Nonlinear dynamic systems

Dissipation as an organizing factor can only occur under certain conditions, necessary prerequisites are energy input (open system), nonlinear dynamics and stabilization far away from thermal equilibrium i.e. beyond the regime of linear irreversible thermodynamics (vid. the requirements of Chapter I). 

The multiple convolutions of the Volterra model involve kernel functions fc,(mi,..., m,) which constitute the descriptors of the system nonlinear dynamics. Consequently, the system identification task is to obtain estimates of these kernels from input-output data. These kernel functions are symmetric with respect to their arguments. 

For modelling conformational transitions and nonlinear dynamics of NA a phenomenological approach is often used. This allows one not just to describe a phenomenon but also to understand the relationships between the basic physical properties of the system. There is a general algorithm for modelling in the frame of the phenomenological approach determine the dominant motions of the system in the time interval of the process treated and theti write... 

In order to treat crystallization systems both dynamically and continuously, a mathematical model has been developed which can correlate the nucleation rate to the level of supersaturation and/or the growth rate. Because the growth rate is more easily determined and because nucleation is sharply nonlinear in the regions normally encountered in industrial crystallization, it has been common to... 

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). 

In the operation of BWRs, especially when operating near the threshold of instability, the stability margin of the stable system and the amplitude of the limit cycle under unstable condition become of importance. A number of nonlinear dynamic studies of BWRs have been reported, notably in an International Workshop on Boiling Water Reactor Stability (1990). The following references are mentioned for further study. 

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). 

Formally the unperturbed Hamiltonian is equivalent to the Hamiltonian of the hydrogen atom in constant homogenious electric field. Chaotic dynamics of hydrogen atom in constant electric field under the influence of time-periodic field was treated earlier (Berman et. al, 1985 Stevens and Sundaraml987). To treat nonlinear dynamics of this system under the influence of periodic perturbations we need to rewrite (1) in action-angle variables. Action can be found using its standard definition ... 

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

Recently there has been an increasing interest in self-oscillatory phenomena and also in formation of spatio-temporal structure, accompanied by the rapid development of theory concerning dynamics of such systems under nonlinear, nonequilibrium conditions. The discovery of model chemical reactions to produce self-oscillations and spatio-temporal structures has accelerated the studies on nonlinear dynamics in chemistry. The Belousov-Zhabotinskii(B-Z) reaction is the most famous among such types of oscillatory chemical reactions, and has been studied most frequently during the past couple of decades [1,2]. The B-Z reaction has attracted much interest from scientists with various discipline, because in this reaction, the rhythmic change between oxidation and reduction states can be easily observed in a test tube. As the reproducibility of the amplitude, period and some other experimental measures is rather high under a found condition, the mechanism of the B-Z reaction has been almost fully understood until now. The most important step in the induction of oscillations is the existence of auto-catalytic process in the reaction network. 

If the most recent available measurements are at time step c, then a history horizon HAt can be defined from (tc — HAt) to tc, where At is the time step size. In order to obtain enough redundant information about the process, it is important to choose a horizon length appropriate to the dynamic of the specific system (Liebman et al., 1992). As shown in Fig. 5, only data measurements within the horizon will be reconciled during the nonlinear dynamic data reconciliation run. 

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). 

It is not possible to discuss highly excited states of molecules without reference to the recent progress in nonlinear dynamics.2 Indeed, the stimulation is mutual. Rovibrational spectra of polyatomic molecules provides both an ideal testing ground for the recent ideas on the manifestation of chaos in Hamiltonian systems and in turn provides many challenges for the theory. 

The notion of chaos is interwoven with the discussion of time evolution, which we do not pursue in this volume. It is worthwhile, however, to note that it is, by now, well understood that a quantum-mechanical system with a finite Hamiltonian matrix cannot satisfy many of the purely mathematical characterizations of chaos. Equally, however, over long periods of time such systems can manifest many of the qualitative features that one associates with classically chaotic systems. It is not our intention to follow this most interesting theme. Instead we seek a more modest aim, namely, to forge a link between the elementary notions of classical nonlinear dynamics and the algebraic approach. This turns out to be possible using the action-angle variables of classical mechanics. In this section we consider only the nonlinear dynamics aspects. We complete the bridge in Chapter 7. 

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. 

A. Isidori, A.R. Teel, and L. Praly. A note on the problem of semiglobal practical stabilization of uncertain nonlinear systems via dynamic ontpnt feedback. Systems Control Lett., 39 165-171, 2000. 

H. Nijmeijer and A.J. van der Schaft. Nonlinear Dynamical Control Systems. Springer Verlag, 1991. 

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. 

MSN. 165.1. Prigogine and D. Driebe, Time, chaos and the laws of nature, in Nonlinear Dynamics, Chaotic and Complex Systems, E. Infeld, R. Zelazny, and A. Galkowski, eds., Cambridge University Press, Cambridge, 1997, pp. 206-223. 

In this section we consider a model of interactions between the Kerr oscillators applied by J. Fiurasek et al. [139] and Perinova and Karska [140]. Each Kerr oscillator is externally pumped and damped. If the Kerr nonlinearity is turned off, the system is linear. This enables us to perform a simple comparison of the linear and nonlinear dynamics of the system, and we have found a specific nonlinear version of linear filtering. We study numerically the possibility of synchronization of chaotic signals generated by the Kerr oscillators by employing different feedback methods. 

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... 

We focus on the nonlinear dynamics for the collinear configuration which we treat as part of the full multidimensional system. This is to be contrasted with two-degree-of-freedom models where the molecule is assumed to be frozen in some angular configuration, such that the bending degree of freedom is excluded from the dynamics. In our analysis, bending is taken into account in terms of linearized dynamics, which allows us to extend the results for the collinear situation to the full three-dimensional system. The restriction we must be aware of is that the three-dimensional system may have periodic orbits that are not of collinear type. 

See, for example, OSA Proceedings on Nonlinear Dynamics in Optical Systems, N. B. Abraham, E. Garmire, and P. Mandel (Eds.), Optical Society of America, Washington, DC, 1991 and subsequent volumes in this proceedings series. 

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

R. G. Harrison, J. S. Uppal, and P. Osborne, (Eds.), Nonlinear Dynamics and Spatial Complexity in Optical Systems, SUSSP/Institute of Physics, Bristol, 1993. 

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