Characterization of chaos

Accuracy of voltage measurements It should be noted that the resolution of the digitizer that transforms an analog voltage signal into a numerical value is a crucial factor so far as accuracy is concerned. 

Sampling of data In this context, it is often recommended that oversampling can seriously bias analysis for chaos. 

of data points Analytical studies for detecting chaos require extremely large data sets. 

Stability of the system Electronic systems are stable systems while biological systems are constantly changing. 

Choice of signal filters For distinguishing chaos from noise. 

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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. 

A quantitative characterization of chaos is possible with the help of Lyapunov exponents. These exponents are numbers which essentially measure the degree of chaoticity of a given chaotic system. With reference to the discussion in Section 1.3, the Lyapunov exponents enable us to calculate quantitatively an estimate for the critical time tc of a chaotic system, which limits our ability to predict the time evolution of the system for t > tc-... 

Since the very beginning non graphical methods for the detection of chaos where implemented, taking into account the exponential divergence of nearby chaotic orbits (Froeschle 1970,b). The introduction, in a comprehensive way, of the Lyapunov Characteristic Exponents (LCEs hereafter) and a method for computing all of them (Benettin et al. 1980) made a major breakthrough for the characterization of chaos. Actually, the largest Lyapunov Exponent was already computed earlier (Froeschle 1970,b) but was called indicator of stochasticity since the works of the Russian mathematician where not known by the author (Froeschle 1984). 

The Liapunov number can be used as a quantitative measure for chaos. The connection between chaos and the Liapunov number is through attractors. An attractor is a set of points S such that for nearly any point surrounding S, the dynamics will approach S as the time approaches infinity. The steady state of a fluid flow can be termed an attractor with dimension zero and a stable limit cycle dimension one. There are attractors that do not have integer dimensirms and are often called strange attractors. There is no tmiversally acceptable definition for strange attractors. The Liapunov number is determined by the principle axes of the ellipsoidal in the phase space, which originates from a ball of points in the phase space. The relatitaiship between the Liapunov number and the characterization of chaos is not universal and is an area of intensive research. 

In Sections 2 and 3, respectively, we will describe the characterization of chaos and some of the routes to chaos. Theory and models of chemical chaos will be briefly reviewed in Section 4, and Section 5 contains concluding remarks. The detailed studies of chemical chaos have all been conducted for the BZ reaction, but some evidence for chaos has also been obtained for several other chemical reactions [29-30]. ... 

Bottiglieri M, C Keel (2006) Characterization of PhlG, a hydrolase that specifically degrades the antifungal compound diacetylphloroglucinol in the biocontrol agent Pseudomonas fluorescens CHAO. Appl Environ Microbiol 72 418-427. 

Chao, J. R., Ni, Y. G., Bolanos, C. A. etal. Characterization of the mouse adenylyl cyclase type VIII gene promoter Regulation by cAMP and CREB. Eur. J. Neurosci. 16 1284— 1294,2002. 

A. S. Verkman, M. C. Sellers, A. C. Chao, T. Leung, and R. Ketcham, Synthesis and characterization of improved chloride-sensitive fluorescent indicators for biological applications, Analyt. Biockem. 178, 355-361 (1989). 

A set of experiments on gas-liquid motion in a vertical column has been carried out to study its d3mamical behavior. Fluctuations volume fraction of the fluid were indirectly measured as time series. Similar techniques that previous section were used to study the system. Time-delay coordinates were used to reconstruct the underl3ung attractor. The characterization of such attractor was carried out via Lyapunov exponents, Poincare map and spectral analysis. The d3mamical behavior of gas-liquid bubbling flow was interpreted in terms of the interactions between bubbles. An important difference between this study case and former is that gas-liquid column is controlled in open-loop by manipulating the superficial velocity. The gas-liquid has been traditionally studied in the chaos (turbulence) context [24]. 

Borderline personality disorder (BPD) is the most common and best described of all the personality disorders. These patients lack stability in their relationships, have a clonded concept of their own identity, and have trouble modulating their mood. Their lives are often characterized by chaos as they frantically seek intensely... 

The mass spectrum of (8) is characterized by an extremely low intensity of the series A, C, E, H, and J, and by the absence of series B and D due to the lower electron-donating power of the CHaCO group compared with that of CHaO. The ions of the series C, D, H, and J could indeed be formed after cleavage of the C-l to C-2-bond leading to an ion-radical of the... 

Xiong W, Chen LM, Chao J. Purification and characterization of a kallikrein-like T-kininogenase. J Biol Chem 1990 265 2822-2827. 

Chen LM, Murray SR, Chai KX, Chao L, Chao J. Molecular cloning and characterization of a novel kallikrein transcript in colon and its distribution in human tissues. Braz J Med Biol Res 1994 27 1829-1838. 

However, a multidimensional system is generally classically nonintegrable, and so the existence of classical chaos, which more or less appears in the (complex) phase space, introduces some intrinsic difficulties to applying the semiclassical method to multidimensional tunneling. Even if we restrict ourselves to the real domain, which means that we don t take into account tunneling phenomena, the existence of chaos is a real obstacle to endowing the semiclassical method with the rigorously mathematical basis, while some practical applications of the semiclassical method work well in prediction of quantal quantities which are used to characterize the quantum chaotic nature of a system under consideration [9,10]. The extension of the phase space to the complex domain will introduce further complexities and difficulties, and there is... 

Especially in the highly excited semiclassical regime the quantum properties and dynamics of atomic and molecular systems are most naturally discussed within the framework of chaos. Not only does chaos theory help to characterize spectra and wave functions, it also makes specific predictions about the existence of new quantum dynamical regimes and hitherto unknown exotic states. Examples are the discovery of frozen planet states in the helium atom by Richter and Wintgen (1990a) and... 

In order to allow for the largest possible class of chaotic systems, the degree of sensitivity is not specified in Devaney s definition of chaos. It turns out that many chaotic systems of practical importance are exponentially sensitive to initial conditions. In this case the sensitivity can be characterized quantitatively with the help of Lyapunov exponents. 

Given the abovementioned bewildering cornucopia of quantum systems that in one way or another all invoke the notion of chaos, we have to ask the question what exactly is quantum chaos We think that quantum chaos comes in three varieties (I) quantized chaos, (II) semi-quantum chaos and (III) quantum chaos. We refer to these three categories as type I, II and III quantum chaos. The division of quantum chaos into these three types arises naturally if quantum systems are characterized according to whether they do or do not show exponential sensitivity and chaos. The three different types of quantum systems are discussed in Sections 4.1, 4.2 and 4.3, respectively. A short preview of the three different types of quantum chaos follows. 

According to Chirikov [23J, the onset of chaos is associated with the overlap of neighboring nonlinear resonances. The overlap criterion, which bears the qualitative significance, uses the model of isolated resonances. Each resonance is characterized by its width, the maximum distance (in the action variable) from the elliptic fixed point The overlap means that the sum of the widths of two neighboring resonances is equal to the distance between two fixed points of these isolated resonances. We start with the pendulum Hamiltonian, which describes an isolated 1 N resonance under the periodic perturbation of frequency Q ... 

Further characterization of P(t) and Pg(t) requires details of the dynamics, here formulated in terms of ergodic-theory-based assumptions. However, since ergodic theory11 considers dynamics on a bounded manifold, it is not directly applicable to the unbounded phase space associated with molecular decay. To resolve this problem we first introduce a related auxiliary bounded system upon which conditions of chaos are imposed, and then determine their effect on the molecular decay details of this construction are provided elsewhere.46 What we show is that adopting this condition leads to a new model for decay, the delayed lifetime gap model (DLGM) for P(t) and Pg(t). The simple statistical theory assumption that Pt(t) and P(t) are exponential with rate ks(E) is shown to arise only as a limiting case. 

G. Lapeyre. Characterization of finite-time Lyapunov exponents and vectors in two-dimensional turbulence. Chaos, 12 688-698, 2002. 

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