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Статті в журналах з теми "Chaos map"

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Автор: Grafiati

Опубліковано: 8 червня 2024

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1

Danca, Marius-F., Michal Fečkan, and Miguel Romera. "Generalized Form of Parrondo's Paradoxical Game with Applications to Chaos Control." International Journal of Bifurcation and Chaos 24, no. 01 (January 2014): 1450008. http://dx.doi.org/10.1142/s0218127414500084.

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In this paper, we show that a generalized form of Parrondo's paradoxical game can be applied to discrete systems, working out the logistic map as a concrete example, to generate stable orbits. Written in Parrondo's terms, this reads: chaos1 + chaos2 + ⋯ + chaosN = order, where chaosi, i = 1, 2, …, N, are denoted as the chaotic behaviors generated by N values of the parameter control, and by order one understands some stable behavior. The numerical results are sustained by quantitative dynamics generated by Parrondo's game. The implementation of the generalized Parrondo's game is realized here via the parameter switching (PS) algorithm for continuous-time systems [Danca, 2013] adapted to the logistic map. Some related results for more general maps on averaging, which represent discrete analogies of the PS method for ODE, are also presented and discussed.

2

Boyarsky, Abraham, Peyman Eslami, Paweł Góra, Zhenyang Li, Jonathan Meddaugh, and Brian E. Raines. "Chaos for successive maxima map implies chaos for the original map." Nonlinear Dynamics 79, no. 3 (November 21, 2014): 2165–75. http://dx.doi.org/10.1007/s11071-014-1802-6.

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3

Begun, Nikita, Pavel Kravetc, and Dmitrii Rachinskii. "Chaos in Saw Map." International Journal of Bifurcation and Chaos 29, no. 02 (February 2019): 1930005. http://dx.doi.org/10.1142/s0218127419300052.

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We consider the dynamics of a scalar piecewise linear “saw map” with infinitely many linear segments. In particular, such maps are generated as a Poincaré map of simple two-dimensional discrete time piecewise linear systems involving a saturation function. Alternatively, these systems can be viewed as a feedback loop with the so-called stop hysteresis operator. We analyze chaotic sets and attractors of the “saw map” depending on its parameters.

4

Gururajan, N., and M. Sambassivame. "Chaos-Logistic Map-Tent Map -Corresponding Cellular Automata." Mapana - Journal of Sciences 9, no. 2 (November 30, 2010): 28–34. http://dx.doi.org/10.12723/mjs.17.4.

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We study the characteristic features of Tent Mop and Logistic Map towards Chaos. Cellular Automata is obtained for Tent Mop and Logistic Map. The behavior of Tent Mop and Logistic Mop is reflected in the Cellular Automata generated.

5

Caranicolas, N. D. "Controlling chaos in map models." Mechanics Research Communications 26, no. 1 (January 1999): 13–20. http://dx.doi.org/10.1016/s0093-6413(98)00094-9.

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6

Han, Xiujing, Chun Zhang, Yue Yu, and Qinsheng Bi. "Boundary-Crisis-Induced Complex Bursting Patterns in a Forced Cubic Map." International Journal of Bifurcation and Chaos 27, no. 04 (April 2017): 1750051. http://dx.doi.org/10.1142/s0218127417500511.

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This paper reports novel routes to complex bursting patterns based on a forced cubic map, in which boundary-crisis-induced novel bursting patterns are investigated. Typically, the cubic map exhibits stable upper and lower branches of fixed points, which may evolve into chaos in opposite parameter directions by a cascade of period-doubling bifurcations. We show that the chaotic attractors on the stable branches may suddenly disappear by boundary crisis, thus leading to fast transitions from chaos to other attractors and giving rise to switchings between the stable branches of solutions of the cubic map. In particular, the attractors that the trajectory switches to by boundary crisis can be fixed points, periodic orbits and chaos, dependent on parameter values of the cubic map, and this helps us to reveal three general types of boundary-crisis-induced bursting, i.e. bursting of chaos-point type, bursting of chaos-cycle type and bursting of chaos-chaos type. Moreover, each bursting type may contain various bursting patterns. For bursting of chaos-cycle type, we see rich bursting patterns, e.g. chaos-period-2 bursting, chaos-period-4 bursting, chaos-period-8 bursting, etc. Our results enrich the possible routes to complex bursting patterns as well as the underlying mechanisms of complex bursting patterns.

7

Yusof, Norliza Muhamad, Muhamad Luqman Sapini, Lidiya Irdeena Az’hari, Nor Akma Hanis Roslee, Siti Noor Afiqah Rahmat, and Siti Hidayah Muhad Salleh. "Chaos Theory of 0-1 Test and Logistic Map in New Confirmed COVID-19 Cases." Science and Technology Indonesia 7, no. 2 (April 19, 2022): 179–85. http://dx.doi.org/10.26554/sti.2022.7.2.179-185.

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This study aims to forecast the new number of positive cases of COVID-19 using the logistic map, where the presence of chaos is determined using the 0-1 Test. There are a small number of investigations on chaos utilizing the 0-1 Test and logistic map in the literature. This study provides a straightforward technique of forecasting and a current way of determining chaos. Information on new confirmed COVID-19 cases and population of ten countries involving China, Vietnam, Korea, Malaysia, Singapore, New Zealand, India, USA, Brazil, and Mexico for six months from January 20 to June 15, 2020, are selected as samples. The logistic map runs through three stages: data training, forecasting, and validation using the Mean Absolute Error method (MAE). The data training procedure is critical for determining the best growth rate, r, for the logistic map. In chaotic investigations of the 0-1 Test, there appears to be an inverse expectation toward a logistic map. The 0-1 Test in the data of new confirmed COVID-19 cases in all the selected countries reveals the presence of non-chaotic. This contrasts with the existence of chaos in the logistic map forecasts for the USA, Brazil, and Mexico. Regardless, the logistic map was found to be capable of forecasting new COVID-19 positive cases with low error instances. Beginning in the middle of May 2020, new COVID-19 positive cases are forecasted to be on the rise in the USA, Brazil, and Mexico.

8

Inoue, Kei, Masanori Ohya, and Igor V. Volovich. "On a Combined Quantum Baker's Map and Its Characterization by Entropic Chaos Degree." Open Systems & Information Dynamics 16, no. 02n03 (September 2009): 179–93. http://dx.doi.org/10.1142/s123016120900013x.

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Quantum Baker's map is a theoretical model that exhibits chaos in a quantum system. In this paper, we introduce a combined map by combining several quantum Baker's maps. Chaos of such a combined dynamics is studied by the entropic chaos degree.

9

Babilonová-Štefánková, Marta. "Extreme Chaos and Transitivity." International Journal of Bifurcation and Chaos 13, no. 07 (July 2003): 1695–700. http://dx.doi.org/10.1142/s0218127403007540.

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In the eighties, Misiurewicz, Bruckner and Hu provided examples of functions chaotic in the sense of Li and Yorke almost everywhere. In this paper we show that similar results are true for distributional chaos, introduced in [Schweizer & Smítal, 1994]. In fact, we show that any bitransitive continuous map of the interval is conjugate to a map uniformly distributionally chaotic almost everywhere. Using a result of A. M. Blokh we get as a consequence that for any map f with positive topological entropy there is a k such that fk is semiconjugate to a continuous map uniformly distributionally chaotic almost everywhere, and consequently, chaotic in the sense of Li and Yorke almost everywhere.

10

Sahid, Sahid, Atmini Dhoruri, Dwi Lestari, Eminugroho Ratna Sari, and Muhammad Fauzan. "Sistem Kriptografi Stream Cipher Berbasis Fungsi Chaos untuk Keamanan Informasi." Jurnal Sains Dasar 8, no. 1 (February 10, 2021): 6–12. http://dx.doi.org/10.21831/jsd.v8i1.38666.

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Tujuan penelitian ini adalah menerapkan Fungsi chaos Logistic Map dalam meningkatkan keamanan pengiriman informasi. Fungsi chaos memiliki tingkah laku yang sangat kompleks, irregular dan random di dalam sebuah sistem yang deterministik. Chaos mempunyai sifat yang kacau atau acak, perubahan sedikit saja akan membangkitkan bilangan yang berbeda, hal ini berguna dalam membangkitkan kunci. Fungsi chaos Logistic Map akan digunakan untuk membangkitkan kunci. Selanjutnya, digunakan fungsi sinus berosilasi tinggi untuk meningkatkan keacakan bilangan. Dalam menentukan pembangkit kunci akan digunakan protokol perjanjian kunci stickel. Selanjutnya pembangkit kunci akan di proses menggunakan fungsi chaos Logistic Map dikombinasikan dengan fungsi sinus berosilasi tinggi dan akan diperoleh kunci yang akan digunakan untuk enkripsi serta dekripsi. Pada proses enkripsi dilakukan perhitungan dengan rumus 15Ci=Ki+Pi" mod 256, sedangkan proses dekripsi dilakukan perhitungan dengan rumus 15Ci=Ki-Pi" mod 256, dengan 15Ci" adalah Ciphertext, 15 Pi" adalah Plaintext, serta 15Ki " adalah Kunci. Dengan menggunakan Logistic Map dan fungsi sinus pada pembangkit kunci diperoleh sifat chaos yang tinggi untuk nilai parameter tertentu, bersifat chaos hanya pada beberapa iterasi awal, selanjutnya error berkaitan dengan nilai 15xi=0" . Untuk nilai-nilai parameter yang lain diperoleh barisan kunci yang konvergen setelah beberapa iterasi.

11

Drwięga, Tomasz. "Dendrites and Chaos." International Journal of Bifurcation and Chaos 28, no. 13 (December 12, 2018): 1850158. http://dx.doi.org/10.1142/s0218127418501584.

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We answer the two questions left open in [Kočan, 2012] i.e. whether there is a relation between [Formula: see text]-chaos and distributional chaos and whether there is a relation between an infinite LY-scrambled set and distributional chaos for dendrite maps. We construct a continuous self-map of a dendrite without any DC3 pairs but containing an uncountable [Formula: see text]-scrambled set. To answer the second question we construct a dendrite [Formula: see text] and a continuous dendrite map without an infinite LY-scrambled set but with DC1 pairs.

12

Aboites, Vicente, David Liceaga, Rider Jaimes-Reátegui, and Juan Hugo García-López. "Bogdanov Map for Modelling a Phase-Conjugated Ring Resonator." Entropy 21, no. 4 (April 10, 2019): 384. http://dx.doi.org/10.3390/e21040384.

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In this paper, we propose using paraxial matrix optics to describe a ring-phase conjugated resonator that includes an intracavity chaos-generating element; this allows the system to behave in phase space as a Bogdanov Map. Explicit expressions for the intracavity chaos-generating matrix elements were obtained. Furthermore, computer calculations for several parameter configurations were made; rich dynamic behavior among periodic orbits high periodicity and chaos were observed through bifurcation diagrams. These results confirm the direct dependence between the parameters present in the intracavity chaos-generating element.

13

YUAN, SHAOLIANG, TAO JIANG, and ZHUJUN JING. "BIFURCATION AND CHAOS IN THE TINKERBELL MAP." International Journal of Bifurcation and Chaos 21, no. 11 (November 2011): 3137–56. http://dx.doi.org/10.1142/s0218127411030581.

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In this paper, the dynamical behaviors of the Tinkerbell map are investigated in detail. Conditions for the existence of fold bifurcation, flip bifurcation and Hopf bifurcation are derived, and chaos in the sense of Marotto is verified by both analytical and numerical methods. Numerical simulations include bifurcation diagrams in two- and three-dimensional spaces, phase portraits, and the maximum Lyapunov exponent and fractal dimension, as well as the distribution of dynamics in the parameter plane, which exhibit new and interesting dynamical behaviors. More specifically, this paper reports the findings of chaos in the sense of Marotto, a route from an invariant circle to transient chaos with a great abundance of periodic windows, including period-2, 7, 8, 9, 10, 13, 17, 19, 23, 26 and so on, and suddenly appearing or disappearing chaos, convergence of an invariant circle to a period-one orbit, symmetry-breaking of periodic orbits, interlocking period-doubling bifurcations in chaotic regions, interior crisis, chaotic attractors, coexisting (2, 10, 13) chaotic sets, two coexisting invariant circles, two attracting chaotic sets coexisting with a non-attracting chaotic set, and so on, all in the Tinkerbell map. In particular, it is found that there is no obvious road from period-doubling bifurcations to chaos, but there is a route from a period-one orbit to an invariant circle and then to transient chaos as the parameters are varied. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the Tinkerbell map is obtained.

14

Brown, Ray, and Leon O. Chua. "Chaos: Generating Complexity from Simplicity." International Journal of Bifurcation and Chaos 07, no. 11 (November 1997): 2427–36. http://dx.doi.org/10.1142/s021812749700162x.

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The most commonly used mapping to illustrate the phenomenon of chaos is the map x → 2x mod (1). This map is known as the 'unilateral shift' because, in the binary number system this map shifts all digits to the left by one decimal place, and truncates the integer. The second most commonly used paradigm of chaos is the Smale horseshoe whose complexity is essentially the bilateral shift obtained when we simply shift without truncation in some symbol system. Neither of these paradigms fully explains chaos since shifts cannot generate complex orbits from simple (rational) initial conditions. How chaos generates complexity from simplicity is an essential part that needs explanation. Providing this explanation is the objective of this paper.

15

Shimizu, Toshihiro, and Nozomi Morioka. "Chaos Induced Transition." International Journal of Bifurcation and Chaos 07, no. 04 (April 1997): 855–67. http://dx.doi.org/10.1142/s0218127497000650.

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To study the coherent nature of chaos, two models are proposed. Model 1 is a simple nonlinear system [Formula: see text] and Model 2 is a linear harmonic oscillator [Formula: see text], which are driven by a chaotic force f(t). The chaotic force f(t) is defined by [Formula: see text] for nτ < t ≤ (n + 1)τ(n = 0, 1, 2, …), where yn+1 is a chaotic sequence of a map F(y; r) with the bifurcation parameter r: yn+1 = F(yn; r) (-0.5 ≤ yn ≤ 0.5) and ŷn = yn - < y0>. In Model 1 the relaxation process of this system and the τ- and r-dependence of the stationary distribution of x are discussed. It is shown that the small change of the bifurcation parameter r causes the drastic change of the stationary distribution. In Model 2, resonance phenomena are investigated near the period 3 window of the logistic map, in particular, in the intermittent chaos region and the period doubling region. The theoretical results are shown to be in a good agreement with numerical ones, which have been done for the logistic map as F(y; r).

16

Chen, Young-Long, and Chung-Ming Cheng. "Combining a chaos system with an Arnold cat map for a secure authentication scheme in wireless communication networks." Engineering Computations 31, no. 2 (February 25, 2014): 317–30. http://dx.doi.org/10.1108/ec-01-2013-0021.

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Purpose – Wu et al.'s scheme has a security problem that is related to anonymity: attackers can determine by interception the identity of a legal user. This paper aims to propose a new secure authentication which combines a chaos system with an Arnold cat map. The scheme improves upon that of the Wu et al.'s scheme. The scheme proposed herein provides for full anonymity and improves the security of authentication messages for wireless communications. Design/methodology/approach – A novel scheme that integrates a chaos sequence is used with an Arnold cat map for authentication messages. Authentication messages are shuffled using an Arnold cat map to improve the security of authentication in wireless communications. An analytic approach based on a chaos sequence with an Arnold cat map is developed to secure authentication. The proposed scheme is presented in this study to overcome the inherent drawbacks of existing designs. Findings – The integrated scheme involves two steps. First, a chaos map is used to generate a set of chaos sequences that is added to the authentication messages. Second, the authentication messages are shuffled using an Arnold cat map. The main feature of the proposed design is such that the chaos systems are sensitive to the initial values of conditions. Sensitivity will lead to long-term behavior unpredictability to reflect the non-linear dynamic systems. Furthermore, to increase the complexity of the authentication message, the authors also use an Arnold cat map. Originality/value – The proposed scheme provides functions that include full anonymity properties, protection of the real identity of the user, one-time password properties, timestamp benefits and sufficient complexity of the password. The analysis shows that the proposed scheme exhibits the advantages of the chaos system and is more secure than previous schemes. Notably, the proposed scheme is effective for wireless communications.

17

BUB, GIL, and LEON GLASS. "BIFURCATIONS IN A DISCONTINUOUS CIRCLE MAP: A THEORY FOR A CHAOTIC CARDIAC ARRHYTHMIA." International Journal of Bifurcation and Chaos 05, no. 02 (April 1995): 359–71. http://dx.doi.org/10.1142/s0218127495000302.

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The dynamics of discontinuous circle maps are investigated in the context of modulated parasystole, a cardiac arrhythmia in which there is an interaction between normal (sinus) and abnormal (ectopic) pacemaking sites in the heart. A class of noninvertible discontinuous circle maps with slope greater than 1 displays banded chaos under certain conditions. Banded chaos in these maps is characterized by a zero rotation interval width in the presence of a positive Lyapunov exponent. The bifurcations of a simple piecewise linear circle map are investigated. Parameters that result in banded chaos are organized into discrete, nonoverlapping zones in the parameter space. We apply these results to analyze a circle map that models modulated parasystole. Analysis of the model is complicated by the fact that the map has slope less than 1 for part of its domain. However, numerical simulations indicate that the modulated parasystole map displays banded chaos over a wide range of parameters. Banded chaos in this map produces rhythms with a relatively constant sinus-ectopic coupling interval, long trains of uninterrupted sinus beats, and patterns of successive sinus beats between ectopic beats characteristic of those found clinically.

18

TSUCHIYA, TAKASHI. "CIRCULAR CHAOS GAME REPRESENTATION OF 1-D CHAOS AND ITS RELATION TO THE COMPLEX WEIERSTRASS FUNCTION." International Journal of Bifurcation and Chaos 09, no. 10 (October 1999): 2069–80. http://dx.doi.org/10.1142/s0218127499001504.

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Barnsley's chaos game which is originally played on a triangular field is generalized to a circular field and is used to visualize one-dimensional (1-D) fully developed chaos. It is shown that when the 1-D map used is the shift map or its extension (the r-adic map) the game point is exactly represented by the complex Weierstrass function.

19

Tricarico, Mariarosaria, and Francesca Visentin. "Logistic map: from order to chaos." Applied Mathematical Sciences 8 (2014): 6819–26. http://dx.doi.org/10.12988/ams.2014.49699.

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20

Kawabe, T., and Y. Kondo. "Intermittent Chaos Generated by Logarithmic Map." Progress of Theoretical Physics 86, no. 3 (September 1, 1991): 581–86. http://dx.doi.org/10.1143/ptp/86.3.581.

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21

Bunimovich, L. A., and Ya G. Sinai. "Spacetime chaos in coupled map lattices." Nonlinearity 1, no. 4 (November 1, 1988): 491–516. http://dx.doi.org/10.1088/0951-7715/1/4/001.

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22

Hu, Tongchun. "Discrete Chaos in Fractional Henon Map." Applied Mathematics 05, no. 15 (2014): 2243–48. http://dx.doi.org/10.4236/am.2014.515218.

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23

Meleshenko, P. A., A. V. Tolkachev, and O. I. Kanishcheva. "HENON MAP WITH HYSTERESIS: CHAOS CONTROL." Вестник ВГУ Серия Системный анализ и информационные технологии, no. 3 (2022): 22–31. http://dx.doi.org/10.17308/sait/1995-5499/2022/3/22-32.

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24

Inoue, Kei. "An Improved Calculation Formula of the Extended Entropic Chaos Degree and Its Application to Two-Dimensional Chaotic Maps." Entropy 23, no. 11 (November 14, 2021): 1511. http://dx.doi.org/10.3390/e23111511.

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The Lyapunov exponent is primarily used to quantify the chaos of a dynamical system. However, it is difficult to compute the Lyapunov exponent of dynamical systems from a time series. The entropic chaos degree is a criterion for quantifying chaos in dynamical systems through information dynamics, which is directly computable for any time series. However, it requires higher values than the Lyapunov exponent for any chaotic map. Therefore, the improved entropic chaos degree for a one-dimensional chaotic map under typical chaotic conditions was introduced to reduce the difference between the Lyapunov exponent and the entropic chaos degree. Moreover, the improved entropic chaos degree was extended for a multidimensional chaotic map. Recently, the author has shown that the extended entropic chaos degree takes the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. However, the author has assumed a value of infinity for some numbers, especially the number of mapping points. Nevertheless, in actual numerical computations, these numbers are treated as finite. This study proposes an improved calculation formula of the extended entropic chaos degree to obtain appropriate numerical computation results for two-dimensional chaotic maps.

25

LAWRANCE, A. J., and N. BALAKRISHNA. "STATISTICAL DEPENDENCY IN CHAOS." International Journal of Bifurcation and Chaos 18, no. 11 (November 2008): 3207–19. http://dx.doi.org/10.1142/s0218127408022366.

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This paper is concerned with the statistical dependency effects in chaotic map processes, both before and after their discretization at branch boundaries. The resulting processes are no longer chaotic but are left with realizable statistical behavior. Such processes have appeared over several years in the electronic engineering literature. Informal but extended mathematical theory that facilitates the practical calculation of autocorrelation of such statistical behavior, is developed. Both the continuous and discretized cases are treated further by using Kohda's notions of equidistribution and constant-sum to maps which are not onto. Some particularly structured chaotic map processes, and also well-known maps are examined for their statistical dependency, with the tailed shift map family from chaotic communications receiving detailed attention. Several parts of the paper form a brief review of existing theory.

26

Mudrika, Mudrika, Suryadi Mt, and Sarifuddin Madenda. "New chaos function of composition function Gauss map and dyadic transformation map for digital image encryption." ITM Web of Conferences 61 (2024): 01004. http://dx.doi.org/10.1051/itmconf/20246101004.

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Encryption algorithms mostly use key-streams generated from random number generators. Several recent studies have shown that the random number generator used is a chaos function. In this paper, a new chaos function will be developed which can be used as a chaotic random number generator. The development is carried out by forming a new chaos function using the function composition method. The function that is composed is the Gauss Map function against the Dyadic Transformation Map. The results of the new chaos function are chaotic, this is based on the results of the analysis obtained from the results of the bifurcation diagram, the Lyapunov Exponent and the National Institute of Standard Technologies Test (NIST) standard randomness test. The results of the bifurcation diagram show that the density is for the value of α ∈ [−30,0] and has periodic properties to choose the values of β ∈ [−1.02, −0.75], β ∈ [−0.60, −0.30], β ∈ [0.10, 0.25] and β ∈ [0.55, 0.75]. A positive value of Lyapunov Exponential diagram will be employed alpha equal to negative value (α < 0). The results of the NIST standard randomness test with values x0 = 0.9, α = −15 and β = 0.7 resulted in 100 % passing the test (16 tests).

27

CHEN, LI-QUN, and YAN-ZHU LIU. "AN OPEN-PLUS-CLOSED-LOOP APPROACH TO SYNCHRONIZATION OF CHAOTIC AND HYPERCHAOTIC MAPS." International Journal of Bifurcation and Chaos 12, no. 05 (May 2002): 1219–25. http://dx.doi.org/10.1142/s0218127402005066.

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In this paper, the open-plus-closed-loop control strategy is adapted to synchronize discrete chaos. Two synchronization problems of chaos are studied: one is to drive a chaotic map with the aim of obtaining desirable chaotic dynamics; the other is to identify chaotic behaviors of a nonlinear map for different initial conditions. It is shown that in the latter case the needed control signal can be arbitrarily small. Two numerical examples, namely, the Gaussian map and a hyperchaotic map, are investigated in detail for demonstration of the effectiveness of the approach. The results show that synchronization of discrete chaos can be realized if the control is activated in the basin of entrainment.

28

BERSHADSKII, A. "MULTIFRACTAL CRITICAL CHAOS GENERATED BY TWO INTERACTING PARTICLES IN A DISORDERED CHAIN: ANDERSON MODEL." Modern Physics Letters B 13, no. 21 (September 10, 1999): 751–57. http://dx.doi.org/10.1142/s0217984999000944.

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It is shown, using data of numerical simulations performed by different authors, that multifractal chaos generated by two interacting particles in a disordered chain (in the frames of the Anderson model) belongs to the same class of multifractal chaotic processes as critical chaos generated by the map xn+1=1-a|xn|z via the Feigenbaum scenario. Moreover, a single parameter — multifractal critical index — characterizing this class, takes the same value for both the Anderson quantum chaos and for the universal critical chaos generated by the map xn+1=1-a|xn|3.

29

Li, Risong, Tianxiu Lu, Jingmin Pi, and Waseem Anwar. "Three Types of Distributional Chaos for a Sequence of Uniformly Convergent Continuous Maps." Advances in Mathematical Physics 2022 (June 18, 2022): 1–7. http://dx.doi.org/10.1155/2022/5481666.

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Let h s s = 1 ∞ be a sequence of continuous maps on a compact metric space W which converges uniformly to a continuous map h on W . In this paper, some equivalence conditions or necessary conditions for the limit map h to be distributional chaotic are obtained (where distributional chaoticity includes distributional chaotic in a sequence, distributional chaos of type 1 (DC1), distributional chaos of type 2 (DC2), and distributional chaos of type 3 (DC3)).

30

Braun, Thomas. "Suppression and Excitation of Chaos: The Example of the Glow Discharge." International Journal of Bifurcation and Chaos 08, no. 08 (August 1998): 1739–42. http://dx.doi.org/10.1142/s0218127498001431.

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I report on the experimental observation of excitation and suppression of chaos through time dependent perturbations in the dynamical variable of a glow discharge. The interaction of the external signal with the dynamical system is explained in terms of the 1D map associated to the glow discharge. Numerical simulations are also performed with the logistic map. The proposed mechanism of exciting and/or suppressing chaos is in accordance with the OGY method of controlling chaos.

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YANG, XIAO-SONG, and QINGDU LI. "HORSESHOES IN A NEW SWITCHING CIRCUIT VIA WIEN-BRIDGE OSCILLATOR." International Journal of Bifurcation and Chaos 15, no. 07 (July 2005): 2271–75. http://dx.doi.org/10.1142/s0218127405011631.

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In this paper we revisit a switching circuit designed by the authors and present a theoretical analysis on the existence of chaos in this circuit. For the ordinary differential equations describing this circuit, we give a computer-aided proof in terms of cross-section and Poincare map, by applying a modern theory of topological horseshoes theory to the obtained Poincare map, that this map is semiconjugate to the two-shift map. This implies that the corresponding differential equations exhibit chaos.

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Furusho, K., S. Iriyama, and M. Ohya. "Chaos Amplification Process Can Be Described by the GKSL Master Equation." Open Systems & Information Dynamics 24, no. 02 (June 2017): 1750008. http://dx.doi.org/10.1142/s1230161217500081.

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In 2000, Ohya et al. proposed a quantum algorithm with the amplification process of success probability, so-called chaos amplifier. They defined the process based on the logistic map, and its chaos behaviour amplifies the probability very fast. In this paper, we construct the chaos amplifier using a lifting map, a master equation and a partial trace. We also calculate the condition on the Lindblad operators in the GKSL master equation to achieve an effective amplification.

33

Ozawa, Kazuya, Kaito Isogai, Hideo Nakano, and Hideaki Okazaki. "Formal Chaos Existing Conditions on a Transmission Line Circuit with a Piecewise Linear Resistor." Applied Sciences 11, no. 20 (October 17, 2021): 9672. http://dx.doi.org/10.3390/app11209672.

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By using one-dimensional (1-D) map methods, some lossless transmission line circuits with a short at one side terminal have been actively studied. Bifurcation results or chaotic states in the circuits have been reported. On the other hand, many weak or strong definitions such that a 1-D map is mathematically chaotic are still being studied. In such definitions, the definition of formal chaos is well known as being the most traditional and most definite. However, formal chaos existences have not been rigorously proven in such circuits. In this paper, a general lossless transmission circuit is considered first with a dc bias voltage source in series with a load resistor at one side terminal and with a three-segment piecewise linear resistor at another side terminal. Secondly, the method for deriving a 1-D map describing the behavior of the circuit is summarized. Thirdly, to provide a basis of chaotic application for the 1-D map, the mathematical definition of formal chaos and the sufficient conditions of the existence of formal chaos are discussed. Furthermore, by using Maple, formal chaos existences and bifurcation behavior of 1-D maps are presented. By using the Lyapunov exponent, the observability of formal chaos in such bifurcation processes is outlined. Finally, the principal results and the future works are summarized.

34

KENNEDY, JUDY, and VAN NALL. "Dynamical properties of shift maps on inverse limits with a set valued function." Ergodic Theory and Dynamical Systems 38, no. 4 (September 22, 2016): 1499–524. http://dx.doi.org/10.1017/etds.2016.73.

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Set-valued functions from an interval into the closed subsets of an interval arise in various areas of science and mathematical modeling. Research has shown that the dynamics of a single-valued function on a compact space are closely linked to the dynamics of the shift map on the inverse limit with the function as the sole bonding map. For example, it has been shown that with Devaney’s definition of chaos the bonding function is chaotic if and only if the shift map is chaotic. One reason for caring about this connection is that the shift map is a homeomorphism on the inverse limit, and therefore the topological structure of the inverse-limit space must reflect in its richness the dynamics of the shift map. In the set-valued case there may not be a natural definition for chaos since there is not a single well-defined orbit for each point. However, the shift map is a continuous single-valued function so it together with the inverse-limit space form a dynamical system which can be chaotic in any of the usual senses. For the set-valued case we demonstrate with theorems and examples rich topological structure in the inverse limit when the shift map is chaotic (on certain invariant sets). We then connect that chaos to a property of the set-valued function that is a natural generalization of an important chaos producing property of continuous functions.

35

Csernák, Gábor, and Gábor Stépán. "Life expectancy calculation of transient chaos in the 2D micro-chaos map." Periodica Polytechnica Mechanical Engineering 51, no. 2 (2007): 59. http://dx.doi.org/10.3311/pp.me.2007-2.03.

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36

HE DA REN, WU SHUN GUANG, QU SHI XIAN, and MAO XIANG YU. "STABILIZATION OF CHAOS OR SUPPRESSION OF CHAOS IN A DISCONTINUOUS CIRCLE MAP." Acta Physica Sinica 46, no. 8 (1997): 1464. http://dx.doi.org/10.7498/aps.46.1464.

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37

Chen, Shaoqiu, Shiya Feng, Wenjun Fu, and Yingying Zhang. "Logistic Map: Stability and Entrance to Chaos." Journal of Physics: Conference Series 2014, no. 1 (September 1, 2021): 012009. http://dx.doi.org/10.1088/1742-6596/2014/1/012009.

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38

WANG, XING-YUAN, QING-YONG LIANG, and JUAN MENG. "CHAOS AND FRACTALS IN C–K MAP." International Journal of Modern Physics C 19, no. 09 (September 2008): 1389–409. http://dx.doi.org/10.1142/s0129183108012935.

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The characteristic of the fixed points of the Carotid–Kundalini (C–K) map is investigated and the boundary equation of the first bifurcation of the C–K map in the parameter plane is given. Based on the studies of the phase graph, the power spectrum, the correlation dimension and the Lyapunov exponents, the paper reveals the general features of the C–K map transforming from regularity. Meanwhile, using the periodic scanning technology proposed by Welstead and Cromer, a series of Mandelbrot–Julia (M–J) sets of the complex C–K map are constructed. The symmetry of M–J set and the topological inflexibility of distributing of periodic region in the Mandelbrot set are investigated. By founding the whole portray of Julia sets based on Mandelbrot set qualitatively, we find out that Mandelbrot sets contain abundant information of structure of Julia sets.

39

Casati, Giulio, and Tomaž Prosen. "Triangle Map: A Model of Quantum Chaos." Physical Review Letters 85, no. 20 (November 13, 2000): 4261–64. http://dx.doi.org/10.1103/physrevlett.85.4261.

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40

Subashini, V. J., and S. Poornachandra. "Chaos Based Image Encryption Using Bogdanov Map." Journal of Computational and Theoretical Nanoscience 14, no. 9 (September 1, 2017): 4508–14. http://dx.doi.org/10.1166/jctn.2017.6768.

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41

Wu, Guo-Cheng, and Dumitru Baleanu. "Discrete fractional logistic map and its chaos." Nonlinear Dynamics 75, no. 1-2 (September 20, 2013): 283–87. http://dx.doi.org/10.1007/s11071-013-1065-7.

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42

Jiang, Tao, and Zhi-yan Yang. "Bifurcations and chaos in Mira 2 map." Acta Mathematicae Applicatae Sinica, English Series 33, no. 4 (October 2017): 967–78. http://dx.doi.org/10.1007/s10255-017-0710-1.

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43

Das, Ruchi. "Chaos of product map on G-spaces." International Mathematical Forum 8 (2013): 647–52. http://dx.doi.org/10.12988/imf.2013.13067.

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44

Rak, R., and E. Rak. "Route to Chaos in Generalized Logistic Map." Acta Physica Polonica A 127, no. 3a (March 2015): A—113—A—117. http://dx.doi.org/10.12693/aphyspola.127.a-113.

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45

Zheng, Wei-mou, and Li-sha Lu. "Boundary of Chaos for the Gap Map." Communications in Theoretical Physics 15, no. 2 (March 1991): 161–68. http://dx.doi.org/10.1088/0253-6102/15/2/161.

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46

Vladimirov, Sergey, and Vladimir Negrul. "Order and chaos in modified logistic map." Izvestiya VUZ. Applied Nonlinear Dynamics 9, no. 4-5 (2002): 64–77. http://dx.doi.org/10.18500/0869-6632-2001-9-4-64-77.

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The topical problem of making the robust sources of chaotic signals is considered. Using of such sources is quite necessary аt working out of modem systems of information transmission, store and processing. Obviously, it must be precise devices, i.e. devices with stable statistical properties regarding to variations of their parameters, and it must be realised sufficiently simply in practice. In this connection both problems of study of modified logistic map dynamics and of comparing given mathematical model with the real physical system are formulated and solved. The borders and attraction basin of the time series attractor generated by the тар were determined. The values of order parameters separating regular, chaotic and strictly chaotic types of motion were found. Analytical expressions for calculating the Kolmogorov entropy and Shannon information entropy were obtained. It was shown, that while generating chaotic motion the order parameter’s behaviour corresponds with the second order phase transition. The roughness of the appearing attractor was discovered. Relation between the map being investigated and the physical system with delayed feedback and with infinite phase space dimension was proved. Obtained results make the prospect for application of the modified logistic map as a basis for operation of the chaotic oscillations sources which can be realised in an analogous and numerical form.

47

ANGULO, FABIOLA, ENRIC FOSSAS, and GERARD OLIVAR. "TRANSITION FROM PERIODICITY TO CHAOS IN A PWM-CONTROLLED BUCK CONVERTER WITH ZAD STRATEGY." International Journal of Bifurcation and Chaos 15, no. 10 (October 2005): 3245–64. http://dx.doi.org/10.1142/s0218127405014015.

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The transition from periodicity to chaos in a DC-DC Buck power converter is studied in this paper. The converter is controlled through a direct Pulse Width Modulation (PWM) in order to regulate the error dynamics at zero. Results show robustness with low output error and a fixed switching frequency. Furthermore, some rich dynamics appear as the constant associated with the first-order error dynamics decreases. Finally, a transition from periodicity to chaos is observed. This paper describes this transition and the bifurcations in the converter. Chaos appears in the system with a stretching and folding mechanism. It can be observed in the one-dimensional Poincaré map of the inductor current. This Poincaré map converges to a tent map with the variation of the system parameter ks.

48

CSERNÁK, GÁBOR, and GÁBOR STÉPÁN. "DIGITAL CONTROL AS SOURCE OF CHAOTIC BEHAVIOR." International Journal of Bifurcation and Chaos 20, no. 05 (May 2010): 1365–78. http://dx.doi.org/10.1142/s0218127410026538.

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In the present paper, we introduce and analyze a mechanical system, in which the digital implementation of a linear control loop may lead to chaotic behavior. The amplitude of such oscillations is usually very small, this is why these are called micro-chaotic vibrations. As a consequence of the digital effects, i.e. the sampling, the processing delay and the round-off error, the behavior of the system can be described by a piecewise linear map, the micro-chaos map. We examine a 2D version of the micro-chaos map and prove that the map is chaotic.

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ANDRECUT, M., and M. K. ALI. "CHAOS IN A SIMPLE BOOLEAN NETWORK." International Journal of Modern Physics B 15, no. 01 (January 10, 2001): 17–23. http://dx.doi.org/10.1142/s021797920100259x.

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We study the complex dynamics of a simple stochastic Boolean network. The investigated system is equivalent to a randomly connected Boolean cellular automaton. The dynamical evolution of the cellular automaton is exactly described by a polynomial map with binomial coefficients. We show that the map is chaotic and the route to chaos is period-doubling bifurcations.

50

Zhou, Xue Song, Man Li, You Jie Ma, and Si Jia Liu. "The Research and Analysis Based on a New Hybrid Chaotic Sequences." Advanced Materials Research 383-390 (November 2011): 2307–12. http://dx.doi.org/10.4028/www.scientific.net/amr.383-390.2307.

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The chaos sequence generated by the chaotic map methods usually has unsatisfactory performances. To solve this problem, this paper proposes a novel method with chaos sequence, which blends the Chebyshev chaotic map and one-dimensional chaotic map with infinite collapse in finite interval. Both MATLAB simulations and performance analyses results show that the proposed method has much improved chaotic performances than the single Chebyshev sequence or the chaotic sequence with infinite collapses, by analyzing and simulating the sequence of traversal feature, period doubling properties, Lyapunov exponents, initial value sensitivity, autocorrelation, and the uniformity of sequence.