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Journal articles on the topic 'Higher-dimensional maps'

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Published: 10 December 2022

Last updated: 28 January 2023

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1

BALOGH, ZOLTAN M., and CHRISTOPH LEUENBERGER. "HIGHER DIMENSIONAL RIEMANN MAPS." International Journal of Mathematics 09, no. 04 (June 1998): 421–42. http://dx.doi.org/10.1142/s0129167x9800018x.

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We consider the notion of Riemann map of Lempert and Semmes. The purpose of this paper is to give an intrinsic and biholomorphically invariant characterization of strictly pseudoconvex domains in Cn which admit a Riemann map. In this sense necessary and sufficient conditions are given for the existence of a Riemann map in terms of Kobayashi discs and the associated Lempert invariants.

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2

Mira, C. "Noninvertible maps and their embedding into higher dimensional invertible maps." Regular and Chaotic Dynamics 15, no. 2-3 (April 27, 2010): 246–60. http://dx.doi.org/10.1134/s1560354710020127.

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3

Góra, P., A. Boyarsky, and Y. S. Lou. "Lyapunov exponents for higher dimensional random maps." Journal of Applied Mathematics and Stochastic Analysis 10, no. 3 (January 1, 1997): 209–18. http://dx.doi.org/10.1155/s1048953397000270.

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A random map is a discrete time dynamical system in which one of a number of transformations is selected randomly and implemented. Random maps have been used recently to model interference effects in quantum physics. The main results of this paper deal with the Lyapunov exponents for higher dimensional random maps, where the individual maps are Jabloński maps on the n-dimensional cube.

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4

Mihailescu, Eugen. "Higher dimensional expanding maps and toral extensions." Proceedings of the American Mathematical Society 141, no. 10 (June 12, 2013): 3467–75. http://dx.doi.org/10.1090/s0002-9939-2013-11597-2.

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5

Balreira, E. Cabral, Saber Elaydi, and Rafael Luís. "Global stability of higher dimensional monotone maps." Journal of Difference Equations and Applications 23, no. 12 (October 12, 2017): 2037–71. http://dx.doi.org/10.1080/10236198.2017.1388375.

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6

Boyarsky, A., W. Byers, and P. Gauthier. "Higher dimensional analogues of the tent maps." Nonlinear Analysis: Theory, Methods & Applications 11, no. 11 (November 1987): 1317–24. http://dx.doi.org/10.1016/0362-546x(87)90048-4.

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7

RICHTER, HENDRIK. "THE GENERALIZED HÉNON MAPS: EXAMPLES FOR HIGHER-DIMENSIONAL CHAOS." International Journal of Bifurcation and Chaos 12, no. 06 (June 2002): 1371–84. http://dx.doi.org/10.1142/s0218127402005121.

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The generalized Hénon maps (GHM) are discrete-time systems with given finite dimension, which show chaotic and hyperchaotic behavior for certain parameter values and initial conditions. A study of these maps is given where particularly higher-dimensional cases are considered.

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8

Sano, Yuki, Pierre Arnoux, and Shunji Ito. "Higher dimensional extensions of substitutions and their dual maps." Journal d'Analyse Mathématique 83, no. 1 (December 2001): 183–206. http://dx.doi.org/10.1007/bf02790261.

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9

Ruan, Huo-Jun, and Robert S. Strichartz. "Covering Maps and Periodic Functions on Higher Dimensional Sierpinski Gaskets." Canadian Journal of Mathematics 61, no. 5 (October 1, 2010): 1151–81. http://dx.doi.org/10.4153/cjm-2009-054-5.

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Abstract.We construct covering maps from infinite blowups of the$n$-dimensional Sierpinski gasket$S{{G}_{n}}$to certain compact fractafolds based on$S{{G}_{n}}$. These maps are fractal analogs of the usual covering maps fromthe line to the circle. The construction extends work of the second author in the case$n=2$, but a differentmethod of proof is needed, which amounts to solving a Sudoku-type puzzle. We can use the covering maps to define the notion of periodic function on the blowups. We give a characterization of these periodic functions and describe the analog of Fourier series expansions. We study covering maps onto quotient fractalfolds. Finally, we show that such covering maps fail to exist for many other highly symmetric fractals.

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10

BOYARSKY, A., and Y. S. LOU. "CHAOTIC BEHAVIOR OF HIGHER DIMENSIONAL TRANSFORMATIONS DEFINED ON COUNTABLE PARTITIONS." International Journal of Bifurcation and Chaos 03, no. 04 (August 1993): 1045–49. http://dx.doi.org/10.1142/s0218127493000866.

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Jablonski maps are higher dimensional maps defined on rectangular partitions with each component a function of only one variable. It is well known that expanding Jablonski maps have absolutely continuous invariant measures. In this note we consider Jablonski maps defined on countable partitions. Such maps occur, for example, in multivariable number theoretic problems. The main result establishes the existence of an absolutely continuous invariant measure for Jablonski maps on a countable partition with the additional condition that the images of all the partition elements form a finite collection. An example is given.

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11

Pintea, Cornel. "Smooth Mappings with Higher Dimensional Critical Sets." Canadian Mathematical Bulletin 53, no. 3 (September 1, 2010): 542–49. http://dx.doi.org/10.4153/cmb-2010-057-8.

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12

Frances, Charles. "Removable and essential singular sets for higher dimensional conformal maps." Commentarii Mathematici Helvetici 89, no. 2 (2014): 405–41. http://dx.doi.org/10.4171/cmh/323.

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13

Anastassiou, Stavros, Anastasios Bountis, and Arnd Bäcker. "Recent Results on the Dynamics of Higher-dimensional Hénon Maps." Regular and Chaotic Dynamics 23, no. 2 (March 2018): 161–77. http://dx.doi.org/10.1134/s156035471802003x.

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14

Fahrenberg, Uli, and Axel Legay. "History-Preserving Bisimilarity for Higher-Dimensional Automata via Open Maps." Electronic Notes in Theoretical Computer Science 298 (November 2013): 165–78. http://dx.doi.org/10.1016/j.entcs.2013.09.012.

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15

Yannacopoulos, A. N., and G. Rowlands. "Diffusion coefficients for higher dimensional symplectic maps on the cylinder." Physica D: Nonlinear Phenomena 57, no. 3-4 (August 1992): 355–74. http://dx.doi.org/10.1016/0167-2789(92)90008-b.

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16

Chen, Chaoqun, and Fangyan Lu. "Nonlinear maps preserving higher-dimensional numerical ranges of Jordan $$*$$-products." Annals of Functional Analysis 11, no. 1 (December 1, 2019): 185–93. http://dx.doi.org/10.1007/s43034-019-00021-4.

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17

Krzempek, Jerzy. "Fully closed maps and non-metrizable higher-dimensional Anderson–Choquet continua." Colloquium Mathematicum 120, no. 2 (2010): 201–22. http://dx.doi.org/10.4064/cm120-2-3.

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18

Boyarsky, A., and Y. S. Lou. "Existence of absolutely continuous invariant measures for higher-dimensional random maps." Dynamics and Stability of Systems 7, no. 4 (January 1992): 233–44. http://dx.doi.org/10.1080/02681119208806141.

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19

ISLAM, MD SHAFIQUL. "INVARIANT MEASURES FOR HIGHER DIMENSIONAL MARKOV SWITCHING POSITION DEPENDENT RANDOM MAPS." International Journal of Bifurcation and Chaos 19, no. 01 (January 2009): 409–17. http://dx.doi.org/10.1142/s021812740902297x.

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A higher dimensional Markov switching position dependent random map is a random map where the probabilities of switching from one higher dimension transformation to another are the entries of a stochastic matrix and the entries of stochastic matrix are functions of positions. In this note, we prove sufficient conditions for the existence of absolutely continuous measures for a class of higher dimensional Markov switching position dependent random maps. Our result is a generalization of the result in [Bahsoun & Góra, 2005; Bahsoun et al., 2005].

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20

Saito, Satoru, and Noriko Saitoh. "Invariant Varieties of Periodic Points for Some Higher Dimensional Integrable Maps." Journal of the Physical Society of Japan 76, no. 2 (February 15, 2007): 024006. http://dx.doi.org/10.1143/jpsj.76.024006.

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21

Koschorke, Ulrich. "A generalization of Milnor's μ-invariants to higher-dimensional link maps." Topology 36, no. 2 (March 1997): 301–24. http://dx.doi.org/10.1016/0040-9383(96)00018-3.

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22

Bunke, Ulrich, and Georg Tamme. "Regulators and cycle maps in higher-dimensional differential algebraic K-theory." Advances in Mathematics 285 (November 2015): 1853–969. http://dx.doi.org/10.1016/j.aim.2015.08.004.

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23

Chen, Chaoqun, and Fangyan Lu. "Maps that preserve higher-dimensional numerical ranges with operator Jordan products." Linear and Multilinear Algebra 65, no. 12 (January 13, 2017): 2530–37. http://dx.doi.org/10.1080/03081087.2016.1278197.

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24

Haapala, Amanda F., and Kathleen C. Howell. "Representations of higher-dimensional Poincaré maps with applications to spacecraft trajectory design." Acta Astronautica 96 (March 2014): 23–41. http://dx.doi.org/10.1016/j.actaastro.2013.11.019.

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25

Lin, Fanghua, and Changyou Wang. "Harmonic Maps in Connection of Phase Transitions with Higher Dimensional Potential Wells." Chinese Annals of Mathematics, Series B 40, no. 5 (September 2019): 781–810. http://dx.doi.org/10.1007/s11401-019-0160-6.

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26

Zhang, Xu. "Chaotic Polynomial Maps." International Journal of Bifurcation and Chaos 26, no. 08 (July 2016): 1650131. http://dx.doi.org/10.1142/s0218127416501315.

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This paper introduces a class of polynomial maps in Euclidean spaces, investigates the conditions under which there exist Smale horseshoes and uniformly hyperbolic invariant sets, studies the chaotic dynamical behavior and strange attractors, and shows that some maps are chaotic in the sense of Li–Yorke or Devaney. This type of maps includes both the Logistic map and the Hénon map. For some diffeomorphisms with the expansion dimension equal to one or two in three-dimensional spaces, the conditions under which there exist Smale horseshoes and uniformly hyperbolic invariant sets on which the systems are topologically conjugate to the two-sided fullshift on finite alphabet are obtained; for some expanding maps, the chaotic region is analyzed by using the coupled-expansion theory and the Brouwer degree theory. For three types of higher-dimensional polynomial maps with degree two, the conditions under which there are Smale horseshoes and uniformly hyperbolic invariant sets are given, and the topological conjugacy between the maps on the invariant sets and the two-sided fullshift on finite alphabet is obtained. Some interesting maps with chaotic attractors and positive Lyapunov exponents in three-dimensional spaces are found by using computer simulations. In the end, two examples are provided to illustrate the theoretical results.

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27

un Chen, Chao, and Fang an Lu. "Nonlinear maps preserving higher-dimensional numerical range of skew Lie product of operators." Operators and Matrices, no. 2 (2016): 335–44. http://dx.doi.org/10.7153/oam-10-18.

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28

Saito, Satoru, and Noriko Saitoh. "Fate of the Julia set of higher dimensional maps in the integrable limit." Journal of Mathematical Physics 51, no. 6 (June 2010): 063501. http://dx.doi.org/10.1063/1.3430554.

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29

Meester, Ronald, and Jeffrey E. Steif. "Higher-dimensional subshifts of finite type, factor maps and measures of maximal entropy." Pacific Journal of Mathematics 200, no. 2 (October 1, 2001): 497–510. http://dx.doi.org/10.2140/pjm.2001.200.497.

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30

Budinich, Marco, and John G. Taylor. "On the Ordering Conditions for Self-Organizing Maps." Neural Computation 7, no. 2 (March 1995): 284–89. http://dx.doi.org/10.1162/neco.1995.7.2.284.

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We present a geometric interpretation of ordering in self-organizing feature maps. This view provides simpler proofs of Kohonen ordering theorem and of convergence to an ordered state in the one-dimensional case. At the same time it explains intuitively the origin of the problems in higher dimensional cases. Furthermore it provides a geometric view of the known characteristics of learning in self-organizing nets.

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31

Choo, Zacky, and Victor Snaith. "p-adic cocycles and their regulator maps." Journal of K-Theory 8, no. 2 (October 19, 2010): 241–49. http://dx.doi.org/10.1017/is010008010jkt125.

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AbstractWe derive a power series formula for the p-adic regulator on the higher dimensional algebraic K-groups of number fields. This formula is designed to be well suited to computer calculations and to reduction modulo powers of p.

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32

Saitoh, Noriko, and Satoru Saito. "Perturbative Changes of the Nature of Invariant Varieties for Some Higher Dimensional Integrable Maps." Journal of the Physical Society of Japan 77, no. 2 (February 15, 2008): 024001. http://dx.doi.org/10.1143/jpsj.77.024001.

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33

SHI, YUMING, and GUANRONG CHEN. "CHAOTIFICATION OF DISCRETE DYNAMICAL SYSTEMS GOVERNED BY CONTINUOUS MAPS." International Journal of Bifurcation and Chaos 15, no. 02 (February 2005): 547–55. http://dx.doi.org/10.1142/s0218127405012351.

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This paper is concerned with chaotification of discrete dynamical systems in finite-dimensional real spaces, via feedback control techniques. A chaotification theorem for one-dimensional discrete dynamical systems and a chaotification theorem for general higher-dimensional discrete dynamical systems are established, respectively. The controlled systems are proved to be chaotic in the sense of Devaney. In particular, the maps corresponding to the original systems and designed controllers are only required to satisfy some mild assumptions on two very small disjoint closed subsets in the domains of interest. This condition is weaker than those in the existing relevant literature.

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34

Komuro, Motomasa, Kyohei Kamiyama, Tetsuro Endo, and Kazuyuki Aihara. "Quasi-Periodic Bifurcations of Higher-Dimensional Tori." International Journal of Bifurcation and Chaos 26, no. 07 (June 30, 2016): 1630016. http://dx.doi.org/10.1142/s0218127416300160.

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We classify the local bifurcations of quasi-periodic [Formula: see text]-dimensional tori in maps (abbr. MT[Formula: see text]) and in flows (abbr. FT[Formula: see text]) for [Formula: see text]. It is convenient to classify these bifurcations into normal bifurcations and resonance bifurcations. Normal bifurcations of MT[Formula: see text] can be classified into four classes: namely, saddle-node, period doubling, double covering, and Neimark–Sacker bifurcations. Furthermore, normal bifurcations of FT[Formula: see text] can be classified into three classes: saddle-node, double covering, and Neimark–Sacker bifurcations. These bifurcations are determined by the type of the dominant Lyapunov bundle. Resonance bifurcations are well known as phase locking of quasi-periodic solutions. These bifurcations are classified into two classes for both MT[Formula: see text] and FT[Formula: see text]: namely, saddle-node cycle and heteroclinic cycle bifurcations of the [Formula: see text]-dimensional tori. The former is reversible, while the latter is irreversible. In addition, we propose a method for analyzing higher-dimensional tori, which uses one-dimensional tori in sections (abbr. ST[Formula: see text]) and zero-dimensional tori in sections (abbr. ST[Formula: see text]). The bifurcations of ST[Formula: see text] can be classified into five classes: saddle-node, period doubling, component doubling, double covering, and Neimark–Sacker bifurcations. The bifurcations of ST[Formula: see text] can be classified into four classes: saddle-node, period doubling, component doubling, and Neimark–Sacker bifurcations. Furthermore, we clarify the relationship between the bifurcations of ST[Formula: see text]/ST[Formula: see text] and the bifurcations of MT[Formula: see text]/FT[Formula: see text]. We present examples of all of these bifurcations.

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35

Doliwa, Adam. "Desargues maps and the Hirota–Miwa equation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2116 (December 11, 2009): 1177–200. http://dx.doi.org/10.1098/rspa.2009.0300.

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We study the Desargues maps , which generate lattices whose points are collinear with all their nearest (in positive directions) neighbours. The multi-dimensional compatibility of the map is equivalent to the Desargues theorem and its higher dimensional generalizations. The nonlinear counterpart of the map is the non-commutative (in general) Hirota–Miwa system. In the commutative case of the complex field we apply the non-local -dressing method to construct Desargues maps and the corresponding solutions of the system. In particular, we identify the Fredholm determinant of the integral equation inverting the non-local -dressing problem with the τ -function. Finally, we establish equivalence between the Desargues maps and quadrilateral lattices provided we take into consideration also their Laplace transforms.

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36

Yao, Wei Feng, and Xiao Bao Jia. "An Improved SVM Based on Feature Extension and Feature Selection." Applied Mechanics and Materials 552 (June 2014): 128–32. http://dx.doi.org/10.4028/www.scientific.net/amm.552.128.

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Support Vector Machine (SVM) implicitly maps samples from the lower-dimensional feature space to a higher-dimensional space, and designs a non-linear classifier via optimize the linear classifier in the higher-dimensional space. This paper proposed an improved SVM method based on feature extension and feature selection. The method explicitly maps the samples to a higher-dimensional feature space, perform the feature selection in the space, and finally design a linear classifier with a selected feature set. This paper illustrated the reason why the generalization ability is improved by this technique. The experiment results on benchmark datasets show that the improved SVM greatly decreases the error rate compared with other classifiers, which proves the feasibility of the proposed SVM.

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37

Pan, Ivan, and Aron Simis. "Cremona Maps of de Jonquières Type." Canadian Journal of Mathematics 67, no. 4 (August 1, 2015): 23–941. http://dx.doi.org/10.4153/cjm-2014-037-3.

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AbstractThis paper is concerned with suitable generalizations of a plane de Jonquières map to higher dimensional space ℙn with n ≥ 3. For each given point of ℙn there is a subgroup of the entire Cremona group of dimension n consisting of such maps. We study both geometric and group-theoretical properties of this notion. In the case where n = 3 we describe an explicit set of generators of the group and give a homological characterization of a basic subgroup thereof.

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38

Araujo, Vitor, Maria Jose Pacifico, and Mariana Pinheiro. "Adapted random perturbations for non-uniformly expanding maps." Stochastics and Dynamics 14, no. 04 (September 22, 2014): 1450007. http://dx.doi.org/10.1142/s0219493714500075.

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We obtain stochastic stability of C2 non-uniformly expanding one-dimensional endomorphisms, requiring only that the first hyperbolic time map be Lp-integrable for p > 3. We show that, under this condition (which depends only on the unperturbed dynamics), we can construct a random perturbation that preserves the original hyperbolic times of the unperturbed map and, therefore, to obtain non-uniform expansion for random orbits. This ensures that the first hyperbolic time map is uniformly integrable for all small enough noise levels, which is known to imply stochastic stability. The method enables us to obtain stochastic stability for a class of maps with infinitely many critical points. For higher dimensional endomorphisms, a similar result is obtained, but under stronger assumptions.

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39

DENG, YANXIA, and ZHIHONG XIA. "Conley–Zehnder index and bifurcation of fixed points of Hamiltonian maps." Ergodic Theory and Dynamical Systems 38, no. 6 (March 14, 2017): 2086–107. http://dx.doi.org/10.1017/etds.2016.118.

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We study the bifurcations of fixed points of Hamiltonian maps and symplectic diffeomorphisms. We are particularly interested in the bifurcations where the Conley–Zehnder index of a fixed point changes. The main result is that when the Conley–Zehnder index of a fixed point increases (or decreases) by one or two, we observe that there are several bifurcation scenarios. Under some non-degeneracy conditions on the one-parameter family of maps, two, four or eight fixed points bifurcate from the original one. We give a relatively detailed analysis of the bifurcation in the two-dimensional case. We also show that higher-dimensional cases can be reduced to the two-dimensional case.

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40

ZHANG, XU, YUMING SHI, and GUANRONG CHEN. "CONSTRUCTING CHAOTIC POLYNOMIAL MAPS." International Journal of Bifurcation and Chaos 19, no. 02 (February 2009): 531–43. http://dx.doi.org/10.1142/s0218127409023172.

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This paper studies the construction of one-dimensional real chaotic polynomial maps. Given an arbitrary nonzero polynomial of degree m (≥ 0), two methods are derived for constructing chaotic polynomial maps of degree m + 2 by simply multiplying the given polynomial with suitably designed quadratic polynomials. Moreover, for m + 2 arbitrarily given different positive constants, a method is given to construct a chaotic polynomial map of degree 2m based on the coupled-expansion theory. Furthermore, by multiplying a real parameter to a special kind of polynomial, which has at least two different non-negative or nonpositive zeros, the chaotic parameter region of the polynomial is analyzed based on the snap-back repeller theory. As a consequence, for any given integer n ≥ 2, at least one polynomial of degree n can be constructed so that it is chaotic in the sense of both Li–Yorke and Devaney. In addition, two natural ways of generalizing the logistic map to higher-degree chaotic logistic-like maps are given. Finally, an illustrative example is provided with computer simulations for illustration.

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41

KOSCHORKE, U. "LINK MAPS IN ARBITRARY MANIFOLDS AND THEIR HOMOTOPY INVARIANTS." Journal of Knot Theory and Its Ramifications 12, no. 01 (February 2003): 79–104. http://dx.doi.org/10.1142/s0218216503002329.

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In this paper we generalize Milnor's μ-invariants of classical links to certain ("κ-Brunnian") higher dimensional link maps into fairly arbitrary manifolds. Our approach involves the homotopy theory of configuration spaces and of wedges of spheres. We discuss the strength of these invariants and their compatibilities e.g. with (Hilton decompositions of) linking coefficients. Our results suggest, in particular, a conjecture about possible new link homotopies.

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42

Bullivant, Alex, Marcos Calçada, Zoltán Kádár, João Faria Martins, and Paul Martin. "Higher lattices, discrete two-dimensional holonomy and topological phases in (3 + 1)D with higher gauge symmetry." Reviews in Mathematical Physics 32, no. 04 (November 4, 2019): 2050011. http://dx.doi.org/10.1142/s0129055x20500117.

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Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we study Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. We show that a construction for higher lattice gauge theory is well-defined, including in particular a Hamiltonian for topological phases of matter in [Formula: see text] dimensions. Our construction builds upon the Kitaev quantum double model, replacing the finite gauge connection with a finite gauge 2-group 2-connection. Our Hamiltonian higher lattice gauge theory model is defined on spatial manifolds of arbitrary dimension presented by slightly combinatorialized CW-decompositions (2-lattice decompositions), whose 1-cells and 2-cells carry discrete 1-dimensional and 2-dimensional holonomy data. We prove that the ground-state degeneracy of Hamiltonian higher lattice gauge theory is a topological invariant of manifolds, coinciding with the number of homotopy classes of maps from the manifold to the classifying space of the underlying gauge 2-group. The operators of our Hamiltonian model are closely related to discrete 2-dimensional holonomy operators for discretized 2-connections on manifolds with a 2-lattice decomposition. We therefore address the definition of discrete 2-dimensional holonomy for surfaces embedded in 2-lattices. Several results concerning the well-definedness of discrete 2-dimensional holonomy, and its construction in a combinatorial and algebraic topological setting are presented.

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43

Ivanov, Vsevolod Ivanov. "Characterizations of nonsmooth generalized polarly cone-monotone maps." Studia Scientiarum Mathematicarum Hungarica 42, no. 4 (October 1, 2005): 445–58. http://dx.doi.org/10.1556/sscmath.42.2005.4.8.

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In this paper we consider different types of generalized cone-mono-tone maps: polarly C-monotone, strictly polarly C-monotone, strongly polarly C-monotone, polarly C-pseudomonotone, strictly polarly C-pseudomonotone and polarly C-quasimonotone maps, where C is a cone in a finite-dimensional space Rm. We characterize these maps in the case when they are radially continuous with respect to the positive polar cone C+ of the cone C, generalizing some well known results. In the obtained theorems we use first and higher-order lower Dini directional derivatives.

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44

Collet, G., M. Drmota, and L. D. Klausner. "Limit laws of planar maps with prescribed vertex degrees." Combinatorics, Probability and Computing 28, no. 4 (February 4, 2019): 519–41. http://dx.doi.org/10.1017/s0963548318000573.

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AbstractWe prove a generalmulti-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integersD. Our results rely on a classical bijection with mobiles (objects exhibiting a tree structure), combined with refined analytic tools to deal with the systems of equations on infinite variables that arise. We also discuss possible extensions to maps of higher genus and to weighted maps.

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45

Fundinger, Danny. "Toward the Calculation of Higher-Dimensional Stable Manifolds and Stable Sets for Noninvertible and Piecewise-Smooth Maps." Journal of Nonlinear Science 18, no. 4 (December 28, 2007): 391–413. http://dx.doi.org/10.1007/s00332-007-9016-4.

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46

LI, HUIMIN, YANGYU FAN, and JING ZHANG. "A NEW ALGORITHM FOR COMPUTING ONE-DIMENSIONAL STABLE AND UNSTABLE MANIFOLDS OF MAPS." International Journal of Bifurcation and Chaos 22, no. 01 (January 2012): 1250018. http://dx.doi.org/10.1142/s0218127412500186.

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A new algorithm is presented to compute one-dimensional stable and unstable manifolds of fixed points for both two-dimensional and higher dimensional diffeomorphism maps. When computing the stable manifold, the algorithm does not require the explicit expression of the inverse map. The global manifold is grown from a local manifold and one point is added at each step. The new point is located with a "prediction and correction" scheme, which avoids searching the computed part of the manifold with a bisection method and accelerates the searching process. By using the fact that the Jacobian transports derivatives along the orbit of the manifold, the tangent component of the manifold is determined and a new accuracy criterion is proposed to check whether the new point that defines the manifold is acceptable. The performance of the algorithm is demonstrated with several numerical examples.

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47

Moysis, Lazaros, Ioannis Kafetzis, Murilo S. Baptista, and Christos Volos. "Chaotification of One-Dimensional Maps Based on Remainder Operator Addition." Mathematics 10, no. 15 (August 7, 2022): 2801. http://dx.doi.org/10.3390/math10152801.

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In this work, a chaotification technique is proposed that can be used to enhance the complexity of any one-dimensional map by adding the remainder operator to it. It is shown that by an appropriate parameter choice, the resulting map can achieve a higher Lyapunov exponent compared to its seed map, and all periodic orbits of any period will be unstable, leading to robust chaos. The technique is tested on several maps from the literature, yielding increased chaotic behavior in all cases, as indicated by comparison of the bifurcation and Lyapunov exponent diagrams of the original and resulting maps. Moreover, the effect of the proposed technique in the problem of pseudo-random bit generation is studied. Using a standard bit generation technique, it is shown that the proposed maps demonstrate increased statistical randomness compared to their seed ones, when used as a source for the bit generator. This study illustrates that the proposed method is an efficient chaotification technique for maps that can be used in chaos-based encryption and other relevant applications.

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JAHED-MOTLAGH, MOHAMMAD R., BEHNAM KIA, WILLIAM L. DITTO, and SUDESHNA SINHA. "FAULT TOLERANCE AND DETECTION IN CHAOTIC COMPUTERS." International Journal of Bifurcation and Chaos 17, no. 06 (June 2007): 1955–68. http://dx.doi.org/10.1142/s0218127407018142.

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We introduce a structural testing method for a dynamics based computing device. Our scheme detects different physical defects, manifesting themselves as parameter variations in the chaotic system at the core of the logic blocks. Since this testing method exploits the dynamical properties of chaotic systems to detect damaged logic blocks, the damaged elements can be detected by very few testing inputs, leading to very low testing time. Further the method does not entail dedicated or extra hardware for testing. Specifically, we demonstrate the method on one-dimensional unimodal chaotic maps. Some ideas for testing higher dimensional maps and flows are also presented.

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Dong, Weihua, and Hua Liao. "EYE TRACKING TO EXPLORE THE IMPACTS OF PHOTOREALISTIC 3D REPRESENTATIONS IN PEDSTRIAN NAVIGATION PERFORMANCE." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLI-B2 (June 8, 2016): 641–45. http://dx.doi.org/10.5194/isprs-archives-xli-b2-641-2016.

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Despite the now-ubiquitous two-dimensional (2D) maps, photorealistic three-dimensional (3D) representations of cities (e.g., Google Earth) have gained much attention by scientists and public users as another option. However, there is no consistent evidence on the influences of 3D photorealism on pedestrian navigation. Whether 3D photorealism can communicate cartographic information for navigation with higher effectiveness and efficiency and lower cognitive workload compared to the traditional symbolic 2D maps remains unknown. This study aims to explore whether the photorealistic 3D representation can facilitate processes of map reading and navigation in digital environments using a lab-based eye tracking approach. Here we show the differences of symbolic 2D maps versus photorealistic 3D representations depending on users’ eye-movement and navigation behaviour data. We found that the participants using the 3D representation were less effective, less efficient and were required higher cognitive workload than using the 2D map for map reading. However, participants using the 3D representation performed more efficiently in self-localization and orientation at the complex decision points. The empirical results can be helpful to improve the usability of pedestrian navigation maps in future designs.

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50

Dong, Weihua, and Hua Liao. "EYE TRACKING TO EXPLORE THE IMPACTS OF PHOTOREALISTIC 3D REPRESENTATIONS IN PEDSTRIAN NAVIGATION PERFORMANCE." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLI-B2 (June 8, 2016): 641–45. http://dx.doi.org/10.5194/isprsarchives-xli-b2-641-2016.

Full text

Abstract:

Despite the now-ubiquitous two-dimensional (2D) maps, photorealistic three-dimensional (3D) representations of cities (e.g., Google Earth) have gained much attention by scientists and public users as another option. However, there is no consistent evidence on the influences of 3D photorealism on pedestrian navigation. Whether 3D photorealism can communicate cartographic information for navigation with higher effectiveness and efficiency and lower cognitive workload compared to the traditional symbolic 2D maps remains unknown. This study aims to explore whether the photorealistic 3D representation can facilitate processes of map reading and navigation in digital environments using a lab-based eye tracking approach. Here we show the differences of symbolic 2D maps versus photorealistic 3D representations depending on users’ eye-movement and navigation behaviour data. We found that the participants using the 3D representation were less effective, less efficient and were required higher cognitive workload than using the 2D map for map reading. However, participants using the 3D representation performed more efficiently in self-localization and orientation at the complex decision points. The empirical results can be helpful to improve the usability of pedestrian navigation maps in future designs.

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