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We study the topological entropy of chaotic repellers formed by those points in a given chaotic attractor that never visit some small forbidden hole-region in the phase space. The hole is a set of points in the phase space that have a sequence $\alpha=(\alpha_0\alpha_1...\alpha_{l-1})$ as the first $l$ letters in their itineraries. We point out that the difference between the topological entropies of the attractor and the embedded repeller is for most choices of $\alpha$ approximately equal to the Parry measure corresponding to $\alpha$, $\mu_P(\alpha)$. When the hole encompasses a point of a short periodic orbit, the entropy difference is significantly smaller than $\mu_P(\alpha)$. This discrepancy is described by the formula which relates the length of the short periodic orbit, the Parry measure $\mu_P(\alpha)$, and the topological entropies of the two chaotic sets., Comment: Submitted to Physica D

We present a method for detecting the dimension of a dynamical system encompassing simultaneously two distinct discrete time series. This method is an extension of the technique introduced by Grassberger and Procaccia for single time series and allows to evaluate the common correlation dimension of the chaotic attractor. The method is applied to some mathematical models and to multiple single-neuron spike trains

We analyze and implement fully discrete schemes for periodic initial value problems for a general class of dispersively modified Kuramoto-Sivashinsky equations. Time discretizations are constructed using linearly implicit schemes and spectral methods are used for the spatial discretization. The general case analyzed covers several physical applications arising in multiphase hydrodynamics and the emerging dynamics arise from a competition of long-wave instability (negative diffusion), short-wave damping (fourth order stabilization), nonlinear saturation (Burgers nonlinearity), and dispersive effects. The solutions of such systems typically converge to compact absorbing sets of finite dimension (i.e., global attractors) and are characterized by chaotic behavior. Our objective is to employ schemes which capture faithfully these chaotic dynamics. In the general case the dispersive term is taken to be a pseudodifferential operator which is allowed to have higher order than the familiar fourth order stabilizing term in the Kuramoto-Sivashinsky equation. In such instances we show that first and second order time-stepping schemes are appropriate and provide convergence proofs for the schemes. In physical situations when the dispersion is of lower order than the fourth order stabilization term (for example, a hybrid Kuramoto-Sivashinsky-Korteweg-deVries equation also known as the Kawahara equation in hydrodynamics), higher order time-stepping schemes can be used and we analyze and implement schemes of order six or less. We derive optimal order error estimates throughout and utilize the schemes to compute the long time dynamics and to characterize the attractors. Various numerical diagnostic tools are implemented, such as the projection of the infinite-dimensional dynamics to one-dimensional return maps that enable us to probe the geometry of the attractors quantitatively. Such results are possible only if computations are carried out for very long times (we provide examples where integrations are carried out for 10 8 time units), and it is shown that the schemes used here are very well suited for such tasks. For illustration, computations are carried out for third order dispersion (the Kawahara equation) as well as fifth order dispersion (the Benney-Lin equation) but the methods developed here are applicable for rather general dispersive terms with similar accuracy characteristics. © 2012 Society for Industrial and Applied Mathematics. 34 2 A792 A813 Cited By :12

Τα πρωτότυπα αποτελέσματα της διατριβής περιέχονται στα κεφ. 2, 3 και 4. Στο κεφ. 2 μελετούμε με αριθμητικές μεθόδους ένα 3-διάστατο διακριτό μοντέλο μακροοικονομίας με σταθερές ισοτιμίες. Χρησιμοποιώντας μια μέθοδο πλέγματος, βρίσκουμε την περιοχή ευστάθειας στον παραμετρικό χώρο, προσδιορίζουμε την καμπύλη διακλαδώσεων Hopf-Neimark και θεωρούμε σύντομα την εμφάνιση “γλωσσών” Arnold. Υπολογίζονται διαγράμματα διακλαδώσεων και εκθέτη Lyapunov που δίνουν πληροφορίες για τους επιχειρηματικούς κύκλους και την πολύπλοκη δυναμική του μοντέλου και. παρουσιάζουμε παραδείγματα κυκλικών και χαοτικών ελκυστών. Στο κεφ. 3 μελετούμε με τις ίδιες μεθόδους ένα διακριτό μοντέλο αλληλεπίδρασης περιοχών με σταθερές ισοτιμίες, επέκταση του προηγούμενου μοντέλου σε 5 διαστάσεις. Στόχος ήταν να δείξουμε πόσο εφικτή και αποτελεσματική είναι μία αριθμητική μελέτη για ηπίως πολυδιάστατα διακριτά δυναμικά συστήματα με πολλές παραμέτρους. Βρήκαμε ότι η κίνηση κεφαλαίων δεν αρκεί για τη δημιουργία κύκλων όταν είναι χαμηλή η εμπορική αλληλεπίδραση. Το κατώφλι εμπορίου προβλέπεται περίπου στο 15% των εμπορικών συναλλαγών. Αντίθετα, το μοντέλο δεν προβλέπει αναγκαίο ελάχιστο επίπεδο κίνησης κεφαλαίων για την εμφάνιση των κύκλων. Δίνουμε παραδείγματα διαγραμμάτων διακλάδωσης και εκθέτη Lyapunov που δείχνουν την εμφάνιση κύκλων ή ακολουθίας διπλασιασμού περιόδου, και παραδείγματα της ανάπτυξής τους. Το κεφ. 4 περιέχει σύντομη περιγραφή συμπληρωματικών αποτελεσμάτων στα παραπάνω μοντέλα, και στα αντίστοιχα μοντέλα μεταβλητής ισοτιμίας συναλλάγματος, καθώς και κατευθύνσεις μελλοντικής έρευνας. Στο κεφ. 5 περιγράφονται σύντομα οι υπολογιστικές τεχνικές που χρησιμοποιήσαμε. Η διατριβή δείχνει την αποτελεσματικότητα της αριθμητικής προσέγγισης για πολυδιάστατα διακριτά μοντέλα. The original results of the dissertation are contained in ch. 2, 3 and 4, and concern mainly the problem of business cycles. In ch. 2 we explore numerically a 3D discrete Kaldorian macrodynamic model of open economy with fixed exchange rates. Using a grid search method we determine the stability region in parameter space, and the Hopf-Neimark bifurcation curve, and discuss briefly the occurrence of Arnold tongues. Bifurcation and Lyapunov exponent diagrams are computed providing information on the business cycles and illustrating the complex dynamics involved. Examples of cycles and chaotic attractors are presented. In ch. 3 we explore a 5D extension of the previous model using the same methods. The aim was to demonstrate the feasibility and effectiveness of the numerical approach for discrete dynamical systems of moderately high dimensionality and several parameters. We found that capital movement is not sufficient to generate interregional business cycles when trade interaction is low. The trade threshold is predicted at about 15% of trade transactions. By contrast, no minimum level of capital mobility exists as a requirement for the emergence of business cycles. Examples of bifurcation and Lyapunov exponent diagrams illustrating the occurrence of cycles or period doubling, and examples of their development, are given. Ch. 4 contains a short description of complementary results on the above models, and on two other models which extend the previous models to the case of flexible exchange rates, as well as some lines of future research. In ch. 5, the computational techniques employed in the present study are briefly described. The dissertation indicates the effectiveness of the numerical approach for high dimensional discrete models.

In this paper a periodic parameter switching scheme is applied to the Hindmarsh-Rose neuronal system to synthesize certain attractors. Results show numerically, via computer graphic simulations, that the obtained synthesized attractor belongs to the class of all admissible attractors for the Hindmarsh-Rose neuronal system and matches the averaged attractor obtained with the control parameter replaced with the averaged switched parameter values. This feature allows us to imagine that living beings are able to maintain vital behavior while the control parameter switches so that their dynamical behavior is suitable for the given environment., Comment: published in Nonlinear Dynamics

This document shows a software that shows different chaotic generator, as continuous as discrete time. The software gives the option for obtain the different signals, using different parameters and initial condition value. The program shows then critical parameter for each model. All theses models are capable of encrypter information, this software show it too., {"references":["L.M... Pechora and T.L. Carroll, \"Synchronization in chaotic\nsystems,\"Phys. Rev. Lett. 64, 821-824 (1990).","L. M. Pecora and T. L. Carroll, \"Circuit implementation of\nsynchronizedchaos with applications to communications,\" Phys. Rev. A\n44, (1991).","Special Issue on Chaos synchronization and control: Theory and\napplications IEEE Trans. Circuits Syst. I, 44, (1997).","Special Issue on Control and synchronization of chaos, Int. J. Bifurc.\nChaos, 10, (2000).","C. Cruz-Hern├índez and H. Nijmeijer, \"Synchronization through\nfiltering,\" Int. J. Bifurc. Chaos, 10, 763-775 (2000). Synchronization\nthrough extended Kalman filtering. In: Nijmeijer H, Fossen TI, editors.\nNew trends in nonlinear observer design. Lecture notes in control and\ninformation sciences, 244 London: Springer; 469-490, (1999).","H. Sira-Ram├¡rez and C. Cruz-Hern├índez, \"Synchronization of\nchaotic systems: a generalized Hamiltonian systems approach,\" Int. J.\nBifurc. Chaos, 11, 1381-1395 (2001). And in: Proceedings of the\nAmerican Control Conference, Chicago, USA, 769-773 (2000).","D. L├│pez-Mancilla and C. Cruz-Hern├índez, \"Output synchronization\nof chaotic systems: model-matching approach with application to secure\ncommunication,\" Nonlinear Dynamics and Systems Theory, 5, 141-15\n(2005).","U. Feldmann, M. Hasler and W. Schwarz, \"Communication by\nchaotic signals: the inverse system approach,\" Int. J. Circuits Theory and\nApplications, 24, 551-579 (1996).","H. Nijmeijer and I. M. Y. Mareels, \"An observer looks\natsynchronization,\" IEEE Trans. Circuits Syst. I, 44, 882-890 (1997).\n[10] Lorenz, E. N. , \"Deterministic nonperiodic flow\". J. Atmos. Sci. 20: 130-\n141. doi:10.1175/1520-0469(1963)0202.0.CO;2., (1963).\n[11] Matsumoto, Takashi , \"A Chaotic Attractor from Chua's Circuit\". IEEE\nTransactions on Circuits and Systems (IEEE) CAS-31 (12): 1055-1058.\n(1984).\n[12] Madan, Rabinder N., \"Chua's circuit: a paradigm for chaos\". River\nEdge, N.J.: World Scientific Publishing Company. ISBN 9810213662,\n(1993).\n[13] O. E. Rössler, \"An Equation for Continuous Chaos\".\nPhysics Letters 57A (5): 397-398M., (1976).\n[14] Hénon, \"A two-dimensional mapping with a strange attractor\".\nCommunications in Mathematical Physics 50: 69-77.\ndoi:10.1007/BF01608556, (1976).\n[15] Eric W. Weisstein, Logistic Equation at MathWorld.\n[16] R.M. May, \"Simple mathematical models with very complicated\ndynamics\". Nature 261: 459. doi:10.1038/261459a0, 1976"]}

Contains fulltext : 56948.pdf (Publisher’s version ) (Closed access) This study assessed the feasibility of online calculation of the correlation integral (C(r)) aiming to apply C(r)-derived statistics. For real-time application it is important to reduce calculation time. It is shown how our method works for EEG time series. Methods: To achieve online calculation of C(r) a non-randomly subset of inter vector distances was chosen and computer code was optimized. The effect of distance exclusion was investigated for both non-randomly and randomly chosen subsets. A C(r)-derived statistic was computed: an effective correlation dimension (CD) following the Grassberger-Procaccia-Takens approach. Results: By taking a subset of the maximum possible number of distances, C(r)-computation time could be easily 100 times reduced with marginal changes and minor variability in the C(r)-derived statistic CD. Applied to the EEG CD gives a good indication of the depth of anesthesia. Conclusions: If applied to the EEG, apparently a large number of distances can be omitted in the calculation of C(r) with minimal consequences. This outcome confirms Hoeffding's theory of U-statistics and hence it is expected to occur generally, also for non-EEG time series.

We study local and global correlations between the naturally invariant measure of a chaotic one-dimensional map f and the conditionally invariant measure of the transiently chaotic map f(H). The two maps differ only within a narrow interval H, while the two measures significantly differ within the images f(l)(H), where l is smaller than some critical number l(c). We point out two different types of correlations. Typically, the critical number l(c) is small. The chi(2) value, which characterizes the global discrepancy between the two measures, typically obeys a power law dependence on the width epsilon of the interval H, with the exponent identical to the information dimension. If H is centered on an image of the critical point, then l(c) increases indefinitely with the decrease of epsilon, and the chi(2) value obeys a modulated power-law dependence on epsilon.

This paper is concerned with synchronization and control of chaotic nonlinear dynamical systems. First, a unified frame for both the synchronization and the control problem is described. Then by modifying a feedback plus feedforward controller, a discontinuous strategy is synthesized which exploits the boundedness of chaotic attractors and limit cycles. Finally, an adaptive approach is investiagted and an adaptive estimation law is implemented. The result, which is both global and not reliant on complete knowledge of the systems involved, is rigorously proved by means of Lyapunov Theory. An application to the synchronization of two chaotic systems is presented

Recent numerical studies suggest that in pipe and related shear flows, the region of phase space separating laminar from turbulent motion is organized by a chaotic attractor, called an edge state, which mediates the transition process. We here confirm the existence of the edge state in laboratory experiments. We observe that it governs the dynamics during the decay of turbulence underlining its potential relevance for turbulence control. In addition we unveil two unstable traveling wave solutions underlying the experimental flow fields. This observation corroborates earlier suggestions that unstable solutions organize turbulence and its stability border.

The continuous-time evolution of many systems is accompanied by striking changes in the physical variables that are repeated more or less regularly. This situation typically arises in the biological sciences, and is encountered in neurobiology ~neuron firings corresponding to voltage spikes @1#!, cardiology ( R peaks of electrocardiograms @2#!, membrane biology ~bursting oscillations of the cell membrane potential @3#!, etc. Systems with this type of dynamics are often analyzed by processing time intervals between the relevant events @for example, interspike intervals ~ISI’s !@ 4##. Different models of spike generation are known. Within the framework of integrate-and-fire~IF! models @5‐8#, a signal S(t), being a function of the variables of a lowdimensional dynamical system ~DS! is integrated from some moment T 0 . The times T i when spikes occur can then be defined by the equation