Journal articles: 'Chaotic modes'

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Zelevinsky, Vladimir. "Chaotic dynamics and collective modes." Nuclear Physics A 649, no. 1-4 (March 1999): 403–11. http://dx.doi.org/10.1016/s0375-9474(99)00090-1.

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YAO, WEIGUANG, PEI YU, CHRISTOPHER ESSEX, and MATT DAVISON. "COMPETITIVE MODES AND THEIR APPLICATION." International Journal of Bifurcation and Chaos 16, no. 03 (March 2006): 497–522. http://dx.doi.org/10.1142/s0218127406014976.

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We investigate nonlinear dynamical systems from the mode competition point of view, and propose the necessary conditions for a system to be chaotic. We conjecture that a chaotic system has at least two competitive modes (CM's). For a general nonlinear dynamical system, we give a simple, dynamically motivated definition of mode suitable for this concept. Since for most chaotic systems it is difficult to obtain the form of a CM, we focus on the competition between the corresponding modulated frequency components of the CM's. Some direct applications result from the explicit form of the frequency functions. One application is to estimate parameter regimes which may lead to chaos. It is shown that chaos may be found by analyzing the frequency function of the CM's without applying a numerical integration scheme. Another application is to create new chaotic systems using custom-designed CM's. Several new chaotic systems are reported.

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Evano, Benjamin, François Lignières, and Bertrand Georgeot. "Regularities in the spectrum of chaotic p-modes in rapidly rotating stars." Astronomy & Astrophysics 631 (November 2019): A140. http://dx.doi.org/10.1051/0004-6361/201936459.

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Context. Interpreting the oscillations of massive and intermediate mass stars remains a challenging task. In fast rotators, the oscillation spectrum of p-modes is a superposition of sub-spectra which corresponds to different types of modes, among which island modes and chaotic modes are expected to be the most visible. This paper is focused on chaotic modes, which have not been thoroughly studied before. Aims. We study the properties of high frequency chaotic p-modes in a polytropic model. Unexpected peaks appear in the frequency autocorrelations of the spectra. Our goal is to find a physical interpretation for these peaks and also to provide an overview of the mode properties. Methods. We used the 2D oscillation code “TOP” to produce the modes and acoustic ray simulations to explore the wave properties in the asymptotic regime. Using the tools developed in the field of quantum chaos (or wave chaos), we derived an expression for the frequency autocorrelation involving the travel time of acoustic rays. Results. Chaotic mode spectra were previously thought to be irregular, that is, described only through their statistical properties. Our analysis shows the existence, in chaotic mode spectra, of a pseudo large separation. This means that chaotic modes are organized in series, such that the modes in each series follow a nearly regular frequency spacing. The pseudo large separation of chaotic modes is very close to the large separation of island modes. Its value is related to the sound speed averaged over the meridional plane of the star. In addition to the pseudo large separation, other correlations appear in the numerically calculated spectra. We explain their origin by the trapping of acoustic rays near the stable islands.

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Peña, M., and E. Kalnay. "Separating fast and slow modes in coupled chaotic systems." Nonlinear Processes in Geophysics 11, no. 3 (July 27, 2004): 319–27. http://dx.doi.org/10.5194/npg-11-319-2004.

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Abstract. We test a simple technique based on breeding to separate fast and slow unstable modes in coupled systems with different time scales of evolution and variable amplitudes. The technique takes advantage of the earlier saturation of error growth rate of the fastest mode and of the lower value of the saturation amplitude of perturbation of either the fast or the slow modes. These properties of the coupled system allow a physically-based selection of the rescaling time interval and the amplitude of initial perturbations in the "breeding" of unstable modes (Toth and Kalnay, 1993, 1996, 1997; Aurell et al., 1997; Boffetta et al., 1998) to isolate the desired mode. We perform tests in coupled models composed of fast and slow versions of the Lorenz (1963) model with different strengths of coupling. As examples we present first a coupled system which we denote "weather with convection", with a slow, large amplitude model coupled with a fast, small amplitude model, second an "ENSO" system with a "tropical atmosphere" strongly coupled with a "tropical ocean", and finally a triply coupled system denoted "tropical-extratropical" in which a fast model (representing the "extratropical atmosphere") is loosely coupled to the "ENSO" system. We find that it is always possible to isolate the fast modes by taking the limit of small amplitudes and short rescaling intervals, in which case, as expected, the results are the same as the local Lyapunov growth obtained with the linear tangent model. In contrast, slow modes cannot be isolated with either Lyapunov or Singular vectors, since the linear tangent and adjoint models are dominated by the fast modes. Breeding is successful in isolating slow modes if rescaling intervals and amplitudes are chosen from physically appropriate scales.

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Lindgren, Kristian, and Bengt Å. G. Månsson. "Entropy Production in a Chaotic Chemical System." Zeitschrift für Naturforschung A 41, no. 9 (September 1, 1986): 1111–17. http://dx.doi.org/10.1515/zna-1986-0904.

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The average rate of entropy production in a homogenous chemical system is investigated in oscillating periodic and chaotic modes as well as in coexisting stationary states. The simulations are based on an abstract model of a chemical reaction system with three freely varying concentrations. Five concentrations are assumed to be kept constant by suitable flows across the boundary. A fixed concentration is used as a control parameter. Second order mass action kinetics with reverse reaction is used. An unexpected result is that periodic modes in some windows in the chaotic interval have higher average rate of entropy production than the surrounding chaotic modes. A chaotic mode coexists with a stable stationary state with smaller entropy production. A unique (unstable) stationary state produces more entropy than the corresponding oscillating mode.

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KUSMARTSEV, FEO V., and KARL E. KÜRTEN. "CHAOTIC MODES IN SCALE FREE OPINION NETWORKS." International Journal of Modern Physics B 23, no. 20n21 (August 20, 2009): 4001–20. http://dx.doi.org/10.1142/s0217979209063225.

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In this paper, we investigate processes associated with formation of public opinion in varies directed random, scale free and small-world social networks. The important factor of the opinion formation is the existence of contrarians which were discovered by Granovetter in various social psychology experiments1,2,3 long ago and later introduced in sociophysics by Galam.4 When the density of contrarians increases the system behavior drastically changes at some critical value. At high density of contrarians the system can never arrive to a consensus state and periodically oscillates with different periods depending on specific structure of the network. At small density of the contrarians the behavior is manifold. It depends primary on the initial state of the system. If initially the majority of the population agrees with each other a state of stable majority may be easily reached. However when originally the population is divided in nearly equal parts consensus can never be reached. We model the emergence of collective decision making by considering N interacting agents, whose opinions are described by two state Ising spin variable associated with YES and NO. We show that the dynamical behaviors are very sensitive not only to the density of the contrarians but also to the network topology. We find that a phase of social chaos may arise in various dynamical processes of opinion formation in many realistic models. We compare the prediction of the theory with data describing the dynamics of the average opinion of the USA population collected on a day-by-day basis by varies media sources during the last six month before the final Obama-McCain election. The qualitative ouctome is in reasonable agreement with the prediction of our theory. In fact, the analyses of these data made within the paradigm of our theory indicates that even in this campaign there were chaotic elements where the public opinion migrated in an unpredictable chaotic way. The existence of such a phase of social chaos reflects a main feature of the human being associated with some doubts and uncertainty and especially associated with contrarians which undoubtly exist in any society.

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Cejnar, Pavel, Pavel Stránský, and Michal Macek. "Regular and Chaotic Collective Modes in Nuclei." Nuclear Physics News 21, no. 4 (October 2011): 22–27. http://dx.doi.org/10.1080/10619127.2011.629919.

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Hellberg, Carl S., and Steven A. Orszag. "Chaotic behavior of interacting elliptical instability modes." Physics of Fluids 31, no. 1 (January 1988): 6–8. http://dx.doi.org/10.1063/1.867010.

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Podvalny, S. L., and E. M. Vasiljev. "Intensification of heat transfer in chaotic modes." IOP Conference Series: Materials Science and Engineering 1035, no. 1 (January 1, 2021): 012046. http://dx.doi.org/10.1088/1757-899x/1035/1/012046.

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Karimov, Timur, Denis Butusov, Valery Andreev, Artur Karimov, and Aleksandra Tutueva. "Accurate Synchronization of Digital and Analog Chaotic Systems by Parameters Re-Identification." Electronics 7, no. 7 (July 20, 2018): 123. http://dx.doi.org/10.3390/electronics7070123.

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The verification of the digital models of chaotic systems and processes is a valuable problem in many practical applications, such as nonlinear control and communications. In our study, we propose a hybrid technique for chaotic systems’ identification, based on the chaotic synchronization of digital and analog counterparts and a numerical optimization method used for the fine tuning of parameters. An analog circuit implementing the Rössler oscillator with digitally controlled parameters was chosen as an identification object, and the FPGA model was used as a digital counterpart for coupling and parameter retrieval. The synchronization between analog and digital chaotic models can be used to estimate the quality of an identification procedure. The results of this study clarify the practical bounds of digital and analog systems’ equivalence. They also contribute to the problem of designing technical systems possessing advantages of both analog and digital chaotic generators (e.g., a high accuracy and protection from quasi-chaotic oscillation modes).

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Mallory, Kristina, and Robert A. Van Gorder. "Competitive Modes for the Detection of Chaotic Parameter Regimes in the General Chaotic Bilinear System of Lorenz Type." International Journal of Bifurcation and Chaos 25, no. 04 (April 2015): 1530012. http://dx.doi.org/10.1142/s0218127415300128.

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We study chaotic behavior of solutions to the bilinear system of Lorenz type developed by Celikovsky and Vanecek [1994] through an application of competitive modes. This bilinear system of Lorenz type is one possible canonical form holding the Lorenz equation as a special case. Using a competitive modes analysis, which is a completely analytical method allowing one to identify parameter regimes for which chaos may occur, we are able to demonstrate a number of parameter regimes which admit a variety of distinct chaotic behaviors. Indeed, we are able to draw some interesting conclusions which relate the behavior of the mode frequencies arising from writing the state variables for the Celikovsky–Vanecek model as coupled oscillators, and the types of emergent chaotic behaviors observed. The competitive modes analysis is particularly useful if all but one of the model parameters are fixed, and the remaining free parameter is used to modify the chaos observed, in a manner analogous to a bifurcation parameter. Through a thorough application of the method, we are able to identify several parameter regimes which give new dynamics (such as specific forms of chaos) which were not observed or studied previously in the Celikovsky–Vanecek model. Therefore, the results demonstrate the advantage of the competitive modes approach for detecting new parameter regimes leading to chaos in third-order dynamical systems.

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VAN GORDER, ROBERT A., and S. ROY CHOUDHURY. "CLASSIFICATION OF CHAOTIC REGIMES IN THE T SYSTEM BY USE OF COMPETITIVE MODES." International Journal of Bifurcation and Chaos 20, no. 11 (November 2010): 3785–93. http://dx.doi.org/10.1142/s0218127410028033.

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We study chaotic behavior of the T system, a three-dimensional autonomous nonlinear system introduced by G. Tigan [Analysis of a dynamical system derived from the Lorenz system, Sci. Bull. Politehnica Univ Timisoara50 (2005) 61–72] which has potential application in secure communications. The recently-developed technique of competitive modes analysis is applied to determine parameter regimes for which the system may exhibit chaotic behavior. We verify that the T system exhibits interesting behaviors in the many parameter regimes thus obtained, thereby demonstrating the great utility of the competitive modes approach in delineating chaotic regimes in multiparemeter systems, where their identification can otherwise involve tedious numerical searches. An additional, novel finding is that one may use competitive modes "at infinity" in order to identify parameter regimes admitting stable equilibria in dynamical models such as the T system.

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Parovik, R. I. "Chaotic modes of a non-linear fractional oscillator." IOP Conference Series: Materials Science and Engineering 919 (September 26, 2020): 052040. http://dx.doi.org/10.1088/1757-899x/919/5/052040.

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Mangin, Laurence, Christine Clerici, Thomas Similowski, and Chi-Sang Poon. "Chaotic dynamics of cardioventilatory coupling in humans: effects of ventilatory modes." American Journal of Physiology-Regulatory, Integrative and Comparative Physiology 296, no. 4 (April 2009): R1088—R1097. http://dx.doi.org/10.1152/ajpregu.90862.2008.

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Cardioventilatory coupling (CVC), a transient temporal alignment between the heartbeat and inspiratory activity, has been studied in animals and humans mainly during anesthesia. The origin of the coupling remains uncertain, whether or not ventilation is a main determinant in the CVC process and whether the coupling exhibits chaotic behavior. In this frame, we studied sedative-free, mechanically ventilated patients experiencing rapid sequential changes in breathing control during ventilator weaning during a switch from a machine-controlled assistance mode [assist-controlled ventilation (ACV)] to a patient-driven mode [inspiratory pressure support (IPS) and unsupported spontaneous breathing (USB)]. Time series were computed as R to start inspiration (RI) and R to the start of expiration (RE). Chaos was characterized with the noise titration method (noise limit), largest Lyapunov exponent (LLE) and correlation dimension (CD). All the RI and RE time series exhibit chaotic behavior. Specific coupling patterns were displayed in each ventilatory mode, and these patterns exhibited different linear and chaotic dynamics. When switching from ACV to IPS, partial inspiratory loading decreases the noise limit value, the LLE, and the correlation dimension of the RI and RE time series in parallel, whereas decreasing intrathoracic pressure from IPS to USB has the opposite effect. Coupling with expiration exhibits higher complexity than coupling with inspiration during mechanical ventilation either during ACV or IPS, probably due to active expiration. Only 33% of the cardiac time series (RR interval) exhibit complexity either during ACV, IPS, or USB making the contribution of the cardiac signal to the chaotic feature of the coupling minimal. We conclude that 1) CVC in unsedated humans exhibits a complex dynamic that can be chaotic, and 2) ventilatory mode has major effects on the linear and chaotic features of the coupling. Taken together these findings reinforce the role of ventilation in the CVC process.

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ZHANG, WEI, and JING LI. "GLOBAL ANALYSIS FOR A NONLINEAR VIBRATION ABSORBER WITH FAST AND SLOW MODES." International Journal of Bifurcation and Chaos 11, no. 08 (August 2001): 2179–94. http://dx.doi.org/10.1142/s0218127401003334.

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A two-degree-of-freedom model of a nonlinear vibration absorber is considered in this paper. Both the global bifurcations and chaotic dynamics of the nonlinear vibration absorber are investigated. The nonlinear equations of motion of this model are derived. The method of multiple scales is used to find the averaged equations. Based on the averaged equations, the theory of normal form is used to obtain the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues by Maple software program. The fast and slow modes may simultaneously exist in the averaged equations. On the basis of the normal form, the global bifurcation and the chaotic dynamics of the nonlinear vibration absorber are analyzed by a global perturbation method developed by Kovacic and Wiggins. The chaotic motion of this model is also found by numerical simulation.

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Jensen, O. E. "Chaotic Oscillations in a Simple Collapsible-Tube Model." Journal of Biomechanical Engineering 114, no. 1 (February 1, 1992): 55–59. http://dx.doi.org/10.1115/1.2895450.

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A steady flow through a segment of externally pressurized, collapsible tube can become unstable to a wide variety of self-excited oscillations of the internal flow and tube walls. A simple, one-dimensional model of the conventional laboratory apparatus, which has been shown previously to predict steady flows and multiple modes of oscillation, is investigated numerically here. Large amplitude oscillations are shown to have a relaxation structure, and the nonlinear interaction between different modes is shown to give rise to quasiperiodic and apparently aperiodic behavior. These predictions are shown to compare favorably with experimental observations.

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Drybin, Y. A., S. V. Sadau, and V. S. Sadau. "Digital model of a pseudo-random number generator based on a continuous chaotic system." Informatics 17, no. 4 (January 3, 2021): 36–47. http://dx.doi.org/10.37661/1816-0301-2020-17-4-36-47.

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It is shown that the choice of the time sampling parameter of the digital model of a continuous dynamic system with chaotic modes based on its dynamics makes it possible to control the characteristics of the output sequence, including avoiding short cycles and periodic behavior modes. On the example of the Lorentz system, the analysis of the law of motion of a chaotic system, linearized in the vicinity of points of stable and unstable equilibrium, is carried out, on the basis of which the parameters of the mathematical model of the generator of pseudo-random numbers are selected. The output sequence of numbers generated in proposed way is subjected to statistical and correlation analysis. Based on the results of the tests carried out, we can say that the obtained pseudo-random sequences based on continuous chaotic systems have statistically random properties and can be used in steganographic and cryptographic systems.

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Drybin, Y. A., S. V. Sadau, and V. S. Sadau. "Digital model of a pseudo-random number generator based on a continuous chaotic system." Informatics 17, no. 4 (January 3, 2021): 36–47. http://dx.doi.org/10.37661/1816-0301-2020-17-4-36-47.

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It is shown that the choice of the time sampling parameter of the digital model of a continuous dynamic system with chaotic modes based on its dynamics makes it possible to control the characteristics of the output sequence, including avoiding short cycles and periodic behavior modes. On the example of the Lorentz system, the analysis of the law of motion of a chaotic system, linearized in the vicinity of points of stable and unstable equilibrium, is carried out, on the basis of which the parameters of the mathematical model of the generator of pseudo-random numbers are selected. The output sequence of numbers generated in proposed way is subjected to statistical and correlation analysis. Based on the results of the tests carried out, we can say that the obtained pseudo-random sequences based on continuous chaotic systems have statistically random properties and can be used in steganographic and cryptographic systems.

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Lochan, Kshetrimayum, Binoy Krishna Roy, and Bidyadhar Subudhi. "Chaotic tip trajectory tracking and deflection suppression of a two-link flexible manipulator using second-order fast terminal SMC." Transactions of the Institute of Measurement and Control 41, no. 12 (May 17, 2019): 3292–308. http://dx.doi.org/10.1177/0142331218819700.

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The problem of chaotic tip trajectory tracking control for a planar assumed modes modelled two-link flexible manipulator is addressed. Tracking of such an apparently random-like (chaotic) desired trajectory is a challenging task. Initially, a PID-type sliding surface is designed in terms of the tip trajectory tracking error, then a second-order integral-type fast terminal sliding mode control is designed using the above-designed sliding surfaces. The desired chaotic trajectory is generated from a four-dimensional chaotic hyperjerk system. The proposed controller guarantees fast tracking performance with lower steady-state error and less control input. The model of a two-link flexible manipulator is obtained using the assumed modes method. The robustness of the proposed control method is evaluated in the presence of matched uncertainty and variability of payload. The performances of the proposed control technique are verified in terms of low tracking error and fast tip deflection suppression. The effectiveness of the proposed technique is validated using numerical simulations, and compared with the normal second-order sliding mode control (SMC) and another controller reported recently in the literature.

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Tsuda, Ichiro. "The plausibility of a chaotic brain theory." Behavioral and Brain Sciences 24, no. 5 (October 2001): 829–40. http://dx.doi.org/10.1017/s0140525x01420097.

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We consider the significance of high-dimensional transitory dynamics in the brain and mind. In particular, we highlight the roles of high-dimensional chaotic dynamical systems as an “adequate language” (Gelfand 1989), which should possess both explanatory and predictive power of description. We discuss the methods of description of dynamic behavior of the brain. These methods have been adopted to capture the averaged or deterministic complexity, and further to allow for discussion of a new approach to capture the complexity of the deviation from such an averaged complexity and also the complexity of interactive modes. We also give arguments in defense of our models for dynamic memory with chaotic itinerancy and Cantor coding. In addition, we discuss the reality that a model of the brain and mind should reflect.

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Zhang, Xiaoqiang, and Xuesong Wang. "Multiple-Image Encryption Algorithm Based on the 3D Permutation Model and Chaotic System." Symmetry 10, no. 11 (November 20, 2018): 660. http://dx.doi.org/10.3390/sym10110660.

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Large numbers of images are produced in many fields every day. The content security of digital images becomes an important issue for scientists and engineers. Inspired by the magic cube game, a three-dimensional (3D) permutation model is established to permute images, which includes three permutation modes, i.e., internal-row mode, internal-column mode, and external mode. To protect the image content on the Internet, a novel multiple-image encryption symmetric algorithm (block cipher) with the 3D permutation model and the chaotic system is proposed. First, the chaotic sequences and chaotic images are generated by chaotic systems. Second, the sender permutes the plain images by the 3D permutation model. Lastly, the sender performs the exclusive OR operation on permuted images. The simulation and algorithm comparisons display that the proposed algorithm possesses desirable encryption images, high security, and efficiency.

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Kai-fen, He. "Statistical Model for a Chaotic System of two Kinds of Interacting Modes." Communications in Theoretical Physics 7, no. 1 (January 1987): 15–26. http://dx.doi.org/10.1088/0253-6102/7/1/15.

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Rysev, P. V., D. V. Rysev, V. K. Fedorov, K. S. Shulga, and S. Yu Pruss. "IDENTIFYING AND MODELING CHAOTIC MODES IN ELECTRICAL POWER SYSTEMS." Dynamics of Systems, Mechanisms and Machines 5, no. 3 (2017): 101–7. http://dx.doi.org/10.25206/2310-9793-2017-5-3-101-107.

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Kosarev, B. A., O. A. Lysenko, V. K. Fedorov, and R. N. Khamitov. "CHAOTIC OPERATING MODES OF POWER SYSTEMS WITH DISTRIBUTED GENERATION." Actual Issues Of Energy 2, no. 1 (2020): 027–31. http://dx.doi.org/10.25206/2686-6935-2020-2-1-27-31.

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Kryzhevich, S. G., and V. A. Pliss. "Chaotic modes of oscillation of a vibro-impact system." Journal of Applied Mathematics and Mechanics 69, no. 1 (January 2005): 13–26. http://dx.doi.org/10.1016/j.jappmathmech.2005.01.002.

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Chattopadhyay, Saranyu, Pranesh Santikellur, Rajat Subhra Chakraborty, Jimson Mathew, and Marco Ottavi. "A Conditionally Chaotic Physically Unclonable Function Design Framework with High Reliability." ACM Transactions on Design Automation of Electronic Systems 26, no. 6 (November 30, 2021): 1–24. http://dx.doi.org/10.1145/3460004.

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Physically Unclonable Function (PUF) circuits are promising low-overhead hardware security primitives, but are often gravely susceptible to machine learning–based modeling attacks. Recently, chaotic PUF circuits have been proposed that show greater robustness to modeling attacks. However, they often suffer from unacceptable overhead, and their analog components are susceptible to low reliability. In this article, we propose the concept of a conditionally chaotic PUF that enhances the reliability of the analog components of a chaotic PUF circuit to a level at par with their digital counterparts. A conditionally chaotic PUF has two modes of operation: bistable and chaotic , and switching between these two modes is conveniently achieved by setting a mode-control bit (at a secret position) in an applied input challenge. We exemplify our PUF design framework for two different PUF variants—the CMOS Arbiter PUF and a previously proposed hybrid CMOS-memristor PUF, combined with a hardware realization of the Lorenz system as the chaotic component. Through detailed circuit simulation and modeling attack experiments, we demonstrate that the proposed PUF circuits are highly robust to modeling and cryptanalytic attacks, without degrading the reliability of the original PUF that was combined with the chaotic circuit, and incurs acceptable hardware footprint.

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Wang, Congqing, and Linfeng Wu. "Chaotic Vibration Prediction of a Free-Floating Flexible Redundant Space Manipulator." Shock and Vibration 2016 (2016): 1–12. http://dx.doi.org/10.1155/2016/6015275.

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The dynamic model of a planar free-floating flexible redundant space manipulator with three joints is derived by the assumed modes method, Lagrange principle, and momentum conservation. According to minimal joint torque’s optimization (MJTO), the state equations of the dynamic model for the free-floating redundant space manipulator are described. The PD control using the tracking position error and velocity error in the manipulator is introduced. Then, the chaotic dynamic behavior of the manipulator is analyzed by chaotic numerical methods, in which time series, phase plane portrait, Poincaré map, and Lyapunov exponents are used to analyze the chaotic behavior of the manipulator. Under certain conditions for the joint torque optimization and initial values, chaotic vibration motion of the space manipulator can be observed. The chaotic time series prediction scheme for the space manipulator is presented based on the theory of phase space reconstruction under Takens’ embedding theorem. The trajectories of phase space can be reconstructed in embedding space, which are equivalent to the original space manipulator in dynamics. The one-step prediction model for the chaotic time series and the chaotic vibration was established by using support vector regression (SVR) prediction model with RBF kernel function. It has been proved that the SVR prediction model has a good performance of prediction. The experimental results show the effectiveness of the presented method.

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COURBAGE, M., and V. I. NEKORKIN. "MAP BASED MODELS IN NEURODYNAMICS." International Journal of Bifurcation and Chaos 20, no. 06 (June 2010): 1631–51. http://dx.doi.org/10.1142/s0218127410026733.

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This tutorial reviews a new important class of mathematical phenomenological models of neural activity generated by iterative dynamical systems: the so-called map-based systems. We focus on 1-D and 2-D maps for the replication of many features of the neural activity of a single neuron. It was shown that such systems can reproduce the basic activity modes such as spiking, bursting, chaotic spiking-bursting, subthreshold oscillations, tonic and phasic spiking, normal excitability, etc. of the real biological neurons. We emphasize on the representation of chaotic spiking-bursting oscillations by chaotic attractors of 2-D models. We also explain the dynamical mechanism of formation of such attractors and transition from one mode to another. We briefly present some synchronization mehanisms of chaotic spiking-bursting activity for two coupled neurons described by 1-D maps.

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Yang, B., C. S. Suh, and A. K. Chan. "Characterization and Detection of Crack-induced Rotary Instability." Journal of Vibration and Acoustics 124, no. 1 (July 1, 2001): 40–48. http://dx.doi.org/10.1115/1.1421053.

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System instability and chaotic response are the failure modes that could significantly impact the reliability and operating safety of high-speed rotor-dynamical machines. Initiation and propagation of surface cracks in rotary shafts are common causes for such failure modes. To be able to detect the onset and progression of these faults will considerably extend the lifetime and improve the reliability of the mechanical system. A wavelet-based algorithm effective in identifying mechanical chaotic response has been applied to determine the nonlinear dynamical characteristics of a model-based, cracked rotor. This investigation confirms reported correlation of surface crack breathing with rotor chaotic motions. The effectiveness of the algorithm in detecting rotor-dynamic instability induced by mechanical faults as contrast to algorithms that are based on nonlinear dynamics is discussed. The results show not just the feasibility of the algorithm in mechanical fault diagnosis but also suggest its applicability to in-line, real-time condition monitoring at both the system and component levels.

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Krot, A. M., and U. A. Sychou. "The analysis of chaotic regimes in Chua’s circuit with smooth nonlinearity based on the matrix decomposition method." Proceedings of the National Academy of Sciences of Belarus, Physical-Technical Series 63, no. 4 (January 12, 2019): 501–12. http://dx.doi.org/10.29235/1561-8358-2018-63-4-501-512.

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The scope of this work are electric circuits or electronic devices with chaotic regimes, in particular the Chua’s circuit. A nonlinear analysis of chaotic attractors based on the Krot’s method of matrix decomposition of vector functions in state-space of complex systems has been used to investigate the Chua’s circuit with smooth nonlinearity. It includes an analysis of linear term of the matrix series as well as an estimation of influence of high order terms of this series on stability of complex system under investigation. Here the method of matrix decomposition has been applied to analysis of the Chua’s attractor. The terms of matrix series have been used to create a simulation model and to reconstruct an attractor of chaotic modes. The proposed simulation model makes it possible to separate an influence of nonlinearities on forming a chaotic regime of the Chua’s circuit. Usage of both the matrix decomposition method and computational experiment has allowed us to find out that the initial turbulence model proposed by L. D. Landau is suitable for set-up description of the chaotic regime of the Chua’s circuit. It is shown that a mode of hard self-excitation in the Chua’s circuit leads to its chaotic regime operating with a double-scroll attractor in the state-space. The results might be used to generate of chaotic oscillations or data encryption.

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Qi, Guoyuan, and Xiyin Liang. "Force Analysis of Qi Chaotic System." International Journal of Bifurcation and Chaos 26, no. 14 (December 30, 2016): 1650237. http://dx.doi.org/10.1142/s0218127416502370.

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The Qi chaotic system is transformed into Kolmogorov type of system. The vector field of the Qi chaotic system is decomposed into four types of torques: inertial torque, internal torque, dissipation and external torque. Angular momentum representing the physical analogue of the state variables of the chaotic system is identified. The Casimir energy law relating to the orbital behavior is identified and the bound of Qi chaotic attractor is given. Five cases of study have been conducted to discover the insights and functions of different types of torques of the chaotic attractor and also the key factors of producing different types of modes of dynamics.

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Zotov, Oleg, Boris Klain, and Nadezhda Kurazhkovskaya. "INFLUENCE OF THE ß SOLAR WIND PARAMETER ON STATISTICAL CHARACTERISTICS OF THE Ap INDEX IN THE SOLAR ACTIVITY CYCLE." Solar-Terrestrial Physics 5, no. 4 (December 17, 2019): 46–52. http://dx.doi.org/10.12737/stp-54201906.

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We have studied the effect of the β solar wind parameter (equal to the ratio of the plasma pressure to the magnetic pressure) on statistical characteristics of the Ap index reflecting the triggering behavior of the activity of Earth’s magnetosphere. The trigger effect of the dynamics of magnetospheric activity consists in the abrupt transition from the periodic mode to the chaotic mode in the solar activity cycle. It is shown that cumulative amplitude distribution functions and power spectra of the Ap index of both the periodic and chaotic modes are well approximated by power and exponential functions respectively. At the same time, the indices of power functions and the indices characterizing the slope of the Ap index spectrum differ significantly in magnitude for the periodic and chaotic modes. We have found that Ap nonlinearly depends on β for both the modes of magnetospheric dynamics. The maximum of the Ap index amplitude for periodic modes is observed when β>1; and for chaotic ones, when β<1. In almost every cycle of solar activity, the energy of the Ap index fluctuations of chaotic modes is higher than that of periodic ones. The results indicate intermittency and its associated turbulence of magnetospheric activity. The exponential character of the spectral density of the Ap index suggests that the behavior of magnetospheric activity is determined by its internal dynamics, which can be described by a finite number of deterministic equations. The trigger effect of magnetospheric activity is assumed to be due to the angle of inclination of the axis of the solar magnetic dipole to the ecliptic plane, on which the dynamics of the β parameter in the solar activity cycle depends.

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Roy Choudhury, S., and Daniel Mandragona. "A Chaotic Chemical Reactor With and Without Delay: Bifurcations, Competitive Modes, and Amplitude Death." International Journal of Bifurcation and Chaos 29, no. 02 (February 2019): 1950019. http://dx.doi.org/10.1142/s0218127419500196.

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Bifurcations in Huang’s chaotic chemical reactor leading from simple dynamics into chaotic regimes are considered. Following the linear stability analysis, the periodic orbit resulting from a Hopf bifurcation of any of the six fixed points is constructed analytically by the method of multiple scales, and its stability is determined from the resulting normal form and verified by numerical simulations. The dynamically rich range of parameters past the Hopf bifurcation is next explored. In order to bring some order to the search for parameter regimes with more complex dynamics, we employ the recent conjecture of Competitive Modes to find chaotic parameter sets in the large multiparameter space for this system. In addition, it is demonstrated that, by changing the point of view, one may tightly localize the chaotic attractor in shape and location in the phase space by mapping the Competitive Modes surfaces geometrically. Finally, we consider the effect of delay on the system, leading to the suppression of the Hopf bifurcation in some regimes, and also all of the subsequent complex dynamics. In modern terminology, this is an example of Amplitude Death, rather than Oscillation Death, as the complex system dynamics is quenched, with all the variables additionally settling to a fixed point of the original system.

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Ruks, Lewis, and Robert A. Van Gorder. "On the Inverse Problem of Competitive Modes and the Search for Chaotic Dynamics." International Journal of Bifurcation and Chaos 27, no. 10 (September 2017): 1730032. http://dx.doi.org/10.1142/s0218127417300324.

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Generalized competitive modes (GCM) have been used as a diagnostic tool in order to analytically identify parameter regimes which may lead to chaotic trajectories in a given first order nonlinear dynamical system. The approach involves recasting the first order system as a second order nonlinear oscillator system, and then checking to see if certain conditions on the modes of these oscillators are satisfied. In the present paper, we will consider the inverse problem of GCM: If a system of second order oscillator equations satisfy the GCM conditions, can we then construct a first order dynamical system from it which admits chaotic trajectories? Solving the direct inverse problem is equivalent to finding solutions to an inhomogeneous form of the Euler equations. As there are no general solutions to this PDE system, we instead consider the problem for restricted classes of functions for autonomous systems which, upon obtaining the nonlinear oscillatory representation, we are able to extract at least two of the modes explicitly. We find that these methods often make finding chaotic regimes a much simpler task; many classes of parameter-function regimes that lead to nonchaos are excluded by the competitive mode conditions, and classical knowledge of dynamical systems then allows us to tune the free parameters or functions appropriately in order to obtain chaos. To find new hyperchaotic systems, a similar approach is used, but more effort and additional considerations are needed. These results demonstrate one method for constructing new chaotic or hyperchaotic systems.

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Baram, Yoram. "Noninvertibility, Chaotic Coding, and Chaotic Multiplexity of Synaptically Modulated Neural Firing." Neural Computation 24, no. 3 (March 2012): 676–99. http://dx.doi.org/10.1162/neco_a_00239.

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Widely accepted neural firing and synaptic potentiation rules specify a cross-dependence of the two processes, which, evolving on different timescales, have been separated for analytic purposes, concealing essential dynamics. Here, the morphology of the firing rates process, modulated by synaptic potentiation, is shown to be described by a discrete iteration map in the form of a thresholded polynomial. Given initial synaptic weights, a firing activity is triggered by conductance. Elementary dynamic modes are defined by fixed points, cycles, and saddles of the map, building blocks of the underlying firing code. Showing parameter-dependent multiplicity of real polynomial roots, the map is proved to be noninvertible. The incidence of chaos is then implied by the parameter-dependent existence of snap-back repellers. The highly patterned geometric and statistical structures of the associated chaotic attractors suggest that these attractors are an integral part of the neural code. It further suggests the chaotic attractor as a natural mechanism for statistical encoding and temporal multiplexing of neural information. The analytic findings are supported by simulation.

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Manfredi, V. R., and L. Salasnich. "A Note on the Toda Criterion for Interacting Dipole–Quadrupole Vibrations." Modern Physics Letters A 12, no. 26 (August 30, 1997): 1951–56. http://dx.doi.org/10.1142/s0217732397001990.

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The Toda criterion of the Gaussian curvature is applied to calculate analytically the transition energy from regular to chaotic motion in a schematic model describing the interaction between collective dipole and quadrupole modes in atomic nuclei.

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Mirza, Arshad M., and P. K. Shukla. "Chaotic behavior of nonlinearly coupled electromagnetic modes in nonuniform magnetoplasmas." Physics Letters A 229, no. 5 (May 1997): 313–16. http://dx.doi.org/10.1016/s0375-9601(97)00208-9.

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Bittner, Stefan, Kyungduk Kim, Yongquan Zeng, Qi Jie Wang, and Hui Cao. "Spatial structure of lasing modes in wave-chaotic semiconductor microcavities." New Journal of Physics 22, no. 8 (August 4, 2020): 083002. http://dx.doi.org/10.1088/1367-2630/ab9e33.

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Rafiq, Tariq, Anisa Qamar, Arshad M. Mirza, and G. Murtaza. "Chaotic behavior of ion-temperature-gradient driven drift-dissipative modes." Physics of Plasmas 7, no. 11 (November 2000): 4499–505. http://dx.doi.org/10.1063/1.1316083.

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Hsieh, S. R., and S. W. Shaw. "The stability of modes at rest in a chaotic system." Journal of Sound and Vibration 138, no. 3 (May 1990): 421–31. http://dx.doi.org/10.1016/0022-460x(90)90596-r.

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Hynönen, V., O. Dumbrajs, A. W. Degeling, T. Kurki-Suonio, and H. Urano. "The search for chaotic edge localized modes in ASDEX Upgrade." Plasma Physics and Controlled Fusion 46, no. 9 (July 22, 2004): 1409–22. http://dx.doi.org/10.1088/0741-3335/46/9/005.

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Barceló Forteza, S., T. Roca Cortés, A. García Hernández, and R. A. García. "Evidence of chaotic modes in the analysis of fourδScuti stars." Astronomy & Astrophysics 601 (April 28, 2017): A57. http://dx.doi.org/10.1051/0004-6361/201628675.

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Elashry, Ibrahim F., Walid El-Shafai, Emad S. Hasan, S. El-Rabaie, Alaa M. Abbas, Fathi E. Abd El-Samie, Hala S. El-sayed, and Osama S. Faragallah. "Efficient chaotic-based image cryptosystem with different modes of operation." Multimedia Tools and Applications 79, no. 29-30 (April 22, 2020): 20665–87. http://dx.doi.org/10.1007/s11042-019-08322-5.

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Chen, Zhi-Min, and W. G. Price. "Onset of Chaotic Kolmogorov Flows Resulting from Interacting Oscillatory Modes." Communications in Mathematical Physics 256, no. 3 (March 8, 2005): 737–66. http://dx.doi.org/10.1007/s00220-005-1290-0.

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Morisue, Mititada, Masayuki Yamadaya, Hiroshi Noguchi, and Akinori Kanasugi. "A digital application of chaotic oscillation modes in Josephson circuit." International Journal of Intelligent Systems 12, no. 4 (April 1997): 267–90. http://dx.doi.org/10.1002/(sici)1098-111x(199704)12:4<267::aid-int2>3.0.co;2-m.

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Tang, D. M., and E. H. Dowell. "On the Threshold Force for Chaotic Motions for a Forced Buckled Beam." Journal of Applied Mechanics 55, no. 1 (March 1, 1988): 190–96. http://dx.doi.org/10.1115/1.3173628.

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The effects of higher modes on the chaotic oscillations of a buckled beam under forced external excitation are studied. Of principal interest are the threshold force required for chaotic motions and the influence of damping on the system response. A comparison is also presented of results from numerical simulations with experimental data.

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Goza, Andres, Tim Colonius, and John E. Sader. "Global modes and nonlinear analysis of inverted-flag flapping." Journal of Fluid Mechanics 857 (October 22, 2018): 312–44. http://dx.doi.org/10.1017/jfm.2018.728.

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An inverted flag has its trailing edge clamped and exhibits dynamics distinct from that of a conventional flag, whose leading edge is restrained. We perform nonlinear simulations and a global stability analysis of the inverted-flag system for a range of Reynolds numbers, flag masses and stiffnesses. Our global stability analysis is based on a linearisation of the fully coupled fluid–structure system of equations. The calculated equilibria are steady-state solutions of the fully coupled nonlinear equations. By implementing this approach, we (i) explore the mechanisms that initiate flapping, (ii) study the role of vorticity generation and vortex-induced vibration (VIV) in large-amplitude flapping and (iii) characterise the chaotic flapping regime. For point (i), we identify a deformed-equilibrium state and show through a global stability analysis that the onset of small-deflection flapping – where the oscillation amplitude is significantly smaller than in large-amplitude flapping – is due to a supercritical Hopf bifurcation. For large-amplitude flapping, point (ii), we confirm the arguments of Sader et al. (J. Fluid Mech., vol. 793, 2016a) that classical VIV exists when the flag is sufficiently light with respect to the fluid. We also show that for heavier flags, large-amplitude flapping persists (even for Reynolds numbers ${<}50$ ) and is not classical VIV. Finally, with respect to point (iii), chaotic flapping has been observed experimentally for Reynolds numbers of $O(10^{4})$ , and here we show that chaos also persists at a moderate Reynolds number of 200. We characterise this chaotic regime and calculate its strange attractor, whose structure is controlled by the above-mentioned deformed equilibria and is similar to a Lorenz attractor.

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Giuliani, Filippo, Marcel Guardia, Pau Martin, and Stefano Pasquali. "Chaotic-Like Transfers of Energy in Hamiltonian PDEs." Communications in Mathematical Physics 384, no. 2 (February 8, 2021): 1227–90. http://dx.doi.org/10.1007/s00220-021-03956-9.

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AbstractWe consider the nonlinear cubic Wave, the Hartree and the nonlinear cubic Beam equations on $${\mathbb {T}}^2$$ T 2 and we prove the existence of different types of solutions which exchange energy between Fourier modes in certain time scales. This exchange can be considered “chaotic-like” since either the choice of activated modes or the time spent in each transfer can be chosen randomly. The key point of the construction of those orbits is the existence of heteroclinic connections between invariant objects and the construction of symbolic dynamics (a Smale horseshoe) for the Birkhoff Normal Form truncation of those equations.

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Sanghi, Sanjeev, and Nadine Aubry. "Mode interaction models for near-wall turbulence." Journal of Fluid Mechanics 247 (February 1993): 455–88. http://dx.doi.org/10.1017/s0022112093000527.

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Intermittent bursting events, similar to those characterizing the dynamics of near-wall turbulence, have been observed in a low-dimensional dynamical model (Aubry et al. 1988) built from eigenfunctions of the proper orthogonal decomposition (Lumley 1967). In the present work, we investigate the persistency of the intermittent behaviour in higher - but still of relatively low-dimensional dynamical systems. In particular, streamwise variations which were not accounted for in an explicit way in Aubry et al.'s model are now considered. Intermittent behaviour persists but can be of a different nature. Specifically, the non-zero streamwise modes become excited during the eruptive events so that rolls burst downstream into smaller scales. When structures have a finite length, they travel at a convection speed approximately equal to the mean velocity at the top of the layer (y+ ≈ 40). In all cases, intermittency seems to be due to homoclinic cycles connecting hyperbolic fixed points or more complex (apparently chaotic) limit sets. While these sets lie in the zero streamwise modes invariant subspace, the connecting orbits consist of nonzero streamwise modes travelling downstream. Chaotic limit sets connected by quasi-travelling waves have also been observed in a spatio-temporal chaotic regime of the Kuramoto–Sivashinsky equation (Aubry & Lian 1992a). When the limit sets lose their steadiness, the elongated rolls become randomly active, as they probably are in the real flow. A coherent structure study in our resulting flow fields is performed in order to relate our findings to experimental observations. It is shown that streaks, streamwise rolls, horseshoe vortical structures and shear layers, present in our models, are all connected to each other. Finally, criteria to determine a realistic value of the eddy viscosity parameter are developed.

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Aikawa, T. "Bifurcation in Hydrodynamic Models of Stellar Pulsation." International Astronomical Union Colloquium 134 (1993): 269–80. http://dx.doi.org/10.1017/s0252921100014317.

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AbstractPhenomena of bifurcation in hydrodynamic stellar models of radial pulsation are reviewed. By changing control parameters of models, we can see qualitatively different pulsation behaviors in hydrodynamic models with transitions due to various types of bifurcation.In weakly dissipative models (classical Cepheids). the bifurcation is induced by modal resonances. Two types of the modal resonances found in models are discussed: The higher-harmonic resonances of the second overtone mode in the fundamental mode pulsator and of the fourth overtone mode in the first overtone pulsator are relevant to observations. The subharmonic resonance between the fundamental and first overtone modes is confirmed in classical Cepheid models.In strongly dissipative models (less-massive supergiant stars), the bifurcation of nonlinear pulsation is induced by the hydrodynamics of ionization zones as well as modal resonances. The sequence of the bifurcation sometimes leads to chaotic behaviors in nonlinear pulsation. The transition routes from regular to the chaotic pulsations found in models are discussed with respect to the theory of chaos in simple dynamical systems: The cascade of period-doubling bifurcation is confirmed to cause chaotic pulsation in W Virginis models. For models of higher luminosity, the tangent bifurcation is found to lead intermittent chaos.Finally, hydrodynamic models for chaotic pulsation with small amplitudes observed in the post-AGB stars are briefly discussed.

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Journal articles on the topic 'Chaotic modes'

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