Narrow Results

In this paper a new deterministic chaotic oscillator with elements equivalent to Hartley's oscillator is presented and described. P-SPICE simulation program and practical measurement are used to prove the deterministic chaotic character of the mentioned circuit.

Noise-free stochastic resonance is investigated in two chaotic maps with periodically modulated control parameter close to a boundary crisis: the Hénon map and the kicked spin model. Response of the maps to the periodic signal at the fundamental frequency and its higher harmonics is examined. The systems show noise-free stochastic multiresonance, i.e. multiple maxima of the signal-to-noise ratio at the fundamental frequency as a function of the control parameter. The maxima are directly related to the fractal structure of the attractors and basins of attraction colliding at the crisis point. The signal-to-noise ratios at higher harmonics show more maxima, as well as dips where the signal-to-noise ratio is zero. This opens a way to use noise-free stochastic resonance to probe the fractal structure of colliding sets by a method which can be called "fractal spectroscopy". Using stochastic resonance at higher harmonics can reveal smaller details of the fractal structures, but the interpretation of results becomes more difficult. Quantitative theory based on a model of a colliding fractal attractor and a fractal basin of attraction is derived which agrees with numerical results for the signal-to-noise ratio at the fundamental frequency and at the first two harmonics, quantitatively for the Hénon map, and qualitatively for the kicked spin model. It is also argued that the maps under study belong to a more general class of threshold-crossing stochastic resonators with a modulated control parameter, and qualitative discussion of conditions under which stochastic multiresonance appears in such systems is given.

A second-order nonlinear differential equation with an aftereffect for the density of a thin homogeneous layer on a liquid and vapor interface is considered. The acts of evaporation and condensation of molecules, which are regarded as periodic "impacts", excite the layer. The mentioned NDE is integrated over a finite time interval to find a 2D (two-dimensional) mapping whose numerical solution describes the chaotic dynamics of density and pressure in time. The algorithms of constructing bifurcation diagrams, Lyapunov's exponents and Kolmogorov's entropy for systems with first-order, second-order phase transitions and Van der Waals' systems were elaborated. This approach allows to associate such concepts as phase transition, deterministic chaos and nonlinear processes. It also allows to answer a question whether deterministic chaos occurs in systems with phase transitions and how fast the information about starting conditions is lost within them.

The results of a study of the bifurcation diagram of the Hindmarsh–Rose neuron model in a two-dimensional parameter space are reported. This diagram shows the existence and extent of complex bifurcation structures that might be useful to understand the mechanisms used by the neurons to encode information and give rapid responses to stimulus. Moreover, the information contained in this phase diagram provides a background to develop our understanding of the dynamics of interacting neurons.

We examine and assess deterministic chaos as an observable. First, we present the development of model ecological systems. We illustrate how to apply the Kolmogorov theorem to obtain limits on the parameters in the system, which assure the existence of either stable equilibrium point or stable limit cycle behavior in the phase space of two-dimensional (2D) dynamical systems. We also illustrate the method of deriving conditions using the linear stability analysis. We apply these procedures on some basic existing model ecological systems. Then, we propose four model ecological systems to study the dynamical chaos (chaos and intermittent chaos) and cycles. Dynamics of two predation and two competition models have been explored. The predation models have been designed by linking two predator–prey communities, which differ from one another in one essential way: the predator in the first is specialist and that in the second is generalist. The two competition models pertain to two distinct competition processes: interference and exploitative competition. The first competition model was designed by linking two predator–prey communities through inter-specific competition. The other competition model assumes that a cycling predator–prey community is successfully invaded by a predator with linear functional response and coexists with the community as a result of differences in the functional responses of the two predators. The main criterion behind the selection of these two model systems for the present study was that they represent diversity of ecological interactions in the real world in a manner which preserves mathematical tractability. For investigating the dynamic behavior of the model systems, the following tools are used: (i) calculation of the basin boundary structures, (ii) performing two-dimensional parameter scans using two of the parameters in the system as base variables, (iii) drawing the bifurcation diagrams, and (iv) performing time series analysis and drawing the phase space diagrams. The results of numerical simulation are used to distinguish between chaotic and cyclic behaviors of the systems.

The conclusion that we obtain from the first two model systems (predation models) is that it would be difficult to capture chaos in the wild because ecological systems appear to change their attractors in response to changes in the system parameters quite frequently. The detection of chaos in the real data does not seem to be a possibility as what is present in ecological systems is not robust chaos but short-term recurrent chaos. The first competition model (interference competition) shares this conclusion with those of predation ones. The model with exploitative competition suggests that deterministic chaos may be robust in certain systems, but it would not be observed as the constituent populations frequently execute excursions to extinction-sized densities. Thus, no matter how good the data characteristics and analysis techniques are, dynamical chaos may continue to elude ecologists. On the other hand, the models suggest that the observation of cyclical dynamics in nature is the most likely outcome.

We present in this paper the first example of chaotic evolutionary dynamics in biology. We consider a Lotka–Volterra tritrophic food chain composed of a resource, its consumer, and a predator species, each characterized by a single adaptive phenotypic trait, and we show that for suitable modeling and parameter choices the evolutionary trajectories approach a strange attractor in the three-dimensional trait space. The study is performed through the bifurcation analysis of the so-called canonical equation of Adaptive Dynamics, the most appropriate modeling approach to long-term evolutionary dynamics.

A new methodology to characterize nonlinear systems is described. It is based on the measurement over the time series of two quantities: the "Dynamical order" and the "Self-correlation". The averaged "Scalar" and "Perpendicular" products are introduced to measure these quantities. While this approach can be applied to general nonlinear systems, the aim of this work is to focus on the characterization and modeling of chaotic systems. In order to illustrate the method, applications to a two-dimensional chaotic system and the modeling of real telephony traffic series are presented. Three important aspects are discussed: the use of the averaged "Scalar" product as supplement of the "Lyapunov exponent", the use of the averaged "Perpendicular" product as a refinement of the "Mutual information" and the reduction of m-dimensional systems to the study of only one dimension. This new conceptual framework introduces a perspective to characterize real and theoretical processes with a unifying method, irrespective of the system classification.

We have investigated the scenarios of transition to chaos in the mathematical model of a genetic system constituted by a single transcription factor-encoding gene, the expression of which is self-regulated by a feedback loop that involves protein isoforms. Alternative splicing results in the synthesis of protein isoforms providing opposite regulatory outcomes — activation or repression. The model is represented by a differential equation with two delayed arguments. The possibility of transition to chaos dynamics via all classical scenarios: a cascade of period-doubling bifurcations, quasiperiodicity and type-I, type-II and type-III intermittencies, has been numerically demonstrated. The parametric features of each type of transition to chaos have been described.

Microbioreactors operated in real environments are often subject to noise from the environment. This is commonly manifested as fluctuations in the flow rates of the feed streams. Previous studies with larger bioreactors have shown that noise can seriously impair the performance. Given this possibility, the effects of noise on the performance of a microbioreactor have been analyzed for the trans-esterification of vinyl butyrate by 1-butanol by immobilized lipase B to produce butyl butyrate. As in previous work for macrobioreactors, the analysis was done with (i) no noise, (ii) unfiltered noise, and (iii) noise filtered by four different methods, and the fractal dimension of the product was used as an index of the performance.

All fractal dimensions decreased with increasing dilution rates, and significant stochastic chaos was likely at low dilution rates. Of the four types of filters, the auto-associative neural filter (ANF) was the most effective in reducing chaos and restoring of smooth, nearly noise-free performance. The ANF also does not require a process model, which is a significant advantage for real systems. Simulations also revealed that even in the absence of noise, deterministic chaos is possible at low dilution rates; this underscores the importance of efficient filtering under such conditions when external noise too is present. The results thus establish the importance of noise in microbioreactor behavior and the usefulness of the fractal dimension in characterizing the effects.

Predicting foreign exchange rates and stock market indices have been a well researched topic in the field of financial engineering. However, most methods suffer from serious drawback due to the inherent uncertainty in the data acquisition process. Here, we have analyzed the very nature of the time series data from a pure dynamic system point of view and explored the deterministic chaotic characteristic in it. In this research, the concept of chaos has been analyzed thoroughly and the relationships among chaos, stability and order have been explained with respect to the concept of time. A method of predicting time series data based on deterministic dynamically system has been presented in this monograph. The present research revolves around the concepts of embedding and fuzzy reconstruction. In this regard, the necessary and sufficient condition for this reconstruction of the state space of the dynamic system in a multi-dimensional Euclidean space has been substantiated in accordance to Theory of embedding. Finally, a fuzzy reconstruction method based on fuzzy multiple regression analysis method has been used to predict the foreign exchange rates with accuracy.

In this paper we try to bring forward evidence on the practical application of the Grassberger–Procaccia algorithm, in particular on the determination of the "delay time" parameter. For this purpose, we analyze the results obtained from applying the main methods proposed to calculate this delay time for series simulated from well-known chaotic systems. As the most relevant result we conclude that, in general, all the methods display inadequate behavior, except for that based on the previous filtering of the series according to singular value decomposition. In a second stage we apply the same study to three financial series, with the results appearing to confirm the advantage of this method.

Our randomized versions of the Sharkovsky-type cycle coexistence theorems on tori and, in particular, on the circle are applied to random impulsive differential equations and inclusions. The obtained effective coexistence criteria for random subharmonics with various periods are formulated in terms of the Lefschetz numbers (in dimension one, in terms of degrees) of the impulsive maps and their iterates w.r.t. the (deterministic) state variables. Otherwise, the forcing properties of certain periods of the given random subharmonics are employed, provided there exists a random harmonic solution. In the single-valued case, the exhibition of deterministic chaos in the sense of Devaney is detected for random impulsive differential equations on the factor space $ℝ/\mathbb{Z}$. Several simple illustrative examples are supplied.

A simple theoretical consideration of the possibility of using deterministic chaos for a quantitative determination of trace amounts of chemical compounds is analyzed in terms of return maps. This allows the development of a general approach to find a calibration plot for a chemical compound determination, i.e. to find a one-to-one correspondence between a measured value and the concentration of a substance to be determined. As experimental verification of these theoretical results, we present a study of the effect of copper (II) ions on the transient chaotic regime in the BZ reaction catalyzed by ferroin in batch as well as of the effect of vanadium (IV) on the deterministic chaotic regime in the BZ reaction catalyzed by ferroin in a CSTR. These studies show that it is possible to quantify the response of transient chaotic regimes to chemical compounds, i.e. to construct a calibration plot for chemical compound determination using their effect on the chaotic dynamics in the BZ reaction. Moreover, these studies shown that vanadium (IV) affects the chaotic regimes at concentration levels of the order of 10^-12g/mL, which gives the best detection limit for the kinetic methods of trace analysis.

The purpose of the present study is to investigate the existence of deterministic chaos in the time series of occurrence or non-occurrence of severe thunderstorms of the pre-monsoon season over the Northeastern part of India. Results from the current study reveal the existence of chaos in the relevant time series. The corresponding predictabilities are also computed quantitatively. The study recommends that the formulation of numerical weather prediction models for forecasting the occurrence of this high frequency meso-scale convective system must take into account the intrinsic chaos.

The methods for constructing "chaotic" nonlinear systems of differential equations modeling gene networks of arbitrary structure and dimensionality with various types of symmetry are considered. It has been shown that an increase in modality of the functions describing the control of gene expression efficiency allows for a decrease in the dimensionality of these systems with retention of their chaotic dynamics. Three-dimensional "chaotic" cyclic systems are considered. Symmetrical and asymmetrical attractors with "narrow" chaos having a Moebius-like structure have been detected in such systems. As has been demonstrated, a complete symmetry of the systems with respect to permutation of variables does not prevent the emergence of their chaotic dynamics.

Alternative splicing is a widespread phenomenon in higher eukaryotes, where it serves as a mechanism to increase the functional diversity of proteins. This phenomenon has been described for different classes of proteins, including transcription regulatory proteins. We demonstrated that in the simplest genetic system model the formation of the alternatively spliced isoforms with opposite functions (activators and repressors) could be a cause of transition to chaotic dynamics. Under the simplest genetic system we understand a system consisting of a single gene encoding the structure of a transcription regulatory protein whose expression is regulated by a feedback mechanism. As demonstrated by numerical analysis of the models, if the synthesized isoforms regulate the expression of their own gene acting through different sites and independently of each other, for the generation of chaotic dynamics it is sufficient that the regulatory proteins have a dimeric structure. If regulatory proteins act through one site, the chaotic dynamics is generated in the system only when the repressor protein is either a tetrameric or a higher-dimensional multimer. In this case the activator can be a dimer. It was also demonstrated that if the transcription factor isoforms exhibit either activating or inhibiting activity and are lower-dimensional multimers (< 4), independently of the regulation type the model demonstrates either cyclic or stationary trajectories.

Today there are examples that prove the existence of chaotic dynamics at all levels of organization of living systems, except intracellular, although such a possibility has been theoretically predicted. The lack of experimental evidence of chaos generation at the intracellular level in vivo may indicate that during evolution the cell got rid of chaos. This work allows the hypothesis that one of the possible mechanisms for avoiding chaos in gene networks can be a negative evolutionary selection, which prevents fixation or realization of regulatory circuits, creating too mild, from the biological point of view, conditions for the emergence of chaos. It has been shown that one of such circuits may be a combination of negative autoregulation of expression of transcription factors at the level of their synthesis and degradation. The presence of such a circuit results in formation of multiple branches of chaotic solutions as well as formation of hyperchaos with equal and sufficiently low values of the delayed argument that can be implemented not only in eukaryotic, but in prokaryotic cells.

The inherent micro-structure (agent-agent/system) of human organizations has been introduced in Chapters 2 and 3. Fundamentally, human organizations are composite complex adaptive systems with human beings as interacting agents (each an intrinsic complex adaptive system). This chapter further analyzes the basic conceptual foundation of the multi-layer structure, including advantages of the intelligent biotic macro-structure (with inherent features similar to that of an intelligent biological being — a structural reform), and its unique and more integrative complex adaptive dynamic in intelligent human organizations (towards iCAD). The necessity of nurturing an intelligent biotic macro-structure with vital characteristics that better synchronize and enhance sophisticated information/knowledge-related activities is highly beneficial — achieving a higher structural capacity. Thus, the attributes, functions and higher structural capacity of the more intelligent biotic macro-structure can reinforced the competitiveness of any categories of human organizations extensively.

In this respect, connecting and engaging of intelligence/consciousness sources (individual and collective), organizing around intelligence, intelligence/ consciousness management, and the intelligent biotic macrostructure are mutually enhancing (towards higher coherency). Apparently, being intelligence/consciousness-centric is a beneficial and critical activator (strategic foundation) of the intelligence paradigmatic shift. In the present context, the roles and integration of intelligence, information and knowledge, as well as nurturing a ‘common’ language and elevating coherency in human organizations (with respect to the macro-structure and micro-structure, as well as their higher collectiveness — collectiveness capacity) must be more deeply scrutinized and utilized. The presence and significance of the individual intelligence enhancer encompassing three entities namely, intelligence, knowledge, and theory in the human thinking systems, and the necessity of nurturing a similar and effective intelligence enhancer at organizational level are analyzed. Subsequently, the supporting roles and contributions of artificial intelligence systems are also examined.

In between the macro-structure and micro-structure are two meso-structures. In the intelligent organization theory, the complexity meso-structure encompasses spaces of complexity and punctuation points. In this respect, complexity is a highly significant focal point, and the exploitation of co-existence of order and complexity is a new necessity (complexity-centricity). Next, the network meso-structure encompassing complex network (network of networks) is also an inherent structure and dynamic in all human organizations. This meso-structure is briefly introduced, and will be more deeply analyzed with respect to governance (network-centricity, network governance).

Hence, it is crucial to lead and manage human organizations with a strategic approach that integrates the above multi-layer structure/ dynamic at all time so that a higher structural capacity, collectiveness capacity, adaptive capacity, self-organizing capacity, and emergence-intelligence capacity can be nurtured. In the current highly competitive context, possessing these positive capabilities to elevate coherency and synergetic characteristics (including social consensus and the construal aspect) and dynamic is also highly crucial — a key focus of the complexity-intelligence strategy (towards achieving higher organizational mental cohesion). Hence, the significance and impact of nurturing intelligent human organizations with the complexity-intelligence-centric and network-centric approach that leads to the emergent of smarter evolvers and emergent strategists must be better understood and adopted. (The conceptual foundation on structural-dynamic coherency and synergy in intelligent human organizations developed in this chapter will be more deeply reviewed and exploited in later chapters.)

The basic foundation of the intelligent organization theory is conceptualized in this chapter. The significance of intelligence and organizing around intelligence, intelligence management and the general biotic structure of an intelligent human organization and its benefits are further discussed. The roles of intelligence, information and language in a human organization are examined. The presence and necessity of nurturing an effective intelligence enhancer encompassing three entities, namely, intelligence, knowledge and theory in the human thinking systems is introduced. The concepts on the space of order and the space of complexity are also shared. Some fundamental aspects of the deliberate strategy and emergent strategy are further examined. The above concepts are developed with respect to the fact that the human minds and human organizations are nonlinear complex adaptive systems. Consequently, the intelligent organizations that emerge are smarter evolvers.

We built basics of the qualitative theory of the continuous-time difference equations x(t + 1) = f(x(t)), t ∈ ℝ⁺, with the method of going to the infinite-dimensional dynamical system induced by the equation. For a study of this system we suggest an approach to analyzing the asymptotic dynamics of general nondissipative systems on continuous functions spaces. The use of this approach allows us to derive properties of the solutions from that of the ω-limit sets of trajectories of the corresponding dynamical system. In particular, typical continuous solutions are shown to tend (in Hausdorff metric for graphs) to upper semicontinuous functions whose graphs are, in wide conditions, fractal; there may exist especially nonregular solutions described asymptotically exactly by random processes. We introduce the notion of self-stochasticity in deterministic systems — a situation when the global attractor contains random functions. Substantiated is a scenario for a spatial-temporal chaos in distributed parameters systems with regular dynamics on attractor: The attractor consists of cycles only and the onset of chaos results from the very complicated structure of attractor “points” which are elements of some function space (different from the space of smooth functions). We develop a method to research into boundary value problems for partial differential equations, that bases on their reduction to difference equations.