Thoughts on modeling complexity

The simplest and probably the most familiar model of statistical processes in the physical sciences is the random walk. This simple model has been applied to all manner of phenomena, ranging from DNA sequences to the firing of neurons. Herein we extend the random walk model beyond that of mimicking simple statistics to include long‐time memory in the dynamics of complex phenomena. We show that complexity can give rise to fractional‐difference stochastic processes whose continuum limit is a fractional Langevin equation, that is, a fractional differential equation driven by random fluctuations. Furthermore, the index of the inverse power‐law spectrum in many complex processes can be related to the fractional derivative index in the fractional Langevin equation. This fractional stochastic model suggests that a scaling process guides the dynamics of many complex phenomena. The alternative to the fractional Langevin equation is a fractional diffusion equation describing the evolution of the probability density for certain kinds of anomalous diffusion. © 2006 Wiley Periodicals, Inc. Complexity 11: 33–43, 2006     

Dynamics in a ratio-dependent predator-prey model with predator harvesting

The objective of this paper is to study systematically the dynamical properties of a ratio-dependent predator-prey model with nonzero constant rate predator harvesting. It is shown that the model has at most two equilibria in the first quadrant and can exhibit numerous kinds of bifurcation phenomena, including the bifurcation of cusp type of codimension 2 (i.e., Bogdanov-Takens bifurcation), the subcritical and supercritical Hopf bifurcations. These results reveal far richer dynamics compared to the model with no harvesting and different dynamics compared to the model with nonzero constant rate prey harvesting in [D. Xiao, L. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM Appl. Math. 65 (2005) 737-753]. Biologically, it is shown that nonzero constant rate predator harvesting can prevent mutual extinction as a possible outcome of the predator prey interaction, and remove the singularity of the origin, which was regarded as “pathological behavior” for a ratio-dependent predator prey model in [P. Yodzis, Predator-prey theory and management of multispecies fisheries, Ecological Applications 4 (2004) 51-58].     

Complexity analysis of dynamic noncooperative game models for closed-loop supply chain with product recovery

In this paper, we consider a closed-loop supply chain (CLSC) with product recovery, which is composed of one manufacturer and one retailer. The retailer is in charge of recollecting and the manufacturer is responsible for product recovery. The system can be regarded as a coupling dynamics of the forward and reverse supply chain. Under different decision criteria, two noncooperative game models: Stackelberg game model and peer-to-peer game model are developed. The dynamic phenomena, such as the bifurcation, chaos and sensitivity to initial values are analyzed through bifurcation diagrams and the largest Lyapunov exponent (LLE). The influences of decision parameters on the complex nonlinear dynamics behaviors of the two models are further analyzed by comparing parameter basin plots, and the results show that with the improvement of retailer’s competitive position, the CLSC system will be more easier to enter into chaos.     

Quasi-steady state model determination for systems with singular perturbations modelled by bond graphs

A bond graph model for a singularly perturbed system is presented. This system is characterized by fast and slow dynamics. In addition, the bond graph can have storage elements with derivative and integral causality assignments for both dynamics. When the singular perturbation method is applied, the fast dynamic differential equation degenerates to an algebraic equation; the real roots of this equation can be determined by using another bond graph called singularly perturbed bond graph (SPBG). This SPBG has the characteristic that storage elements of the fast state and slow state have a derivative and integral causality assignment, respectively. Thus, a quasi-steady state model by using SPBG is obtained. A Lemma to get the junction structure from SPBG is proposed. Finally, the proposed methodology is applied to two examples.     

Perturbation analysis of entrainment in a micromechanical limit cycle oscillator

We study the dynamics of a thermo-mechanical model for a forced disc shaped, micromechanical limit cycle oscillator. The forcing can be accomplished either parametrically, by modulating the laser beam incident on the oscillator, or non-parametrically, using inertial driving. The system exhibits both 2:1 and 1:1 resonances, as well as quasiperiodic motions and hysteresis. A perturbation method is used to derive slow flow equations, which are then studied using the software packages AUTO and pplane7. Results show that the model agrees well with experiments. Details of the slow flow behavior explain how and where transitions into and out of entrainment occur.     

Emergence and Bifurcations of Lyapunov Manifolds in Nonlinear Wave Equations

Persistence and bifurcations of Lyapunov manifolds can be studied by a combination of averaging-normalization and numerical bifurcation methods. This can be extended to infinite-dimensional cases when using suitable averaging theorems. The theory is applied to the case of a parametrically excited wave equation. We find fast dynamics in a finite, resonant part of the spectrum and slow dynamics elsewhere. The resonant part corresponds with an almost-invariant manifold and displays bifurcations into a wide variety of phenomena among which are 2- and 3-tori.     

Slow foliation of a slow–fast stochastic evolutionary system

This work is concerned with the dynamics of a slow–fast stochastic evolutionary system quantified with a scale parameter. An invariant foliation decomposes the state space into geometric regions of different dynamical regimes, and thus helps understand dynamics. A slow invariant foliation is established for this system. It is shown that the slow foliation converges to a critical foliation (i.e., the scale parameter is zero) in probability distribution, as the scale parameter tends to zero. The approximation of slow foliation is also constructed with error estimate in distribution. Furthermore, the geometric structure of the slow foliation is investigated: every fiber of the slow foliation parallels each other, with the slow manifold as a special fiber. In fact, when an arbitrarily chosen point of a fiber falls in the slow manifold, the fiber must be the slow manifold itself.     

A model of an age-structured population in a multipatch environment

We propose a model of an age-structured population divided into N geographical patches. We distinguish two time scales, at the fast time scale we have the migration dynamics and at the slow time scale the demographic dynamics. The demographic process is described using the classical McKendrick-von Foerster model for each patch, and a simple matrix model including the transfer rates between patches depicts the migration process.Assuming that 0 is a simple strictly dominant eigenvalue for the migration matrix, we transform the model (an e.d.p. problem with N state variables) into a classical McKendrick-von Foerster model (scalar e.d.p. problem) for the global variable: total population density. We prove, under certain assumptions, that the semigroup associated to our problem has the property of positive asynchronous exponential growth and so we compare its asymptotic behaviour to that of the transformed scalar model. This type of study can be included in the so-called aggregation methods, where a large scale dynamical system is approximately described by a reduced system. Aggregation methods have been already developed for systems of ordinary differential equations and for discrete time models.An application of the results to the study of the dynamics of the Sole larvae is also provided.     

Aggregation and emergence in systems of ordinary differential equations

The aim of this article is to present aggregation methods for a system of ordinary differential equations (ODE's) involving two time scales. The system of ODE's is composed of the sum of fast parts and a perturbation. The fast dynamics are assumed to be conservative. The corresponding first integrals define a few global variables. Aggregation corresponds to the reduction of the dimension of the dynamical system which is replaced by an aggregated system governing the global variables at the slow time scale. The centre manifold theorem is used in order to get the slow reduced model as a Taylor expansion of a small parameter. We particularly look for the conditions necessary to get emerging properties in the aggregated model with respect to the nonaggregated one. We define two different types of emergences, functional and dynamical. Functional emergence corresponds to different functions for the two dynamics, aggregated and nonaggregated. Dynamical emergence means that both dynamics are qualitatively different. We also present averaging methods for aggregation when the fast system converges towards a stable limit cycle.     

On the controllability of some singularly perturbed nonlinear systems

The controllability of a large scale dynamic system which depends singularly upon a small parameter λ is considered. When λ = 0, the large scale system degenerates into a reduced order subsystem representing its slow dynamics while neglecting the fast phenomena. Another subsystem, often called a boundary layer system, represents the fast dynamics. In this paper sufficient conditions are established under which the controllability of the overall large scale system is inferred from the same property of the two subsystems.     

Time-scale analysis nonlocal diffusion systems,applied to disease models

The objective of the present paper is to use the well-known Ross–Macdonald models as a prototype, incorporating spatial movements, identifying different time scales and proving a singular perturbation result using a system of local and nonlocal diffusion. This results can be applied to the prototype model, where the vector has a fast dynamics, local in space, and the host has a slow dynamics, nonlocal in space.     

Approximate inertial manifolds of Burgers equation approached by nonlinear Galerkin’s procedure and its application

The approximate inertial manifolds (AIMs) of Burgers equation is approached by nonlinear Galerkin methods, and it can be used to capture and study the shock wave numerically in a reduced system with low dimension. Following inertial manifolds, the asymptotic behavior of Burgers equation, an infinite dimensional dissipative dynamic systems, will evolve to a compact set known as a global attractor, which is finite-dimensional, and the nonlinear phenomena are included and captured in such global attractor. In the application, nonlinear Galerkin methods is introduced to approach such inertial manifolds. By this method, the solution of the original system is projected onto the complete space spanned by the eigenfunctions or the modes of the linear operator of Burgers equation, and nonlinear Galerkin method splits the infinite-dimensional phase space into two complementary subspaces: a finite-dimensional one and its infinite-dimensional complement. Then, the post-processed Galerkin’s procedure is used to approximate the solution of the reduced system, with the introduction of the interaction between lower and higher modes. Additionally, some numerical examples are presented to make a comparison between the traditional Galerkin method and nonlinear Galerkin method, in particular, some sharp jumping phenomena, which are related to the shock wave, have been captured by the numerical method presented. As the conclusion, it can be drawn that it is possible to completely describe the dynamics on the attractor of a nonlinear partial differential equation (PDE) with a finite-dimensional dynamical system, and the study can provide a numerical method for the analysis of the nonlinear continuous dynamic systems and complicated nonlinear phenomena in finite-dimensional dynamic system, whose nonlinear dynamics has been developed completely compared with infinite-dimensional dynamic system.     

Causes for “ghost” manifolds

The paper concerns intrinsic low-dimensional manifold (ILDM) method suggested in [Maas U, Pope SB. Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space, combustion and flame 1992;88:239–64] for dimension reduction of models describing kinetic processes. It has been shown in a number of publications [Goldfarb I, Gol’dshtein V, Maas U. Comparative analysis of two asymptotic approaches based on integral manifolds. IMA J Appl Math 2004;69:353–74; Kaper HG, Kaper TJ, Asymptotic analysis of two reduction methods for systems of chemical reactions. Phys D 2002;165(1–2):66–93; Rhodes C, Morari M, Wiggins S. Identification of the low order manifolds: validating the algorithm of Maas and Pope. Chaos 1999;9(1):108–23] that the ILDM-method works successfully and the intrinsic low-dimensional manifolds belong to a small vicinity of invariant slow manifolds. The ILDM-method has a number of disadvantages. One of them is appearance of so-called “ghost”-manifolds, which do not have connection to the system dynamics [Borok S, Goldfarb I, Gol’dshtein V. “Ghost” ILDM – manifolds and their discrimination. In: Twentieth Annual Symposium of the Israel Section of the Combustion Institute, Beer-Sheva, Israel; 2004. p. 55–7; Borok S, Goldfarb I, Gol’dshtein V. About non-coincidence of invariant manifolds and intrinsic low-dimensional manifolds (ILDM). CNSNS 2008;71:1029–38; Borok S, Goldfarb I, Gol’dshtein V, Maas U. In: Gorban AN, Kazantzis N, Kevrekidis YG, Ottinger HC, Theodoropoulos C, editors. “Ghost” ILDM-manifolds and their identification: model reduction and coarse-graining approaches for multiscale phenomena. Berlin–Heidelberg–New York: Springer; 2006. p. 55–80; Borok S, Goldfarb I, Gol’dshtein V. On a modified version of ILDM method and its asymptotic analysis. IJPAM 2008; 44(1): 125–50; Bykov V, Goldfarb I, Gol’dshtein V, Maas U. On a modified version of ILDM approach: asymptotic analysis based on integral manifolds. IMA J Appl Math 2006;71:359–82; Flockerzi D. Tutorial: intrinsic low-dimensional manifolds and slow attractors. Magdeburg: Max-Planck-Institut; 2001–2005. <www.mpi-magdeburg.mpg.de/people/flocke/tutorial-ildm.pdf>; Flockerzi D, Heineken W. Comment on “Identification of low order manifolds: validating the algorithm of Maas and Pope”. Chaos 1999;9:108–23; Flockerzi D, Heineken W. Comment on “Identification of low order manifolds: validating the algorithm of Maas and Pope”. Chaos 2006;16:048101]. The present work studies the causes for the “ghost” manifolds appearance for the case of a two-dimensional singularly perturbed system.     

Contracting over multiple parameters: Capacity allocation in semiconductor manufacturing

This paper is a generalization of Mallik and Harker [Mallik, S., Harker, P.T., 2004. Coordinating supply chains with competition: Capacity allocation in semiconductor manufacturing. European Journal of Operational Research 159, 330–347] that presented an integrated model of incentive problems arising in forecasting and capacity allocation. In that model, multiple product managers and multiple manufacturing managers forecast the means of their respective demand and capacity distributions, and a central coordinator allocates capacities based on these forecasts. A mechanism that elicits truthful information from the managers was the main contribution of that paper. The objective of this paper is to generalize our previous results to multiple statistics reporting. This work assumes that the central coordinator can ask the managers to report multiple statistics (mean and variance, for example) about their respective distributions. We propose a game theoretic model and design a mechanism (a bonus scheme and an allocation rule) that elicits truthful reporting of all statistics by all managers. It turns out that the structure of the optimal bonus schemes are rather simple with easily calculable parameters. We also show that a large class of allocation rules are manipulable. A bonus is often required for elicitation of truthful information. We compare our results of multiple statistics reporting with those from Mallik and Harker (2004). We also characterize under what conditions the reporting of the extra information is of limited use.     

Predictive control of uncertain nonlinear parabolic PDE systems using a Galerkin/neural-network-based model

In this paper, a model predictive control (MPC) scheme for a class of parabolic partial differential equation (PDE) systems with unknown nonlinearities, arising in the context of transport-reaction processes, is proposed. A spatial operator of a parabolic PDE system is characterized by a spectrum that can be partitioned into a finite slow and an infinite fast complement. In this view, first, Galerkin method is used to derive a set of finite dimensional slow ordinary differential equation (ODE) system that captures the dominant dynamics of the initial PDE system. Then, a Multilayer Neural Network (MNN) is employed to parameterize the unknown nonlinearities in the resulting finite dimensional ODE model. Finally, a Galerkin/neural-network-based ODE model is used to predict future states in the MPC algorithm. The proposed controller is applied to stabilize an unstable steady-state of the temperature profile of a catalytic rod subject to input and state constraints.     

Non-parametric estimation of lifetime and repair time criteria for a semi-Markov process

In this Note, we model an industrial system by a semi-Markov process where failure and repair phenomena are in mutual competition. A non-parametric estimation method for system component lifetime and repair time distributions and for associated hazard rate functions is proposed. The lifetime and repair time empirical distributions are reduced to two Kaplan–Meier estimators. A numerical example from an industrial system with three components and one repair man modeled by a birth and death process is provided to illustrate the previous results. To cite this article: A.-L. Afchain, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Functional integral method for investigating the dynamics of a Dicke type model

Conclusions The approach to the investigation of nonequilibrium systems proposed in this paper has made it possible to treat fairly readily the dynamics of a Dicke type model in the thermodynamic limit. A number of other model systems can be investigated similarly. The results obtained from the expression for the generating functional of the temperature-time Green's functions generalize the corresponding result of [4]. Exact study of the dynamical behavior of the Dicke type model at large times has shown that it possesses different nonlinear dynamical regimes, the conditions for the existence of which depend strongly on the initial values of the parameters of the system and the statistics of the fields. The point of bifurcation of the dynamics is an analog of the critical temperature for an equilibrium phase transition.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 69, No. 2, pp. 279–286, November, 1987.     

TheH model of critical dynamics: A proof of the multiplicative renormalizability of the simplifiedH 0 model

For the H₀ model of critical dynamics, which is obtained from the usual H model by omitting the term with the velocity derivative ∂_tv, renormalization multiplicity is proved, and its connection with the statistics is shown. In the proof, a universal general scheme is formulated that can be used to prove similar statements for any model of critical dynamics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 3, pp. 385–399, March, 2000     

A novel nonlinear viscoelastic solid model

The standard linear solid (SLS) model has been widely used to describe linear viscoelastic materials. Although the SLS can be extended to describe more complex phenomena through the inclusion of additional components, a compact model of nonlinear viscoelasticity remains elusive. Here, using the framework of the three component SLS model, the stress-strain relationships of the individual components have been generalised. This work describes the derivation of the new generalised model, termed nonlinear viscoelastic solid model, or NVS, that can potentially describe nonlinear viscoelastic phenomena in a compact differential form and reduces to the familiar SLS model if linear components are selected. As a proof of concept, exponential functions of strain and strain rate were selected for the three components and major viscoelastic phenomena such as stress relaxation, creep and sawtooth strain loading were simulated. Finally, to demonstrate its efficacy in describing biological tissue, the NVS model was used to simulate the cyclic loading of mammalian stomach and cardiac muscle tissues.     

A statistical study of the dynamics of the voltage membrane in protozoa Euplotes vannus

We consider the dynamics of the membrane voltage of Euplotes vannus and we suppose that behind this behaviour there is a system that generates Stochastic Resonance phenomena (Benzi R, Sutera A, Vulpiani A. J Phys A 1981;14:L453). We observe the statistical distributions of the resident times in the polarised and depolarised states. The experimental data are compared with the results obtained by an analogue device that simulates a model that gives an Autonomous Stochastic Resonance Process.