Dynamical nonlinear systems

Hernandez, E., and Arkun, Y., A study of the control relevant properties of backpropagation neural net models of nonlinear dynamical systems. Comput. Chem. Eng. 16, 227 (1992). 

Analytical approaches applicable for small and large amplitudes (for weak and strong nonlinearity) of the oscillations in a nonlinear dynamic system subjected to the influence of a wave has been developed (Damgov, 2004 Damgov, Trenchev and Sheiretsky, 2003). 

Vol. 1483 E. Reithmeier, Periodic Solutions of Nonlinear Dynamical Systems. VI, 171 pages. 1991. 

The established tools of nonlinear dynamics provide an elaborate and versatile mathematical framework to examine the dynamic properties of metabolic systems. In this context, the metabolic balance equation (Eq. 5) constitutes a deterministic nonlinear dynamic system, amenable to systematic formal analysis. We are interested in the asymptotic, the linear stability of metabolic states, and transitions between different dynamic regimes (bifurcations). For a more detailed account, see also the monographs of Strogatz [290], Kaplan and Glass [18], as well as several related works on the topic [291 293],... 

T. Gross and U. Feudel, Generalized models as an universal approach to the analysis of nonlinear dynamical systems. Phys. Rev. E 73, 016205 (2006). 

Before focusing in the controller design, it is important to review some basic concepts of the geometric control theory. The control tools based in differential geometry are proposed for those nonlinear dynamical systems called affine systems. So, let s star by its definition. 

S. Wiggins. Introduction to Applied Nonlinear Dynamical System and Chaos. Springer, New York, 1990. 

In terms of nonlinear dynamical systems, the second waveguide of the junction can be considered as a system that is initially more or less far from its stable point. The global dynamics of the system is directly related to the spatial transfomation of the total field behind the plane of junction. In structure A, the initial linear mode transforms into a nonlinear mode of the waveguide with the same width and refractive index. In the structure B, the initial filed distribution corresponds to a nonlinear mode of the first waveguide it differs from nonlinear mode of the second waveguide, however. The dynamics in both cases is complicated and involves nonlinear modes as well as radiation. Global dynamics of a non-integrable system usually requires numerical simulations. For the junctions, the Cauchy problem also cannot be solved analytically. 

As it is well known, stationary solutions to Eq.(3.2) occur at the extrema of the Hamiltonian for a given power. The solutions that correspond to global or local minimum of H for a family of solitons are stable. The representation of the output nonlinear waveguide as a nonlinear dynamical system by the Hamiltonian allows to predict, to some extent, the dynamics of the total field behind the waveguide junction. 

The prototype potential surface invoked in chemical kinetics is a two-dimensional surface with a saddle equilibrium point and two exit channels at lower energies. The classical and quantal dynamics of such surfaces has been the object of many studies since the pioneering works by Wigner and Polanyi. Recent advances in nonlinear dynamical systems theory have provided powerful tools, such as the concepts of bifurcations and chaos, to investigate the classical dynamics from a new point of view and to perform the semiclassical... 

Noise in Nonlinear Dynamical Systems, Cambridge Univ. Press, Cambridge, UK, 1989, Vol. 3, pp. 119-158. 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

Whereas the operation of batch reactors is intrinsically unsteady, the continuous reactors, as any open system, allow for at least one reacting steady-state. Thus, the control problem consists in approaching the design steady-state with a proper startup procedure and in maintaining it, irrespective of the unavoidable changes in the operating conditions (typically, flow rate and composition of the feed streams) and/or of the possible failures of the control devices. When the reaction scheme is complex enough, the continuous reactors behave as a nonlinear dynamic system and show a complex dynamic behavior. In particular, the steady-state operation can be hindered by limit cycles, which can result in a marked decrease of the reactor performance. The analysis of the above problem is outside the purpose of the present text ... 

The perpendicular slice through the phase portrait provides the stroboscopic phase portrait or Poincare section (e.g. [3]). This is in the case of the harmonic oscillator one point in the phase portrait. A further powerful method for the analysis of nonlinear dynamical systems is the determination of the Fourier spectrum of the response function >2. 

Two typical properties of nonlinear dynamical systems are responsible for the realization of controlling chaos. Firstly, nonlinear systems show a sensitive dependence on initial conditions. This is represented in Table 14.1 by the nonlinear equation... 

The above scenario is typical of nonlinear dynamical systems when the amplitude of the internally generated oscillations becomes sufficiently large. In the bifurcation diagram of Fig. 12.5 this occurs when the slope of the feedback characteristics exceeds a critical value. However, similar scenarios can be produced through variation of other parameters such as, for instance, the damping of the arteriolar oscillator. 

Figure 13.4 illustrates three aspects of the basal ganglia network. First, a reciprocal control exists between GPe and STN. Second the SNc acts not only on the striatum but also on the cortex, the STN and the GPi. Finally the location of the STN at the intersection between vertical and horizontal feedback loops is crucial. Reference [50] concludes that the BG can no longer be considered as a unidirectional linear system that transfers information based solely on a firing-rate code and must rather be seen as a highly organized network with operational characteristics that simulate a nonlinear dynamical system (Fig. 13.4). 

The book targets graduate students and researchers interested in dynamics and control, as well as practitioners involved in advanced control in industry. It can serve as a reference text in an advanced process systems engineering or process control course and as a valuable resource for the researcher or practitioner. Written at a basic mathematical level (and largely self-contained from a mathematical point of view), the material assumes some familiarity with process modeling and an elementary background in nonlinear dynamical systems and control. 

Chechin G.M., Sakhnenko V.P. (1998) Interactions between normal modes in nonlinear dynamical systems with discrete symmetry. Exact results. PhysicaD. 117 43. 

It is my opinion that recent developments in the mathematical description of nonlinear dynamical systems have the potential for an enormous impact in the fields of fluid mechanics and transport phenomena. However, an attempt to assess this potential, based upon research accomplishments to date, is premature in any case, there are others better qualified than myself to undertake the task. Instead, I will offer a few general observations concerning the nature of the changes that may occur as the mathematical concepts of nonlinear dynamics become better known, better understood, more highly developed, and, lastly, applied to transport problems of interest to chemical engineers. 

A second, even more speculative point is that the mathematical framework of nonlinear dynamics may provide a basis to begin to bridge the gap between local microstructural features of a fluid flow or transport system and its overall meso- or macroscale behavior. On the one hand, a major failure of researchers and educators alike has been the inability to translate increasingly sophisticated fundamental studies to the larger-scale transport systems of traditional interest to chemical engineers. On the other hand, a basic result from theoretical studies of nonlinear dynamical systems is that there is often an intimate relationship between local solution structure and global behavior. Unfortunately, I am presently unable to improve upon the necessarily vague notion of a connection between these two apparently disparate statements. 

Guay M. and Zang T., Adaptive extremum seeking control of nonlinear dynamic systems with parametric uncertainty , Automatica 39 1283-1294, 2003. 

From a Bayesian interpretation, MHE and the extended Kalman filter assume normal or uniform distributions for the prior and the likelihood. Unfortunately, these assumptions are easily violated by nonlinear dynamic systems in which the conditional density is generally asymmetric, potentially multimodal and can vary significantly with time. 

Suppose that we have a nonlinear dynamical system, that is, first-order ordinary differential equations,... 

B. M. Deb, P. K. Chattaraj, in Solitons Introduction and Applications , Ed. M. Lakshmanan, Springer Verlag, Berlin, pp. 392-398, 1988 P. K. Chattaraj, in Symmetries and Singularity Structures Integrability and Chaos in Nonlinear Dynamical Systems , Eds. M. Lakshmanan, M. Daniel, Springer-Verlag, Berlin, pp. 172-182, 1990. 

On the theoretical physics side, the Kolmogorov-Arnold-Moser (KAM) theory for conservative dynamical systems describes how the continuous trajectories of a particle break up into a chaotic sea of randomly disconnected points. Furthermore, the strange attractors of dissipative dynamical systems have a fractal dimension in phase space. Both these developments in classical dynamics—KAM theory and strange attractors—emphasize the importance of nonanalytic functions in the description of the evolution of deterministic nonlinear dynamical systems. We do not discuss the details of such dynamical systems herein, but refer the reader to a number of excellent books on the... 

R Haber and H Unbehauen. Structure identification of nonlinear dynamic systems - A survey on input/output approaches. Automatica, 26 651-677, 1990. 

Significance of these models is that the complicated solutions are shown to exist even for simple nonlinear dynamic systems. Recently some of these models have been applied to explain the oscillations in experimental systems, e.g. Olsen and Degn (1977). 

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