Chaos control infinity series#

The Karhunen-Loeve dimension of the turbulent flow is found to vary from 49 at Rayleigh number Ra = 4 10 to 127 at Ra = 10. The nonlinear dynamics of the turbulent convection is studied using tools of Dynamical Systems Theory. « lessĭirect numerical simulation of two dimensional turbulent thermal convection of an incompressible fluid (Prandtl number = 0.71) in a laterally heated rigid box has been carried out for a range of Rayleigh numbers. The presentation includes the necessary background on the Wiener chaos and optimal nonlinear filtering. The more » paper summarizes the main existing results about these Wiener chaos algorithms and resolves some open questions concerning the convergence of the algorithms in the noise-correlated setting. For real-time applications, computing the coefficients in advance improves the performance of the algorithms in comparison with most other existing methods of nonlinear filtering. The coefficients in the expansion depend only on the deterministic dynamics of the state and observation processes. Both algorithms rely on the Cameron-Martin version of the Wiener chaos expansion, so that the approximate filter is a finite linear combination of the chaos elements generated by the observation process. The paper discusses two algorithms for solving the Zakai equation in the time-homogeneous diffusion filtering model with possible correlation between the state process and the observation noise. Our question is whether the data`s deviations from an annual cyclic trend (which would reflect global nonlinearities) are described by a linear, noise-driven stochastic process. When we ask ``are epidemics nonlinear?``, we are not questioning the existence of global nonlinearities in epidemic dynamics, such as nonlinear transmission rates. We also identify features of epidemic data that create problems for the older, better known methods of detecting chaos.

Our goals here are to outline some newer methods for detecting nonlinearity and chaos that have a solid statistical basis and are suited to epidemic data, and to begin a re-evaluation of the claims for nonlinear dynamics and chaos in epidemics using these newer methods. noise in any data set have given way to a recognition that a variety of methods must be used, and interpretation of results must take into account the limitations of each method and the imperfections of the data.

Early hopes of a ``one size fits all`` algorithm to diagnose chaos vs.

The initial enthusiasm for methods of detecting chaos in data has been followed by critical re-evaluations of their limitations. The proposition that the precise timing and magnitude of more » epidemic outbreaks are deterministic but chaotic is appealing, since it raises the hope of finding determinism and simplicity beneath the apparently stochastic and complicated surface of the data.

Chaos control infinity series#

Using attractor reconstruction, estimates of dynamical invariants, and comparisons between data and simulation of SEIR models, the ``case for chaos in childhood epidemics`` was made through a series of influential papers beginning in the mid 1980`s. Schaffer and Kot who first recognized that the abundance and accuracy of disease incidence data opened the door to applying a range of methods for detecting chaos that had been devised in the early 1980`s.

Historical data on recurrent epidemics have been central to the debate about the prevalence of chaos in biological population dynamics.