On the choice of the non-trainable internal weights in random feature maps for forecasting chaotic dynamical systems

Modeling diversity by strange attractors with application to temporal pattern recognition

Adaptive Synchronization of Memristor - based Chaotic Neural Systems

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.).

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.

A new chaos control method is proposed to take advantage of chaos or avoid it. The hybrid Internal Model Control and Proportional Control learning scheme are introduced. In order to gain the desired robust performance and ensure the system's stability, Adaptive Momentum Algorithms are also developed. Through properly designing the neural network plant model and neural network controller, the chaotic dynamical systems are controlled while the parameters of the BP neural network are modified. Taking the Lorenz chaotic system as example, the results show that chaotic dynamical systems can be stabilized at the desired orbits by this control strategy.

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.

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.

Cryptanalyzing chaotic secure communications using return maps

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The characteristics of the existing cipher algorithms for chaotic system were briefly analyzed in this paper.In order to provide a more effective encryption technique,the concept of random chaotic dynamical systems group was given.The random chaotic systems group can generate complex chaotic systems sequence under given conditions.Therefore,a stream cipher algorithm based on the random chaotic dynamical systems group was presented directly.Its encryption process could be influenced by secret keys,plaintext and specialty of chaotic systems group.Compared with other algorithms,the algorithm has been proved of higher security.The results of experiments show stream cipher algorithm based on the random chaotic dynamical systems group is easy to realize.Meanwhile,it has been tested with good encryption result and large secret space.

— The problem of chaos synchronization is to design a coupling between two chaotic systems (master-slave/drive-response systems configuration) such that the chaotic time evaluation becomes ideal and the output of the slave (response) system asymptotically follows the output of the master (drive) system. This paper has addressed the chaos synchronization problem of two chaotic systems using the Nonlinear Control Techniques, based on Lyapunov stability theory. It has been shown that the proposed schemes have outstanding transient performances and that analytically as well as graphically, synchronization is asymptotically globally stable. Suitable feedback controllers are designed to stabilize the closed-loop system at the origin. All simulation results are carried out to corroborate the effectiveness of the proposed methodologies by using Mathematica 9. Keywords-Synchronization; Lyapunov Stability Theory; Nonlinear Control; Routh-Hurwitz Criterion I. I NTRODUCTION Synchronization of chaotic systems is a process where two (or many) chaotic systems eventually progress identically for different initial conditions in all future states. This means that the dynamical state of one of the system is completely dictated by the dynamical state of the other system [1]. Chaos Synchronization between two chaotic systems is one of the most primary procedures in complex systems’ control and has wide potential applications in different fields [2-6]. After a pioneering work on chaos synchronization [1], synchronization of chaotic dynamical systems has received a great interest among researchers in nonlinear sciences for more than two decades [7]. Until now, diverse techniques have been proposed and applied successfully to synchronize two identical (or nearly identical) as well as nonidentical chaotic systems [8-13]. Notable among those, the Nonlinear control algorithm [7, 9] is one of the effectual techniques for synchronizing two chaotic systems [7]. Nonlinear control techniques take the advantage of the given nonlinear system dynamics to produce high-performance designs. No Lyapunov exponents or gain matrix are required for its execution. These qualities allow the designer to focus on the synchronization problem, leaving troublesome model manipulations [9]. Edward Lorenz, a meteorologist and mathematician, is known to be the pioneer of chaos theory. In the 1960s, Lorenz made his historical discovery by observing weather phenomena particularly in convections of fluids [14]. Lorenz took different mathematical models of fluid convection and simplified them into a system of ordinary differential equations and came up with a 3-D chaotic attractor for the first time, what is now known as the popular Lorenz equations [14]. After the exceptional discovery of E. Lorenz on chaotic attractor, chaos has become an interesting topic for many researchers. During the last three decades, remarkable research has been done on chaos which explored its different applications, features and fundamental properties [15]. The significance of the 3-D differential equations is that relatively simple systems could exhibit rather complex or specifically chaotic behavior. The 3-dimensional chaotic systems have many potential applications in different scientific fields such as chemical reactions, secure communications, biological systems and nonlinear circuits [15]. Due to a wide range of applications of 3-D chaotic systems, various systems such as the Chen system, Rossler system, Liu system, Qi system, Tigan system and Lu system [16-19] have been proposed and applied successfully to many practical systems and have shown some effective outcomes. Recently, a new 3-D autonomous chaotic system based on a quadratic exponential nonlinear term and a quadratic cross product term has been proposed and studied [20]. A quadratic exponential nonlinear term was added to the third equation while eliminating the second term from the second equation and a nonlinear term from the third equation of the Lorenz System [20]. The new 3-D chaotic system is topologically different from the Lorenz System. The two-scroll attractor from the new system exhibits multiplex chaotic dynamics. The nonlinear dynamical properties of the new

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

In this paper, the problem of Q-S synchronisation for arbitrary dimensional chaotic dynamical systems in continuous-time is investigated. Based on nonlinear control method, we would like to present a constructive scheme to study the Q-S synchronisation between n-dimensional master chaotic system and m-dimensional slave chaotic system in arbitrary dimension. The new derived synchronisation result is proved using Lyapunov stability theory. In order to verify the effectiveness of the proposed method, our approach is applied to some typical chaotic systems and numerical simulations are given to validate the derived results.