Our efforts in this area include stochastic processes and nonlinear dynamics.
Complex processes in physics, chemistry, biology, engineering and finance, often exhibit uncertainties, or fluctuations, or noises, in their structures as a rule rather than as an exception. It is known that low dimension nonlinear dynamical systems can exert chaotic phenomena that appear random in nature. Nevertheless, stochastic processes involve an unmanageably large number of variables and their temporal fractal dimensions approach infinite. To interpret, analyze, model and simulate these noises, serval stochastic algorithms developed in the last century yield not only the structure or pattern of these processes, but also the statistical characteristics of the underlined uncertainties that the conventional deterministic models can not offer.
Algorithms have been developed for analyzing two distinct types of noises, internal and external noises. The internal noise is caused by the fact that the system itself consist of discrete particles and many variables associated with the particles are ignored; it is inherent in the very mechanism by which the process evolves. Small discrete systems governed by a large number of variables often exhibit notable internal fluctuations. The energy state of an electron in a molecule, the spread of an epidemic within a population, the mutation of a gene, and diffusion of a molecule into a medium can not be predicted precisely due to their inherent complexities, thus, these processes exert internal fluctuations. The discrete state, continuous time stochastic processes have been analyzed by the master-equation approach. The efficacy of this modern methodology of stochastic processes is mainly attributable to its ability to solve the nonlinear master equations through the system-size expansion. This renders it possible to circumvent the simplifying assumption of linearity imposed in the majority, if not all, of the past works. The modern methodology based on the formulation of the master equation of the phenomenon or process of concern has been rigorously established by Nicolis and Prigogine (1977), Hill (1977), Oppenheim et al. (1977), van Kampen (1992), and Risken (1996).
External noises are the fluctuations created in an otherwise deterministic system by the application of a random force, whose stochastic properties are supposed to be known. The displacement of a mass attached to a spring with a fluctuating external force, the animal populations under the influence of a fluctuating environment, the response of an electronic circuit with equipped with a noise generator, random loading of a bridge and the yield from a chemical reactor with fluctuating feed rate are examples of external noises. The Langevin approach is widely used for finding the effect of external noises in macroscopically known systems. Langevin equation can be either solved directly or transformed to a Fokker-Planck equation. Modern techniques for solving these equations, including eigenfunction expansion, variational methods, matrix continued-fraction method, simulation, and numerical integration, are under investigation. These methodologies have been rigorously established by Horsthemke and Lefever (1984), Gardiner (1985), Soize (1994), Risken (1996), and Grasman and Herwaarden (1999).
We have been exploring complex systems by various means discussed above. Emphases in the last decade have been on stochastic analysis and modeling of nonlinear processes governed by either internal or external noises. Various physical, biological and chemical processes have been modeled and simulated, including spread of infectious diseases, controlled drug release, tumor growth, adsorption, carbon oxidation, crystallization, aerosol dynamics, coal fragmentation, dynamics of molecular weight distribution of coal tar during pyrolysis, and bed-load dispersion in alluvial streams. We are currently applying this highly versatile algorithm in the analyses of a number of biological processes, including gene switching, epidemiology, and virus dynamics.
We have also adopted Monte Carlo procedures for the simulation of fluctuations governed by multi-step processes which are too complex to be approached by the algorithms discussed above. Linear or nonlinear dynamic processes have been simulated either deterministically or stochastically by Monte Carlo procedures. A well-developed class of Monte Carlo simulation procedures, the Metropolis method, essentially shares identical computational bases with the master-equation algorithm mentioned above. Specifically, the assumption of Markov, or semi-Markov for the time-dependent processes, property of the random variables leads to the definitions of transition-intensity functions (see, e.g., Gillespie, 1977; 1992). The probability balances of various events on the basis of these intensity functions lead to the master, or Fokker-Planck equations. In the Monte Carlo simulation, the system's state is simulated by a step-wise, random-walk scheme based on the same intensity functions as those established for the master-or Fokker-Planck-equation algorithm. In the near future, we plan to conduct Monte Carlo simulations for various nanocatalysis processes, particularly for those during biological and fossil fuel processing.
We were recently invited to contribute a review on Stochastic Processes in Encyclopedia on Nonlinear Science , Routledge, Francis-Taylor, London. We were also invited to give a plenary lecture on Stochastic Analysis of Nonlinear Biological and Ecological Processes and an oral lecture on tumor growth at a World Scientific and Engineering Academy and Society's International Conference on Mathematical Biology and Ecology at Corfu, Greece. The slides of my lectures at Corfu have been uploaded to:
http://www.worldses.org/plenary/index.html
In the area of nonlinear dynamics, we have recently completed a bifurcation analysis of HIV-1 dynamics in vivo . The work takes into account the effectiveness of the drugs and the populations of uninfected and productively infected macrophages. The dynamical system includes six equations for the populations of six species and four parameters: the effectiveness of protease and reverse transcriptase on each of T and M (macruphages) cells. In the near future, we plan to study the bifurcation phenomena of neuron dynamics through the FitzHugh-Nagumo equations and protein-regulated gene expressions.
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