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Abstract

Dynamics and control of discrete chaotic systems of fractional-order have received considerable attention over the last few years. So far, nonlinear control laws have been mainly used for stabilizing at zero the chaotic dynamics of fractional maps. This article provides a further contribution to such research field by presenting simple linear control laws for stabilizing three fractional chaotic maps in regard to their dynamics. Specifically, a one-dimensional linear control law and a scalar control law are proposed for stabilizing at the origin the chaotic dynamics of the Zeraoulia-Sprott rational map and the Ikeda map, respectively. Additionally, a two-dimensional linear control law is developed to stabilize the chaotic fractional flow map. All the results have been achieved by exploiting new theorems based on the Lyapunov method as well as on the properties of the Caputo h-difference operator. The relevant simulation findings are implemented to confirm the validity of the established linear control scheme.
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Bibliography

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[20] A-A. Khennaoui, A. Ouannas, S. Bendoukha, G. Grassi, X. Wang, V-T. Pham, and F.E. Alsaadi: Chaos, control, and synchronization in some fractional-order difference equations. Advances in Difference Equations, 412 (2019), DOI: 10.1186/s13662-019-2343-6.
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Authors and Affiliations

A. Othman Almatroud
1
Adel Ouannas
2
Giuseppe Grassi
3
Iqbal M. Batiha
4
Ahlem Gasri
5
M. Mossa Al-Sawalha
1

  1. Department of Mathematics, Faculty of Science, University of Ha'il, Ha'il 81451, Saudi Arabia
  2. Department of Mathematics and Computer Science, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, Algeria
  3. Dipartimento Ingegneria Innovazione, Universita del Salento, 73100 Lecce, Italy
  4. Department of Mathematics, Faculty of Science and Information Technology, Irbid National University, Irbid, Jordan and Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE
  5. Department of Mathematics, University of Larbi Tebessi, Tebessa 12002, Algeria
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Abstract

The discussion in this article is based on the assumption that the sociocultural dynamics of quixotism is common to many cultures and, as a consequence, each of them should produce its own version of the emblematic Don Quixote. The formula of this concept of quixotism comes from Magdalena Barbaruk’s studies in the field of theory and cultural practice, in which she probes into vast stretches of history, including the centuries after the publication of Cervantes’ novel as well as the epochs that preceded it. Accordingly, the circle of Quixote-like figures should include Ignatius Loyola, Saint James, Christopher Columbus, the Polish Romantic poet Juliusz Słowacki and Prince Myshkin from Fyodor Dostoevsky’s novel The Idiot. The principal criterion for inclusion in this category is “to be a reader who walked out of the library so as to act in accordance with the books” (Magdalena Bodnaruk, ‘Don Kichote w naukach o kulturze’ [Don Quixote in Cultural Studies], in: Wieczna krucjata. Szkice o Don Kichocie [The Eternal Wandering: Essays on Don Quixote], Poznań 2016, p. 164). Taking that step results, as a matter of necessity, in a clash with the generally accepted rules and conventions. Moreover, while doing so the quixotic individual has to face the risk of having his heroism held up to ridicule or dismissed as folly.
This article puts up some additional candidates to Barbaruk’s short list of ‘Quixotes’ and considers the way in which their distinctive qualities may modify her quixotic formula. The first is the protagonist of the 1955 stage/screen adaptation of Cervantes’ novel by the Soviet Russian author Evgeny Schwartz. His Quixote is a knight errant who knows all too well that he defies people’s routines and expectations and yet remains true to himself and his ideals. He is aware that to ‘save the world’ he has to live and act in the boundary area between the profanum and the sacred, or the real world and a kind of fairyland. Therefore, what marks the timeless Quixote is the deliberate overstepping of a role sanctioned by the ruling consensus, and making a stand against the powers that be. The Middle Ages certainly produced many figures cast in that mould, among them Saint Gerald of Aurillac (whose Vita was written by Odo of Cluny). If a sharp, uncompromising view of reality is a distinct character trait of a quixotic personality, another figure that need to be added to the short list of is Buono, the good alter ego of Viscount Medardo, the protagonist of Italo Calvino’s novel The Cloven Viscount (1952). Finally, the article argues that a character who sets off on a journey (quest) which gives him the opportunity to perform noble (chivalric) deeds represents another version of a Quixotic knight errant. The case in point is Tristran from Neil Gaiman’s fairy tale fantasy Stardust.
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Authors and Affiliations

Elwira Buszewicz
1
ORCID: ORCID

  1. Wydział Polonistyki UJ

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