Dynamic behavior, attractors

Attractor states arc tyirically stable with respect to 80 - 95% of the possible minimal perturbations, and mutations (by which sites are deleted or Boolean functions of single sites are altered) affect the dynamical behavior only slightly. 

Controlled chaos may also factor into the generation of rhythmic behavior in living systems. A recently proposed modeL describes the central circadian oscillator as a chaotic attractor. Limit cycle mechanisms have been previously offered to explain circadian clocks and related phenomena, but they are limited to a single stable periodic behavior. In contrast, a chaotic attractor can generate rich dynamic behavior. Attractive features of such a model include versatility of period selection as well as use of control elements of the type already well known for metabolic circuitry. 

Phase plane plots are not the best means of investigating these complex dynamics, for in such cases (which are at least 3-dimensional) the three (or more) dimensional phase planes can be quite complex as shown in Figures 16 and 17 (A-2) for two of the best known attractors, the Lorenz strange attractor [80] and the Rossler strange attractor [82, 83]. Instead stroboscopic maps for forced systems (nonautonomous) and Poincare maps for autonomous systems are better suited for investigating these types of complex dynamic behavior. 

It is necessary to emphasize one principal peculiarity of the copolymerization dynamics which arises under the transition from the three-component to the four-component systems. While the attractors of the former systems are only SPs and limit cycles (see Fig. 5), for the latter ones we can also expect the realization of other more complex attractors [202]. Two-dimensional surfaces of torus on which the system accomplishes the complex oscillations (which are superpositions of the two simple oscillations with different periods) ate regarded to be trivial examples of such attractors. Other similar attractors are fitted by the superpositions of few simple oscillations, the number of which is arbitrary. And, finally, the most complicated type of dynamic behavior of the system when m 4 is fitted by chaotic oscillations [16], for which a so-called strange attractor is believed to be a mathematical image [206]. 

Q.3.11 List three types of attractors that describe dynamic behavior. 

Our initial studies of dynamics in biochemical networks included spatially localized components [32]. As a consequence, there will be delays involved in the transport between the nuclear and cytoplasmic compartments. Depending on the spatial structure, different dynamical behaviors could be faciliated, but the theoretical methods are useful to help understand the qualitative features. In other (unpublished) work, computations were carried out in feedback loops with cyclic attractors in which a delay was introduced in one of the interactions. Although the delay led to an increase of the period, the patterns of oscillation remained the same. However, delays in differential equations that model neural networks and biological control systems can introduce novel dynamics that are not present without a delay (for example, see Refs. 57 and 58). 

Whichever qualitative features are examined, the apparently rich chemical literature reduces to a few simple classes. It is difficult to judge the reason why so few dynamic classes have appeared so far. Perhaps there are certain (still unanalyzed) features of kinetic equations that lead to simple dynamics. Another possibility is that chemists have tended to study only a small subclass of chemical kinetics and have ignored the rest. For example, a type of dynamical behavior that it is hard to imagine not existing is one in which there are two stable attractors. A limit cycle oscillation and a stable steady state where transitions can occur between the two as a result of large perturbations of concentrations. An example has been previously given in which this type of... 

This model with only three variables, whose only nonlinearities are xy and xz, exhibited dynamic behavior of unexpected complexity (Fig. 7.2). It was especially surprising that this deterministic model was able to generate chaotic oscillations. The corresponding limit set was called the Lorenz attractor and limit sets of similar type are called strange attractors. Trajectories within a strange attractor appear to hop around randomly but, in fact, are organized by a very complex type of stable order, which keeps the system within certain ranges. 

The variety of dynamical behavior observed in this apparently simple system is truly remarkable. As r is increased, there are Hopf bifurcations to simple periodic behavior, then period-doubling sequences to chaos, as well as several types of multistability. One of the most interesting findings, shown in Figure 10.7, is the first experimental observation (in a chemical system) of crisis (Grebogi et al., 1982), the sudden expansion of the size of a chaotic attractor as a parameter (in this case, r) is varied. Simulations with the delayed feedback of eq. (10.48) taken into account give qualitative agreement with the bifurcation sequences observed in the experiments. 

Another important property of an attractor is its dimension. It is, loosely speaking, the number of independent degrees of freedom relevant to the dynamical behavior. There are several different definitions that differ mainly in the measure used [35, 39] these definitions all yield the Euclidean value of dimension for Euclidean objects, but for strange attractors the dimension is in general fractional ("fractal [40]). As an example of computing the dimension from experimental data, we will describe a procedure for computing the information dimension, d [41]. Let N(e) be the number of points in a ball of radius e about a point x on an attractor. For a uniform density of points one would have... 

The strange attractors have small fractional dimension ( 2). An unexpected bonus of the determinations of the dimension is that the procedure separates the deterministic dynamical behavior from the stochastic noise (see Section 2.4). Thus the procedure provides an estimate of the experimental noise arising from incomplete mixing, fluctuations in stirring rate, etc., as will be described elsewhere [44]. 

Figures 3.38 and 3.39 show typical space-time patterns generated by a few r = 1 reversible rules starting from both simple and disordered initial states. Although analogs of the four generic classes of behavior may be discerned, there are important dynamical differences. The most important difference being the absence of attractors, since there can never be a merging of trajectories in a reversible system for finite lattices this means that the state transition graph must consist exclusively of cyclic states. We make a few general observations.

The typical strategy employed in studying the behavior of nonlinear dissipative dynamical systems consists of first identifying all of the periodic solutions of the system, followed by a detailed characterization of the chaotic motion on the attractors. 

Essentially, MLE is a measure on time-evolution of the distance between orbits in an attractor. When the dynamics are chaotic, a positive MLE occurs which quantifies the rate of separation of neighboring (initial) states and give the period of time where predictions are possible. Due to the uncertain nature of experimental data, positive MLE is not sufficient to conclude the existence of chaotic behavior in experimental systems. However, it can be seen as a good evidence. In [50] an algorithm to compute the MLE form time series was proposed. Many authors have made improvements to the Wolf et al. s algorithm (see for instance [38]). However, in this work we use the original algorithm to compute the MLE values. 

Let us now consider the behavior of the system when the Kerr coupling constant is switched on (e12 / 0). For brevity and clarity, we restrict our discussion to the question of how the attractors in Fig. 20 change when both oscillators interact with each other. To answer this question, let us have a look at the joint auto-nomized spectrum of Lyapunov exponents for the two oscillators A,j, A,2, L3, A-4, L5 versus the interaction parameter 0 < ( 2 < 0.7. The spectrum is seen in Fig. 32 and describes the dynamical properties of our oscillators in a global sense. The dynamics of individual oscillators can be glimpsed at the appropriate phase portraits. Let us now fix our attention on a detailed analysis of Fig. 32. For the limit value ei2 = 0, the dynamics of the uncoupled oscillators has already been presented in Fig. 20. In the case of very weak interaction 0 < C 2 < 0.0005, the system of coupled oscillators manifests chaotic behavior. For C 2 = 0.0005 we obtain the spectrum 0.06,0.00, —0.21, 0.54, 0.89. It is interesting to... 

The profile and phase plots show oscillatory behavior of Figure 4.56, but no pattern or periods can be seen therein. This is called a strange attractor in modern nonlinear dynamics theory. A strange attractor can be chaotic or nonchaotic (high-dimensional torus). Differentiating between chaotic and nonchaotic strange attractors is beyond the scope of this undergraduate book. 

In this fermentation process, sustained oscillations have been reported frequently in experimental fermentors and several mathematical models have been proposed. Our approach in this section shows the rich static and dynamic bifurcation behavior of fermentation systems by solving and analyzing the corresponding nonlinear mathematical models. The results of this section show that these oscillations can be complex leading to chaotic behavior and that the periodic and chaotic attractors of the system can be exploited for increasing the yield and productivity of ethanol. The readers are advised to investigate the system further. 

This phenomenon of increased conversion, yield and productivity through deliberate unsteady-state operation of a fermentor has been known for some time. Deliberate unsteady-state operation is associated with nonautonomous or externally forced systems. The unsteady-state operation of the system (periodic operation) is an intrinsic characteristic of this system in certain regions of the parameters. Moreover, this system shows not only periodic attractors but also chaotic attractors. This static and dynamic bifurcation and chaotic behavior is due to the nonlinear coupling of the system which causes all of these phenomena. And this in turn gives us the ability to achieve higher conversion, yield and productivity rates. 

There are many other interesting and complex dynamic phenomena besides oscillation and chaos which have been observed but not followed in depth both theoretically and experimentally. One example is the wrong directional behavior of catalytic fixed-bed reactors, for which the dynamic response to input disturbances is opposite of that suggested by the steady-state response [99, 100], This behavior is most probably connected to the instability problems in these catalytic reactors as shown crudely by Elnashaie and Cresswell [99]. Recently Elnashaie and co-workers [102-105] have also shown rich bifurcation and chaotic behavior of an anaerobic fermentor for producing ethanol. They have shown that the periodic and chaotic attractors may give higher ethanol yield and productivity than the optimal steady states. These results have been confirmed experimentally [105],... 

The trajectories of dissipative dynamic systems, in the long run, are confined in a subset of the phase space, which is called an attractor [32], i.e., the set of points in phase space where the trajectories converge. An attractor is usually an object of lower dimension than the entire phase space (a point, a circle, a torus, etc.). For example, a multidimensional phase space may have a point attractor (dimension 0), which means that the asymptotic behavior of the system is an equilibrium point, or a limit cycle (dimension 1), which corresponds to periodic behavior, i.e., an oscillation. Schematic representations for the point, the limit cycle, and the torus attractors, are depicted in Figure 3.2. The point attractor is pictured on the left regardless of the initial conditions, the system ends up in the same equilibrium point. In the middle, a limit cycle is shown the system always ends up doing a specific oscillation. The torus attractor on the right is the 2-dimensional equivalent of a circle. In fact, a circle can be called a 1-torus,... 

A dynamic system may exhibit qualitatively different behavior for different values of its control parameters 0. Thus, a system that has a point attractor for some value of a parameter may oscillate (limit cycle) for some other value. The critical value where the behavior changes is called a bifurcation point, and the event a bifurcation [32]. More specifically, this kind of bifurcation, i.e., the transition from a point attractor to a limit cycle, is referred to as Hopf bifurcation. 

Pharmacokinetic studies are in general less variable than pharmacodynamic studies. This is so since simpler dynamics are associated with pharmacokinetic processes. According to van Rossum and de Bie [234], the phase space of a pharmacokinetic system is dominated by a point attractor since the drug leaves the body, i.e., the plasma drug concentration tends to zero. Even when the system is as simple as that, tools from dynamic systems theory are still useful. When a system has only one variable a plot referred to as a phase plane can be used to study its behavior. The phase plane is constructed by plotting the variable against its derivative. The most classical, quoted even in textbooks, phase plane is the c (f) vs. c (t) plot of the ubiquitous Michaelis-Menten kinetics. In the pharmaceutical literature the phase plane plot has been used by Dokoumetzidis and Macheras [235] for the discernment of absorption kinetics, Figure 6.21. The same type of plot has been used for the estimation of the elimination rate constant [236]. 

Although the detailed features of the interactions involved in cortisol secretion are still unknown, some observations indicate that the irregular behavior of cortisol levels originates from the underlying dynamics of the hypothalamic-pituitary-adrenal process. Indeed, Ilias et al. [514], using time series analysis, have shown that the reconstructed phase space of cortisol concentrations of healthy individuals has an attractor of fractal dimension dj = 2.65 0.03. This value indicates that at least three state variables control cortisol secretion [515]. A nonlinear model of cortisol secretion with three state variables that takes into account the simultaneous changes of adrenocorticotropic hormone and corticotropin-releasing hormone has been proposed [516]. 

The given sustained thermodynamic solution can be considered as the appearance of a temporary dissipative system. Emphasize that Y and Z always remain localized around point Y,Z. This behavior of the dynamic system is commonly considered as a manifestation of an attractor that is, the point of "attraction" of even unstable solutions. 

Thus in the oscillator death state (Mirollo and Strogatz, 1990), the stirring completely inhibits the chemical oscillations, transforming the unstable steady state of the local dynamics into a stable attractor of the reaction-advection-diffusion system. This arises from the competition between the non-uniform frequencies that lead to the dispersion of the phase of the oscillations, while mixing tends to homogenize the system and bring it back to the unstable steady state. Therefore this behavior is only possible in oscillatory media with spatially non-uniform frequencies. The phase diagram of the non-uniform oscillatory system in the plane of the two main control parameters (v, 5) is shown in Fig. 8.4. 

The influence of noise on a dynamical system may have two counteracting effects. On the one hand if the underlying deterministic systems is already oscillatory, like a limit cycle oscillator or a chaotic oscillator, one expects these oscillations to become less regular due to the influence of the noise. On the other hand oscillatory behavior can also be generated by the noise in systems which deterministically do not show any oscillations. A prominent example are excitable systems but also the noise induced hopping between the attractors in a bistable system can be considered as oscillations [1]. 

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