Hybrid internal model control and proportional control of chaotic dynamical systems.

Modeling diversity by strange attractors with application to temporal pattern recognition

The subject of the research is the steganographic method of embedding information in digital images. Steganography is able to hide not only the content of information, but also the fact of its existence. The paper presents a method of embedding and extracting information into digital images using a chaotic dynamic system. Chaotic systems are sensitive to certain signals and at the same time immune to noise. These properties allow the use of chaotic systems for embedding information with small image distortions in statistical and visual terms. The methodological basis of the study is the methods of the theory of dynamical systems, mathematical statistics, as well as the theory of image processing. The novelty of the study lies in the development of a new method of embedding information in static images. The author examines in detail the problem of using a chaotic dynamic Duffing system for embedding and extracting information in digital still images. It is shown that the proposed method allows you to embed information in digital images without significant distortion.

Chaotic neural networks consisting of a great number of chaotic neurons are able to reproduce the rich dynamics observed in biological nervous systems. In recent years, the memristor has attracted much interest in the efficient implementation of artificial synapses and neurons. This work addresses adaptive synchronization of a class of memristor-based neural chaotic systems using a novel adaptive backstepping approach. A systematic design procedure is presented. Simulation results have demonstrated the effectiveness of the proposed adaptive synchronization method and its potential in practical application of memristive chaotic oscillators in secure communication.

A new image encryption model using different chaotic dynamical system is proposed and investigated. The proposed encryption model uses an efficient pixel shuffling method based on chaotic dynamical systems under study. Multiple performance measure have been computed for each of the chaotic dynamical systems used in the algorithm for encrypting various images. A detailed comparison table and list of control parameters is given in the paper. Various tests for the proposed encryption method were also carried out and corresponding results are included. The test results for dynamical systems clearly indicates the most suitable candidate for image crypto systems among chaotic systems taken in consideration.

In this thesis, a new class of codes on graphs based on chaotic dynamical systems are proposed. In particular, trellis coded modulation and iteratively decodable codes on graphs are studied. The codes are designed by controlling symbolic dynamics of chaotic systems and using linear convolutional codes. The relation between symbolic dynamics of chaotic systems and trellis aspects to minimum distance properties of coded modulations is explained. Our arguments are supported by computer simulations and results of search procedures for more powerful modulations. Ensembles of codes in systematic forms based on high dimensional (couple of hundreds and thousands) are developed generalizing lower dimensional systems. Analyzing the complex structure of chaotic systems a particular kind of factor graphs is developed. A forward-backward decoding method based on belief propagation on factor graphs of codes based on chaotic systems is proposed. The communication performance with signaling over additive white Gaussian noise (AWGN) channel and 8- and 16-PSK modulations is studied and convergence analysis of iterative decoding system is presented. An important property of our schemes relies in their low encoding complexity. Hence, comparison and some advantages over Low Density Generator Matrix (LDGM) block codes in terms of encoding complexity and bit error rate (BER) performance are described and possible applications of our codes are discussed.

Blowout bifurcation in chaotic dynamical systems occurs when a chaotic attractor, lying in some invariant subspace, becomes transversely unstable. We establish quantitative characterization of the blowout bifurcation by unstable periodic orbits embedded in the chaotic attractor. We argue that the bifurcation is mediated by changes in the transverse stability of an infinite number of unstable periodic orbits . There are two distinct groups of periodic orbits: one transversely stable and another transversely unstable. The bifurcation occurs when some properly weighted transverse eigenvalues of these two groups are balanced. @S1063-651X~97!50902-0# PACS number~s!: 05.45.1b Recently, a novel type of bifurcation has been discovered in chaotic dynamical systems @1,2#. This is the so-called ‘‘blowout bifurcation’’ that occurs in dynamical systems with a symmetric invariant subspace. Let S be the invariant subspace in which there is a chaotic attractor. Since S is invariant, initial conditions in S result in trajectories that remain in S forever. Whether the chaotic attractor in S is also an attractor in the full phase space depends on the sign of the largest Lyapunov exponent L’ computed for trajectories in S with respect to perturbations in the subspace T which is transverse to S. When L’ is negative, S attracts trajectories transversely in the vicinity of S and, hence, the chaotic attractor in S is an attractor in the full phase space. If L’ is positive, trajectories in the neighborhood of S are repelled away from it and, consequently, the attractor in S is transversely unstable and it is hence not an attractor in the full phase space. Blowout bifurcation occurs when L’ changes from negative to positive values. There are distinct physical phenomena associated with the blowout bifurcation. For example, near the bifurcation point where L ’ is negative, if there are other attractors in the phase space, then typically, the basin of the chaotic attractor in S is riddled @3#. When L ’ is slightly positive, if there are no other attractors in the phase space, the dynamics in the transverse subspace T exhibits an extreme type of temporally intermittent bursting behavior, the on-off intermittency @4,5#. Recent study has also revealed that blowout bifurcation can lead to symmetry breaking in chaotic systems @6#. In the study of chaos theory, it is important to be able to understand a bifurcation in terms of unstable periodic orbits of the system because the knowledge of periodic orbits usually yields a great deal of information about the dynamics @7‐9#. Periodic orbits are known to be responsible for many different types of bifurcations in chaotic systems. For example, the period-doubling bifurcation @10# and the saddlenode bifurcation are bifurcations of periodic orbits. Catastrophic events in chaotic systems such as crises @11# and basin boundary metamorphoses @12# are triggered by collision of periodic orbits, usually of low period, embedded in different dynamical invariant sets. The birth of Wada basin boundaries, meaning common boundaries of more than two basins of attraction, is caused by a saddle-node bifurcation on the basin boundary @13#. More recent study indicates that the riddling bifurcation, bifurcation that gives birth to a riddled basin, is triggered by the loss of transverse stability of some periodic orbit of low period embedded in the chaotic attractor in S @14#. In view of the role of periodic orbits played in these major bifurcations, it is desirable to study the blowout bifurcation by periodic orbits. In this regard, Ashwin, Buescu, and Stewart have noticed that as a system parameter changes towards the blowout bifurcation point, more and more atypical invariant measures become transversely unstable @2#. At the bifurcation, the natural measure of the chaotic attractor in S becomes unstable. In this paper, we establish a quantitative characterization of the blowout bifurcation by unstable periodic orbits embedded in the chaotic attractor in the invariant subspace S .I n particular, we argue that near the bifurcation, there exist two groups of periodic orbits S s and S u , each having an infinite number of members, one transversely stable and another transversely unstable, respectively. The sign of the largest transverse Lyapunov exponent L’ is determined by the relative weights of S s and S u : L’ is negative ~positive! when S s ~S u! weighs over S u ~S s!. ~A precise definition of the ‘‘weights’’ will be described in the sequel. ! At the bifurcation, the weights of S s and S u are balanced. In contrast to most known bifurcations in chaotic systems that usually involve only one or a few periodic orbits @10‐14#, blowout bifurcation is induced by a change in the transverse stability of an infinite number of unstable periodic orbits . The num

Pseudorandom number generators (PRNGs) have always been a central research topic in data science, and chaotic dynamical systems are one of the means to obtain scientifically proven data. Chaotic dynamical systems have the property that they have a seemingly unpredictable and random behavior obtained by making use of deterministic laws. The current paper will show how several notions used in the study of chaotic systems—statistical independence, singularity, and observability—can be used together as a suite of test methods for chaotic systems with high potential of being used in the PRNG or cryptography fields. In order to address these topics, we relied on the adaptation of the observability coefficient used in previous papers of the authors, we calculated the singularity areas for the chaotic systems considered, and we evaluated the selected chaotic maps from a statistical independence point of view. By making use of the three notions above, we managed to find strong correlations between the methods proposed, thus supporting the idea that the resulting test procedure is consistent. Future research directions consist of applying the proposed test procedure to other chaotic systems in order to gather more data and formalize the approach in a test suite that can be used by the data scientist when selecting the best chaotic system for a specific use (PRNG, cryptography, etc.).

In the present decade, chaotic systems are used and appeared in many fields like in information security, communication systems, economics, bioengineering, mathematics, etc. Thus, developing of chaotic dynamical systems is most interesting and desirable in comparison with dynamical systems with regular behaviour. The chaotic systems are categorised into two groups. These are (i) system with self-excited attractors and (ii) systems with hidden attractors. A self-excited attractor is generated depending on the location of its unstable equilibrium point and in such case, the basin of attraction touches the equilibria. But, in the case of hidden attractors, the basin of attraction does not touch the equilibria and also finding of such attractors is a difficult task. The systems with (i) no equilibrium point and (ii) stable equilibrium points belong to the category of hidden attractors. Recently chaotic systems with infinitely many equilibria/a line of equilibria are also considered under the cattegory of hidden attractors. Higher dimensional chaotic systems have more complexity and disorders compared with lower dimensional chaotic systems. Recently, more attention is given to the development of higher dimensional chaotic systems with hidden attractors. But, the development of higher dimensional chaotic systems having both hidden attractors and self-excited attractors is more demanding. This chapter reports three hyperchaotic and two chaotic, 5-D new systems having the nature of both the self-excited and hidden attractors. The systems have non-hyperbolic equilibria, hence, belong to the category of self-excited attractors. Also, the systems have many equilibria, and hence, may be considered under the category of a chaotic system with hidden attractors. A systematic procedure is used to develop the new systems from the well-known 3-D Lorenz chaotic system. All the five systems exhibit multistability with the change of initial conditions. Various theoretical and numerical tools like phase portrait, Lyapunov spectrum, bifurcation diagram, Poincare map, and frequency spectrum are used to confirm the chaotic nature of the new systems. The MATLAB simulation results of the new systems are validated by designing their circuits and realising the same.

These dates are widely applied to crustal movement monitoring,navigation and earth physics,and studying on other related scientific filed.Especially,applying to crustal movement application,for earthquakes often happen in some areas in China have caused heavy casualties and property losses.This paper first uses the PanTa method to eliminate outliers and based on chaotic dynamical system inspection.We apply to amplitude adjustment Fourier algorithm of constraint to generate alternative and Lyapunov index inspection time series nonlinear time series. Through the analysis,the elevation of time series Lyapunov index in the range of 0.0618～ 0.0618,and biggest Lyapunov index is positive,so the elevation time series for unsteady time series.According to the nonlinear model of the test,the elevation of time series Lyapunov index in the range of 151.3914～191.2036,so the time series was nonlinear.According to System uncertainty test,eesamein the range of-0.1637～0.0759,so the time series observation system for part of the deterministic mechanism loworder chaotic dynamical system.Through the analysis,Beijing Fangshan magmatic station elevation for the nonlinear and unsteady time series,time series observation system for part of the deterministic mechanism low-order chaotic dynamical system. So date was phased space reconstruction by regularization RBF and wavelet neural network filter.According to windowing spectrum estimation, minimum distance decoding fitting and fitting of FFT cycle.Calculation results show that spectral power of time series of Beijing Fangshan was 7694 W,periodic year was range from 0.994to1.109(year),periodic half year is range from 0.438 to 0.617(year)and periodic season was range from 0.247 to 0.378(year),we can know this time series tendency is not obvious,randomness is not obvious,however exist in periodic year,periodic half year and periodic season,among them,year periodic is the most obvious.

Detecting unstable periodic orbits (UPOs) from chaotic dynamic systems is a challenging problem. For a large number of complex systems, we can collect some experimental time series data but cannot find theoretical models to describe them. Thus, detecting unstable periodic orbits from experimental data can help us understand the chaotic properties of physical phenomenon without using theoretical models. We, in this paper, first use the dynamical transformation (DT) algorithm to detect unstable periodic orbits from chaotic systems, and find that the original DT algorithm can detect the UPOs from the time series of chaotic discrete map, but it is infeasible for the time series from continuous chaotic flow. In this regard, we then propose an improved DT algorithm that is based on the Poincare section method to detect the UPOs from continuous chaotic flow. In particular, we transform the continuous flow data into discrete map time series in terms of Poincare section, and then detect unstable periodic orbits from the transformed discrete map time series. In addition, we take Rössler and Lorenz chaotic systems as examples to demonstrate the effectiveness of our proposed method.

Chaotic dynamical systems may be characterised by a positive Lyapunov exponent, which measures the exponential rate of separation of nearby trajectories. However in a wide range of so-called weakly chaotic systems, the separation of nearby trajectories is sub-exponential, for example stretched exponential, in time; and therefore in such cases the Lyapunov exponent vanishes. When a hole is introduced in chaotic systems, the Lyapunov exponent on the system’s fractal repeller can be related to the generation of entropy and the escape rate from the system via the escape rate formalism, but no suitable generalisation exists to weakly chaotic systems. In this work we show that in a paradigmatic one-dimensional weakly chaotic iterated map, the Pomeau-Manneville map, the generation of generalised Lyapunov stretching is completely suppressed in the presence of a hole. These results are shown based on numerical evidence, and explained with a fully analytic stochastic model.

It is well known that chaotic dynamic systems (such as three-body system, turbulent flow and so on) have the sensitive dependence on initial conditions (SDIC). Unfortunately, numerical noises (such as truncation error and round-off error) always exist in practice. Thus, due to the SDIC, long-term accurate prediction of chaotic dynamic systems is practically impossible. In this paper, a new strategy for chaotic dynamic systems, i.e. the Clean Numerical Simulation (CNS), is briefly described, together with its applications to a few Hamiltonian chaotic systems. With negligible numerical noises, the CNS can provide convergent (reliable) chaotic trajectories in a long enough interval of time. This is very important for Hamiltonian systems such as three-body problem, and thus should have many applications in various fields. We find that the traditional numerical methods in double precision cannot give not only reliable trajectories but also reliable Fourier power spectra and autocorrelation functions. In addition, it is found that even statistic properties of chaotic systems can not be correctly obtained by means of traditional numerical algorithms in double precision, as long as these statistics are time-dependent. Thus, our CNS results strongly suggest that one had better to be very careful on DNS results of statistically unsteady turbulent flows, although DNS results often agree well with experimental data when turbulent flows are in a statistical stationary state.

Back to table of contents Previous article Next article LETTERFull AccessGlobal Versus Local Perspectives on SchizophreniaSareh ZendehrouhFatemeh BakouieShahriar Gharibzadeh,Sareh ZendehrouhSearch for more papers by this authorFatemeh BakouieSearch for more papers by this authorShahriar GharibzadehSearch for more papers by this author,Published Online:1 Apr 2009AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InEmail To the Editor: Recent neurobiological studies indicate that schizophrenia may be a neurodevelopmental and progressive disorder with multiple biochemical abnormalities involving dopamine, serotonin, glutamate, and gamma-aminobutyric acid secretion. In postmortem tissue studies, structural abnormalities and alterations in synaptic connectivity have been observed in the intracortical circuitry of the prefrontal dorsal cortex of schizophrenia patients. These morphological changes could be the sequelae of earlier environmental insults and genetic processes. There are probably multiple susceptibility genes, each of small effect, which act in conjunction with environmental factors as obstetric abnormalities, intrauterine infection, and abnormal nutrition. Candidate identified genes could influence neurodevelopment, synaptic plasticity, and neurotransmission. 1 The recent findings are consistent with notions that epigenetic factors play a major role in the disease process and epigenetic factors may continue to influence the expression of the affected genes in adulthood. 2 It is also demonstrated that schizophrenia is a dynamical disease—i.e., that important aspects of schizophrenia can be understood on the basis of concepts of the theory of nonlinear dynamical systems. 3 Chaotic dynamical systems are characterized by a lawful but delicate sensitivity to initial conditions. This leads to the observation that initially similar behaviors evolve into a striking divergence of behavioral patterns over time. Therefore, small differences in the input can result in an entirely different sequence of outputs. 4 We believe that genes act as initial conditions and environmental factors are the control parameters of a chaotic human brain system, which can force the system to special states with particular characteristics (e.g., schizophrenia). Since in chaotic dynamical systems, control parameters can direct the system to special states, environmental factors play a key role in aggravation or decline of schizophrenia symptoms. This complies with some reports that showed high relapse rate of schizophrenia in families that expressed high emotion, 5 which can be considered as a positive control factor. Based on our hypothesis, we propose that schizophrenia could be considered a chaotic model and the global features of the disease could be extracted instead of paying attention to local detailed features. In such a perspective, the manner of managing the disease will be changed. For example, we think that it is not necessary to include the effects of each part of the brain circuits and to correct any change in the amount of neurotransmitters; instead it is better to model the disease state as a chaotic system and recognize the interaction of different environmental factors, such as maternal health, birth complications, emotional states, familial relationships, etc., on it and try to minimize the pathological effects of them. We stress that it is not the compartments but the global interactions of different elements that play major roles in the disease and deserve attention. Modeling the behavior of schizophrenia on the basis of chaos theory to control the disease seems to be a good place to start.Biomedical Engineering, Amirkabir University of Technology, Tehran, Iran

Comment

A new chaotic discrete dynamical system, built on trigonometric functions, is proposed. With intent to use this system within cryptographic applications, we proved with the aid of specific tools from chaos theory (e.g., Lyapunov exponent, attractor’s fractal dimension, and Kolmogorov-Smirnov test) and statistics (e.g., NIST suite of tests) that the newly proposed dynamical system has a chaotic behavior, for a large parameter’s value space, and very good statistical properties, respectively. Further, the proposed chaotic dynamical system is used, in conjunction with a binary operation, in the designing of a new pseudorandom bit generator (PRBG) model. The PRBG is subjected, by turns, to an assessment of statistical properties. Theoretical and practical arguments, rounded by good statistical results, confirm viability of the proposed chaotic dynamical system and newly designed PRBG, recommending them for usage within cryptographic applications.