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Journal articles on the topic 'Dynamic chaos'

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Author: Grafiati

Published: 30 May 2022

Last updated: 31 May 2022

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1

Fang, Pan, Liming Dai, Yongjun Hou, Mingjun Du, and Wang Luyou. "The Study of Identification Method for Dynamic Behavior of High-Dimensional Nonlinear System." Shock and Vibration 2019 (March 7, 2019): 1–9. http://dx.doi.org/10.1155/2019/3497410.

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The dynamic behavior of nonlinear systems can be concluded as chaos, periodicity, and the motion between chaos and periodicity; therefore, the key to study the nonlinear system is identifying dynamic behavior considering the different values of the system parameters. For the uncertainty of high-dimensional nonlinear dynamical systems, the methods for identifying the dynamics of nonlinear nonautonomous and autonomous systems are treated. In addition, the numerical methods are employed to determine the dynamic behavior and periodicity ratio of a typical hull system and Rössler dynamic system, respectively. The research findings will develop the evaluation method of dynamic characteristics for the high-dimensional nonlinear system.

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2

Kumar, Deepak, and Mamta Rani. "Alternated Superior Chaotic Biogeography-Based Algorithm for Optimization Problems." International Journal of Applied Metaheuristic Computing 13, no. 1 (January 2022): 1–39. http://dx.doi.org/10.4018/ijamc.292520.

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In this study, we consider a switching strategy that yields a stable desirable dynamic behaviour when it is applied alternatively between two undesirable dynamical systems. From the last few years, dynamical systems employed “chaos1 + chaos2 = order” and “order1 + order2 = chaos” (vice-versa) to control and anti control of chaotic situations. To find parameter values for these kind of alternating situations, comparison is being made between bifurcation diagrams of a map and its alternate version, which, on their own, means independent of one another, yield chaotic orbits. However, the parameter values yield a stable periodic orbit, when alternating strategy is employed upon them. It is interesting to note that we look for stabilization of chaotic trajectories in nonlinear dynamics, with the assumption that such chaotic behaviour is not desirable for a particular situation. The method described in this paper is based on the Parrondo’s paradox, where two losing games can be alternated, yielding a winning game, in a superior orbit.

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3

Albers, D. J., J. C. Sprott, and W. D. Dechert. "Routes to Chaos in Neural Networks with Random Weights." International Journal of Bifurcation and Chaos 08, no. 07 (July 1998): 1463–78. http://dx.doi.org/10.1142/s0218127498001121.

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Neural networks are dense in the space of dynamical system. We present a Monte Carlo study of the dynamic properties along the route to chaos over random dynamical system function space by randomly sampling the neural network function space. Our results show that as the dimension of the system (the number of dynamical variables) is increased, the probability of chaos approaches unity. We present theoretical and numerical results which show that as the dimension is increased, the quasiperiodic route to chaos is the dominant route. We also qualitatively analyze the dynamics along the route.

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4

Kaneko, Kunihiko. "Chaos as a Source of Complexity and Diversity in Evolution." Artificial Life 1, no. 1_2 (October 1993): 163–77. http://dx.doi.org/10.1162/artl.1993.1.1_2.163.

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The relevance of chaos to evolution is discussed in the context of the origin and maintenance of diversity and complexity. Evolution to the edge of chaos is demonstrated in an imitation game. As an origin of diversity, dynamic clustering of identical chaotic elements, globally coupled each to the other, is briefly reviewed. The clustering is extended to nonlinear dynamics on hypercubic lattices, which enables us to construct a self-organizing genetic algorithm. A mechanism of maintenance of diversity, “homeochaos,” is given in an ecological system with interaction among many species. Homeochaos provides a dynamic stability sustained by high-dimensional weak chaos. A novel mechanism of cell differentiation is presented, based on dynamic clustering. Here, a new concept—“open chaos”—is proposed for the instability in a dynamical system with growing degrees of freedom. It is suggested that studies based on interacting chaotic elements can replace both top-down and bottom-up approaches.

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5

Egorov, Vladimir V. "Dynamic Symmetry in Dozy-Chaos Mechanics." Symmetry 12, no. 11 (November 11, 2020): 1856. http://dx.doi.org/10.3390/sym12111856.

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All kinds of dynamic symmetries in dozy-chaos (quantum-classical) mechanics (Egorov, V.V. Challenges 2020, 11, 16; Egorov, V.V. Heliyon Physics 2019, 5, e02579), which takes into account the chaotic dynamics of the joint electron-nuclear motion in the transient state of molecular “quantum” transitions, are discussed. The reason for the emergence of chaotic dynamics is associated with a certain new property of electrons, consisting in the provocation of chaos (dozy chaos) in a transient state, which appears in them as a result of the binding of atoms by electrons into molecules and condensed matter and which provides the possibility of reorganizing a very heavy nuclear subsystem as a result of transitions of light electrons. Formally, dozy chaos is introduced into the theory of molecular “quantum” transitions to eliminate the significant singularity in the transition rates, which is present in the theory when it goes beyond the Born–Oppenheimer adiabatic approximation and the Franck–Condon principle. Dozy chaos is introduced by replacing the infinitesimal imaginary addition in the energy denominator of the full Green’s function of the electron-nuclear system with a finite value, which is called the dozy-chaos energy γ. The result for the transition-rate constant does not change when the sign of γ is changed. Other dynamic symmetries appearing in theory are associated with the emergence of dynamic organization in electronic-vibrational transitions, in particular with the emergence of an electron-nuclear-reorganization resonance (the so-called Egorov resonance) and its antisymmetric (chaotic) “twin”, with direct and reverse transitions, as well as with different values of the electron–phonon interaction in the initial and final states of the system. All these dynamic symmetries are investigated using the simplest example of quantum-classical mechanics, namely, the example of quantum-classical mechanics of elementary electron-charge transfers in condensed media.

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6

Russell, David W. "Dynamic Systems & Chaos." IFAC Proceedings Volumes 31, no. 29 (October 1998): 6. http://dx.doi.org/10.1016/s1474-6670(17)38316-7.

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7

Hide, R. "Chaos in Dynamic Systems." Physics Bulletin 37, no. 9 (September 1986): 390. http://dx.doi.org/10.1088/0031-9112/37/9/034.

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8

Kana, L. K., A. Fomethe, H. B. Fotsin, E. T. Wembe, and A. I. Moukengue. "Complex Dynamics and Synchronization in a System of Magnetically Coupled Colpitts Oscillators." Journal of Nonlinear Dynamics 2017 (April 10, 2017): 1–13. http://dx.doi.org/10.1155/2017/5483956.

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We propose the use of a simple, cheap, and easy technique for the study of dynamic and synchronization of the coupled systems: effects of the magnetic coupling on the dynamics and of synchronization of two Colpitts oscillators (wireless interaction). We derive a smooth mathematical model to describe the dynamic system. The stability of the equilibrium states is investigated. The coupled system exhibits spectral characteristics such as chaos and hyperchaos in some parameter ranges of the coupling. The numerical exploration of the dynamics system reveals various bifurcations scenarios including period-doubling and interior crisis transitions to chaos. Moreover, various interesting dynamical phenomena such as transient chaos, coexistence of solution, and multistability (hysteresis) are observed when the magnetic coupling factor varies. Theoretical reasons for such phenomena are provided and experimentally confirmed with practical measurements in a wireless transfer.

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9

Field, Richard J. "Chaos in the Belousov–Zhabotinsky reaction." Modern Physics Letters B 29, no. 34 (December 20, 2015): 1530015. http://dx.doi.org/10.1142/s021798491530015x.

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The dynamics of reacting chemical systems is governed by typically polynomial differential equations that may contain nonlinear terms and/or embedded feedback loops. Thus the dynamics of such systems may exhibit features associated with nonlinear dynamical systems, including (among others): temporal oscillations, excitability, multistability, reaction-diffusion-driven formation of spatial patterns, and deterministic chaos. These behaviors are exhibited in the concentrations of intermediate chemical species. Bifurcations occur between particular dynamic behaviors as system parameters are varied. The governing differential equations of reacting chemical systems have as variables the concentrations of all chemical species involved, as well as controllable parameters, including temperature, the initial concentrations of all chemical species, and fixed reaction-rate constants. A discussion is presented of the kinetics of chemical reactions as well as some thermodynamic considerations important to the appearance of temporal oscillations and other nonlinear dynamic behaviors, e.g., deterministic chaos. The behavior, chemical details, and mechanism of the oscillatory Belousov–Zhabotinsky Reaction (BZR) are described. Furthermore, experimental and mathematical evidence is presented that the BZR does indeed exhibit deterministic chaos when run in a flow reactor. The origin of this chaos seems to be in toroidal dynamics in which flow-driven oscillations in the control species bromomalonic acid couple with the BZR limit cycle.

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10

Yuan, Ying Cai, Yan Li, and Yi Ming Wang. "Robust Design to Control the Chaos of Fold Mechanism with Clearance." Applied Mechanics and Materials 312 (February 2013): 153–57. http://dx.doi.org/10.4028/www.scientific.net/amm.312.153.

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With the increasing of web offset printing machines working speed, the nonlinear dynamics responses are more significant, even the fold mechanism with clearances appears some chaos phenomenon. Based on the dynamic model of fold mechanism, the nonlinear dynamics responses and the chaos movement in pair are studied. Used the performance parameters and dynamics response sensitivities as the goal values, the robust design model is established. By the robust design model, the nonlinear dynamic responses and chaos phenomenon can be under controlled in the same clearance degree. In this way, the performance of fold mechanism may be improved.

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11

Podchukaev, Vladimir Anatolievich. "Mathematical Model of Dynamic Chaos." Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics 12, no. 4 (2012): 27–31. http://dx.doi.org/10.18500/1816-9791-2012-12-4-27-31.

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12

Tuhua, MA, ZHU Xingsheng, and LI Changjiang. "Chaos Dynamic Behaviour of Mineralization." Acta Geologica Sinica - English Edition 72, no. 4 (September 7, 2010): 392–98. http://dx.doi.org/10.1111/j.1755-6724.1998.tb00417.x.

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13

Ouellette, Nicholas T., and J. P. Gollub. "Dynamic topology in spatiotemporal chaos." Physics of Fluids 20, no. 6 (June 2008): 064104. http://dx.doi.org/10.1063/1.2948849.

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14

Giles, Ronald, and M. Martelli. "Discrete Dynamic Systems and Chaos." Statistician 43, no. 3 (1994): 462. http://dx.doi.org/10.2307/2348594.

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15

Vaganova, N. I., and E. N. Rumanov. "Frequency locking of dynamic chaos." Doklady Physics 58, no. 10 (October 2013): 421–23. http://dx.doi.org/10.1134/s1028335813100091.

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16

V’yun, V. A. "Dynamic chaos in acoustoelectronic systems." Technical Physics Letters 23, no. 5 (May 1997): 408–9. http://dx.doi.org/10.1134/1.1261850.

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17

Zhang, Yu, and Longsuo Li. "Chaos Analysis and Control of Relative Rotation System with Mathieu-Duffing Oscillator." Mathematical Problems in Engineering 2015 (2015): 1–5. http://dx.doi.org/10.1155/2015/348462.

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Chaos analysis and control of relative rotation nonlinear dynamic system with Mathieu-Duffing oscillator are investigated. By using Lagrange equation, the dynamics equation of relative rotation system has been established. Melnikov’s method is applied to predict the chaotic behavior of this system. Moreover, the chaotic dynamical behavior can be controlled by adding the Gaussian white noise to the proposed system for the sake of changing chaos state into stable state. Through numerical calculation, the Poincaré map analysis and phase portraits are carried out to confirm main results.

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18

Bahi, Jacques M., Christophe Guyeux, and Antoine Perasso. "Chaos in DNA evolution." International Journal of Biomathematics 09, no. 05 (June 13, 2016): 1650076. http://dx.doi.org/10.1142/s1793524516500765.

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In this paper, we explain why the chaotic mutation (CM) model of J. M. Bahi and C. Michel (2008) simulates the genes mutations over time with good accuracy. It is firstly shown that the CM model is a truly chaotic one, as it is defined by Devaney. Then, it is established that mutations occurring in genes mutations have indeed a same chaotic dynamic, thus making relevant the use of chaotic models for genomes evolution. Transposition and inversion dynamics are finally investigated.

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19

SUN, KEHUI, and J. C. SPROTT. "DYNAMICS OF A SIMPLIFIED LORENZ SYSTEM." International Journal of Bifurcation and Chaos 19, no. 04 (April 2009): 1357–66. http://dx.doi.org/10.1142/s0218127409023688.

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A simplified Lorenz system with one bifurcation parameter is investigated by a detailed theoretical analysis as well as dynamic simulation, including some basic dynamical properties, Lyapunov exponent spectra, fractal dimension, bifurcations and routes to chaos. The results show that this system has complex dynamics with interesting characteristics.

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20

Li, Chunbiao, Yuxuan Peng, Ze Tao, Julien Clinton Sprott, and Sajad Jafari. "Coexisting Infinite Equilibria and Chaos." International Journal of Bifurcation and Chaos 31, no. 05 (April 2021): 2130014. http://dx.doi.org/10.1142/s0218127421300147.

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Equilibria are a class of attractors that host inherent stability in a dynamic system. Infinite number of equilibria and chaos sometimes coexist in a system with some connections. Hidden chaotic attractors exist independent of any equilibria rather than being excited by them. However, the equilibria can modify, distort, eliminate, or even instead coexist with the chaotic attractor depending on the distance between the equilibria and chaotic attractor. In this paper, chaotic systems with infinitely many equilibria are considered and explored. Extra surfaces of equilibria are introduced into the chaotic flows, showing that a chaotic system can maintain its basic dynamics if the newly added equilibria do not intersect the original attractor. The offset-boostable plane of equilibria rescales the frequency of the chaotic oscillation with an almost linearly modified largest Lyapunov exponent or conversely drives the system into periodic oscillation, even ending in a divergent state. Furthermore, additional infinite number of equilibria or even a solid space of equilibria are safely nested into the chaotic system without destroying the original dynamics, which provides an alternate permanent location for a dynamical system. A circuit simulation agrees with the numerical calculation.

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21

Handelman, Don. "How dynamic is the anthropology of chaos?" Focaal 2007, no. 50 (December 1, 2007): 155–65. http://dx.doi.org/10.3167/foc.2007.500112.

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22

Gong, Chibing. "Dynamic search fireworks algorithm with chaos." Journal of Algorithms & Computational Technology 13 (January 2019): 174830261988955. http://dx.doi.org/10.1177/1748302619889559.

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As a relatively new algorithm for swarm intelligence, fireworks algorithm imitates the explosion process of fireworks. A different amplitude in dynamic search fireworks algorithm is presented for an improvement of enhanced fireworks algorithm. This paper integrates chaos with the dynamic search fireworks algorithm so as to further improve the performance and achieve global optimization. Three different variants of dynamic search fireworks algorithm with chaos are introduced and 10 chaotic maps are used to tune either the amplification coefficient [Formula: see text] or the reduction coefficient [Formula: see text]. Twelve benchmark functions are verified in use of the dynamic search fireworks algorithm with chaos (dynamic search fireworks algorithm). The dynamic search fireworks algorithm significantly outperformed the Fireworks Algorithm, enhanced fireworks algorithm, and dynamic search fireworks algorithm based on solution accuracy. The highest performance was seen when dynamic search fireworks algorithm was used with a Gauss/mouse map to tune Ca. Additionally, the dynamic search fireworks algorithm was compared with the firefly algorithm, harmony search, bat algorithm, and standard particle swarm optimization (SPSO2011). Study results indicated that the dynamic search fireworks algorithm has the highest accuracy solution among the five algorithms.

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23

QIN, WEIYANG, and GUANG MENG. "NONLINEAR DYNAMIC RESPONSE AND CHAOS OF A CRACKED ROTOR WITH TWO DISKS." International Journal of Bifurcation and Chaos 13, no. 11 (November 2003): 3425–36. http://dx.doi.org/10.1142/s021812740300865x.

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In this paper, the nonlinear response and chaos of a cracked rotor with two disks are studied. Considering the breadth of crack in one rotor revolution, the motion equations of the system are derived and then solved. The results show that the rotor response is sensitive to the crack depth, rotating speed, damping ratio and imbalance. When a crack occurs, the frequency of swing vibration is a multiple of rotating speed (NΩ,N=2,3,…). There are three main routes for response to chaos, that is from quasi-periodic to chaos, from quasi-periodic to quasi-periodic bifurcation and then to chaos and the intermittence to chaos. The intermittence chaos occurs even for a small crack. With the intermittence chaos range there exists the periodic-doubling bifurcation with time. Larger imbalance parameter and damping ratio can suppress chaos. The diagram of time-phase is a useful way to analyze the nonlinear response.

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24

Wang Guang-Yi and Yuan Fang. "Cascade chaos and its dynamic characteristics." Acta Physica Sinica 62, no. 2 (2013): 020506. http://dx.doi.org/10.7498/aps.62.020506.

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25

FENG, Yanhong, Jianqin LIU, and Yichao HE. "Chaos-based dynamic population firefly algorithm." Journal of Computer Applications 33, no. 3 (September 26, 2013): 796–99. http://dx.doi.org/10.3724/sp.j.1087.2013.00796.

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26

Escultura, Edgar E. "Dynamic modeling of chaos and turbulence." Nonlinear Analysis: Theory, Methods & Applications 63, no. 5-7 (November 2005): e519-e532. http://dx.doi.org/10.1016/j.na.2005.02.052.

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27

Lambert, Philip A. "The Order - Chaos Dynamic of Creativity." Creativity Research Journal 32, no. 4 (October 1, 2020): 431–46. http://dx.doi.org/10.1080/10400419.2020.1821562.

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28

Jamitzky, F., M. Stark, W. Bunk, W. M. Heckl, and R. W. Stark. "Chaos in dynamic atomic force microscopy." Nanotechnology 17, no. 7 (March 10, 2006): S213—S220. http://dx.doi.org/10.1088/0957-4484/17/7/s19.

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29

Kurdyumov, S. P., G. G. Malinetskii, and A. B. Potapov. "Nonstationary Structures, Dynamic Chaos, Cellular Automata." International Journal of Fluid Mechanics Research 22, no. 5-6 (1995): 75–133. http://dx.doi.org/10.1615/interjfluidmechres.v22.i5-6.30.

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30

Howe, Mark L., and F. Michael Rabinowitz. "Dynamic Modeling, Chaos, and Cognitive Development." Journal of Experimental Child Psychology 58, no. 2 (October 1994): 184–99. http://dx.doi.org/10.1006/jecp.1994.1032.

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31

Laskin, N. V., S. P. Fomin, and N. F. Shul'ga. "Dechanneling kinetics under dynamic chaos conditions." Physics Letters A 138, no. 6-7 (July 1989): 309–12. http://dx.doi.org/10.1016/0375-9601(89)90284-3.

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32

Majumdar, Mukul, and Tapan Mitra. "Robust chaos in dynamic optimization models." Ricerche Economiche 48, no. 3 (September 1994): 225–40. http://dx.doi.org/10.1016/0035-5054(94)90026-4.

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33

Hover, F. S. "Gradient dynamic optimization with Legendre chaos." Automatica 44, no. 1 (January 2008): 135–40. http://dx.doi.org/10.1016/j.automatica.2007.06.001.

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34

Lu, Ming-Chi, Yen-Wen Tseng, Yan-Lin Zhong, Chen-An Chan, Che-An Chiu, Wen-I. Huang, Chia-Ju Liu, and Ming-Chung Ho. "Chaos Synchronization via Ameliorated Dynamic Control." Sensors and Materials 34, no. 3 (March 24, 2022): 1211. http://dx.doi.org/10.18494/sam3490.

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35

TABORDA, JOHN ALEXANDER, FABIOLA ANGULO, and GERARD OLIVAR. "CHARACTERIZATION OF CHAOS SCENARIOS WITH PERIODIC INCLUSIONS FOR ONE CLASS OF PIECEWISE-SMOOTH DYNAMICAL MAPS." International Journal of Bifurcation and Chaos 21, no. 09 (September 2011): 2427–66. http://dx.doi.org/10.1142/s0218127411029975.

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In this paper, we study bifurcation scenarios characterized by period-adding cascades with alternating chaos in one class of piecewise-smooth maps (PWS). In this class, the state space is separated in three smooth zones defined by a saturation function. Some power converters controlled by Digital Pulse-Width Modulation (PWM) are physical applications of this class of PWS systems denoted by PWS3. Chaos has virtually been detected and studied in all disciplines, however the characterization problem of chaos scenarios has many open problems, mainly in nonsmooth dynamical systems. Novel bifurcation scenarios have recently been reported such as bandcount adding and bandcount increment scenarios based on the numerical detection of bands (where bands are considered as strongly connected components). However, this approach known as Bandcounter cannot be applied to detect bifurcations in chaos scenarios without crisis bifurcations or to identify topological changes inside of one-band chaos. We have proposed a novel framework named Dynamic Linkcounter approach to characterize chaos and torus breakdown scenarios in PWS systems. In this paper, we report overlapping period-adding cascades interspersed with a dynamic linkcount adding cascade. Each complex dynamic link (CDL) structure is a fingered strange attractor increasing in an arithmetic progression the number of CDL or fingers when a bifurcation parameter is varied. Alternative point of view based on tent-map-like structures is given to illustrate the formation of fingered strange attractors.

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36

Song, Tingting, Yiyuan Xie, Yichen Ye, Bocheng Liu, Junxiong Chai, Xiao Jiang, and Yanli Zheng. "Numerical Analysis of Nonlinear Dynamics Based on Spin-VCSELs with Optical Feedback." Photonics 8, no. 1 (January 4, 2021): 10. http://dx.doi.org/10.3390/photonics8010010.

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In this paper, the nonlinear dynamics of a novel model based on optically pumped spin-polarized vertical-cavity surface-emitting lasers (spin-VCSELs) with optical feedback is investigated numerically. Due to optical feedback being the external disturbance component, the complex nonlinear dynamical behaviors can be enhanced and the regions of different nonlinear dynamics in size can be extended with appropriate parameters of spin-VCSELs. According to the equations of the modified spin-flip model (SFM), the comparison of bifurcation diagrams is first presented for the clear presentation of different routes to chaos. Meanwhile, numerous bifurcation diagrams in color are illustrated to demonstrate the rich dynamical regimes intuitively, and the crucial effects of optical feedback strength, feedback delay, linewidth enhancement factor, and spin-flip relaxation rate on the region evolvement of complex dynamics of the proposed model are revealed to investigate the dependence of dynamical behaviors on external and internal parameters when the optical feedback scheme is introduced. These parameters play a remarkable role in enhancing the mechanism of complex dynamic oscillations. Furthermore, utilizing combination with time series, power spectra, and phase portraits, the various dynamical behaviors observed in the bifurcation diagram are simulated numerically. Correspondingly, the powerful measure 0–1 test is employed to distinguish between chaos and non-chaos.

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37

Nan, Guofang, Yujie Zhu, Yang Zhang, and Wei Guo. "Nonlinear Dynamic Analysis of Rotor-Bearing System with Cubic Nonlinearity." Shock and Vibration 2021 (May 25, 2021): 1–11. http://dx.doi.org/10.1155/2021/8878319.

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Nonlinear dynamic characteristics of a rotor-bearing system with cubic nonlinearity are investigated. The comprehensive effects of the unbalanced excitation, the internal clearance, the nonlinear Hertzian contact force, the varying compliance vibration, and the nonlinear stiffness of support material are considered. The expression with the linear and the cubic nonlinear terms is adopted to characterize the synthetical nonlinearity of the rotor-bearing system. The effects of nonlinear stiffness, rotating speed, and mass eccentricity on the dynamic behaviors of the system are studied using the rotor trajectory diagrams, bifurcation diagrams, and Poincaré map. The complicated dynamic behaviors and types of routes to chaos are found, including the periodic doubling bifurcation, sudden transition, and quasiperiodic from periodic motion to chaos. The research results show that the system has complex nonlinear dynamic behaviors such as multiple period, paroxysmal bifurcation, inverse bifurcation, jumping phenomena, and chaos; the nonlinear characteristics of the system are significantly enhanced with the increase of the nonlinear stiffness, and the material with lower nonlinear stiffness is more conducive to the stable operation of the system. The research will contribute to a comprehensive understanding of the nonlinear dynamics of the rotor-bearing system.

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38

Hajipour, Ahmad, and Hamidreza Tavakoli. "Dynamic Analysis and Adaptive Sliding Mode Controller for a Chaotic Fractional Incommensurate Order Financial System." International Journal of Bifurcation and Chaos 27, no. 13 (December 15, 2017): 1750198. http://dx.doi.org/10.1142/s021812741750198x.

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In this study, the dynamic behavior and chaos control of a chaotic fractional incommensurate-order financial system are investigated. Using well-known tools of nonlinear theory, i.e. Lyapunov exponents, phase diagrams and bifurcation diagrams, we observe some interesting phenomena, e.g. antimonotonicity, crisis phenomena and route to chaos through a period doubling sequence. Adopting largest Lyapunov exponent criteria, we find that the system yields chaos at the lowest order of [Formula: see text]. Next, in order to globally stabilize the chaotic fractional incommensurate order financial system with uncertain dynamics, an adaptive fractional sliding mode controller is designed. Numerical simulations are used to demonstrate the effectiveness of the proposed control method.

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39

Ji, Wei Zhuo. "Complexity of Nonlinear Dynamic Outputs Model in the Electricity Triopoly." Advanced Materials Research 1008-1009 (August 2014): 1395–98. http://dx.doi.org/10.4028/www.scientific.net/amr.1008-1009.1395.

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A dynamic repeated model has been established in electric power Triopoly. The chaos and density cycling of the nonlinear dynamic model are investigated in detail. The nonlinear feedback chaos control method is successfully applied to the dynamic repeated game model.

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40

Medio, Alfredo. "NONLINEAR DYNAMICS AND CHAOS PART I: A GEOMETRICAL APPROACH." Macroeconomic Dynamics 2, no. 4 (December 1998): 505–32. http://dx.doi.org/10.1017/s1365100598009079.

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This paper is the first part of a two-part survey reviewing some basic concepts and methods of the modern theory of dynamical systems. The survey is introduced by a preliminary discussion of the relevance of nonlinear dynamics and chaos for economics. We then discuss the dynamic behavior of nonlinear systems of difference and differential equations such as those commonly employed in the analysis of economically motivated models. Part I of the survey focuses on the geometrical properties of orbits. In particular, we discuss the notion of attractor and the different types of attractors generated by discrete- and continuous-time dynamical systems, such as fixed and periodic points, limit cycles, quasiperiodic and chaotic attractors. The notions of (noninteger) fractal dimension and Lyapunov characteristic exponent also are explained, as well as the main routes to chaos.

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41

Ni, Hong Mei, Zhian Yi, and Jin Yue Liu. "A Hybrid Particle Swarm Optimization Algorithm for Dynamic Environments." Advanced Materials Research 926-930 (May 2014): 3338–41. http://dx.doi.org/10.4028/www.scientific.net/amr.926-930.3338.

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Chaos is a non-linear phenomenon that widely exists in the nature. Due to the ease of implementation and its special ability to avoid being trapped in local optima, chaos has been a novel optimization technique and chaos-based searching algorithms have aroused intense interests. Many real world optimization problems are dynamic in which global optimum and local optima change over time. Particle swarm optimization has performed well to find and track optima in static environments. When the particle swarm optimization (PSO) algorithm is used in dynamic multi-objective problems, there exist some problems, such as easily falling into prematurely, having slow convergence rate and so on. To solve above problems, a hybrid PSO algorithm based on chaos algorithm is brought forward. The hybrid PSO algorithm not only has the efficient parallelism but also increases the diversity of population because of the chaos algorithm. The simulation result shows that the new algorithm is prior to traditional PSO algorithm, having stronger adaptability and convergence, solving better the question on moving peaks benchmark.

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42

SPROTT, J. C., and KONSTANTINOS E. CHLOUVERAKIS. "LABYRINTH CHAOS." International Journal of Bifurcation and Chaos 17, no. 06 (June 2007): 2097–108. http://dx.doi.org/10.1142/s0218127407018245.

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A particularly simple and mathematically elegant example of chaos in a three-dimensional flow is examined in detail. It has the property of cyclic symmetry with respect to interchange of the three orthogonal axes, a single bifurcation parameter that governs the damping and the attractor dimension over most of the range 2 to 3 (as well as 0 and 1) and whose limiting value b = 0 gives Hamiltonian chaos, three-dimensional deterministic fractional Brownian motion, and an interesting symbolic dynamic.

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Pellegrini, L., C. Tablino Possio, and G. Biardi. "An Example of How Nonlinear Dynamics Tools Can be Successfully Applied to A Chemical System." Fractals 05, no. 03 (September 1997): 531–47. http://dx.doi.org/10.1142/s0218348x97000425.

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The nonlinear dynamic behavior of a Proportional-Integral controlled Continuously Stirred Tank Reactor (CSTR) is analyzed in depth progressing from chaos characterization, through the high codimension bifurcation theory, up to the application of Controlling Chaos techniques. All these tools can be successfully applied to recognize, to avoid and to use chaos in practical applications, so that the nonlinear dynamic theory turns out to be an indispensable science to constrain dynamic systems to work in the most suitable operative conditions.

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Ge, Z.-M., C.-C. Lin, and Y.-S. Chen. "Chaos, chaos control and synchronization of the vibrometer system." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 218, no. 9 (September 1, 2004): 1001–20. http://dx.doi.org/10.1243/0954406041991206.

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The dynamic system of the vibrometer is shown to produce regular and chaotic behaviour as the parameters are varied. When the system is non-autonomous, the periodic and chaotic motions are obtained by numerical methods. Many effective methods have been used in chaos synchronization. It has been shown that chaos can be synchronized using special feedback control and that external excitations affect the synchronization.

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TIAN, LIXIN, and GANG XU. "SUDDEN OCCURRENCE OF CHAOS IN NONSMOOTH MAPS." International Journal of Bifurcation and Chaos 17, no. 01 (January 2007): 271–82. http://dx.doi.org/10.1142/s021812740701732x.

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This paper introduces a type of one-dimensional nonsmooth nonlinear discrete dynamic system. We find a direct route to chaos from stable period-two point, and this is called Sudden Occurrence of Chaos. It is completely different from the three routes from regular motion to chaos — period-doubling bifurcation chaos, intermittency and quasi-periodicity chaos. Furthermore, we present some examples of sudden occurrence of chaos from m-period directly to chaos.

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El-Sayed, Ahmed M. A., and Mohamed E. Nasr. "Dynamic Properties of the Predator–Prey Discontinuous Dynamical System." Zeitschrift für Naturforschung A 67, no. 1-2 (February 1, 2012): 57–60. http://dx.doi.org/10.5560/zna.2011-0051.

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In this work, we study the dynamic properties (equilibrium points, local and global stability, chaos and bifurcation) of the predator-prey discontinuous dynamical system. The existence and uniqueness of uniformly Lyapunov stable solution will be proved

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Kengne, J. "Coexistence of Chaos with Hyperchaos, Period-3 Doubling Bifurcation, and Transient Chaos in the Hyperchaotic Oscillator with Gyrators." International Journal of Bifurcation and Chaos 25, no. 04 (April 2015): 1550052. http://dx.doi.org/10.1142/s0218127415500522.

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In this paper, the dynamics of the paradigmatic hyperchaotic oscillator with gyrators introduced by Tamasevicius and co-workers (referred to as the TCMNL oscillator hereafter) is considered. This well known hyperchaotic oscillator with active RC realization of inductors is suitable for integrated circuit implementation. Unlike previous literature based on piecewise-linear approximation methods, I derive a new (smooth) mathematical model based on the Shockley diode equation to explore the dynamics of the oscillator. Various tools for detecting chaos including bifurcation diagrams, Lyapunov exponents, frequency spectra, phase portraits and Poincaré sections are exploited to establish the connection between the system parameters and various complex dynamic regimes (e.g. hyperchaos, period-3 doubling bifurcation, coexistence of attractors, transient chaos) of the hyperchaotic oscillator. One of the most interesting and striking features of this oscillator discovered/revealed in this work is the coexistence of a hyperchaotic attractor with a chaotic one over a broad range of system parameters. This phenomenon was not reported previously and therefore represents a meaningful contribution to the understanding of the behavior of nonlinear dynamical systems in general. A close agreement is observed between theoretical and experimental analyses.

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Tu, Hongliang, Xueli Zhan, and Xiaobing Mao. "Complex Dynamics and Chaos Control on a Kind of Bertrand Duopoly Game Model considering R&D Activities." Discrete Dynamics in Nature and Society 2017 (2017): 1–13. http://dx.doi.org/10.1155/2017/7384150.

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We study a dynamic research and development two-stage input competition game model in the Bertrand duopoly oligopoly market with spillover effects on cost reduction. We investigate the stability of the Nash equilibrium point and local stable conditions and stability region of the Nash equilibrium point by the bifurcation theory. The complex dynamic behaviors of the system are shown by numerical simulations. It is demonstrated that chaos occurs for a range of managerial policies, and the associated unpredictability is solely due to the dynamics of the interaction. We show that the straight line stabilization method is the appropriate management measure to control the chaos.

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Zhang, Chunrui, and Huifeng Zheng. "Dynamic Properties of Coupled Maps." Discrete Dynamics in Nature and Society 2010 (2010): 1–10. http://dx.doi.org/10.1155/2010/905102.

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Dynamic properties are investigated in the coupled system of three maps with symmetric nearest neighbor coupling and periodic boundary conditions. The dynamics of the system is controlled by certain coupling parameters. We show that, for some values of the parameters, the system exhibits nontrivial collective behavior, such as multiple bifurcations, and chaos. We give computer simulations to support the theoretical predictions.

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Rodica, Bulai, and Victor Fanari. "Cryptography Chaos Theory." Central and Eastern European eDem and eGov Days 331 (July 11, 2018): 447–58. http://dx.doi.org/10.24989/ocg.v331.37.

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The development of information society, which has led to an impressive increase in the volume of information, mainly economic, circulated in computer networks, accelerated the development and mostly the use of modern cryptography tools. In the last years, researchers have pointed out that there is a possible similarity between chaos and cryptography, many of the properties of chaotic dynamic systems having correlation among the cryptographic systems that are based on computational methods. Studies carried out on chaotic dynamic systems usage in digital crypto-systems have determined the occurrence of similar to classic techniques, but also of some specific techniques and methods that have been analyzed and evaluated. The attempts to develop new encryption ?lgorithms based on chaos theory have evolved gradually from simple solutions, which suppose the iteration of a din?mic system to obtain binary sequence used for text masking, to methods that imply coupled din?mic systems and hybrid techniques that would combine the chaos advantages with classical methods. In this article there are presented 3 encryption algorithms based on chaos theory: RC4, Fractal Encryption and Cellular Automata, implemented in a system of encryption and operation mode analysis for each algorithm separately.

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