Dynamical non-linear systems

According to Stuart A. Kauffman (1991) there is no generally accepted definition for the term complexity . However, there is consensus on certain properties of complex systems. One of these is deterministic chaos, which we have already mentioned. An ordered, non-linear dynamic system can undergo conversion to a chaotic state when slight, hardly noticeable perturbations act on it. Even very small differences in the initial conditions of complex systems can lead to great differences in the development of the system. Thus, the theory of complex systems no longer uses the well-known cause and effect principle. 

N. G. Rambidi and D. S. Chernavskii, Towards a biomolecular computer 2. Information processing and computing devices based on biochemical non-linear dynamic systems, J. Mol. Electron., 1, 115-125 (1991). 

Other approaches to genetic networks include study of small circuits with either differential equations or stochastic differential equations. The use of stochastic equations emphasizes the point that noise is a central factor in the dynamics. This is of conceptual importance as well as practical importance. In all the families of models studied, the non-linear dynamical systems typically exhibit a number of dynamical attractors. These are subregions of the system s state space to which the system flows and in which it thereafter remains. A plausible interpretation is that these attractors correspond to the cell types of the organism. However, in the presence of noise, attractors can be destabilized. 

As highlighted in Sidebox 6.1, Kauffman also stresses the link between biological systems and non-linear dynamic systems. This is a good introduction to the next section, which concerns emergence in some more complex biological systems. 

Bilinear approximation of general non-linear dynamic systems with linear inputs (with S. Svoronos and G.S. Stephanopoulos). Int. J. Corn. 31,109-126 (1980). 

In the modern theory of fluid dynamic systems the term turbulence is accepted to mean a state of spatiotemporal chaos (e.g., [155], chap 5). That is, the fluid exhibits chaos on all scales in both space and time. Chaos theory involves the behavior of non-linear dynamical systems and their response to initial and boundary conditions. Using such methods it can be shown that although the solution of the Navier-Stokes is apparently random for turbulent flows, its behavior presents some orderly structures. In addition, the numerical solution of the Navier-Stokes equations is sometimes strongly dependent on the initial conditions, thus even very small inaccuracies in the initial conditions may be fatal providing completely erroneous results. ... 

Johansen, T.A. and Foss, B.A. (1995). Semi-empirical modeling of non-linear dynamic systems through identification of operating regimes and locals models. In Neural Network Engineering in Control Systems, K Hunt, G Irwin and K Warwick, Eds., pp. 105-126, Springer-Verlag. 

A. Neiman, P. I. Saparin, and L. Sone. Coherence resonance at noisy precursors of bifurcations in non linear dynamical systems. Phys. Rev. E, 56 270, 1997. 

Parameter estimation problem of the presented non-linear dynamic system is stated as the minimization of the distance measure J between the experimental and the model predicted values of the considered state variables ... 

The systems discussed above belong to the class of non-linear dynamical systems. In such cases, the non-linear equation represents evolution of a solution with time or some variable like time. Such non-linear equations may be (i) algebraic, (ii) functional, (iii) ordinary partial differential equations and (iv) integral equations or a combination of these. Non-equilibrium systems can be defined by the type of equations as defined above involving the bifurcation parameter. The solution may change depending on the particular values of parameter. The solutions change at bifurcation points. Such situations do occur in the form of bistability and oscillations. 

Any chemical reaction mechanism may be considered as a system of n differential equations of the general form dy/dt = f(y. A) where y and / are vectors with n components. If /(y. A) represents a non-linear dynamical system, a large multitude of interesting phenomena may occur. 

Prom the standpoint of thermodynamics, the system electrolyte-film-electrode is open and far from equilibrium state. In this study we use the theoretical approach to the description of such systems created by H. Poincare and further developed later by Andronov and others. This method is called bifurcation analysis or, alternatively, theory of non-linear dynamic systems [7]. It has been applied to the studies of macrokinetics (dynamics) of the processes in electrode film systems. 

For certain types of non linear dynamical systems subject to stochastic loading, a method has been proposed based on the free-vibration response of the structure (Koo et al. 2005). In several cases, the output of this method will be a rough approximation of the design point which can be used as a starting point (a so-called warm solution) for a specific optimization algorithm. 

Socha, L. 1986. The sensitivity analysis of stochastic non-linear dynamical systems. Journal of... 

Wiggins, S. (1997) Introduction to Applied Non-linear Dynamical Systems and Chaos, Springer, New York. 

Sequential estimation of states and parameters in noisy, non-linear dynamical systems, Trans. ASME, J. Bas. Eng., 1966, 88, 362-368. 

Reprinted from de Bmin, H.A.E. and Roffel, B. (1996) A new identification method for fuzzy linear models of non-linear dynamic systems. Journal of Process Control, 6 (5), 277-93, with permission from Elsevier. 

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