Yet Another Chaotic Attractor

Chen, Guanrong; Ueta, Tetsushi

doi:10.1142/s0218127499001024

Cited by 2,490 publications

(951 citation statements)

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“…Here, as an example, we state our results for the polynomial differential system in R 3 ,ẋ = a 1 x + a 2 y,ẏ = a 3 x + a 4 y + a 5 xz,ż = a 6 xy + a 7 z, see Example 3.5 in Section 3. It includes the Lorenz system [26], the Lü system [27] and the Chen system [6], and its Darboux integrability has been studied in [25,23,28], respectively. The differential system studied in [39] also belongs to this family.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Seeking Darboux Polynomials

Ferragut

Gasull

2014

Acta Appl Math

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We introduce several techniques which allow to simplify the expression of the cofactor of Darboux polynomials of polynomial differential systems in R n . We apply these techniques to some well-known systems when n = 2, 3, 4. We also propose a general method for computing Darboux polynomials in the plane. As an application we prove that a family of potential systems, that includes the van der Pol one, has no Darboux polynomials, giving in particular a new simple proof that the van der Pol limit cycle is not algebraic.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Seeking Darboux Polynomials

Ferragut

Gasull

2014

Acta Appl Math

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“…is generated by iterating Lorenz [75] and Chen [35] Systems and doing some postprocessing on all the variables of state. When a variation of stream cipher is created, as in the case under study, obtaining the keystream is totally equivalent to obtaining the key whenever different plain-images are encrypted using the same key.…”

Section: • Masking Proceduresmentioning

confidence: 99%

“…This cryptosystem is intended to encrypt images through the concatenation of a shuffling and masking procedures. The shuffling procedure is based on the iteration of the logistic map, whereas masking is built upon the iteration of two continuous-time chaotic systems: Lorenz [75] and Chen Systems [35]. The Lorenz System is given by…”

Section: Case Study 826 ([28])mentioning

confidence: 99%

Lessons Learnt from the Cryptanalysis of Chaos-Based Ciphers

Álvarez

Amigó

Arroyo

et al. 2011

Studies in Computational Intelligence

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The idea of using chaotic transformations in cryptography is explicit in the foundational papers of Shannon on secrecy systems (e.g., [96]). Although the word "chaos" was not minted till the 1970s [71], Shannon clearly refers to this very concept when he proposes the construction of secure ciphers by means of measure-preserving, mixing maps which depend 'sensitively' on their parameters. The implementation of Shannon's intuitions had to wait till the development of Chaos Theory in the 1980s. Indeed, it was around 1990 when the first chaos-based ciphers were proposed (e.g., [78], [46]). Moreover, in 1990 chaos synchronization [91] entered the scene and shortly thereafter, the first applications to secure communications followed [56,37]. The idea is remarkably simple: mask the message with a chaotic signal and use synchronization at the receiver to filter out the chaotic signal. The realization though had to overcome the desynchronization induced by the message itself. After this initial stage, the number of proposals which exploited the properties of chaotic maps for cryptographical purposes, grew in a spectacular way.

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“…Since a description of the general case is rather messy and uneasy, at least notationally, to facilitate the description and discussion, we use a specific chaotic system as an example. This system is still not too special, in the sense that it unifies both the Lorenz [2] and the Chen systems [9,10], where the latter is the dual system of the former in a sense defined in [11]: The Lorenz system satisfies the condition a 12 a 21 > 0 while the Chen system satisfies a 12 a 21 < 0, where A ¼ ½a ij 3Â3 are the matrix of the linear part of the chaotic systems. Recently, L€ u u et al [12][13][14] found a unified chaotic system that not only includes the case of a 12 a 21 ¼ 0 [12] but also the Lorenz and Chen systems as two extreme cases.…”

Section: The Problem Formulationmentioning

confidence: 99%