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Journal articles on the topic 'Symbolic dynamics. Topological dynamics'

Author: Grafiati

Published: 4 June 2021

Last updated: 16 February 2022

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1

Morales, C. A., and Jumi Oh. "Symbolic topological dynamics in the circle." Proceedings of the American Mathematical Society 147, no. 8 (March 26, 2019): 3413–24. http://dx.doi.org/10.1090/proc/14472.

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2

Galias, Zbigniew. "Automatized Search for Complex Symbolic Dynamics with Applications in the Analysis of a Simple Memristor Circuit." International Journal of Bifurcation and Chaos 24, no. 07 (July 2014): 1450104. http://dx.doi.org/10.1142/s0218127414501041.

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An automatized method to search for complex symbolic dynamics is proposed. The method can be used to show that a given dynamical system is chaotic in the topological sense. Application of this method in the analysis of a third-order memristor circuit is presented. Several examples of symbolic dynamics are constructed. Positive lower bounds for the topological entropy of an associated return map are found showing that the system is chaotic in the topological sense.

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3

Correia Ramos, Carlos. "Kinematics in Biology: Symbolic Dynamics Approach." Mathematics 8, no. 3 (March 4, 2020): 339. http://dx.doi.org/10.3390/math8030339.

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Motion in biology is studied through a descriptive geometrical method. We consider a deterministic discrete dynamical system used to simulate and classify a variety of types of movements which can be seen as templates and building blocks of more complex trajectories. The dynamical system is determined by the iteration of a bimodal interval map dependent on two parameters, up to scaling, generalizing a previous work. The characterization of the trajectories uses the classifying tools from symbolic dynamics—kneading sequences, topological entropy and growth number. We consider also the isentropic trajectories, trajectories with constant topological entropy, which are related with the possible existence of a constant drift. We introduce the concepts of pure and mixed bimodal trajectories which give much more flexibility to the model, maintaining it simple. We discuss several procedures that may allow the use of the model to characterize empirical data.

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4

Xu, Haiyun, Fangyue Chen, and Weifeng Jin. "Topological Conjugacy Classification of Elementary Cellular Automata with Majority Memory." International Journal of Bifurcation and Chaos 27, no. 14 (December 30, 2017): 1750217. http://dx.doi.org/10.1142/s0218127417502170.

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The topological conjugacy classification of elementary cellular automata with majority memory (ECAMs) is studied under the framework of symbolic dynamics. In the light of the conventional symbolic sequence space, the compact symbolic vector space is identified with a feasible metric and topology. A slight change is introduced to present that all global maps of ECAMs are continuous functions, thereafter generating the compact dynamical systems. By exploiting two fundamental homeomorphisms in symbolic vector space, all ECAMs are furthermore grouped into 88 equivalence classes in the sense that different mappings in the same global equivalence are mutually topologically conjugate.

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5

Ramos, Carlos. "Animal movement: symbolic dynamics and topological classification." Mathematical Biosciences and Engineering 16, no. 5 (2019): 5464–89. http://dx.doi.org/10.3934/mbe.2019272.

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6

Peng, Shou-Li, and Xu-Sheng Zhang. "Second topological conjugate transformation in symbolic dynamics." Physical Review E 57, no. 5 (May 1, 1998): 5311–24. http://dx.doi.org/10.1103/physreve.57.5311.

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7

Pei, Yangjun, Qi Han, Chao Liu, Dedong Tang, and Junjian Huang. "Chaotic Behaviors of Symbolic Dynamics about Rule 58 in Cellular Automata." Mathematical Problems in Engineering 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/834268.

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The complex dynamical behaviors of rule 58 in cellular automata are investigated from the viewpoint of symbolic dynamics. The rule is Bernoulliστ-shift rule, which is members of Wolfram’s class II, and it was said to be simple as periodic before. It is worthwhile to study dynamical behaviors of rule 58 and whether it possesses chaotic attractors or not. It is shown that there exist two Bernoulli-measure attractors of rule 58. The dynamical properties of topological entropy and topological mixing of rule 58 are exploited on these two subsystems. According to corresponding strongly connected graph of transition matrices of determinative block systems, we divide determinative block systems into two subsets. In addition, it is shown that rule 58 possesses rich and complicated dynamical behaviors in the space of bi-infinite sequences. Furthermore, we prove that four rules of global equivalence classε43of CA are topologically conjugate. We use diagrams to explain the attractors of rule 58, where characteristic function is used to describe that some points fall into Bernoulli-shift map after several times iterations, and we find that these attractors are not global attractors. The Lameray diagram is used to show clearly the iterative process of an attractor.

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8

Chen, Fangyue, Bo Chen, Junbiao Guan, and Weifeng Jin. "A Symbolic Dynamics Perspective of Conway’s Game of Life." International Journal of Bifurcation and Chaos 26, no. 02 (February 2016): 1650035. http://dx.doi.org/10.1142/s0218127416500358.

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An interesting question is whether the intrinsic complexity of the gliders in [Formula: see text]-dimensional cellular automata could be quantitatively analyzed in rigorously mathematical sense. In this paper, by introducing the [Formula: see text]-dimensional symbolic space, some fundamental dynamical properties of [Formula: see text]-dimensional shift map are explored in a subtle way. The purpose of this article is to present an accurate characterization of complex symbolic dynamics of gliders in Conway’s game of life. A series of dynamical properties of gliders on their concrete subsystems are investigated by means of the directed graph representation and transition matrix. More specifically, the gliders here are topologically mixing and possess the positive topological entropy on their subsystems. Finally, it is worth mentioning that the method presented in this paper is also applicable to other gliders in different [Formula: see text] dimensions.

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9

COLLINS, PIETER. "SYMBOLIC DYNAMICS FROM HOMOCLINIC TANGLES." International Journal of Bifurcation and Chaos 12, no. 03 (March 2002): 605–17. http://dx.doi.org/10.1142/s0218127402004565.

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We present a method for finding symbolic dynamics for a planar diffeomorphism with a homoclinic tangle. The method only requires a finite piece of tangle, which can be computed with available numerical techniques. The symbol space is naturally given by components of the complement of the stable and unstable manifolds. The shift map defining the dynamics is a factor of a subshift of finite type, and is obtained from a graph related to the tangle. The entropy of this shift map is a lower bound for the topological entropy of the planar diffeomorphism. We give examples arising from the Hénon family.

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10

HOCHMAN, MICHAEL. "Genericity in topological dynamics." Ergodic Theory and Dynamical Systems 28, no. 1 (February 2008): 125–65. http://dx.doi.org/10.1017/s0143385707000521.

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AbstractWe study genericity of dynamical properties in the space of homeomorphisms of the Cantor set and in the space of subshifts of a suitably large shift space. These rather different settings are related by a Glasner–King type correspondence: genericity in one is equivalent to genericity in the other. By applying symbolic techniques in the shift-space model we derive new results about genericity of dynamical properties for transitive and totally transitive homeomorphisms of the Cantor set. We show that the isomorphism class of the universal odometer is generic in the space of transitive systems. On the other hand, the space of totally transitive systems displays much more varied dynamics. In particular, we show that in this space the isomorphism class of every Cantor system without periodic points is dense and the following properties are generic: minimality, zero entropy, disjointness from a fixed totally transitive system, weak mixing, strong mixing and minimal self joinings. The latter two stand in striking contrast to the situation in the measure-preserving category. We also prove a correspondence between genericity of dynamical properties in the measure-preserving category and genericity of systems supporting an invariant measure with the same property.

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11

Wu–ming, Liu, Ying Yang–jun, Chen Shi–gang, and He Xian–tu. "Symbolic Dynamics and Topological Entropy of Henon Map." Communications in Theoretical Physics 15, no. 1 (January 1991): 1–8. http://dx.doi.org/10.1088/0253-6102/15/1/1.

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12

SAMUEL, TONY, NINA SNIGIREVA, and ANDREW VINCE. "Embedding the symbolic dynamics of Lorenz maps." Mathematical Proceedings of the Cambridge Philosophical Society 156, no. 3 (March 3, 2014): 505–19. http://dx.doi.org/10.1017/s0305004114000061.

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AbstractNecessary and sufficient conditions for the symbolic dynamics of a given Lorenz map to be fully embedded in the symbolic dynamics of a piecewise continuous interval map are given. As an application of this embedding result, we describe a new algorithm for calculating the topological entropy of a Lorenz map.

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13

GOETZ, AREK. "Sofic subshifts and piecewise isometric systems." Ergodic Theory and Dynamical Systems 19, no. 6 (December 1999): 1485–501. http://dx.doi.org/10.1017/s0143385799151964.

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We study the natural symbolic dynamics associated with piecewise continuous, non-invertible, dynamical systems. Our study is centered primarily on the relationship between the point-set topological properties of the partition of the system and the symbolic coding. We prove that for a class of maps locally preserving distances with regular partition, the associated symbolic dynamics cannot embed subshifts of finite type of positive entropy. Hence, in particular, almost sofic subshifts obtained from the symbolic dynamics have zero entropy. However, there are examples in Euclidean spaces of systems with non-regular partitions for which the coding maps can be surjective, particularly embedding all subshifts. For all such examples, the associated group of isometries is a subgroup of $O(\mathbb{R}, N)$.

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14

GALIAS, ZBIGNIEW. "ROBUSTNESS OF SYMBOLIC DYNAMICS AND SYNCHRONIZATION PROPERTIES." International Journal of Bifurcation and Chaos 10, no. 04 (April 2000): 811–18. http://dx.doi.org/10.1142/s0218127400000578.

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In this paper we introduce the method for investigation of coupled chaotic systems using topological methods. We show that if the coupling is small then there exists independent symbolic dynamics for every coupled subsystem and in consequence the systems are not synchronized. As an example we consider coupled Hénon maps. Using computer interval arithmetic we find parameter mismatch and perturbation range for which the symbolic dynamics in the Hénon system is sustained. For coupled Hénon maps we compute the value of coupling strength for which the symbolic dynamics in every subsystem survives.

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15

BUNDFUSS, STEFAN, TYLL KRÜGER, and SERGE TROUBETZKOY. "Topological and symbolic dynamics for hyperbolic systems with holes." Ergodic Theory and Dynamical Systems 31, no. 5 (November 17, 2010): 1305–23. http://dx.doi.org/10.1017/s0143385710000556.

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AbstractWe consider an axiom A diffeomorphism or a Markov map of an interval and the invariant set Ω* of orbits which never falls into a fixed hole. We study various aspects of the symbolic representation of Ω* and of its non-wandering set Ωnw. Our results are on the cardinality of the set of topologically transitive components of Ωnw and their structure. We also prove that Ω* is generically a subshift of finite type in several senses.

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16

KRIEGER, WOLFGANG. "On a syntactically defined invariant of symbolic dynamics." Ergodic Theory and Dynamical Systems 20, no. 2 (April 2000): 501–16. http://dx.doi.org/10.1017/s0143385700000249.

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A partially ordered set that is invariantly associated to a subshift is constructed. A property of subshifts, also an invariant of topological conjugacy, is described. If this property is present in a subshift then the constructed partially ordered set is a partially ordered semigroup (with zero). In the description of these invariants the notion of context is instrumental.

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17

Dong, Chengwei, Lian Jia, Qi Jie, and Hantao Li. "Symbolic Encoding of Periodic Orbits and Chaos in the Rucklidge System." Complexity 2021 (August 27, 2021): 1–16. http://dx.doi.org/10.1155/2021/4465151.

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To describe and analyze the unstable periodic orbits of the Rucklidge system, a so-called symbolic encoding method is introduced, which has been proven to be an efficient tool to explore the topological properties concealed in these periodic orbits. In this work, the unstable periodic orbits up to a certain topological length in the Rucklidge system are systematically investigated via a proposed variational method. The dynamics in the Rucklidge system are explored by using phase portrait analysis, Lyapunov exponents, and Poincaré first return maps. Symbolic encodings of the periodic orbits with two and four letters based on the trajectory topology in the phase space are implemented under two sets of parameter values. Meanwhile, the bifurcations of the periodic orbits are explored, significantly improving the understanding of the dynamics of the Rucklidge system. The multiple-letter symbolic encoding method could also be applicable to other nonlinear dynamical systems.

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18

LIMA, YURI. "ℤd-actions with prescribed topological and ergodic properties." Ergodic Theory and Dynamical Systems 32, no. 1 (April 12, 2011): 191–209. http://dx.doi.org/10.1017/s0143385710000908.

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AbstractWe extend constructions of Hahn and Katznelson [On the entropy of uniquely ergodic transformations. Trans. Amer. Math. Soc.126 (1967), 335–360] and Pavlov [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28 (2008), 1291–1322] to ℤd-actions on symbolic dynamical spaces with prescribed topological and ergodic properties. More specifically, we describe a method to build ℤd-actions which are (totally) minimal, (totally) strictly ergodic and have positive topological entropy.

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19

HOCHMAN, MICHAEL. "On notions of determinism in topological dynamics." Ergodic Theory and Dynamical Systems 32, no. 1 (September 6, 2011): 119–40. http://dx.doi.org/10.1017/s0143385710000738.

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AbstractWe examine the relations between topological entropy, invertibility, and prediction in topological dynamics. We show that topological determinism in the sense of Kamińsky, Siemaszko, and Szymański imposes no restriction on invariant measures except zero entropy. Also, we develop a new method for relating topological determinism and zero entropy, and apply it to obtain a multidimensional analog of this theory. We examine prediction in symbolic dynamics and show that while the condition that each past admits a unique future only occurs in finite systems, the condition that each past has a bounded number of futures imposes no restriction on invariant measures except zero entropy. Finally, we give a negative answer to a question of Eli Glasner by constructing a zero-entropy system with a globally supported ergodic measure in which every point has multiple preimages.

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20

DUARTE, JORGE, CRISTINA JANUÁRIO, CARLA RODRIGUES, and JOSEP SARDANYÉS. "TOPOLOGICAL COMPLEXITY AND PREDICTABILITY IN THE DYNAMICS OF A TUMOR GROWTH MODEL WITH SHILNIKOV'S CHAOS." International Journal of Bifurcation and Chaos 23, no. 07 (July 2013): 1350124. http://dx.doi.org/10.1142/s0218127413501241.

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Dynamical systems modeling tumor growth have been investigated to determine the dynamics between tumor and healthy cells. Recent theoretical investigations indicate that these interactions may lead to different dynamical outcomes, in particular to homoclinic chaos. In the present study, we analyze both topological and dynamical properties of a recently characterized chaotic attractor governing the dynamics of tumor cells interacting with healthy tissue cells and effector cells of the immune system. By using the theory of symbolic dynamics, we first characterize the topological entropy and the parameter space ordering of kneading sequences from one-dimensional iterated maps identified in the dynamics, focusing on the effects of inactivation interactions between both effector and tumor cells. The previous analyses are complemented with the computation of the spectrum of Lyapunov exponents, the fractal dimension and the predictability of the chaotic attractors. Our results show that the inactivation rate of effector cells by the tumor cells has an important effect on the dynamics of the system. The increase of effector cells inactivation involves an inverse Feigenbaum (i.e. period-halving bifurcation) scenario, which results in the stabilization of the dynamics and in an increase of dynamics predictability. Our analyses also reveal that, at low inactivation rates of effector cells, tumor cells undergo strong, chaotic fluctuations, with the dynamics being highly unpredictable. Our findings are discussed in the context of tumor cells potential viability.

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21

Lima, Yuri, and Omri Sarig. "Symbolic dynamics for three-dimensional flows with positive topological entropy." Journal of the European Mathematical Society 21, no. 1 (September 25, 2018): 199–256. http://dx.doi.org/10.4171/jems/834.

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22

Chen, Zhong-Xuan, Ke-Fei Cao, and Shou-Li Peng. "Symbolic dynamics analysis of topological entropy and its multifractal structure." Physical Review E 51, no. 3 (March 1, 1995): 1983–88. http://dx.doi.org/10.1103/physreve.51.1983.

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23

Breymann, Wolfgang, and Jürgen Vollmer. "Symbolic dynamics and topological entropy at the onset of pruning." Zeitschrift für Physik B Condensed Matter 103, no. 3 (April 1997): 539–46. http://dx.doi.org/10.1007/s002570050408.

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24

Duarte, Jorge, Luís Silva, and J. Sousa Ramos. "Computation of the topological entropy in chaotic biophysical bursting models for excitable cells." Discrete Dynamics in Nature and Society 2006 (2006): 1–18. http://dx.doi.org/10.1155/ddns/2006/60918.

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One of the interesting complex behaviors in many cell membranes is bursting, in which a rapid oscillatory state alternates with phases of relative quiescence. Although there is an elegant interpretation of many experimental results in terms of nonlinear dynamical systems, the dynamics of bursting models is not completely described. In the present paper, we study the dynamical behavior of two specific three-variable models from the literature that replicate chaotic bursting. With results from symbolic dynamics, we characterize the topological entropy of one-dimensional maps that describe the salient dynamics on the attractors. The analysis of the variation of this important numerical invariant with the parameters of the systems allows us to quantify the complexity of the phenomenon and to distinguish different chaotic scenarios. This work provides an example of how our understanding of physiological models can be enhanced by the theory of dynamical systems.

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25

Duarte, Jorge, Cristina Januário, and Nuno Martins. "Chaos in Ecology: The Topological Entropy of a Tritrophic Food Chain Model." Discrete Dynamics in Nature and Society 2008 (2008): 1–12. http://dx.doi.org/10.1155/2008/541683.

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An ecosystem is a web of complex interactions among species. With the purpose of understanding this complexity, it is necessary to study basic food chain dynamics with preys, predators and superpredators interactions. Although there is an elegant interpretation of ecological models in terms of chaos theory, the complex behavior of chaotic food chain systems is not completely understood. In the present work we study a specific food chain model from the literature. Using results from symbolic dynamics, we characterize the topological entropy of a family of logistic-like Poincaré return maps that replicates salient aspects of the dynamics of the model. The analysis of the variation of this numerical invariant, in some realistic system parameter region, allows us to quantify and to distinguish different chaotic regimes. This work is still another illustration of the role that the theory of dynamical systems can play in the study of chaotic dynamics in life sciences.

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26

Fried, David. "Finitely presented dynamical systems." Ergodic Theory and Dynamical Systems 7, no. 4 (December 1987): 489–507. http://dx.doi.org/10.1017/s014338570000417x.

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AbstractWe extend results of Bowen and Manning on systems with good symbolic dynamics. In particular we identify the class of dynamical systems that admit Markov partitions. For these systems the Manning-Bowen method of counting periodic points is explained in terms of topological coincidence numbers. We show, in particular, that an expansive system with a finite cover by rectangles has a rational zeta function.

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27

Rashidinia, Jalil, Mehri Sajjadian, Jorge Duarte, Cristina Januário, and Nuno Martins. "On the Dynamical Complexity of a Seasonally Forced Discrete SIR Epidemic Model with a Constant Vaccination Strategy." Complexity 2018 (December 2, 2018): 1–11. http://dx.doi.org/10.1155/2018/7191487.

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In this article, we consider the discretized classical Susceptible-Infected-Recovered (SIR) forced epidemic model to investigate the consequences of the introduction of different transmission rates and the effect of a constant vaccination strategy, providing new numerical and topological insights into the complex dynamics of recurrent diseases. Starting with a constant contact (or transmission) rate, the computation of the spectrum of Lyapunov exponents allows us to identify different chaotic regimes. Studying the evolution of the dynamical variables, a family of unimodal-type iterated maps with a striking biological meaning is detected among those dynamical regimes of the densities of the susceptibles. Using the theory of symbolic dynamics, these iterated maps are characterized based on the computation of an important numerical invariant, the topological entropy. The introduction of a degree (or amplitude) of seasonality, ε, is responsible for inducing complexity into the population dynamics. The resulting dynamical behaviors are studied using some of the previous tools for particular values of the strength of the seasonality forcing, ε. Finally, we carry out a study of the discrete SIR epidemic model under a planned constant vaccination strategy. We examine what effect this vaccination regime will have on the periodic and chaotic dynamics originated by seasonally forced epidemics.

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28

Grassberger, Peter. "On Symbolic Dynamics of One-Humped Maps of the Interval." Zeitschrift für Naturforschung A 43, no. 7 (July 1, 1988): 671–80. http://dx.doi.org/10.1515/zna-1988-0710.

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Abstract We present an explicit construction of minimal deterministic automata which accept the languages of L-R symbolic sequences of unimodal maps resp. arbitrarily close approximations thereof. They are used to study a recently introduced complexity measure of this language which we conjecture to be a new invariant under diffeomorphisms. On each graph corresponding to such an automaton, the evolution is a topological Markov chain which does not seem to correspond to a partition of the interval into a countable number of intervals.

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29

PAVLOV, RONNIE. "A characterization of topologically completely positive entropy for shifts of finite type." Ergodic Theory and Dynamical Systems 34, no. 6 (June 3, 2013): 2054–65. http://dx.doi.org/10.1017/etds.2013.18.

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AbstractA topological dynamical system was defined by Blanchard [Fully Positive Topological Entropy and Topological Mixing (Symbolic Dynamics and Applications (in honor of R. L. Adler), 135). American Mathematical Society Contemporary Mathematics, Providence, RI, 1992, pp. 95–105] to have topologically completely positive entropy (or TCPE) if its only zero entropy factor is the dynamical system consisting of a single fixed point. For ${ \mathbb{Z} }^{d} $ shifts of finite type, we give a simple condition equivalent to having TCPE. We use our characterization to derive a similar equivalent condition to TCPE for the subclass of ${ \mathbb{Z} }^{d} $ group shifts, which was proved by Lind and Schmidt in the abelian case [Homoclinic points of algebraic ${ \mathbb{Z} }^{d} $-actions. J. Amer. Math. Soc. 12(4) (1999), 953–980] and by Boyle and Schraudner in the general case [${ \mathbb{Z} }^{d} $ group shifts and Bernoulli factors. Ergod. Th. & Dynam. Sys. 28(2) (2008), 367–387]. We also give an example of a ${ \mathbb{Z} }^{2} $ shift of finite type which has TCPE but is not even topologically transitive, and prove a result about block gluing ${ \mathbb{Z} }^{d} $ SFTs motivated by our characterization of TCPE.

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30

Dong, Chengwei. "Topological classification of periodic orbits in the Kuramoto–Sivashinsky equation." Modern Physics Letters B 32, no. 15 (May 24, 2018): 1850155. http://dx.doi.org/10.1142/s0217984918501555.

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In this paper, we systematically research periodic orbits of the Kuramoto–Sivashinsky equation (KSe). In order to overcome the difficulties in the establishment of one-dimensional symbolic dynamics in the nonlinear system, two basic periodic orbits can be used as basic building blocks to initialize cycle searching, and we use the variational method to numerically determine all the periodic orbits under parameter [Formula: see text] = 0.02991. The symbolic dynamics based on trajectory topology are very successful for classifying all short periodic orbits in the KSe. The current research can be conveniently adapted to the identification and classification of periodic orbits in other chaotic systems.

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31

WU, ZUO-BING. "ROTATION NUMBERS OF INVARIANT MANIFOLDS AROUND UNSTABLE PERIODIC ORBITS FOR THE DIAMAGNETIC KEPLER PROBLEM." Fractals 16, no. 01 (March 2008): 11–23. http://dx.doi.org/10.1142/s0218348x08003831.

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In this paper, a method to construct topological template in terms of symbolic dynamics for the diamagnetic Kepler problem is proposed. To confirm the topological template, rotation numbers of invariant manifolds around unstable periodic orbits in a phase space are taken as an object of comparison. The rotation numbers are determined from the definition and connected with symbolic sequences encoding the periodic orbits in a reduced Poincaré section. Only symbolic codes with inverse ordering in the forward mapping can contribute to the rotation of invariant manifolds around the periodic orbits. By using symbolic ordering, the reduced Poincaré section is constricted along stable manifolds and a topological template, which preserves the ordering of forward sequences and can be used to extract the rotation numbers, is established. The rotation numbers computed from the topological template are the same as those computed from their original definition.

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32

LABARCA, R., C. MOREIRA, A. PUMARIÑO, and J. A. RODRÍGUEZ. "ON BIFURCATION SETS FOR SYMBOLIC DYNAMICS IN THE MILNOR–THURSTON WORLD." Communications in Contemporary Mathematics 14, no. 04 (August 2012): 1250024. http://dx.doi.org/10.1142/s0219199712500241.

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We show the continuity of the topological entropy for the Milnor–Thurston world of interval maps and we compute the minimum and the maximum values for the entropy of a maximal sequence of any given period. We also study (fractal) geometric properties of the bifurcation set in the parameter space and in the associated phase spaces Σ[a, b], and we compare these results with the previously known results about the lexicographic world of interval maps (related to Lorenz-like maps).

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Dong, Chengwei, and Lian Jia. "Periodic orbits analysis for the Zhou system via variational approach." Modern Physics Letters B 33, no. 19 (July 8, 2019): 1950212. http://dx.doi.org/10.1142/s0217984919502129.

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We proposed a general method for the systematic calculation of unstable cycles in the Zhou system. The variational approach is employed for the cycle search, and we establish interesting symbolic dynamics successfully based on the orbits circuiting property with respect to different fixed points. Upon the defined symbolic rule, cycles with topological length up to five are sought and ordered. Further, upon parameter changes, the homotopy evolution of certain selected cycles are investigated. The topological classification methodology could be widely utilized in other low-dimensional dissipative systems.

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34

Cao, Ke-Fei, and Shou-Li Peng. "Homology of Vertex and Edge Shift Matrices in Symbolic Dynamics and Entropy Invariants." International Journal of Modern Physics B 17, no. 22n24 (September 30, 2003): 4308–15. http://dx.doi.org/10.1142/s0217979203022362.

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By using the homological notion, we analyze the vertex and edge shift matrices in symbolic dynamics. The former is the mathematical basis of the general star product which is transformed into a star direct product of vertex shift matrices, the latter a basis of the calculation of topological entropy. We show that in a general case the first entropy invariant holds, but the second one is broken.

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35

Arioli, Gianni. "Periodic Orbits, Symbolic Dynamics and Topological Entropy for the Restricted 3-Body Problem." Communications in Mathematical Physics 231, no. 1 (August 1, 2002): 1–24. http://dx.doi.org/10.1007/s00220-002-0666-7.

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36

YURI, MICHIKO. "Zeta functions for certain non-hyperbolic systems and topological Markov approximations." Ergodic Theory and Dynamical Systems 18, no. 6 (December 1998): 1589–612. http://dx.doi.org/10.1017/s0143385798117972.

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We study dynamical (Artin–Mazur–Ruelle) zeta functions for piecewise invertible multi-dimensional maps. In particular, we direct our attention to non-hyperbolic systems admitting countable generating definite partitions which are not necessarily Markov but satisfy the finite range structure (FRS) condition. We define a version of Gibbs measure (weak Gibbs measure) and by using it we establish an analogy with thermodynamic formalism for specific cases, i.e. a characterization of the radius of convergence in terms of pressure. The FRS condition leads us to nice countable state symbolic dynamics and allows us to realize it as towers over Markov systems. The Markov approximation method then gives a product formula of zeta functions for certain weighted functions.

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37

Rezagholi, Sharwin. "Subshifts on Infinite Alphabets and Their Entropy." Entropy 22, no. 11 (November 13, 2020): 1293. http://dx.doi.org/10.3390/e22111293.

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We analyze symbolic dynamics to infinite alphabets by endowing the alphabet with the cofinite topology. The topological entropy is shown to be equal to the supremum of the growth rate of the complexity function with respect to finite subalphabets. For the case of topological Markov chains induced by countably infinite graphs, our approach yields the same entropy as the approach of Gurevich We give formulae for the entropy of countable topological Markov chains in terms of the spectral radius in l2.

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KOHDA, TOHRU, YOSHIHIKO HORIO, YOICHIRO TAKAHASHI, and KAZUYUKI AIHARA. "BETA ENCODERS: SYMBOLIC DYNAMICS AND ELECTRONIC IMPLEMENTATION." International Journal of Bifurcation and Chaos 22, no. 09 (September 2012): 1230031. http://dx.doi.org/10.1142/s0218127412300315.

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A new class of analog-to-digital (A/D) and digital-to-analog (D/A) converters that uses a flaky quantizer, known as a β-encoder, has been shown to have exponential bit rate accuracy and a self-correcting property for fluctuations of the amplifier factor β and quantizer threshold ν. The probabilistic behavior of this flaky quantizer is explained by the deterministic dynamics of a multivalued Rényi–Parry map on the middle interval, as defined here. This map is eventually locally onto map of [ν - 1, ν], which is topologically conjugate to Parry's (β, α)-map with α = (β - 1)(ν - 1). This viewpoint allows us to obtain a decoded sample, which is equal to the midpoint of the subinterval, and its associated characteristic equation for recovering β, which improves the quantization error by more than 3 dB when β > 1.5. The invariant subinterval under the Rényi–Parry map shows that ν should be set to around the midpoint of its associated greedy and lazy values. Furthermore, a new A/D converter referred to as the negative β-encoder is introduced, and shown to further improve the quantization error of the β-encoder. Then, a switched-capacitor (SC) electronic circuit technique is proposed for implementing A/D converter circuits based on several types of β-encoders. Electric circuit experiments were used to verify the validity of these circuits against deviations and mismatches of the circuit parameters. Finally, we demonstrate that chaotic attractors can be observed experimentally from these β-encoder circuits.

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Keller, Karsten. "Symbolic dynamics for angle-doubling on the circle III. Sturmian sequences and the quadratic map." Ergodic Theory and Dynamical Systems 14, no. 4 (December 1994): 787–805. http://dx.doi.org/10.1017/s0143385700008154.

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AbstractBy the theory of Douady and Hubbard, the structure of Julia sets of quadratic maps is tightly connected with the angle-doubling maphon the circleT. In particular, a connected and locally connected Julia set can be considered as a topological factorT/ ≈ ofTwith respect to a specialh-invariant equivalence relation ≈ onT, which is called Julia equivalence by Keller. Following an idea of Thurston, Bandt and Keller have investigated a map α →αfromTonto the set of all Julia equivalences, which gives a natural abstract description of the Mandelbrot set. By the use of a symbol sequence called the kneading sequence of the point α, they gave a topological classification of the abstract Julia setsT/α. It turns out thatT/αcontains simple closed curves iff the point α has a periodic kneading sequence. The present article characterizes the set of points possessing a periodic kneading sequence and discusses this set in relation to Julia sets and to the Mandelbrot set.

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Lefranc, M., P. Glorieux, F. Papoff, F. Molesti, and E. Arimondo. "Combining Topological Analysis and Symbolic Dynamics to Describe a Strange Attractor and Its Crises." Physical Review Letters 73, no. 10 (September 5, 1994): 1364–67. http://dx.doi.org/10.1103/physrevlett.73.1364.

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Wu, Xiaoying, Yuanlong Chen, Liangliang Li, and Fen Wang. "Complex Dynamics of Discrete-Time Ring Neural Networks." International Journal of Bifurcation and Chaos 31, no. 08 (June 26, 2021): 2150116. http://dx.doi.org/10.1142/s0218127421501169.

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This paper mainly considers the complex dynamics of a discrete-time ring neural network with multiple time delays. By transforming the network system into a system of difference equations, one can obtain a chaotic neural network. More specifically, by projection method one can determine a closed invariant set of the system governed by some difference equations, and prove that some invariant subsystem is topologically conjugate to the two-sided symbolic dynamical system. Moreover, numerical simulations are presented to verify the theoretical results.

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ZHENG, JIONGXUAN, JOSEPH D. SKUFCA, and ERIK M. BOLLT. "THE BUNDLE PLOT: EVOLUTION OF SYMBOLIC SPACE UNDER THE SYSTEM PARAMETER CHANGES." International Journal of Bifurcation and Chaos 23, no. 08 (August 2013): 1330028. http://dx.doi.org/10.1142/s0218127413300280.

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This paper provides a topological dynamics perspective on the full bifurcation unfolding in unimodal mappings. We present a bundle structure, visualized as a bundle plot, to show the evolution of symbolic space as we vary a system parameter. The bundle plot can be viewed as a limit process of an assignment plot, which are line assignments between points from two dynamical systems. Such line assignments are determined by a commuter, which is a coordinates transformation function that satisfies a commuting relationship but not necessarily a homeomorphism. The bundle structure is studied by understanding the implication of the system's qualitative changes. In addition, the case of the bundle plot with higher dimensional parameter variation is also considered. A main concern in the bundle plot is a special structure, called "joint", which determines a critical value of the parameter where the kneading sequence becomes periodic.

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COUTINHO, RICARDO, and BASTIEN FERNANDEZ. "Extensive bounds on the topological entropy of repellers in piecewise expanding coupled map lattices." Ergodic Theory and Dynamical Systems 33, no. 3 (April 17, 2012): 870–95. http://dx.doi.org/10.1017/s0143385712000144.

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AbstractBeyond the uncoupled regime, the rigorous description of the dynamics of (piecewise) expanding coupled map lattices remains largely incomplete. To address this issue, we study repellers of periodic chains of linearly coupled Lorenz-type maps which we analyze by means of symbolic dynamics. Whereas all symbolic codes are admissible for sufficiently small coupling intensity, when the interaction strength exceeds a chain length independent threshold, we prove that a large bunch of codes is pruned and an extensive decay follows suit for the topological entropy. This quantity, however, does not immediately drop off to 0. Instead, it is shown to be continuous at the threshold and to remain extensively bounded below by a positive number in a large part of the expanding regime. The analysis is firstly accomplished in a piecewise affine setting where all calculations are explicit and is then extended by continuation to coupled map lattices based on$C^1$-perturbations of the individual map.

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GALIAS, ZBIGNIEW, and WARWICK TUCKER. "VALIDATED STUDY OF THE EXISTENCE OF SHORT CYCLES FOR CHAOTIC SYSTEMS USING SYMBOLIC DYNAMICS AND INTERVAL TOOLS." International Journal of Bifurcation and Chaos 21, no. 02 (February 2011): 551–63. http://dx.doi.org/10.1142/s021812741102857x.

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We show that, for a certain class of systems, the problem of establishing the existence of periodic orbits can be successfully studied by a symbolic dynamics approach combined with interval methods. Symbolic dynamics is used to find approximate positions of periodic points, and the existence of periodic orbits in a neighborhood of these approximations is proved using an interval operator. As an example, the Lorenz system is studied; a theoretical argument is used to prove that each periodic orbit has a distinct symbol sequence. All periodic orbits with the period p ≤ 16 of the Poincaré map associated with the Lorenz system are found. Estimates of the topological entropy of the Poincaré map and the flow, based on the number and flow-times of short periodic orbits, are calculated. Finally, we establish the existence of several long periodic orbits with specific symbol sequences.

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Devaney, Robert L., and Xavier Jarque. "Misiurewicz Points for Complex Exponentials." International Journal of Bifurcation and Chaos 07, no. 07 (July 1997): 1599–615. http://dx.doi.org/10.1142/s0218127497001242.

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In this paper we examine the structure of the chaotic regime or Julia set of certain complex exponential maps Eλ(z) = λez. In the case where λ is a Misiurewicz point (i.e. the singular value 0 is eventually periodic), it is known that the Julia set for the map is the entire plane. In this case the Julia set also possesses certain curves or "hairs" that are permuted by the map. We examine the dynamics on these hairs in detail. We describe a certain extended symbolic dynamics by which the topological structure of the hairs may be determined completely.

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KNUDSEN, CARSTEN. "TOPOLOGICAL WINDING NUMBERS FOR PERIOD-DOUBLING CASCADES." International Journal of Bifurcation and Chaos 06, no. 01 (January 1996): 185–87. http://dx.doi.org/10.1142/s0218127496001934.

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We define the topological winding number for unimodal maps that share the essential properties of that of winding numbers for forced oscillators exhibiting period-doubling cascades. It is demonstrated how this number can be computed for any of the periodic orbits in the first period-doubling cascade. The limiting winding number at the accumulation point of the first period-doubling cascade is also derived. It is shown that the limiting value for the winding number ω∞ can be computed as the Farey sum of any two neighbouring topological winding numbers in the period-doubling cascade. The derivations are all based on symbolic dynamics and simple combinatorics.

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Matsumoto, Kengo. "On C*-Algebras Associated with Subshifts." International Journal of Mathematics 08, no. 03 (May 1997): 357–74. http://dx.doi.org/10.1142/s0129167x97000172.

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We construct and study C*-algebras associated with subshifts in symbolic dynamics as a generalization of Cuntz–Krieger algebras for topological Markov shifts. We prove some universal properties for the C*-algebras and give a criterion for them to be simple and purely infinite. We also present an example of a C*-algebra coming from a subshift which is not conjugate to a Markov shift.

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Dong, Chengwei. "Organization of the periodic orbits in the Rössler flow." International Journal of Modern Physics B 32, no. 21 (August 6, 2018): 1850227. http://dx.doi.org/10.1142/s0217979218502272.

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In this paper, we systematically investigate the periodic solutions of the Rössler equations up to certain topological length. To overcome the difficulties for a return map that is multivalued and non-invertible in the nonlinear system, we propose a new approach that establishes one-dimensional symbolic dynamics based on the topological structure of the orbit. A newly designed variational method is numerically stable for cycle searching, and two-orbit fragments can be used as basic building blocks to initialize the system. The topological classification based on the whole orbit structure seems more effective than partitioning the Poincaré surface of section. The current research supplies an interesting framework for a systematic classification of periodic orbits in a chaotic flow.

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Peng, Shou-Li, and Xu-Sheng Zhang. "The Generalized Milnor–Thurston Conjecture and Equal Topological Entropy Class in Symbolic Dynamics of Order Topological Space of Three Letters." Communications in Mathematical Physics 213, no. 2 (September 2000): 381–411. http://dx.doi.org/10.1007/s002200000245.

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JANUÁRIO, CRISTINA, CLARA GRÁCIO, DIANA A. MENDES, and JORGE DUARTE. "MEASURING AND CONTROLLING THE CHAOTIC MOTION OF PROFITS." International Journal of Bifurcation and Chaos 19, no. 11 (November 2009): 3593–604. http://dx.doi.org/10.1142/s021812740902502x.

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The study of economic systems has generated deep interest in exploring the complexity of chaotic motions in economy. Due to important developments in nonlinear dynamics, the last two decades have witnessed strong revival of interest in nonlinear endogenous business chaotic models. The inability to predict the behavior of dynamical systems in the presence of chaos suggests the application of chaos control methods, when we are more interested in obtaining regular behavior. In the present article, we study a specific economic model from the literature. More precisely, a system of three ordinary differential equations gather the variables of profits, reinvestments and financial flow of borrowings in the structure of a firm. Firstly, using results of symbolic dynamics, we characterize the topological entropy and the parameter space ordering of kneading sequences, associated with one-dimensional maps that reproduce significant aspects of the model dynamics. The analysis of the variation of this numerical invariant, in some realistic system parameter region, allows us to quantify and to distinguish different chaotic regimes. Finally, we show that complicated behavior arising from the chaotic firm model can be controlled without changing its original properties and the dynamics can be turned into the desired attracting time periodic motion (a stable steady state or into a regular cycle). The orbit stabilization is illustrated by the application of a feedback control technique initially developed by Romeiras et al. [1992]. This work provides another illustration of how our understanding of economic models can be enhanced by the theoretical and numerical investigation of nonlinear dynamical systems modeled by ordinary differential equations.

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