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Journal articles on the topic 'Piecewise linear map'

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Published: 4 June 2021
Last updated: 11 February 2022

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1

White,, Marvin S., and Patricia Griffin. "Piecewise Linear Rubber-Sheet Map Transformation." American Cartographer 12, no. 2 (January 1985): 123–31. http://dx.doi.org/10.1559/152304085783915135.

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2

BAILLIF, MATHIEU, and ANDRÉ DE CARVALHO. "PIECEWISE LINEAR MODEL FOR TREE MAPS." International Journal of Bifurcation and Chaos 11, no. 12 (December 2001): 3163–69. http://dx.doi.org/10.1142/s0218127401004108.

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We generalize to tree maps the theorems of Parry and Milnor–Thurston about the semi-conjugacy of a continuous piecewise monotone map f to a continuous piecewise linear map with constant slope, equal to the exponential of the entropy of f.
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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.
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4

KOLLÁR, LÁSZLÓ E., GÁBOR STÉPÁN, and JÁNOS TURI. "DYNAMICS OF PIECEWISE LINEAR DISCONTINUOUS MAPS." International Journal of Bifurcation and Chaos 14, no. 07 (July 2004): 2341–51. http://dx.doi.org/10.1142/s0218127404010837.

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In this paper, the dynamics of maps representing classes of controlled sampled systems with backlash are examined. First, a bilinear one-dimensional map is considered, and the analysis shows that, depending on the value of the control parameter, all orbits originating in an attractive set are either periodic or dense on the attractor. Moreover, the dense orbits have sensitive dependence on initial data, but behave rather regularly, i.e. they have quasiperiodic subsequences and the Lyapunov exponent of every orbit is zero. The inclusion of a second parameter, the processing delay, in the model leads to a piecewise linear two-dimensional map. The dynamics of this map are studied using numerical simulations which indicate similar behavior as in the one-dimensional case.
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5

Roberts, John A. G., Asaki Saito, and Franco Vivaldi. "Critical curves of a piecewise linear map." Chaos: An Interdisciplinary Journal of Nonlinear Science 31, no. 7 (July 2021): 073134. http://dx.doi.org/10.1063/5.0054334.

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6

Zhusubaliyev, Z. T., D. S. Kuzmina, and O. O. Yanochkina. "Bifurcation Analysis of Piecewise Smooth Bimodal Maps Using Normal Form." Proceedings of the Southwest State University 24, no. 3 (December 6, 2020): 137–51. http://dx.doi.org/10.21869/2223-1560-2020-24-3-137-151.

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Purpose of reseach. Studyof bifurcations in piecewise-smooth bimodal maps using a piecewise-linear continuous map as a normal form. Methods. We propose a technique for determining the parameters of a normal form based on the linearization of a piecewise-smooth map in a neighborhood of a critical fixed point. Results. The stability region of a fixed point is constructed numerically and analytically on the parameter plane. It is shown that this region is limited by two bifurcation curves: the lines of the classical period-doubling bifurcation and the “border collision” bifurcation. It is proposed a method for determining the parameters of a normal form as a function of the parameters of a piecewise smooth map. The analysis of "border-collision" bifurcations using piecewise-linear normal form is carried out. Conclusion. A bifurcation analysis of a piecewise-smooth irreversible bimodal map of the class Z1–Z3–Z1 modeling the dynamics of a pulse–modulated control system is carried out. It is proposed a technique for calculating the parameters of a piecewise linear continuous map used as a normal form. The main bifurcation transitions are calculated when leaving the stability region, both using the initial map and a piecewise linear normal form. The topological equivalence of these maps is numerically proved, indicating the reliability of the results of calculating the parameters of the normal form.
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7

Wang, Bin, Shihua Zhou, Changjun Zhou, and Xuedong Zheng. "A Novel Joint for Image Encryption and Coding Using Piecewise Linear Chaotic Map." Journal of Computational and Theoretical Nanoscience 13, no. 10 (October 1, 2016): 7137–43. http://dx.doi.org/10.1166/jctn.2016.5683.

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Due to the features of chaotic maps, they are widely used into encrypting and coding information. Inspired by the tent map which is used to code and encrypt binary data, a novel joint for image encryption and coding based on piecewise linear chaotic map is proposed in this paper. We divide piecewise linear chaotic map into 256 parts according to the property of gray level image. In order to enhance the security of image, the image is subsequently encrypted by the piecewise linear chaotic map in which the secret key of image encryption is determined by the initial of chaotic map. This stage of image encryption possesses high key and plain-image sensitivities which results from the secret key related to plain-image. Finally, the encrypted image is coded by the piecewise linear chaotic map with a different initial value. The experimental results validate the effect of the proposed system and demonstrate that the encrypted and coded image is secure for transmission.
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8

Avrutin, Viktor, Michael Schanz, and Björn Schenke. "Breaking the continuity of a piecewise linear map." ESAIM: Proceedings 36 (April 2012): 73–105. http://dx.doi.org/10.1051/proc/201236008.

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9

Weimou, Zheng. "Symbolic Dynamics of the Piecewise Linear Standard Map." Communications in Theoretical Physics 29, no. 3 (April 30, 1998): 369–76. http://dx.doi.org/10.1088/0253-6102/29/3/369.

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10

Just, Wolfram. "Analytical Approach for Piecewise Linear Coupled Map Lattices." Journal of Statistical Physics 90, no. 3-4 (February 1998): 727–48. http://dx.doi.org/10.1023/a:1023272819435.

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11

Bullett, Shaun. "Invariant circles for the piecewise linear standard map." Communications in Mathematical Physics 107, no. 2 (June 1986): 241–62. http://dx.doi.org/10.1007/bf01209394.

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12

Elhadj, Zeraoulia, and Julien Clinton Sprott. "A new simple 2-D piecewise linear map." Journal of Systems Science and Complexity 23, no. 2 (April 2010): 379–89. http://dx.doi.org/10.1007/s11424-010-7184-z.

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13

Zhao, Juan, Zheng-Ming Gao, and Yu-Jun Zhang. "Piecewise Linear map enabled Harris Hawk optimization algorithm." Journal of Physics: Conference Series 1994, no. 1 (August 1, 2021): 012038. http://dx.doi.org/10.1088/1742-6596/1994/1/012038.

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14

Jenkinson, O., M. Pollicott, and P. Vytnova. "How Many Inflections are There in the Lyapunov Spectrum?" Communications in Mathematical Physics 386, no. 3 (July 26, 2021): 1383–411. http://dx.doi.org/10.1007/s00220-021-04161-4.

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AbstractIommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.
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15

Aharonov, D., R. L. Devaney, and U. Elias. "The Dynamics of a Piecewise Linear Map and its Smooth Approximation." International Journal of Bifurcation and Chaos 07, no. 02 (February 1997): 351–72. http://dx.doi.org/10.1142/s0218127497000236.

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The paper describes the dynamics of a piecewise linear area preserving map of the plane, F: (x, y) → (1 - y - |x|, x), as well as that portion of the dynamics that persists when the map is approximated by the real analytic map Fε: (x, y) → (1 - y - fε(x), x), where fε(x) is real analytic and close to |x| for small values of ε. Our goal in this paper is to describe in detail the island structure and the chaotic behavior of the piecewise linear map F. Then we will show that these islands do indeed persist and contain infinitely many invariant curves for Fε, provided that ε is small.
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16

Alves, J. F., and J. Sousa Ramos. "One-Dimensional Semiconjugacy Revisited." International Journal of Bifurcation and Chaos 13, no. 07 (July 2003): 1657–63. http://dx.doi.org/10.1142/s0218127403007503.

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Let f be a piecewise monotone interval map with positive topological entropy h(f)= log (s). Milnor and Thurston showed that f is topological semiconjugated to a piecewise linear map having slope s. Here we prove that these semiconjugacies are the eigenvectors of a certain linear endomorphism associated to f. Using this characterization, we prove a conjecture presented by those authors.
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17

Provatas, Nicholas, and Michael C. Mackey. "Noise-induced asymptotic periodicity in a piecewise linear map." Journal of Statistical Physics 63, no. 3-4 (May 1991): 585–612. http://dx.doi.org/10.1007/bf01029201.

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18

Gardini, L., I. Sushko, and K. Matsuyama. "2D discontinuous piecewise linear map: Emergence of fashion cycles." Chaos: An Interdisciplinary Journal of Nonlinear Science 28, no. 5 (May 2018): 055917. http://dx.doi.org/10.1063/1.5018588.

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19

Avrutin, Viktor, Michael Schanz, and Soumitro Banerjee. "Multi-parametric bifurcations in a piecewise–linear discontinuous map." Nonlinearity 19, no. 8 (July 13, 2006): 1875–906. http://dx.doi.org/10.1088/0951-7715/19/8/007.

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20

Sinha, Sitabhra, and Bikas K. Chakrabarti. "Deterministic stochastic resonance in a piecewise linear chaotic map." Physical Review E 58, no. 6 (December 1, 1998): 8009–12. http://dx.doi.org/10.1103/physreve.58.8009.

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21

Hogan, S. J., L. Higham, and T. C. L. Griffin. "Dynamics of a piecewise linear map with a gap." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2077 (July 26, 2006): 49–65. http://dx.doi.org/10.1098/rspa.2006.1735.

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In this paper, we consider periodic solutions of discontinuous non-smooth maps. We show how the fixed points of a general piecewise linear map with a discontinuity (‘a map with a gap’) behave under parameter variation. We show in detail all the possible behaviours of period 1 and period 2 solutions. For positive gaps, we find that period 2 solutions can exist independently of period 1 solutions. Conversely, for negative gaps, period 1 and period 2 solutions can coexist. Higher periodic orbits can also exist and be stable and we give several examples of how these solutions behave under parameter variation. Finally, we compare our results with those of Jain & Banerjee (Jain & Banerjee 2003 Int. J. Bifurcat. Chaos 13 , 3341–3351) and Banerjee et al . (Banerjee et al . 2004 IEEE Trans. Circ. Syst. II 51 , 649–654) and explain their numerical simulations.
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22

Du, Ru-Hai, Sheng-Jun Wang, Tao Jin, and Shi-Xian Qu. "Phase order in one-dimensional piecewise linear discontinuous map." Chinese Physics B 27, no. 10 (October 2018): 100502. http://dx.doi.org/10.1088/1674-1056/27/10/100502.

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23

Kluiving, R., H. W. Capel, and R. A. Pasmanter. "Phase-transition-like phenomenon in a piecewise linear map." Physica A: Statistical Mechanics and its Applications 164, no. 3 (April 1990): 593–624. http://dx.doi.org/10.1016/0378-4371(90)90225-h.

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24

BARANOVSKY, A., and D. DAEMS. "DESIGN OF ONE-DIMENSIONAL CHAOTIC MAPS WITH PRESCRIBED STATISTICAL PROPERTIES." International Journal of Bifurcation and Chaos 05, no. 06 (December 1995): 1585–98. http://dx.doi.org/10.1142/s0218127495001198.

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The statistical properties of a wide class of 1D piecewise linear Markov maps are compiled. The method used enables one to address analytically the inverse problem of designing a map with a prescribed correlation function. This class of piecewise linear maps is then used as a system of reference to analyze non-Markov piecewise linear maps and to design maps with given invariant measure and correlation function.
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25

Avrutin, Viktor, and Zhanybai T. Zhusubaliyev. "Piecewise-Linear Map for Studying Border Collision Phenomena in DC/AC Converters." International Journal of Bifurcation and Chaos 30, no. 07 (June 15, 2020): 2030015. http://dx.doi.org/10.1142/s0218127420300153.

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Recently, while studying the dynamics of power electronic DC/AC converters we have demonstrated that the behavior of these systems can be modeled by piecewise-smooth maps which belong to a specific class of models not investigated before. The characteristic feature of these maps is the presence of a very high number of switching manifolds (border points in 1D). Obviously, the multitude of control strategies applied in the modern power electronics leads to different maps belonging to this class of models. However, in this paper we show that several models can be studied using the same piecewise-linear approximation, so that the bifurcation phenomena which can be observed in this approximation are generic for many models. Based on the results obtained before for piecewise-smooth models with different kinds of nonlinearities resulting from the corresponding control strategies, in the present paper we discuss the generic bifurcation patterns in the underlying piecewise-linear map.
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26

Panchuk, Anastasiia, Iryna Sushko, and Viktor Avrutin. "Bifurcation Structures in a Bimodal Piecewise Linear Map: Chaotic Dynamics." International Journal of Bifurcation and Chaos 25, no. 03 (March 2015): 1530006. http://dx.doi.org/10.1142/s0218127415300062.

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In this work, we investigate the bifurcation structure of the parameter space of a generic 1D continuous piecewise linear bimodal map focusing on the regions associated with chaotic attractors (cyclic chaotic intervals). The boundaries of these regions corresponding to chaotic attractors with different number of intervals are identified. The results are obtained analytically using the skew tent map and the map replacement technique.
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27

Avrutin, Viktor, Iryna Sushko, and Fabio Tramontana. "Bifurcation Structure in a Bimodal Piecewise Linear Business Cycle Model." Abstract and Applied Analysis 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/401319.

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We study the bifurcation structure of the parameter space of a 1D continuous piecewise linear bimodal map which describes dynamics of a business cycle model introduced by Day-Shafer. In particular, we obtain the analytical expression of the boundaries of several periodicity regions associated with attracting cycles of the map (principal cycles and related fin structure). By crossing these boundaries the map displays robust chaos.
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28

PANCHUK, ANASTASIIA, IRYNA SUSHKO, BJÖRN SCHENKE, and VIKTOR AVRUTIN. "BIFURCATION STRUCTURES IN A BIMODAL PIECEWISE LINEAR MAP: REGULAR DYNAMICS." International Journal of Bifurcation and Chaos 23, no. 12 (December 2013): 1330040. http://dx.doi.org/10.1142/s0218127413300401.

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We consider a family of piecewise linear bimodal maps having the outermost slopes positive and less than one. Three types of bifurcation structures incorporated into the parameter space of the map are described. These structures are formed by periodicity regions related to attracting cycles, namely, the skew tent map structure is associated with periodic points on two adjacent branches of the map, the period adding structure is related to periodic points on the outermost branches, and the fin structure is contiguous with the period adding structure and is associated with attracting cycles having at most one point on the middle branch. Analytical expressions for the periodicity region boundaries are obtained using the map replacement technique.
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29

MIRA, CHRISTIAN, CHRISTINE RAUZY, YURI MAISTRENKO, and IRINA SUSHKO. "SOME PROPERTIES OF A TWO-DIMENSIONAL PIECEWISE-LINEAR NONINVERTIBLE MAP." International Journal of Bifurcation and Chaos 06, no. 12a (December 1996): 2299–319. http://dx.doi.org/10.1142/s021812749600148x.

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Properties of a piecewise-linear noninvertible map of the plane are studied by using the method of critical curves (two-dimensional extension of the notion of critical point in the one-dimensional case). This map is of (Z0–Z2) type, i.e. the plane consists of a region without preimage, and a region giving rise to two rank one preimages. For the considered parameter values, the map has two saddle fixed points. The characteristic features of the “mixed chaotic area” generated by this map, and its bifurcations (some of them being of homoclinic and heteroclinic type) are examined. Such an area is bounded by the union of critical curves segments and segments of the unstable set of saddle cycles.
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30

CARMONA, V., E. FREIRE, E. PONCE, and F. TORRES. "BIFURCATION OF INVARIANT CONES IN PIECEWISE LINEAR HOMOGENEOUS SYSTEMS." International Journal of Bifurcation and Chaos 15, no. 08 (August 2005): 2469–84. http://dx.doi.org/10.1142/s0218127405013423.

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Invariant surfaces in three-dimensional continuous piecewise linear homogeneous systems with two pieces separated by a plane are detected. The Poincaré map associated to this plane transforms half-straight lines passing through the origin into half-straight lines of the same type. The invariant half-straight lines under this map determine invariant cones for which the existence, stability and bifurcation are studied. This analysis lead us to consider some questions about the topological type and stability of the origin.
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31

KALLE, CHARLENE. "Isomorphisms between positive and negative -transformations." Ergodic Theory and Dynamical Systems 34, no. 1 (November 9, 2012): 153–70. http://dx.doi.org/10.1017/etds.2012.127.

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AbstractWe compare a piecewise linear map with constant slope $\beta \gt 1$ and a piecewise linear map with constant slope $-\beta $. These maps are called the positive and negative $\beta $-transformations. We show that for a certain set of $\beta $s, the multinacci numbers, there exists a measurable isomorphism between these two maps. We further show that for all other values of $\beta $between 1 and 2 the two maps cannot be isomorphic.
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32

TRAMONTANA, FABIO, LAURA GARDINI, VIKTOR AVRUTIN, and MICHAEL SCHANZ. "PERIOD ADDING IN PIECEWISE LINEAR MAPS WITH TWO DISCONTINUITIES." International Journal of Bifurcation and Chaos 22, no. 03 (March 2012): 1250068. http://dx.doi.org/10.1142/s021812741250068x.

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In this work we consider the border collision bifurcations occurring in a one-dimensional piecewise linear map with two discontinuity points. The map, motivated by an economic application, is written in a generic form and considered in the stable regime, with all slopes between zero and one. We prove that the period adding structures occur in maps with more than one discontinuity points and that the Leonov's method to calculate the bifurcation curves forming these structures is applicable also in this case. We demonstrate the existence of particular codimension-2 bifurcation (big-bang bifurcation) points in the parameter space, from which infinitely many bifurcation curves are issuing associated with cycles involving several partitions. We describe how the bifurcation structure of a map with one discontinuity is modified by the introduction of a second discontinuity point, which causes orbits to appear located on three partitions and organized again in a period-adding structure. We also describe particular codimension-2 bifurcation points which represent limit sets of doubly infinite sequences of bifurcation curves and appear due to the existence of two discontinuities.
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33

Coven, Ethan M., and Melissa C. Hidalgo. "On the topological entropy of transitive maps of the interval." Bulletin of the Australian Mathematical Society 44, no. 2 (October 1991): 207–13. http://dx.doi.org/10.1017/s0004972700029634.

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The topological entropy of a continuous map of the interval is the supremum of the topological entropies of the piecewise linear maps associated to its finite invariant sets. We show that for transitive maps, this supremum is attained at some finite invariant set if and only if the map is piecewise monotone and the set contains the endpoints of the interval and the turning points of the map.
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34

ISLAM, MD SHAFIQUL. "INVARIANT MEASURES FOR RANDOM MAPS VIA INTERPOLATION." International Journal of Bifurcation and Chaos 23, no. 02 (February 2013): 1350025. http://dx.doi.org/10.1142/s0218127413500259.

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Let T = {τ1, τ2, …, τK; p1, p2, …, pK} be a position dependent random map on [0, 1], where {τ1, τ2, …, τK} is a collection of nonsingular maps on [0, 1] into [0, 1] and {p1, p2, …, pK} is a collection of position dependent probabilities on [0, 1]. We assume that the random map T has a unique absolutely continuous invariant measure μ with density f*. Based on interpolation, a piecewise linear approximation method for f* is developed and a proof of convergence of the piecewise linear method is presented. A numerical example for a position dependent random map is presented.
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35

MORI, Makoto. "On the Intermittency of a Piecewise Linear Map (Takahashi Model)." Tokyo Journal of Mathematics 16, no. 2 (December 1993): 411–28. http://dx.doi.org/10.3836/tjm/1270128495.

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36

Qin, Zhiying, Jichen Yang, Soumitro Banerjee, and Guirong Jiang. "Border-collision bifurcations in a generalized piecewise linear-power map." Discrete & Continuous Dynamical Systems - B 16, no. 2 (2011): 547–67. http://dx.doi.org/10.3934/dcdsb.2011.16.547.

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37

Scharf, Rainer, and Bala Sundaram. "Classical and quantal morphology of a piecewise-linear standard map." Physical Review A 43, no. 6 (March 1, 1991): 3183–86. http://dx.doi.org/10.1103/physreva.43.3183.

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38

Glendinning, Paul. "Milnor attractors and topological attractors of a piecewise linear map." Nonlinearity 14, no. 2 (December 19, 2000): 239–57. http://dx.doi.org/10.1088/0951-7715/14/2/304.

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39

Avrutin, Viktor, and Michael Schanz. "On multi-parametric bifurcations in a scalar piecewise-linear map." Nonlinearity 19, no. 3 (January 13, 2006): 531–52. http://dx.doi.org/10.1088/0951-7715/19/3/001.

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40

Kelly, Matthew, Jacobus H. M. van Amerongen, Marleen Balvert, and David Craft. "Dynamic fluence map sequencing using piecewise linear leaf position functions." Biomedical Physics & Engineering Express 5, no. 2 (February 5, 2019): 025036. http://dx.doi.org/10.1088/2057-1976/aaffe7.

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41

Mammeri, M. "A NOVEL PIECEWISE LINEAR VERSION OF THE 3D HÉNON MAP." Far East Journal of Mathematical Sciences (FJMS) 96, no. 7 (April 7, 2015): 843–53. http://dx.doi.org/10.17654/fjmsapr2015_843_853.

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42

Patra, Mahashweta, and Soumitro Banerjee. "Bifurcation of Quasiperiodic Orbit in a 3D Piecewise Linear Map." International Journal of Bifurcation and Chaos 27, no. 10 (September 2017): 1730033. http://dx.doi.org/10.1142/s0218127417300336.

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Earlier investigations have demonstrated how a quasiperiodic orbit in a three-dimensional smooth map can bifurcate into a quasiperiodic orbit with two disjoint loops or into a quasiperiodic orbit of double the length in the shape of a Möbius strip. Using a three-dimensional piecewise smooth (PWS) normal form map, we show that in a piecewise smooth system, in addition to the mechanisms reported earlier, new pathways of creation of tori with multiple loops may result from border collision bifurcations. We also illustrate the occurrence of multiple attractor bifurcations due to the interplay between the stable and the unstable manifolds. Two techniques of analyzing bifurcations of ergodic tori are available in literature: the second Poincaré section method and the Lyapunov bundle method. We have shown that these methods can explain the period-doubling and double covering bifurcations in PWS systems, but fail in some cases — especially those which result from nonsmoothness of the system. We have shown that torus bifurcations due to border collision can be explained by change in eigenvalues of the unstable fixed points.
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43

Hu, Yuping, Congxu Zhu, and Zhijian Wang. "An Improved Piecewise Linear Chaotic Map Based Image Encryption Algorithm." Scientific World Journal 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/275818.

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An image encryption algorithm based on improved piecewise linear chaotic map (MPWLCM) model was proposed. The algorithm uses the MPWLCM to permute and diffuse plain image simultaneously. Due to the sensitivity to initial key values, system parameters, and ergodicity in chaotic system, two pseudorandom sequences are designed and used in the processes of permutation and diffusion. The order of processing pixels is not in accordance with the index of pixels, but it is from beginning or end alternately. The cipher feedback was introduced in diffusion process. Test results and security analysis show that not only the scheme can achieve good encryption results but also its key space is large enough to resist against brute attack.
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44

Wang, Xizhong, and Deyun Chen. "A Parallel Encryption Algorithm Based on Piecewise Linear Chaotic Map." Mathematical Problems in Engineering 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/537934.

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We introduce a parallel chaos-based encryption algorithm for taking advantage of multicore processors. The chaotic cryptosystem is generated by the piecewise linear chaotic map (PWLCM). The parallel algorithm is designed with a master/slave communication model with the Message Passing Interface (MPI). The algorithm is suitable not only for multicore processors but also for the single-processor architecture. The experimental results show that the chaos-based cryptosystem possesses good statistical properties. The parallel algorithm provides much better performance than the serial ones and would be useful to apply in encryption/decryption file with large size or multimedia.
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45

Ding, E. J., and P. C. Hemmer. "Exact treatment of mode locking for a piecewise linear map." Journal of Statistical Physics 46, no. 1-2 (January 1987): 99–110. http://dx.doi.org/10.1007/bf01010333.

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46

Bououden, R., M. S. Abdelouahab, F. Jarad, and Z. Hammouch. "A novel fractional piecewise linear map: regular and chaotic dynamics." International Journal of General Systems 50, no. 5 (May 10, 2021): 501–26. http://dx.doi.org/10.1080/03081079.2021.1919102.

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47

Tikjha, Wirot, and Laura Gardini. "Bifurcation Sequences and Multistability in a Two-Dimensional Piecewise Linear Map." International Journal of Bifurcation and Chaos 30, no. 06 (May 2020): 2030014. http://dx.doi.org/10.1142/s0218127420300141.

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Bifurcation mechanisms in piecewise linear or piecewise smooth maps are quite different with respect to those occurring in smooth maps, due to the role played by the borders. In this work, we describe bifurcation mechanisms associated with the appearance/disappearance of cycles, which may be related to several cases: (A) fold border collision bifurcations, (B) degenerate flip bifurcations, supercritical and subcritical, (C) degenerate transcritical bifurcations and (D) supercritical center bifurcations. Each of these is characterized by a particular dynamic behavior, and may be related to attracting or repelling cycles. We consider different bifurcation routes, showing the interplay between all these kinds of bifurcations, and their role in the phase plane in determining attracting sets and basins of attraction.
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48

BARANOVSKI, ALEXANDER L., and ANTHONY J. LAWRANCE. "SENSITIVE PARAMETER DEPENDENCE OF AUTOCORRELATION FUNCTION IN PIECEWISE LINEAR MAPS." International Journal of Bifurcation and Chaos 17, no. 04 (April 2007): 1185–97. http://dx.doi.org/10.1142/s0218127407017768.

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This paper is principally concerned with the effect of splitting and permuting, or shifting, the branches of an onto piecewise linear map, with particular regard to the effect on the autocorrelation of the associated chaotic sequence. This is shown to be a chaotic function of the shifting parameter of the map, and its sensitivity with respect to minute changes in this parameter is termed autocorrelation chaos. This paper presents both analytical and computational studies of the phenomenon.
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49

Ganatsiou, C. "On a Gauss-Kuzmin type problem for piecewise fractional linear maps with explicit invariant measure." International Journal of Mathematics and Mathematical Sciences 24, no. 11 (2000): 753–63. http://dx.doi.org/10.1155/s0161171200003872.

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A random system with complete connections associated with a piecewise fractional linear map with explicit invariant measure is defined and its ergodic behaviour is investigated. This allows us to obtain a variant of Gauss-Kuzmin type problem for the above linear map.
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50

LLIBRE, JAUME, and ANTONIO E. TERUEL. "EXISTENCE OF POINCARÉ MAPS IN PIECEWISE LINEAR DIFFERENTIAL SYSTEMS IN ℝN." International Journal of Bifurcation and Chaos 14, no. 08 (August 2004): 2843–51. http://dx.doi.org/10.1142/s0218127404010874.

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In this paper we present a relationship between the algebraic notion of proper system, the geometric notion of contact point and the dynamic notion of Poincaré map for piecewise linear differential systems. This allows to present sufficient conditions (which are also necessary under additional hypotheses) for the existence of Poincaré maps in piecewise linear differential systems. Moreover, an adequate parametrization of the Poincaré maps make such maps invariant under linear transformations.
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