the TCP–RED congestion control. Unfortunately, a drawback of TCP–RED is that package dropping and congestion are induced. The dynamics of the DDEs are considered for when congestion does not take place and the averaged queue length is between the minimum threshold and the maximum one. Stability and Hopf bifurcation of the DDEs are considered. We find that if the time delays are sufficiently large, Hopf bifurcation of the two equilibria will appear, and thus stationary motions with approximately constant rates of arrival, averaged queue length and oscillations with periodically varying forms will arise. Simulations illustrate the richness of the dynamics of the DDEs.     

Stochastic piecewise affine control with application to pitch control of helicopter

In this paper, at first the stability condition which gives an upper stochastic bound for a class of Stochastic Hybrid Systems (SHS) with deterministic jumps is derived. Here, additive noise signals are considered that do not vanish at equilibrium points. The presented theorem gives an upper bound for the second stochastic moment or variance of the system trajectories. Then, the linear case of SHS is investigated to show the application of the theorem. For the linear case of such stochastic hybrid systems, the stability criterion is obtained in terms of Linear Matrix Inequality (LMI) and an upper bound on state covariance is obtained for them. Then utilizing the stability theorem, an output feedback controller design procedure is proposed which requires the Bilinear Matrix Inequalities (BMI) to be solved. Next, the pitch dynamics of a helicopter is approximated with a set of linear stochastic systems, and the proposed controller is designed for the approximated model and implemented on the main nonlinear system to demonstrate the effectiveness of the proposed theorem and the control design method.     

Stationary analysis of the shortest queue problem

In this paper, we derive an approximation for throughput of TCP Compound connections under random losses. Throughput expressions for TCP Compound under a deterministic loss model exist in the literature. These are obtained assuming that the window sizes are continuous, i.e., a fluid behavior is assumed. We validate this model theoretically. We show that under the deterministic loss model, the TCP window evolution for TCP Compound is asymptotically periodic and is independent of the initial window size. We then consider the case when packets are lost randomly and independently of each other. We discuss Markov chain models to analyze performance of TCP in this scenario. We use insights from the deterministic loss model to get an appropriate scaling for the window size process and show that these scaled processes, indexed by p, the packet error rate, converge to a limit Markov chain process as p goes to 0. We show the existence and uniqueness of the stationary distribution for this limit process. Using the stationary distribution for the limit process, we obtain approximations for throughput, under random losses, for TCP Compound when packet error rates are small. We compare our results with ns2 simulations which show a good match and a better approximation than the fluid model at low p.     

Dimensionality reduction and greedy learning of convoluted stochastic dynamics

Complex natural systems may present interaction dynamics among random variables whose stochastic laws are in part or completely unknown. Statistical inference techniques applied to study such complex systems often require building suitable models that approximately describe the latent stochastic dynamics. When the observability of the variables of interest is limited by the convolution of such dynamics and noise, deconvolution techniques are needed either to estimate statistical characteristics or to decompose mixed signals. A good application field is offered by speculative financial market and their volatility stochastic dynamics. Typically, return generating stochastic processes show nonlinear, multiscale and non-stationary dynamics, especially when observed at very high frequencies. We explore the performance of computational techniques that combine the nonlinear approximation power of wavelets and associated structures with the ability of greedy learning algorithms to recover latent volatility structure by iteratively reducing the signal search space dimensionality across the most informative scales.     

Noise,nonlinear dynamics and the timecourse of affective disorders

Affective disorders, such as depression or mania, tend to be recurrent and progressive. Typically, disease patterns evolve from isolated episodes to rhythmic and finally accelerated “chaotic” mood patterns during the longitudinal course. Concepts from dynamical systems have been considered to explain this progression. However, most natural systems are not only nonlinear but also affected by noise. For this reason it seems important to incorporate cooperative stochastic–dynamic effects into current conceptional models for the course and neurobiology of such disorders. We use a computational perspective and describe behaviors of a simple mathematical model which result from interactions between random and deterministic dynamics. In particular, we focus on a scenario for illness progression that relies on noise enhancement of feedback instabilities. We suggest that noise amplification of subclinical neurobiological vulnerabilities could represent a relevant disease mechanism.     

A self-dual variational approach to stochastic partial differential equations

Unlike many of their deterministic counterparts, stochastic partial differential equations are not amenable to the methods of calculus of variations à la Euler–Lagrange. In this paper, we show how self-dual variational calculus leads to variational solutions of various stochastic partial differential equations driven by monotone vector fields. We construct solutions as minima of suitable non-negative and self-dual energy functionals on Itô spaces of stochastic processes. We show how a stochastic version of Bolza's duality leads to solutions for equations with additive noise. We then use a Hamiltonian formulation to construct solutions for non-linear equations with non-additive noise such as the stochastic Navier–Stokes equations in dimension two.     

Singular perturbations to semilinear stochastic heat equations

We consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof. The interesting feature of these perturbations is that, as the small parameter ε tends to zero, their solutions converge to the ‘wrong’ limit, i.e. they do not converge to the solution obtained by simply setting ε?=?0. A similar effect is also observed for some (formally) small stochastic perturbations of a deterministic semilinear parabolic PDE. Our proofs are based on a detailed analysis of the spatially rough component of the equations, combined with a judicious use of Gaussian concentration inequalities.     

Operator-theoretic solution of stochastic systems

A (stochastic) operator-theoretic approach leads to expresssions for inverses of linear and nonlinear stochastic operators—useful for the solution of linear or nonlinear stochastic differential equations. Operator equations are developed for inverses of linear or nonlinear stochastic operators. Series expressions are obtained which allow writing the solution y=$\text{F}$^?1x of the operator equation $\text{F}$y=x. Special cases are studied in which $\text{F}$ may be linear or nonlinear, deterministic or stochastic in various combinations.