Nonlinear system phenomenon

The chaotic behavior is an interesting nonlinear phenomenon which has been intensively studied during last two decades. The deterministic techniques have been used to understand the d3mamical structure in several nonlinear systems [5], [6], [42]. Particularly, the two-phase flow systems present nonlinear d mamical behavior which can be studied by means of chaos criteria behavior [2], [17], [25], [31], [45], [51]. Two-phase flows they provide a rich variety of cases whose dynamics lead to oscillatory patterns. The following published results are example ... 

Another phenomenon of highly nonlinear systems is parametric sensitivity. We illustrated this behavior for the temperature profile in the plug-flow reactor. Nonideal distillation systems can also show this sensitivity. For example, in Fig. 6.5 a small change in the feed composition or organic reflux flow can dramatically change the composition ( and t emperature) profile in the column. Instead of a vinyl acetate-rich profile in the top section, a water-rich profile can be present. 

As indicated in fig. 2.29, subharmonic entrainment at a fraction of the fundamental frequency, which is characteristic of nonlinear systems (Hayashi, 1964), occurs in smaller and smaller domains of the external period as the value of that fraction diminishes. A similar phenomenon is observed in the entrainment of circadian rhythms by light-dark cycles (Pittendrigh, 1960,1965) the latter property imderlies the adaptation of most living organisms to periodic variations in their environment. 

Because the assumption of homogeneity can be made more rigorously for well-stirred chemical reactors than it can for, say, interacting predator and prey populations in an ecological system, chemical systems have provided important experimental tests of the theoretically predicted features of homogeneous nonlinear systems. One of the universal phenomena to emerge from these studies is that of bistability, a particular case of the phenomenon of multiple steady states. 

A system in which the dependent variables are constant in time is said to be in a steady or stationary state. In a chemical system, the dependent variables are typically densities or concentrations of the component species. Two fundamentally different types of stationary states occur, depending on whether the system is open or closed. There is only one stationary state in a closed system, the state of thermodynamic equilibrium. Open systems often exhibit only one stationary state as well however, multistability may occur in systems with appropriate elements of feedback if they are sufficiently far from equilibrium. This phenomenon of multistability, that is, the existence of multiple steady states in which more than one such state may be simultaneously stable, is our first example of the universal phenomena that arise in dissipative nonlinear systems. 

As mentioned previously, bistability is an example of a universal phenomenon that arises in dissipative nonlinear systems. Its existence is largely independent of the identity of the interacting parts but strongly dependent on the type of... 

Attractive in its simplicity, yet complex in its behavior, the Continuous Stirred Tank Reactor has, for the better part of a century, presented the research community with a rich paradigm for nonlinear dynamics and complexity. The root of complex behavior in this system stems from the combination of its open system feature of maintaining a state far from equilibrium and the nonlinear non-monotonic feedback of various variables on the rate of reaction. Its behavior has been studied under various designs, chemistries and configurations and has exhibited almost every known nonlinear dynamics phenomenon. The polymerization chemistry has especially proven fruitful as concerns complex dynamics in a CSTR, as attested to by the numerous studies reviewed in this chapter. All indications are that this simple paradigm will continue to surprise us with many more complex discoveries to come. 

It is often better to avoid any manipulation of the system if the aim is to reduce its dimensions. In fact, it is advisable to leave the nonlinear system in its original form since it is derived from the modeling of a physical phenomenon. In doing so, there is greater certainty that the numerical system is well conditioned because it describes a real problem. 

It is vital to avoid making false dynamic characterizations of the phenomenon as there is no guarantee that the steady-state condition is a useful starting point for a nonlinear system solution. 

The kinetic behavior of eomplex chemical reactions may, in some temporal or parametric domains, be approximated by A B C, or by an evolution of sueh simple mechanisms first A B C, then C D E, and so on. The discovered patterns ean be used for recognizing submechanisms, and for estimating their parameter values and the evolution of the submechanisms. Generally, sueh properties of simple linear and nonlinear systems in fact reflect their unexpected eomplexily. Constales et al. (2013) have proposed to use a special term for defining this phenomenon simplexity. 

Nonlinear systems often exhibit a collective phenomenon called synchronization when coupled in some fashion or under the influence of an external field. The study of the behavior of large populations of nonlinear oscillators is a topic of central research since the late 1960s, following the pioneering work of Winfree [1]. The topic of synchronization in coupled systems, including chaotic systems, is interesting... 

The coupling of flow and thermal effects with a temperature-dependent viscosity can lead to situations in which more than one solution to the model equations is possible, suggesting that the process can exist in more than one state for a given set of flow conditions. This phenomenon of multiplicity is often a surprise when first encountered, but it is a common characteristic of nonlinear systems. In combustion, for example, it is known that there is a finite range of flow conditions for which the system can exist in either of two states a low-temperature, low-conversion state, and... 

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... 

L. Mandelstam and N. Papalexi performed an interesting experiment of this kind with an electrical oscillatory circuit. If one of the parameters (C or L) is made to oscillate with frequency 2/, the system becomes self-excited with frequency/ this is due to the fact that there are always small residual charges in the condenser, which are sufficient to produce the cumulative phenomenon of self-excitation. It was found that in the case of a linear oscillatory circuit the voltage builds up beyond any limit until the insulation is ultimately punctured if, however, the system is nonlinear, the amplitude reaches a stable stationary value and oscillation acquires a periodic character. In Section 6.23 these two cases are represented by the differential equations (6-126) and (6-127) and the explanation is given in terms of their integration by the stroboscopic method. 

When located at opposite ends (or at conjugated positions) in a molecular system, a donor and an acceptor do more than simply add up their separate effects. A cooperative phenomenon shows up, involving the entire disubstituted molecule, known as charge transfer (C.T.). Such compounds are colored (from pale yellow to red, absorption from 3,000 to 5,000 A) and show high U.V. absorption oscillator strength. "Figure 2 helps understand the enhancement of optical nonlinearity in such a system. 

A class of kick-excited self-adaptive dynamical systems is formed and proposed. The class is characterized by a nonlinear (inhomogeneous) external periodic excitation (as regards the coordinates of the excited system) and is remarkable for the occurrence of the following objective regularities the phenomenon of discrete oscillation excitation in macro-dynamical systems having multiple branch attractors and strong self-adaptive stability. 

The main goal of this report is to present a phenomenon of highly general nature manifested in various dynamical systems. We present the occurrence of peculiar quantization by the parameter of intensity of the excited oscillations and we show that given unchanging conditions it is possible to excite oscillations with a strictly defined discrete set of amplitudes the rest of the amplitudes being forbidden . The realization of oscillations with a specific amplitude from the permitted discrete set of amplitudes is determined by the initial conditions. The occurrence of this unusual property is predetermined by the new general initial conditions, i.e. the nonlinear action of the external excited force with respect to the coordinate of the system subjected to excitation. 

Oscillations have been observed in chemical as well as electrochemical systems [Frl, Fi3, Wol]. Such oscillatory phenomena usually originate from a multivariable system with extremely nonlinear kinetic relationships and complicated coupling mechanisms [Fr4], Current oscillations at silicon electrodes under potentio-static conditions in HF were already reported in one of the first electrochemical studies of silicon electrodes [Tul] and ascribed to the presence of a thin anodic silicon oxide film. In contrast to the case of anodic oxidation in HF-free electrolytes where the oscillations become damped after a few periods, the oscillations in aqueous HF can be stable over hours. Several groups have studied this phenomenon since this early work, and a common understanding of its basic origin has emerged, but details of the oscillation process are still controversial. 

In a linear system, the phenomenon of multiple steadystates cannot occur. It is the nonlinearity of the process, the exponential temperature dependence of the reaction rate, that can lead to more than one steadystate. 

We will explore this phenomenon quantitatively in the s plane. We will discuss linear systems in which instability means that the reactor temperature would theoretically go off to infinity. Actually, in any real system, reactor temperature will not go to infinity because the real system is nonlinear. The nonlinearity makes the reactor temperature climb to some high temperature at which it levels out. The concentration of reactant becomes so low that the reaction rate is hmited. 

Remark. From the linear integro-differential equation for P(y, t) we have derived a nonlinear equation for y(t). Thus the essentially linear master equation may well correspond to a physical process that in the laboratory would be regarded as a nonlinear phenomenon inasmuch as its macroscopic equation is nonlinear. This is not paradoxical provided one bears in mind that the distinction between linear and nonlinear is defined for equations. It is wrong to apply it to a physical phenonemon, unless one has agreed upon a specific mathematical description of it. Newton s equations for the motion of the planets are nonlinear, but the Liouville equation of the solar system is linear. This connection between linear and nonlinear equations is not a matter of approximation the linear Liouville equation is rigorously equivalent with the nonlinear equations of motion of the particles. Generally any linear partial... 

Amplification of Chirality. Perhaps the most striking of the nonclas-sical aspects that emerge from the enantioselective alkylation is the phenomenon illustrated in Scheme 22 (3, 14, 16, 20k, 40). A prominent nonlinear relation that allows for catalytic chiral amplification exists between the enantiomeric purity of the chiral auxiliary and the enantiomeric purity of the methylation or ethylation product (Scheme 23). Typically, when benzaldehyde and diethylzinc react in the presence of 8 mol % of (-)-DAIB of only 15% ee [(-) (+) = 57.5 42.5], the S ethylation product is obtained in 95% ee. This enantiomeric excess is close to that obtained with enantiomerically pure (—)-DAIB (98%). Evidently, chiral and achiral catalyst systems compete in the same reaction. The extent of the chiral amplification is influenced by many factors including the concentration of dialkylzincs, benzaldehyde, and chiral... 

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