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Journal articles on the topic 'Chaos of Devaney'

Author: Grafiati

Published: 4 June 2021

Last updated: 18 February 2022

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1

Lu, Tianxiu, Peiyong Zhu, and Xinxing Wu. "The Retentivity of Chaos under Topological Conjugation." Mathematical Problems in Engineering 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/817831.

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The definitions of Devaney chaos (DevC), exact Devaney chaos (EDevC), mixing Devaney chaos (MDevC), and weak mixing Devaney chaos (WMDevC) are extended to topological spaces. This paper proves that these chaotic properties are all preserved under topological conjugation. Besides, an example is given to show that the Li-Yorke chaos is not preserved under topological conjugation if the domain is extended to a general metric space.

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2

Wang, Xiaoyi, and Yu Huang. "Devaney chaos revisited." Topology and its Applications 160, no. 3 (2013): 455–60. http://dx.doi.org/10.1016/j.topol.2012.12.002.

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3

Li, Jian, Jie Li, and Siming Tu. "Devaney chaos plus shadowing implies distributional chaos." Chaos: An Interdisciplinary Journal of Nonlinear Science 26, no. 9 (2016): 093103. http://dx.doi.org/10.1063/1.4962131.

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4

Fadel, Asmaa, and Syahida Che Dzul-Kifli. "Some Chaos Notions on Dendrites." Symmetry 11, no. 10 (2019): 1309. http://dx.doi.org/10.3390/sym11101309.

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Transitivity is a key element in a chaotic dynamical system. In this paper, we present some relations between transitivity, stronger and alternative notions of it on compact and dendrite spaces. The relation between Auslander and Yorke chaos and Devaney chaos on dendrites is also discussed. Moreover, we prove that Devaney chaos implies strong dense periodicity on dendrites while the converse is not true.

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5

Kwietniak, Dominik, and Michal Misiurewicz. "Exact Devaney chaos and entropy." Qualitative Theory of Dynamical Systems 6, no. 1 (2005): 169–79. http://dx.doi.org/10.1007/bf02972670.

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6

Wu, Xinxing, and Peiyong Zhu. "Devaney chaos and Li-Yorke sensitivity for product systems." Studia Scientiarum Mathematicarum Hungarica 49, no. 4 (2012): 538–48. http://dx.doi.org/10.1556/sscmath.49.2012.4.1226.

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This paper mainly discusses how Devaney chaos and Li-Yorke sensitivity carry over to product systems. First, two results on the periodic points of product systems are obtained. By using them, the following two results are Proved: (1) A finite product system is mixing and Devaney chaotic if and only if each factor system is mixing and Devaney chaotic. (2) An infinite product map Π i=1∞fi is mixing and Devaney chaotic if and only if each factor map fi is mixing and Devaney chaotic and sup {min P(fi): i ∈ ℕ} < + ∞, where P(fi) is the set of all periods of fi. Besides, we obtain that the product system is Li-Yorke sensitive (sensitive) if and only if there exists a factor system that is Li-Yorke sensitive (sensitive).

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7

Tian, Chuanjun. "Chaos in the Sense of Devaney for Two-Dimensional Time-Varying Generalized Symbolic Dynamical Systems." International Journal of Bifurcation and Chaos 27, no. 04 (2017): 1750060. http://dx.doi.org/10.1142/s0218127417500602.

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This paper studies chaos in the sense of Devaney for a class of two-dimensional time-varying generalized symbol systems of the form [Formula: see text] where [Formula: see text] for [Formula: see text], [Formula: see text] is an integer, [Formula: see text] and [Formula: see text] are two well-defined functions. By introducing a more restrictive concept of chaos in the sense of Devaney, some sufficient conditions for this system to be completely chaotic in the sense of Devaney are derived.

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8

Oprocha, Piotr. "Relations between distributional and Devaney chaos." Chaos: An Interdisciplinary Journal of Nonlinear Science 16, no. 3 (2006): 033112. http://dx.doi.org/10.1063/1.2225513.

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9

Zhu, Hao, Yuming Shi, and Hua Shao. "Devaney Chaos in Nonautonomous Discrete Systems." International Journal of Bifurcation and Chaos 26, no. 11 (2016): 1650190. http://dx.doi.org/10.1142/s021812741650190x.

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This paper is concerned with Devaney chaos in nonautonomous discrete systems. It is shown that in its definition, the two former conditions, i.e. transitivity and density of periodic points, in a set imply the last one, i.e. sensitivity, in the case that the set is unbounded, while a similar result holds under two additional conditions in the other case that the set is bounded. Some chaotic behavior is studied for a class of nonautonomous discrete systems, each of which is governed by a convergent sequence of continuous maps. In addition, the concepts of some pseudo-orbits and shadowing properties are introduced for nonautonomous discrete systems, and it is shown that some shadowing properties of the system and density of periodic points imply that the system is Devaney chaotic under the condition that the sequence of continuous maps is uniformly convergent in a compact metric space.

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10

Barrachina, Xavier, and J. Alberto Conejero. "Devaney Chaos and Distributional Chaos in the Solution of Certain Partial Differential Equations." Abstract and Applied Analysis 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/457019.

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The notion of distributional chaos has been recently added to the study of the linear dynamics of operators andC0-semigroups of operators. We will study this notion of chaos for some examples ofC0-semigroups that are already known to be Devaney chaotic.

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11

Harańczyk, Grzegorz, and Dominik Kwietniak. "When lower entropy implies stronger Devaney chaos." Proceedings of the American Mathematical Society 137, no. 06 (2008): 2063–73. http://dx.doi.org/10.1090/s0002-9939-08-09756-6.

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12

Wang, Lidong, Yingnan Li, and Li Liao. "3-Adic System and Chaos." Journal of Applied Mathematics 2011 (2011): 1–9. http://dx.doi.org/10.1155/2011/838639.

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Let(Z(3),τ)be a 3-adic system. we prove in(Z(3),τ)the existence of uncountable distributional chaotic set ofA(τ), which is an almost periodic points set, and further come to a conclusion thatτis chaotic in the sense of Devaney and Wiggins.

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13

WANG, XIAO FAN, and GUANRONG CHEN. "ON FEEDBACK ANTICONTROL OF DISCRETE CHAOS." International Journal of Bifurcation and Chaos 09, no. 07 (1999): 1435–41. http://dx.doi.org/10.1142/s0218127499000985.

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In this Letter, we prove that the algorithm of Chen and Lai [1996, 1997, 1998] for the anticontrol of chaos leads to chaos not only in the sense of Devaney but also in the sense of Li and Yorke, for both linear and nonlinear autonomous systems of any dimensionalities.

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14

WU, XINXING, and PEIYONG ZHU. "CHAOS IN THE WEIGHTED BIEBUTOV SYSTEMS." International Journal of Bifurcation and Chaos 23, no. 08 (2013): 1350133. http://dx.doi.org/10.1142/s0218127413501332.

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In this paper, various notions of chaos in the Biebutov system are studied. First, the Biebutov system is generalized to the weighted Biebutov system. Meanwhile, we introduce the concept of continuous distributional chaos for the weighted Biebutov system and prove that it exhibits continuous distributional chaos and Li–Yorke sensitivity. Finally, we prove that the weighted Biebutov system is mixing and Devaney chaotic.

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15

Dai, Xiongping, and Xinjia Tang. "Devaney chaos, Li–Yorke chaos, and multi-dimensional Li–Yorke chaos for topological dynamics." Journal of Differential Equations 263, no. 9 (2017): 5521–53. http://dx.doi.org/10.1016/j.jde.2017.06.021.

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16

N. S, Vincent, and Vinod Kumar. "Devaney Chaos Induced by Turbulent and Erratic Functions." IOSR Journal of Mathematics 13, no. 01 (2017): 19–21. http://dx.doi.org/10.9790/5728-1301041921.

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17

Hou, Bingzhe, Xianfeng Ma, and Gongfu Liao. "Difference between Devaney chaos associated with two systems." Nonlinear Analysis: Theory, Methods & Applications 72, no. 3-4 (2010): 1616–20. http://dx.doi.org/10.1016/j.na.2009.08.042.

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18

Bahi, Jacques M., Christophe Guyeux, and Antoine Perasso. "Chaos in DNA evolution." International Journal of Biomathematics 09, no. 05 (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

Li, Zongcheng, Qingli Zhao, and Di Liang. "Chaos in a Discrete Delay Population Model." Discrete Dynamics in Nature and Society 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/482459.

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This paper is concerned with chaos in a discrete delay population model. The map of the model is proved to be chaotic in the sense of both Devaney and Li-Yorke under some conditions, by employing the snap-back repeller theory. Some computer simulations are provided to visualize the theoretical result.

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20

LIN, WEI, and GUANRONG CHEN. "HETEROCLINICAL REPELLERS IMPLY CHAOS." International Journal of Bifurcation and Chaos 16, no. 05 (2006): 1471–89. http://dx.doi.org/10.1142/s021812740601543x.

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In this paper, we prove that chaos in the sense of Li–Yorke and of Devaney is prevalent in discrete systems admitting the so-called heteroclinical repellers, which are similar to the transversely heteroclinical orbits in both continuous and discrete systems and are corresponding to the snap-back repeller proposed by Marotto for proving the existence of chaos in higher-dimensional systems. In addition, the concept of heteroclinical repellers is generalized to be applicable to the case with degenerate transformations. In the end, some illustrative examples are provided to illustrate the theoretical results.

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21

Syahida Che Dzul-Kifli and Chris Good. "On Devaney Chaos and Dense Periodic Points: Period 3 and Higher Implies Chaos." American Mathematical Monthly 122, no. 8 (2015): 773. http://dx.doi.org/10.4169/amer.math.monthly.122.8.773.

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22

Arai, Tatsuya, and Naotsugu Chinen. "P-chaos implies distributional chaos and chaos in the sense of Devaney with positive topological entropy." Topology and its Applications 154, no. 7 (2007): 1254–62. http://dx.doi.org/10.1016/j.topol.2005.11.016.

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23

Špitalský, Vladimír. "Entropy and exact Devaney chaos on totally regular continua." Discrete & Continuous Dynamical Systems - A 33, no. 7 (2013): 3135–52. http://dx.doi.org/10.3934/dcds.2013.33.3135.

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24

Baloush, Malouh, Syahida Che Dzul-Kifli, and Chris Good. "Dense periodicity property and Devaney chaos on shifts spaces." International Journal of Mathematical Analysis 10 (2016): 1019–29. http://dx.doi.org/10.12988/ijma.2016.6574.

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25

MARTÍNEZ-GIMÉNEZ, F., and A. PERIS. "CHAOTIC POLYNOMIALS ON SEQUENCE AND FUNCTION SPACES." International Journal of Bifurcation and Chaos 20, no. 09 (2010): 2861–67. http://dx.doi.org/10.1142/s0218127410027416.

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We study chaos in the sense of Devaney for two models of polynomials (homogeneous and non-homogeneous) of arbitrary degree, defined on certain sequence spaces. Consequences are also obtained for the chaotic dynamics of the corresponding polynomials on some function spaces.

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26

Yin, Zongbin. "Chaotic Dynamics of Composition Operators on the Space of Continuous Functions." International Journal of Bifurcation and Chaos 27, no. 06 (2017): 1750084. http://dx.doi.org/10.1142/s0218127417500845.

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In this paper, the chaotic dynamics of composition operators on the space of real-valued continuous functions is investigated. It is proved that the hypercyclicity, topologically mixing property, Devaney chaos, frequent hypercyclicity and the specification property of the composition operator are equivalent to each other and are stronger than dense distributional chaos. Moreover, the composition operator [Formula: see text] exhibits dense Li–Yorke chaos if and only if it is densely distributionally chaotic, if and only if the symbol [Formula: see text] admits no fixed points. Finally, the long-time behaviors of the composition operator with affine symbol are classified in detail.

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27

MARTÍNEZ-GIMÉNEZ, F., and A. PERIS. "CHAOS FOR BACKWARD SHIFT OPERATORS." International Journal of Bifurcation and Chaos 12, no. 08 (2002): 1703–15. http://dx.doi.org/10.1142/s0218127402005418.

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Backward shift operators provide a general class of linear dynamical systems on infinite dimensional spaces. Despite linearity, chaos is a phenomenon that occurs within this context. In this paper we give characterizations for chaos in the sense of Auslander and Yorke [1980] and in the sense of Devaney [1989] of weighted backward shift operators and perturbations of the identity by backward shifts on a wide class of sequence spaces. We cover and unify a rich variety of known examples in different branches of applied mathematics. Moreover, we give new examples of chaotic backward shift operators. In particular we prove that the differential operator I + D is Auslander–Yorke chaotic on the most usual spaces of analytic functions.

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28

Zhu, Hegui, Wentao Qi, Jiangxia Ge, and Yuelin Liu. "Analyzing Devaney Chaos of a Sine–Cosine Compound Function System." International Journal of Bifurcation and Chaos 28, no. 14 (2018): 1850176. http://dx.doi.org/10.1142/s0218127418501766.

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The one-dimensional Sine map and Chebyshev map are classical chaotic maps, which have clear chaotic characteristics. In this paper, we establish a chaotic framework based on a Sine–Cosine compound function system by analyzing the existing one-dimensional Sine map and Chebyshev map. The sensitive dependence on initial conditions, topological transitivity and periodic-point density of this chaotic framework is proved, showing that the chaotic framework satisfies Devaney’s chaos definition. In order to illustrate the chaotic behavior of the chaotic framework, we propose three examples, called Cosine–Polynomial (C–P) map, Sine–Tangent (S–T) map and Sine–Exponent (S–E) map, respectively. Then, we evaluate the chaotic behavior with Sine map and Chebyshev map by analyzing bifurcation diagrams, Lyapunov exponents, correlation dimensions, Kolmogorov entropy and [Formula: see text] complexity. Experimental results show that the chaotic framework has better unpredictability and more complex chaotic behaviors than the classical Sine map and Chebyshev map. The results also verify the effectiveness of the theoretical analysis of the proposed chaotic framework.

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29

Xu, Zhengjie, Wei Lin, and Jiong Ruan. "Decay of correlation implies chaos in the sense of Devaney." Chaos, Solitons & Fractals 22, no. 2 (2004): 305–10. http://dx.doi.org/10.1016/j.chaos.2004.01.006.

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30

ZHANG, HUAGUANG, ZHILIANG WANG, and DERONG LIU. "CHAOTIFYING FUZZY HYPERBOLIC MODEL USING IMPULSIVE AND NONLINEAR FEEDBACK CONTROL APPROACHES." International Journal of Bifurcation and Chaos 15, no. 08 (2005): 2603–10. http://dx.doi.org/10.1142/s021812740501354x.

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In this paper, the problem of chaotifying the continuous-time fuzzy hyperbolic model (FHM) is considered. We use impulsive and nonlinear feedback control methods to chaotify the FHM and show that chaos produced by the present methods satisfy the three criteria of Devaney. Computer simulations will be used to verify the present results.

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31

Tian, Chuanjun. "Chaos of Discrete Spatiotemporal Systems on a Bidirectional Metric Space." International Journal of Bifurcation and Chaos 24, no. 05 (2014): 1450074. http://dx.doi.org/10.1142/s0218127414500746.

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This paper is concerned with chaos of discrete spatiotemporal systems of the form [Formula: see text] where k is a positive integer, I is a bounded subset of R = (-∞, ∞), and f : Ik+1→ I is a function, m, n ∈ N0= {0, 1, 2, …}. Some new sufficient conditions for this system to be chaotic in the sense of Devaney on I∞with a bidirectional metric are derived.

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32

Wu, Xiaoying. "Heteroclinic Cycles Imply Chaos and Are Structurally Stable." Discrete Dynamics in Nature and Society 2021 (May 29, 2021): 1–7. http://dx.doi.org/10.1155/2021/6647132.

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This paper is concerned with the chaos of discrete dynamical systems. A new concept of heteroclinic cycles connecting expanding periodic points is raised, and by a novel method, we prove an invariant subsystem is topologically conjugate to the one-side symbolic system. Thus, heteroclinic cycles imply chaos in the sense of Devaney. In addition, if a continuous differential map h has heteroclinic cycles in ℝ n , then g has heteroclinic cycles with h − g C 1 being sufficiently small. The results demonstrate C 1 structural stability of heteroclinic cycles. In the end, two examples are given to illustrate our theoretical results and applications.

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33

Li, Zongcheng, Shutang Liu, Wei Li, and Qingli Zhao. "Chaotification for a Class of Delay Difference Equations Based on Snap-Back Repellers." Mathematical Problems in Engineering 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/917137.

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We study the chaotification problem for a class of delay difference equations by using the snap-back repeller theory and the feedback control approach. We first study the stability and expansion of fixed points and establish a criterion of chaos. Then, based on this criterion of chaos and the feedback control approach, we establish a chaotification scheme such that the controlled system is chaotic in the sense of both Devaney and Li-Yorke when the parameters of this system satisfy some mild conditions. For illustrating the theoretical result, we give some computer simulations.

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34

XIE, LINGLI, YI ZHOU, and YI ZHAO. "CRITERION OF CHAOS FOR SWITCHED LINEAR SYSTEMS WITH CONTROLLERS." International Journal of Bifurcation and Chaos 20, no. 12 (2010): 4103–9. http://dx.doi.org/10.1142/s0218127410028215.

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In this paper, a sufficient criterion for time-invariant switched linear systems with controllers to be chaotic in the sense of Li–Yorke and Devaney is presented. The switched linear systems consist of an unstable subsystem with expanding flows and a controllable subsystem. It is exposed that the controllability of dynamic systems, instead of asymptotic stability, plays an important role in generating chaos. Finally, we give a numerical simulation for an example with some variable parameters to illustrate the validity of the result.

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35

Akhmet, Marat, and Ejaily Milad Alejaily. "Domain-Structured Chaos in a Hopfield Neural Network." International Journal of Bifurcation and Chaos 29, no. 14 (2019): 1950205. http://dx.doi.org/10.1142/s0218127419502055.

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In this paper, we provide a new method for constructing chaotic Hopfield neural networks. Our approach is based on structuring the domain to form a special set through the discrete evolution of the network state variables. In the chaotic regime, the formed set is invariant under the system governing the dynamics of the neural network. The approach can be viewed as an extension of the unimodality technique for one-dimensional map, thereby generating chaos from higher-dimensional systems. We show that the discrete Hopfield neural network considered is chaotic in the sense of Devaney, Li–Yorke, and Poincaré. Mathematical analysis and numerical simulation are provided to confirm the presence of chaos in the network.

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36

Liang, Wei, and Zihan Zhang. "Anti-Control of Chaos for First-Order Partial Difference Equations via Sine and Cosine Functions." International Journal of Bifurcation and Chaos 29, no. 10 (2019): 1950140. http://dx.doi.org/10.1142/s0218127419501402.

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In this paper, anti-control of chaos for first-order partial difference equations with nonperiod boundary condition is studied. Three new chaotification schemes for first-order partial difference equations with sine and cosine functions are established, respectively. It is proved that all the systems are chaotic in the sense of both Devaney and Li–Yorke by applying coupled-expanding theory of general discrete dynamical systems. Two illustrative examples are provided with computer simulations

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37

Li, Zongcheng. "Anticontrol of Chaos for a Class of Delay Difference Equations Based on Heteroclinic Cycles Connecting Repellers." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/260150.

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This paper is concerned with anticontrol of chaos for a class of delay difference equations via the feedback control technique. The controlled system is first reformulated into a high-dimensional discrete dynamical system. Then, a chaotification theorem based on the heteroclinic cycles connecting repellers for maps is established. The controlled system is proved to be chaotic in the sense of both Devaney and Li-Yorke. An illustrative example is provided with computer simulations.

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38

Chen, Chung-Chuan, J. Alberto Conejero, Marko Kostić, and Marina Murillo-Arcila. "Dynamics of multivalued linear operators." Open Mathematics 15, no. 1 (2017): 948–58. http://dx.doi.org/10.1515/math-2017-0082.

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Abstract We introduce several notions of linear dynamics for multivalued linear operators (MLO’s) between separable Fréchet spaces, such as hypercyclicity, topological transitivity, topologically mixing property, and Devaney chaos. We also consider the case of disjointness, in which any of these properties are simultaneously satisfied by several operators. We revisit some sufficient well-known computable criteria for determining those properties. The analysis of the dynamics of extensions of linear operators to MLO’s is also considered.

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Yin, Zongbin, Yuming Chen, and Shengnan He. "Disjoint Hypercyclicity and Topological Entropy of Composition Operators." International Journal of Bifurcation and Chaos 28, no. 04 (2018): 1850053. http://dx.doi.org/10.1142/s0218127418500530.

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In a previous paper, we characterized the Devaney chaos, frequent hypercyclicity and dense distributional chaos of composition operators induced by continuous self-maps on the real line. The present paper further investigates the disjoint hypercyclicity and topological entropy of these operators. It is shown that the composition operator is [Formula: see text]-transitive if and only if it is Cesàro-hypercyclic, if and only if it is supercyclic, if and only if it has the specification property on the whole space. Furthermore, sufficient and necessary conditions for a pair of composition operators to be disjoint hypercyclic (disjoint mixing, respectively) are obtained. Finally, sufficient conditions for the composition operator to admit infinite topological entropy are provided.

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40

Thakur, Rahul, and Ruchi Das. "Devaney chaos and stronger forms of sensitivity on the product of semiflows." Semigroup Forum 98, no. 3 (2019): 631–44. http://dx.doi.org/10.1007/s00233-019-10008-1.

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41

O'Cairbre, Fiacre, Gian Mario Maggio, and Michael Peter Kennedy. "Devaney Chaos in an Approximate One-Dimensional Model of the Colpitts Oscillator." International Journal of Bifurcation and Chaos 07, no. 11 (1997): 2561–68. http://dx.doi.org/10.1142/s0218127497001710.

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In this paper we consider an approximate one-dimensional model for describing the non-linear dynamics of the Colpitts oscillator. It has already been shown that this model preserves the qualitative dynamical behavior of the Colpitts oscillator. Using the theory of snap-back repellors we prove that the one-dimensional model is chaotic for certain values of its parameters.

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42

ZHENG, YONGAI, GUANRONG CHEN, and ZENGRONG LIU. "ON CHAOTIFICATION OF DISCRETE SYSTEMS." International Journal of Bifurcation and Chaos 13, no. 11 (2003): 3443–47. http://dx.doi.org/10.1142/s0218127403008661.

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In this paper, the problem of making a nonlinear system chaotic by using state-feedback control is studied. The feedback controller uses a simple sine function of the system state, but only one component in each dimension. It is proved, by using the anti-integrable limit method, that the designed control system generates chaos in the sense of Devaney. In fact, the controlled system so designed is a perturbation of the original system, which turns out to be a simple Bernoulli shift.

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43

Alberto Conejero, J., Carlos Lizama, Marina Murillo-Arcila, and Alfredo Peris. "Linear dynamics of semigroups generated by differential operators." Open Mathematics 15, no. 1 (2017): 745–67. http://dx.doi.org/10.1515/math-2017-0065.

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Abstract During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others. These notions have been extended, as far as possible, to the setting of C0-semigroups of linear and continuous operators. We will review some of these notions and we will discuss basic properties of the dynamics of C0-semigroups. We will also study in detail the dynamics of the translation C0-semigroup on weighted spaces of integrable functions and of continuous functions vanishing at infinity. Using the comparison lemma, these results can be transferred to the solution C0-semigroups of some partial differential equations. Additionally, we will also visit the chaos for infinite systems of ordinary differential equations, that can be of interest for representing birth-and-death process or car-following traffic models.

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44

SHI, YUMING, PEI YU, and GUANRONG CHEN. "CHAOTIFICATION OF DISCRETE DYNAMICAL SYSTEMS IN BANACH SPACES." International Journal of Bifurcation and Chaos 16, no. 09 (2006): 2615–36. http://dx.doi.org/10.1142/s021812740601629x.

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This paper is concerned with chaotification of discrete dynamical systems in Banach spaces via feedback control techniques. A criterion of chaos in Banach spaces is first established. This criterion extends and improves the Marotto theorem. Discussions are carried out in general and some special Banach spaces. All the controlled systems are proved to be chaotic in the sense of both Devaney and Li–Yorke. As a consequence, a controlled system described in a finite-dimensional real space studied by Wang and Chen is shown chaotic not only in the sense of Li–Yorke but also in the sense of Devaney. The original system can be driven to be chaotic by using an arbitrarily small-amplitude state feedback control in a certain space. In addition, the Chen–Lai anti-control algorithm via feedback control with mod-operation in a finite-dimensional real space is extended to a certain infinite-dimensional Banach space, and the controlled system is shown chaotic in the sense of Devaney as well as in the sense of both Li–Yorke and Wiggins. Differing from many existing results, it is not here required that the map corresponding to the original system has a fixed point in some cases. An application of the theoretical results to a class of first-order partial difference equations is given with some numerical simulations.

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Liang, Wei, and Haihong Guo. "Chaotification of First-Order Partial Difference Equations." International Journal of Bifurcation and Chaos 30, no. 15 (2020): 2050229. http://dx.doi.org/10.1142/s0218127420502296.

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This paper is concerned with chaotification of first-order partial difference equations. Two criteria of chaos for the difference equations with general controllers are established, and all the controlled systems are proved to be chaotic in the sense of Li–Yorke or of both Li–Yorke and Devaney by applying the coupled-expanding theory of general discrete dynamical systems. The controllers used in this paper can be easily constructed, facilitating the chaotification of first-order partial difference equations. For illustration, two illustrative examples are provided.

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WU, XINXING, and PEIYONG ZHU. "CHAOS IN A CLASS OF NONCONSTANT WEIGHTED SHIFT OPERATORS." International Journal of Bifurcation and Chaos 23, no. 01 (2013): 1350010. http://dx.doi.org/10.1142/s0218127413500107.

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In this paper, chaos generated by a class of nonconstant weighted shift operators is studied. First, we prove that for the weighted shift operator Bμ : Σ(X) → Σ(X) defined by Bμ(x0, x1, …) = (μ(0)x1, μ(1)x2, …), where X is a normed linear space (not necessarily complete), weak mix, transitivity (hypercyclity) and Devaney chaos are all equivalent to separability of X and this property is preserved under iterations. Then we get that [Formula: see text] is distributionally chaotic and Li–Yorke sensitive for each positive integer N. Meanwhile, a sufficient condition ensuring that a point is k-scrambled for all integers k > 0 is obtained. By using these results, a simple example is given to show that Corollary 3.3 in [Fu & You, 2009] does not hold. Besides, it is proved that the constructive proof of Theorem 4.3 in [Fu & You, 2009] is not correct.

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Li, Risong. "A note on decay of correlation implies chaos in the sense of Devaney." Applied Mathematical Modelling 39, no. 21 (2015): 6705–10. http://dx.doi.org/10.1016/j.apm.2015.02.019.

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CHEN, YUANLONG, YU HUANG, and LIANGLIANG LI. "THE PERSISTENCE OF SNAP-BACK REPELLER UNDER SMALL C1 PERTURBATIONS IN BANACH SPACES." International Journal of Bifurcation and Chaos 21, no. 03 (2011): 703–10. http://dx.doi.org/10.1142/s0218127411028702.

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In this paper, we consider the persistence of snap-back repellers under small C1 perturbations in Banach spaces. Let X be a Banach space and f be a C1-map from X into itself. We show that if f has a snap-back repeller, then any small C1 perturbations of f has a snap-back repeller and exhibits chaos in the sense of Devaney. The obtained results further extend the existing relevant results in finite-dimensional Euclidean spaces. As applications, we will discuss the chaotic behavior of two nonlocal population models.

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Salman, Mohammad, and Ruchi Das. "Dynamics of Weakly Mixing Nonautonomous Systems." International Journal of Bifurcation and Chaos 29, no. 09 (2019): 1950123. http://dx.doi.org/10.1142/s0218127419501232.

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For a commutative nonautonomous dynamical system we show that topological transitivity of the nonautonomous system induced on probability measures (hyperspaces) is equivalent to the weak mixing of the induced systems. Several counter examples are given for the results which are true in autonomous but need not be true in nonautonomous systems. Wherever possible sufficient conditions are obtained for the results to hold true. For a commutative periodic nonautonomous system on intervals, it is proved that weak mixing implies Devaney chaos. Given a periodic nonautonomous system, it is shown that sensitivity is equivalent to some stronger forms of sensitivity on a closed unit interval.

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LI, PING, ZHONG LI, and WOLFGANG A. HALANG. "CHAOTIFICATION OF SPATIOTEMPORAL SYSTEMS." International Journal of Bifurcation and Chaos 20, no. 07 (2010): 2193–202. http://dx.doi.org/10.1142/s0218127410027027.

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Spatiotemporal chaotic systems have been applied to design pseudo-random-bit generators or ciphers for better cryptographic performance, where spatiotemporal chaos is desirable. This paper presents a chaotification method (or anti-chaos control) for creating spatiotemporal systems, which are originally either nonchaotic or chaotic, strongly chaotic. Sufficient conditions on the system parameters for chaotification are obtained. A mathematically rigorous proof shows that the chaotified system satisfying the parameter conditions is chaotic in the sense of Li–Yorke and Devaney, respectively. Moreover, the chaotification method is applicable to other spatiotemporal systems even with different configurations. Simulation results have illustrated the effectiveness of the proposed chaotification method. Additionally, the statistical properties of the spatiotemporal systems and their chaotified counterparts are analyzed and compared, showing that the chaotification method endows the spatiotemporal systems with good statistical properties. Therefore, the chaotification method can be used for applications of spatiotemporal systems in cryptography.

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