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Journal articles on the topic 'Topological chaos'

Author: Grafiati

Published: 4 June 2021

Last updated: 12 February 2022

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1

DOWNAROWICZ, TOMASZ, and YVES LACROIX. "Measure-theoretic chaos." Ergodic Theory and Dynamical Systems 34, no. 1 (2012): 110–31. http://dx.doi.org/10.1017/etds.2012.117.

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AbstractWe define new isomorphism invariants for ergodic measure-preserving systems on standard probability spaces, called measure-theoretic chaos and measure-theoretic$^+$ chaos. These notions are analogs of the topological chaos DC2 and its slightly stronger version (which we denote by $\text {DC}1\frac 12$). We prove that: (1) if a topological system is measure-theoretically (measure-theoretically$^+$) chaotic with respect to at least one of its ergodic measures then it is topologically DC2 $(\text {DC}1\frac 12)$ chaotic; (2) every ergodic system with positive Kolmogorov–Sinai entropy is measure-theoretically$^+$ chaotic (even in a slightly stronger uniform sense). We provide an example showing that the latter statement cannot be reversed, that is, of a system of entropy zero with uniform measure-theoretic$^+$chaos.

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2

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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3

Qian, Yun, and Peng Guan. "Li-York Chaos of Set-Valued Discrete Dynamical Systems Based on Semi-Group Actions." Applied Mechanics and Materials 380-384 (August 2013): 1778–82. http://dx.doi.org/10.4028/www.scientific.net/amm.380-384.1778.

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t is well known that a semi-groups action on a space could appear chaos phenomenon, like Li-York chaos and so on. Li-York chaos has important relations with topological transitivity and periodic point. This study analyzed metric space and its dinduced Hausdorff metric space. Letis a semi-group. We make continuously act on space. We study topological transitivity and betweenand. Some important results are presented which show that if is topological transitivity and periodicity (which means Li-York chaos at the same time), then the action of semi-grouponis Li-York chaos.

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4

Li, Shihai. "ω-Chaos and Topological Entropy". Transactions of the American Mathematical Society 339, № 1 (1993): 243. http://dx.doi.org/10.2307/2154217.

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5

Tomaschitz, Roman. "Topological evolution and cosmic chaos." Reports on Mathematical Physics 40, no. 2 (1997): 359–65. http://dx.doi.org/10.1016/s0034-4877(97)85933-2.

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6

Tan, Amanda J., Eric Roberts, Spencer A. Smith, et al. "Topological chaos in active nematics." Nature Physics 15, no. 10 (2019): 1033–39. http://dx.doi.org/10.1038/s41567-019-0600-y.

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7

Li, Shi Hai. "$\omega$-chaos and topological entropy." Transactions of the American Mathematical Society 339, no. 1 (1993): 243–49. http://dx.doi.org/10.1090/s0002-9947-1993-1108612-8.

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8

Leboeuf, P., J. Kurchan, M. Feingold, and D. P. Arovas. "Topological aspects of quantum chaos." Chaos: An Interdisciplinary Journal of Nonlinear Science 2, no. 1 (1992): 125–30. http://dx.doi.org/10.1063/1.165915.

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9

Liu, Xin, Huoyun Wang, and Heman Fu. "Topological Sequence Entropy and Chaos." International Journal of Bifurcation and Chaos 24, no. 07 (2014): 1450100. http://dx.doi.org/10.1142/s0218127414501004.

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A dynamical system is called a null system, if the topological sequence entropy along any strictly increasing sequence of non-negative integers is 0. Given 0 ≤ p ≤ q ≤ 1, a dynamical system is [Formula: see text] chaotic, if there is an uncountable subset in which any two different points have trajectory approaching time set with lower density p and upper density q. It shows that, for any 0 ≤ p < q ≤ 1 or p = q = 0 or p = q = 1, a dynamical system which is null and [Formula: see text] chaotic can be realized.

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10

CÁNOVAS, JOSE S., and MARÍA MUÑOZ. "REVISITING PARRONDO'S PARADOX FOR THE LOGISTIC FAMILY." Fluctuation and Noise Letters 12, no. 03 (2013): 1350015. http://dx.doi.org/10.1142/s0219477513500156.

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The aim of this paper is to investigate the existence of Parrondo's paradox for the logistic family fa(x) = ax(1 - x), x ∈ [0, 1], when the parameter value a ranges over the interval [1, 4]. We find that a paradox of type "order + order = chaos" arises for both physically observable and topological chaos, while a "chaos + chaos = order" paradox can be only detected for the case of physically observable chaos. In addition, we raise the question of whether the paradox "chaos + chaos = order" can appear in the topological sense or whether, as our computations seem to show, it is impossible for the logistic family.

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11

Finn, Matthew D., Jean-Luc Thiffeault, and Emmanuelle Gouillart. "Topological chaos in spatially periodic mixers." Physica D: Nonlinear Phenomena 221, no. 1 (2006): 92–100. http://dx.doi.org/10.1016/j.physd.2006.07.018.

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12

Banasiak, J., M. Lachowicz, and M. Moszyński. "Topological chaos: When topology meets medicine." Applied Mathematics Letters 16, no. 3 (2003): 303–8. http://dx.doi.org/10.1016/s0893-9659(03)80048-4.

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13

Degond, P., and M. Pulvirenti. "Propagation of chaos for topological interactions." Annals of Applied Probability 29, no. 4 (2019): 2594–612. http://dx.doi.org/10.1214/19-aap1469.

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14

Blanchard, François. "Topological chaos: what may this mean?" Journal of Difference Equations and Applications 15, no. 1 (2009): 23–46. http://dx.doi.org/10.1080/10236190802385355.

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15

Downarowicz, T. "Positive topological entropy implies chaos DC2." Proceedings of the American Mathematical Society 142, no. 1 (2013): 137–49. http://dx.doi.org/10.1090/s0002-9939-2013-11717-x.

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16

Tomaschitz, Roman. "Chaos and Topological Evolution in Cosmology." International Journal of Bifurcation and Chaos 07, no. 08 (1997): 1847–53. http://dx.doi.org/10.1142/s0218127497001412.

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An elementary account on the chaoticity of galactic world-lines in an open universe is given. A new type of cosmic evolution by global metrical deformations, unpredicted by Einstein's equations, is pointed out. Physical effects of this evolution are backscattering of electromagnetic fields and particle creation in quantum fields. We review in an untechnical way how global metrical deformations of the open and multiply connected spacelike slices induce angular fluctuations in the temperature of the cosmic microwave background radiation.

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17

Ma, You Jie, Shuang Song, and Xue Song Zhou. "Introduction of Topological Horseshoe Theory in Chaotic Research." Advanced Materials Research 811 (September 2013): 716–19. http://dx.doi.org/10.4028/www.scientific.net/amr.811.716.

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Over the past 10 years, nonlinear dynamics and chaotic theory attracted scholars and people got a deeper understanding of chaos. There are many methods for chaos research, and the method of using topological horseshoe is an important branch of those methods. So far, this is one of the core methods with mathematical rigor for chaos research. Based on simple thinking of geometric space, topological horseshoe build a bridge for numerical and theoretical studies of complex behavior of nonlinear systems so that people can carry out a series of studies for chaotic behavior. This paper introduces the basic content of topological horseshoe theory and the application to a simple power system.

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18

Doleželová-Hantáková, Jana, Zuzana Roth, and Samuel Roth. "On the Weakest Version of Distributional Chaos." International Journal of Bifurcation and Chaos 26, no. 14 (2016): 1650235. http://dx.doi.org/10.1142/s0218127416502357.

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The aim of the paper is to correct and improve some results concerning distributional chaos of type 3. We show that in a general compact metric space, distributional chaos of type 3, denoted DC3, even when assuming the existence of an uncountable scrambled set, is a very weak form of chaos. In particular, (i) the chaos can be unstable (it can be destroyed by conjugacy), and (ii) such an unstable system may contain no Li–Yorke pair. However, the definition can be strengthened to get DC[Formula: see text] which is a topological invariant and implies Li–Yorke chaos, similarly as types DC1 and DC2; but unlike them, strict DC[Formula: see text] systems must have zero topological entropy.

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19

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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20

CHEN, AN, and XUETING TIAN. "Distributional chaos in multifractal analysis, recurrence and transitivity." Ergodic Theory and Dynamical Systems 41, no. 2 (2019): 349–78. http://dx.doi.org/10.1017/etds.2019.57.

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There is much research on the dynamical complexity on irregular sets and level sets of ergodic average from the perspective of density in base space, the Hausdorff dimension, Lebesgue positive measure, positive or full topological entropy (and topological pressure), etc. However, this is not the case from the viewpoint of chaos. There are many results on the relationship of positive topological entropy and various chaos. However, positive topological entropy does not imply a strong version of chaos, called DC1. Therefore, it is non-trivial to study DC1 on irregular sets and level sets. In this paper, we will show that, for dynamical systems with specification properties, there exist uncountable DC1-scrambled subsets in irregular sets and level sets. Meanwhile, we prove that several recurrent level sets of points with different recurrent frequency have uncountable DC1-scrambled subsets. The major argument in proving the above results is that there exists uncountable DC1-scrambled subsets in saturated sets.

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21

Mohtashamipour, Maliheh, and Alireza Zamani Bahabadi. "Chaos in Iterated Function Systems." International Journal of Bifurcation and Chaos 30, no. 12 (2020): 2050177. http://dx.doi.org/10.1142/s0218127420501771.

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In the present paper, we study chaos in iterated function systems (IFS), namely dynamical systems with several generators. We introduce weak Li–Yorke chaos, chaos in branch, and weak topological chaos to perceive the role of branches to create chaos in an IFS. Moreover, we define another type of chaos, [Formula: see text]-chaos, on an IFS. Further, we find the necessary conditions to create the chaotic iterated function systems.

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22

Zamani Bahabadi, Alireza. "Controlled shadowing property." Applied General Topology 19, no. 1 (2018): 91. http://dx.doi.org/10.4995/agt.2018.7731.

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In this paper we introduce a new notion, named controlled shadowing property and we relate it to some notions in dynamical systems such as topological ergodicity, topologically mixing and specication properties. The relation between the controlled shadowing and chaos in sense of Li-Yorke is studied. At the end we give some examples to investigate the controlled shadowing property.

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23

PROTOPOPESCU, V., and Y. Y. AZMY. "TOPOLOGICAL CHAOS FOR A CLASS OF LINEAR MODELS." Mathematical Models and Methods in Applied Sciences 02, no. 01 (1992): 79–90. http://dx.doi.org/10.1142/s0218202592000065.

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We construct an example of linear rate equation in the Banach space of summable sequences, l1, that exhibits the three properties required as signature of topological chaos, namely: (i) topological transitivity, (ii) dense periodic orbits, and (iii) positive Lyapunov exponents. The example is based on the properties of the backward shift operator on the Banach space l1. Since linear chaos in the sense described above can occur only in an infinite-dimensional setting, possible finite-dimensional approximate manifestations are investigated. The relationship between the linear backward shift and the nonlinear Bernoulli shift is also discussed.

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24

FINN, M. D., S. M. COX, and H. M. BYRNE. "Topological chaos in inviscid and viscous mixers." Journal of Fluid Mechanics 493 (October 25, 2003): 345–61. http://dx.doi.org/10.1017/s0022112003005858.

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25

Vikhansky, A. "Simulation of topological chaos in laminar flows." Chaos: An Interdisciplinary Journal of Nonlinear Science 14, no. 1 (2004): 14–22. http://dx.doi.org/10.1063/1.1621092.

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26

Berblinger, Michael, and Christoph Schlier. "The double-morse well and topological chaos." Chemical Physics Letters 145, no. 4 (1988): 299–304. http://dx.doi.org/10.1016/0009-2614(88)80011-3.

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27

Barnsley, Michael F., Krzysztof Leśniak, and Miroslav Rypka. "Chaos game for IFSs on topological spaces." Journal of Mathematical Analysis and Applications 435, no. 2 (2016): 1458–66. http://dx.doi.org/10.1016/j.jmaa.2015.11.022.

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28

Du, Bau-Sen. "Topological entropy and chaos of interval maps." Nonlinear Analysis: Theory, Methods & Applications 11, no. 1 (1987): 105–14. http://dx.doi.org/10.1016/0362-546x(87)90029-0.

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29

Mackay, R. S., and C. Tresser. "Transition to topological chaos for circle maps." Physica D: Nonlinear Phenomena 19, no. 2 (1986): 206–37. http://dx.doi.org/10.1016/0167-2789(86)90020-5.

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30

Mackay, R. S., and C. Tresser. "Transition to topological chaos for circle maps." Physica D: Nonlinear Phenomena 29, no. 3 (1988): 427. http://dx.doi.org/10.1016/0167-2789(88)90042-5.

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31

LETELLIER, C., G. GOUESBET, and N. F. RULKOV. "TOPOLOGICAL ANALYSIS OF CHAOS IN EQUIVARIANT ELECTRONIC CIRCUITS." International Journal of Bifurcation and Chaos 06, no. 12b (1996): 2531–55. http://dx.doi.org/10.1142/s0218127496001624.

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Chaotic oscillations in an electronic circuit are studied by recording two time series simultaneously. The chaotic dynamics is characterized by using topological analysis. A comparison with two models is also discussed. Some prescriptions are given in order to take into account the symmetry properties of the experimental system to perform the topological analysis.

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32

Wang, Lidong, Nan Li, Fengchun Lei, and Zhenyan Chu. "Topological Entropy and Mixing Invariant Extremal Distributional Chaos." International Journal of Bifurcation and Chaos 27, no. 09 (2017): 1750139. http://dx.doi.org/10.1142/s0218127417501395.

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We show that there exists a mixing dynamical system with an invariant, extremal and transitive distributionally scrambled set. Meanwhile, we prove that such a complex dynamical system has zero topological entropy. Finally, we extend the method of constructing the “strong” distributionally scrambled set and show that the new dynamical system has a positive topological entropy.

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33

Kwabi, Prince Amponsah, William Obeng Denteh, and Richard Kena Boadi. "On the dynamics of the Tent function - Phase diagrams." Journal of Advanced Studies in Topology 7, no. 4 (2016): 261. http://dx.doi.org/10.20454/jast.2016.1011.

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This paper focuses on the study of a one-dimensional topological dynamical system, the tent function. We give a good background to the theory of dynamical systems while establishing the important asymptotic properties of topological dynamical systems that distinguishes it from other fields by way of an example - the tent function. A precise definition of the tent function is given and iterates are clearly shown using the phase diagrams. By this same method, chaos in the tent map is shown as iterates become higher. We also show that the tent map has infinitely many chaotic orbits and also express some important features of chaos such as topological transitivity, boundedness and sensitivity to change in initial conditions from a topological viewpoint.

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34

Bernardes, Nilson C., and Rômulo M. Vermersch. "Hyperspace Dynamics of Generic Maps of the Cantor Space." Canadian Journal of Mathematics 67, no. 2 (2015): 330–49. http://dx.doi.org/10.4153/cjm-2014-005-5.

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AbstractWe study the hyperspace dynamics induced fromgeneric continuous maps and fromgeneric homeomorphisms of the Cantor space, with emphasis on the notions of Li– Yorke chaos, distributional chaos, topological entropy, chain continuity, shadowing, and recurrence.

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35

Ovchinnikov, Igor V., and Massimiliano Di Ventra. "Chaos as a symmetry-breaking phenomenon." Modern Physics Letters B 33, no. 24 (2019): 1950287. http://dx.doi.org/10.1142/s0217984919502877.

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Chaos is an ubiquitous and fundamental phenomenon with a wide range of features pointing to a similar phenomenology. Although apparently distinct, it is natural to ask if all these features emerge from a unifying principle. Recently, it was realized that all continuous-time stochastic dynamical systems (DSs) — the most relevant in physics because natural DSs are always subject to noise and time is continuous — possess a topological supersymmetry (TS). It was then suggested that its spontaneous breakdown could be interpreted as the stochastic generalization of deterministic chaos. This conclusion stems from the fact that such phenomenon encompasses features that are traditionally associated with chaotic dynamics such as non-integrability, positive topological entropy, sensitivity to initial conditions and the Poincaré–Bendixson theorem. Here, we strengthen and complete this picture by showing that the remaining hallmarks of deterministic chaos — topological transitivity/mixing and dense periodic orbits — while being consistent with the TS breaking, do not actually admit a stochastic generalization. This is a major limitation, since all physical systems are always noisy to some extent. We, therefore, conclude that spontaneous TS breaking can be considered as the most general definition of continuous-time dynamical chaos. Contrary to the common perception and semantics of the word “chaos,” this phenomenon should then be truly interpreted as the low-symmetry, or ordered phase of the DSs that manifest it. We also argue that the concept of chaos in low-dimensional, discrete-time DSs that do not obey the Poincaré–Bendixson theorem, is related to the explicit TS breaking.

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36

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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37

Lefranc, Marc, Pierre-Emmanuel Morant, and Michel Nizette. "Topological characterization of deterministic chaos: enforcing orientation preservation." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366, no. 1865 (2007): 559–67. http://dx.doi.org/10.1098/rsta.2007.2110.

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The determinism principle, which states that dynamical state completely determines future time evolution, is a keystone of nonlinear dynamics and chaos theory. Since it precludes that two state space trajectories intersect, it is a core ingredient of a topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits embedded in a strange attractor. However, knot theory can be applied only to three-dimensional systems. Still, determinism applies in any dimension. We propose an alternative framework in which this principle is enforced by constructing an orientation-preserving dynamics on triangulated surfaces and find that in three dimensions our approach numerically predicts the correct topological entropies for periodic orbits of the horseshoe map.

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38

HURDER, STEVEN, and ANA RECHTMAN. "Aperiodicity at the boundary of chaos." Ergodic Theory and Dynamical Systems 38, no. 7 (2017): 2683–728. http://dx.doi.org/10.1017/etds.2016.144.

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We consider the dynamical properties of $C^{\infty }$-variations of the flow on an aperiodic Kuperberg plug $\mathbb{K}$. Our main result is that there exists a smooth one-parameter family of plugs $\mathbb{K}_{\unicode[STIX]{x1D716}}$ for $\unicode[STIX]{x1D716}\in (-a,a)$ and $a<1$, such that: (1) the plug $\mathbb{K}_{0}=\mathbb{K}$ is a generic Kuperberg plug; (2) for $\unicode[STIX]{x1D716}<0$, the flow in the plug $\mathbb{K}_{\unicode[STIX]{x1D716}}$ has two periodic orbits that bound an invariant cylinder, all other orbits of the flow are wandering, and the flow has topological entropy zero; (3) for $\unicode[STIX]{x1D716}>0$, the flow in the plug $\mathbb{K}_{\unicode[STIX]{x1D716}}$ has positive topological entropy, and an abundance of periodic orbits.

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39

Jacobs, Joeri, Edward Ott, and Brian R. Hunt. "Calculating topological entropy for transient chaos with an application to communicating with chaos." Physical Review E 57, no. 6 (1998): 6577–88. http://dx.doi.org/10.1103/physreve.57.6577.

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40

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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41

Cattaneo, Gianpiero, Michele Finelli, and Luciano Margara. "Investigating topological chaos by elementary cellular automata dynamics." Theoretical Computer Science 244, no. 1-2 (2000): 219–41. http://dx.doi.org/10.1016/s0304-3975(98)00345-4.

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42

Leboeuf, P., J. Kurchan, M. Feingold, and D. P. Arovas. "Phase-space localization: Topological aspects of quantum chaos." Physical Review Letters 65, no. 25 (1990): 3076–79. http://dx.doi.org/10.1103/physrevlett.65.3076.

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43

Stremler, Mark A., and Jie Chen. "Generating topological chaos in lid-driven cavity flow." Physics of Fluids 19, no. 10 (2007): 103602. http://dx.doi.org/10.1063/1.2772881.

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44

Wu, Xinxing, Xin Ma, Zhu Zhu, and Tianxiu Lu. "Topological Ergodic Shadowing and Chaos on Uniform Spaces." International Journal of Bifurcation and Chaos 28, no. 03 (2018): 1850043. http://dx.doi.org/10.1142/s0218127418500438.

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This paper firstly proves that every dynamical system defined on a Hausdorff uniform space with topologically ergodic shadowing is topologically mixing, thus topologically chain mixing. Then, the following is proved: (1) every weakly mixing dynamical system defined on a second countable Baire–Hausdorff uniform space is chaotic in the sense of both Li–Yorke and Auslander–Yorke; (2) every point transitive dynamical system defined on a Hausdorff uniform space is either almost equicontinuous or sensitive.

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45

Li, Jian, and Xiang Dong Ye. "Recent development of chaos theory in topological dynamics." Acta Mathematica Sinica, English Series 32, no. 1 (2015): 83–114. http://dx.doi.org/10.1007/s10114-015-4574-0.

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46

Jin, Weifeng, and Fangyue Chen. "Topological chaos of universal elementary cellular automata rule." Nonlinear Dynamics 63, no. 1-2 (2010): 217–22. http://dx.doi.org/10.1007/s11071-010-9798-z.

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47

Wang, Hong Qing. "Property of Conjugacy between Two Chaos Maps." Applied Mechanics and Materials 444-445 (October 2013): 771–74. http://dx.doi.org/10.4028/www.scientific.net/amm.444-445.771.

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48

Andres, Jan. "Chaos for Differential Equations with Multivalued Impulses." International Journal of Bifurcation and Chaos 31, no. 07 (2021): 2150113. http://dx.doi.org/10.1142/s0218127421501133.

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The deterministic chaos in the sense of a positive topological entropy is investigated for differential equations with multivalued impulses. Two definitions of topological entropy are examined for three classes of multivalued maps: [Formula: see text]-valued maps, [Formula: see text]-maps and admissible maps in the sense of Górniewicz. The principal tool for its lower estimates and, in particular, its positivity are the Ivanov-type inequalities in terms of the asymptotic Nielsen numbers. The obtained results are then applied to impulsive differential equations via the associated Poincaré translation operators along their trajectories. The main theorems for chaotic differential equations with multivalued impulses are formulated separately on compact subsets of Euclidean spaces and on tori. Several illustrative examples are supplied.

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49

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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50

YU, PEI, WEIGUANG YAO, and GUANRON CHEN. "ANALYSIS ON TOPOLOGICAL PROPERTIES OF THE LORENZ AND THE CHEN ATTRACTORS USING GCM." International Journal of Bifurcation and Chaos 17, no. 08 (2007): 2791–96. http://dx.doi.org/10.1142/s0218127407018762.

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This letter reports a study on some topological properties of chaos using a generalized competitive mode (GCM). The Lorenz system and the Chen system are used as examples for comparison. It is shown that for typical parameter values used in the two systems, the Lorenz attractor has one pair of GCMs in competition, while the Chen attractor has two pairs of GCMs in competition. This explains why the two attractors are topologically different, and furthermore indicates that the Chen attractor is more complex than the Lorenz attractor from the dynamics point of view.

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