In this paper we use the emergent property of the ultra weak multidimensional coupling of p 1-dimensional dynamical **chaotic** **systems** for which **complexity** **leads** from chaos to **randomness**. Efficient **Chaotic** Pseudo Random Number Generators (CPRNG) have been recently introduced (Lozi [3, 4, 5, 6]) and their properties analyzed (Hénaff et al. [7, 8, 9, 10]). The idea of applying discrete **chaotic** dynamical **systems**, intrinsically, exploits the property of extreme sensitivity of trajectories to small changes of initial conditions. The ultra weak multidimensional coupling of p 1-dimensional dynamical **systems** preserves the **chaotic** properties of the continuous models in numerical experiments. The process of **chaotic** sampling and mixing of **chaotic** sequences, which is pivotal for these families, works perfectly in numerical simulation when floating point (or any multi-precision) numbers are handled by a computer.

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Most implementations of the EnKF involve random pertur- bations of the model states and observations. This introduces **randomness** in the likelihood function (27): two evaluations with the same parameter value give different likelihood val- ues. As noted by Dowd (2011), this complicates the param- eter inference, and one has to resort to stochastic optimiza- tion methods that can handle noise in the target function (see Shapiro et al., 2009, for an introduction). Note that some recent variants of EnKF, such as many of the so called en- semble square root filters (Tippett et al., 2003) do not involve random components, and they might be more suitable for pa- rameter estimation purposes. We test a variant called Local Ensemble Transform Kalman Filter (LETKF, Hunt et al., 2007) in the experiments of Sect. 5.

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Random matrix theory provides a powerful paradigm for studying late-time chaos. We have leveraged the technology of random matrix theory and Haar-invariance to study correlation functions like OTOCs which diagnose early-time chaos, and frame potentials which diagnose **randomness** and **complexity**. The salient feature of the GUE which gave us computational traction is its Haar-invariance, namely that the ensemble looks the same in any basis. As a result, the dynamics induced by GUE Hamiltonians is non-local ( O (N)-local) with respect to any tensor factor decomposition of the Hilbert space, and so the dynamics immediately de- localizes quantum information and other more subtle forms of correlations. Accordingly, the GUE captures features of the long-time physics of a local system that has been delocalized. In a **chaotic** quantum system described by a local Hamiltonian, there are two temporal regimes of interest: times before the system scrambles and thus has mostly local correla- tions, and times after the system scrambles when correlations have effectively delocalized. We suggested that the transition between these two regimes may be due to the onset of approximate Haar-invariance, and we defined k-invariance as a precise characterization. A careful understanding of Haar-invariance for ensembles of local quantum **systems** could yield precise insights into the apparent breakdown of locality, and tell us in what time regimes we can use Haar-invariance to calculate late-time physics (i.e., correlation functions, frame potentials, **complexity**, etc.) A concrete way of studying delocalization of operators and

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When a small, time dependent periodic perturbation is added to the pen dulum. the smooth separatrix orbit is disrupted (because o f the transverse in tersection o f the stable and unstable manifolds that form it ) and a **chaotic** layer is created around the old separatrix solution (figure 2.1) . Inside the **chaotic** layer the **systems** dynamics are very complicated and look as if they are gen erated by a random process. The reason why such behaviour occurs near the separatrix is that near the separatrix and especially near the hyperbolic fixed point, the force experienced by a particle from the unperturbed system is very small and so the time dependent perturbation becomes dominant. So the o r bit can switch from librations to rotations and back again under the influence o f the perturbing force. Since the periods o f the motion near the separatrix approach infinity (this can be obtained simply by studying more carefully the example o f the pendulum) the switches from one type o f motion to another will be uncorrelated and so it will be very irregular (AMH).

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The problem on chaos synchronization for a class of **chaotic** system is addressed. Based on impulsive control theory and by constructing a novel Lyapunov functional, new impulsive synchronization strategies are presented and possess more practical application value. Finally some typical numerical simulation examples are included to demonstrate the effectiveness of the theoretical results.

Abstract: This paper describes the relationship between Lyapunov index and the chaos of the dynamics system. Mainly describe the Lorenz system and Rossler system, and calculated the Lyapunov index of these two **systems** with nonlinear dynamics method. Analyses the iterative **chaotic** characteristic of Logistic system. And then calculate the Lyapunov index with the discrete system Lyapunov index computing method, and then calculated the correlation dimension.

Abstract. We introduce two new algorithms for creating an exponentially biased sample over a possibly infinite data stream. Such an algorithm exists in the literature and uses O(log n) random bits per stream element, where n is the number of ele- ments in the sample. In this paper we present algorithms that use O(1) random bits per stream element. In essence, what we achieve is to be able to choose an element at random, out of n elements, by sparing O(1) random bits. Although in general this is not possible, the exact problem we are studying makes it possible. The needed **randomness** for this task is provided through a random walk. To prove the correct- ness of our algorithms we use a model also introduced in this paper, the limited **randomness** model. It is based on the fact that survival probabilities are assigned to the stream elements before they start to arrive.

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The basic problem addressed here is that of decomposition of the overall complex system. This is essentially a problem of organisation of processes (physical, operational, market related). Note that neither hierarchical nor heterarchical system organisations cope adequately with the multitude of demands introduced by the need to disaggregate the processes of a modern Railway System. Hierarchical **systems** typically have a rigid structure that impedes them to react to these disturbances in an agile way. Heterarchical **systems** handle disturbances very well and can continuously adapt themselves to their environment; however, heterarchical control does not guarantee high performance or predictable behaviour. The actual challenge lies in the requirement that future Railway **Systems** need both performance and reactivity. The answer to this challenge is sought in deploying theories on complex adaptive **systems**. Looking at living organisms and social organizations, Koestler made the observation that complex system can only arise if they consist of stable, autonomous subsystems, each of them capable of surviving disturbances, but that are meanwhile able to cooperate to form a more complex, stable system. This has led to the development of the principles of Holonic Manufacturing [39]. The Holonic Manufacturing Paradigm implies a highly distributed organization of the overall system, where intelligence is distributed over the individual entities. These entities are cooperative, intelligent, autonomous modules, called "holons." The new element in the holonic organisation is the fact that the individual entities work together in temporary hierarchies (called "holarchies") to achieve a global goal. The holonic concept combines the best features of hierarchical and heterarchical organization). It preserves the stability of hierarchy while providing the dynamic flexibility of a heterarchy. In this way, a holonic organisation combines high performance with robustness against changes and disturbances.

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Abstract—We address the problem of master-slave synchro- nization of **chaotic** **systems** under parameter uncertainty and with partial measurements. Our approach is based on observer-design theory hence, we view the master dynamics as a system of differen- tial equations with a state and a measurable output and we design an observer (tantamount to the slave system) which reconstructs the dynamic behavior of the master. The main technical condition that we impose is persistency of excitation (PE), a property well studied in the adaptive control literature. In the case of unknown parameters and partial measurements we show that synchroniza- tion is achievable in a practical sense, that is, with “small” error. We also illustrate our methods on particular examples of **chaotic** oscillators such as the Lorenz and the Lü oscillators. Theoretical proofs are provided based on recent results on stability theory for time-varying **systems**.

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Chaos synchronization is the dynamical process which means making two or more os- cillators keep the same rhythms under a weak interaction []. Since Pecora and Carroll [] proposed a pioneering method to synchronize two identical **chaotic** **systems**, synchro- nization of fractional-order **chaotic** dynamical **systems** has gained a lot of popularity for its potential applications in secure communication and cryptography, telecommunica- tion, signal and control processing, chaos synchronization [–]. Several types of syn- chronization techniques and methods, such as adaptive control, sliding mode control [, ], complete synchronization, projective synchronization (PS), and function projective synchronization (FPS) [–], have been proposed for fractional-order dynamical sys- tems. Among those existing synchronization methods, FPS, which has been introduced by Chen and Li [, ], was widely employed for synchronizing **chaotic** **systems**. Some scaling function matrices, which can be given with one’s need, are used in FPS. In fact, the scaling function matrix usually exhibits ﬂexibility and unpredictability. By using error feedback control scheme, FPS of complex dynamical networks with or without external disturbances was discussed in []. Ref. [] investigated adaptive switched modiﬁed FPS

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rate of change, and t the branch length/time. Different to the algorithm used with nucleotide sequences in which scores are adapted to varying base composition along sequences and among sequences, the frequency distribu- tion of scores of randomly similar sequences is only pro- duced once for amino acid data. The frequency distribution is generated by: 1) collecting frequencies of amino acids of the complete observed data set, 2) gener- ating 100 bootstrap resamples of this amino acid fre- quency distribution and 100 delete-half bootstrap resamples of each of the 100 complete bootstrap resam- ples, and 3) by using these 10,000 delete-half bootstrap resamples to generate 1,000,000 scores of randomly simi- lar amino acid sequences with length given by the win- dow size. This complex resampling **leads** to an even distribution of scores of randomly similar sequences. The frequency distribution of randomly similar sequences is used to define a cutoff c(α = 0.95) to assess **randomness** of the observed sequence similarity within the sliding win-

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Complex dynamical changes in humans at different level of spatial organization. A. Examples of chromosomal alterations (mutations): a) deletion of a tract of DNA; b) duplication of a tract of DNA sequence. B. The progressive changes occurring in the nucleus and cytoplasm that accompany the death of a cell. a) Normal cell; b) The nucleus becomes contracted and stains intensely. The cytoplasm is pinker, showing that it binds eosin (a common histochemical stain) more avidly. c) The nucleus dis- integrates, appearing as a more or less central area of dispersed chromatin. This phase is called karyorrhexis. d) All nuclear material has now disappeared (kariolysis) and the cytoplasm stains an intense red colour. C. The final appearance of the liver (a) when it assumes the state of cirrhosis (b). Cirrhosis is the final stage of several pathogenic mechanisms operating either alone or in concert to produce a liver diffusely involved by fibrosis (abnormal extra-cellular matrix deposition) and the forma- tion of structurally abnormal parenchymal nodules. D. Human life: from the embryonic stage of morula (a), through that of foe- tus (b), to the adult being (c). The times elapsing in the variousdynamical processes exemplified (A-D) are very different (simplified by green bars), ranging from nanoseconds to years. It is interesting to highlight the inverse relationship between the level of anatomical **complexity** and timescale.

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Введенная структурная сложность системы является универсальной и определяется через ее подструктуры, множество операций на них и аксиоматику правил сигнатуры и сложности.. Рассмотрены пр[r]

Abstract. A semi-measure is a generalization of a probability measure obtained by relaxing the additivity requirement to super-additivity. We introduce and study several **randomness** notions for left-c.e. semi- measures, a natural class of effectively approximable semi-measures induced by Turing functionals. Among the **randomness** notions we consider, the generalization of weak 2-**randomness** to left-c.e. semi-measures is the most compelling, as it best reflects Martin-L¨ of **randomness** with respect to a computable measure. Additionally, we analyze a question of Shen from [BBDM12], a positive answer to which would also have yielded a reasonable **randomness** notion for left-c.e. semi-measures. Unfortunately though, we find a negative answer, except for some special cases.

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Normally high order ordinary differential equations are transformed to dynamical **systems** (**systems** of 1st order equations), suitable for further analytical treatment and numerical solution. Here we followed the opposite procedure. Owing to the appropriate structure (1.1) of the equations in the original Lorenz low-order system (2.1) and its modifications (3.1), (3.11) and (3.17), single oscillatory type equations with forcing endogenous ‘‘memory’’ terms are derived, valid at t → ∞. In this connection linear and quadratic memories were defined by (1.4) and (3.16). For particular relationships among the **systems**’ parameters (b = 2 σ for (3.1) and b = 2 for (3.11)), the memory terms in (2.11) and (3.9) vanish. The latter result is new, while the former one is already known. In both cases, the results are autonomous Duffing type equations and invariant ratios (2.10) and (3.10). Moreover, other new invariant ratios (2.9) and (3.8) are rigorously derived from (2.1) and (3.1). Section 4 demonstrates the possibilities of the statistical approach in case of final (t → ∞) **chaotic** behaviour. The results of the previous sections may prove to be useful for various applications of the Lorenz and Lorenz-like **systems**, including meteorological ones.

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The form of the design used to attempt to control a given system can be motivated by many factors. In general, controlling nonlinear high-dimensional **chaotic** dynamical **systems** can be a formidable problem, Musielak and Musielak, [1]. Viera and Lichtenberg, [5], illustrate several examples of controlling chaos using a nonlinear feedback with delay. On the other hand, Tan et al., [4], develop a controller using a backstopping design.

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In the case of **chaotic** dynamics, however, the geometrical structure of the attractor (and also the boundary in cases where this is fractal) is very complex. And it is not known a priori if the boundary conditions in **chaotic** **systems** can be specified uniquely or, indeed, whether a unique MPEP exists at all. Consequently, the fluctuational escape problem in **chaotic** **systems** has attracted a great deal of attention during the last two decades (see e.g. [Beale, 1989; Grassberger, 1989; Kifer, 1990; Graham et al., 1991; Reimann et al. , 1994; Reimann & Talkner, 1995; Kraut & Feudel, 2003]). Here we present a brief review of recent progress in understanding the boundary conditions for escape from a CA, both with and without fractal boundaries [Luchinsky & Khovanov, 1999; Khovanov et al. , 2000; Luchinsky, 2002; Silchenko et al. , 2003a; Khovanov et al. , 2003; Kraut & Grebogi, 2004; Beri et al., 2005; Silchenko et al., 2005] It turns out that the boundary conditions at the attractor and on its fractal boundary can be identified uniquely with certain unstable periodic orbits. This in turn opens up the possibility of developing a general approach to the solution of the stability and control problems by application of a WKB-like approximation. To identify boundary conditions on the attractor a method based on the prehistory probability distribution Dykman et al. [1992] can conveniently be employed [Dykman et al., 1996; Luchinsky et al., 1997; Luchinsky, 2002; Silchenko et al., 2005]. The exact locations of the unstable periodic orbits can be found by use of the standard methods of nonlinear dynamics [Schmelcher & Diakonos, 1997]. It has also been found recently that in some quite general situations the boundary conditions can be identified directly from the known structure of non-hyperbolic attractors and fractal boundaries [Silchenko et al. , 2003a; Kraut & Grebogi, 2004].

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This paper presents fuzzy model-based designs for synchronization of new **chaotic** system. The T–S fuzzy models for new **chaotic** **systems** are exactly derived. Then utilizing an intelligent controller which based on brain emotional learning (BELBIC), this fuzzy **chaotic** system is synchronized. Numerical simulation results are presented to show the effectiveness of the proposed method.

Temperature and SAR measurements we performed showed also the importance of the morphology of the phantom: the heating induced in the same implant con- figurations when placed in the rectangular phantom was higher than in the human-shaped phantom. The irregular surface of the human-shaped phantom implies a distance of the implant from the edges that varies point to point. This may partially explain such differences. Furthermore, the currents that are induced in the rectangular phantom, follow straight regular paths that provide a better coupling with straight metallic wires or **leads**. Our findings high- light the factors that have significant effects on RF-induced heating of implanted wires and **leads**. These factors must be taken into account by those who plan to study, model MRI heating of implants. Also our data should help those who wish to develop guidelines for defining safe medical implants for MRI patients. In addition, our database of the entire set of measurements can help those who wish to

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This paper studies the problem of isochronal synchronization of **chaotic** **systems** with time-de- layed mutual coupling. Based on the invariance principle of differential equations, an adaptive feedback scheme is proposed for the stability of isochronal synchronization between two identical **chaotic** **systems**. Unlike the usual linear feedback, the variable feedback strength is automatically adapted to isochronally synchronize two identical **chaotic** **systems** with delay-coupled, so this scheme is analytical, and simple to implement in practice. Simulation results show that the iso- chronal synchronization behavior is determined by time delay.

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