Abstract Hyperbolic Chaos

The abstract hyperbolic sets are introduced. Continuous and differentiable mappings as well as rate of convergence and transversal manifolds are not under discussion, and the symbolic dynamics paradigm is realized in a new way. Our suggestions are for more neat comprehension of chaos in the domain. The novelties can serve for revisited models as well as motivate new ones.
Discontinuity, Nonlinearity, and Complexity


Chaos generation in hyperbolic systems
Akhmet, Marat; FEN, MEHMET ONUR (2012-01-01)
© 2012 L & H Scientific Publishing, LLC.In the present paper, we consider extension of chaos in hyperbolic systems with arbitrary large dimensions. Our investigations comprise chaos in the sense of both Devaney and Li-Yorke. We provide a mechanism for unidirectionally coupled systems through the insertion of chaos from one system to another, where the latter is initially nonchaotic. In our procedure for the chaos extension, we take advantage of chaotic sets of functions to provide mathematically approved re...
Multiple linear regression model with stochastic design variables
İslam, Muhammed Qamarul (Informa UK Limited, 2010-01-01)
In a simple multiple linear regression model, the design variables have traditionally been assumed to be non-stochastic. In numerous real-life situations, however, they are stochastic and non-normal. Estimators of parameters applicable to such situations are developed. It is shown that these estimators are efficient and robust. A real-life example is given.
Almost periodicity in chaos
Akhmet, Marat (2018-01-01)
Periodicity plays a significant role in the chaos theory from the beginning since the skeleton of chaos can consist of infinitely many unstable periodic motions. This is true for chaos in the sense of Devaney [1], Li-Yorke [2] and the one obtained through period-doubling cascade [3]. Countable number of periodic orbits exist in any neighborhood of a structurally stable Poincaré homoclinic orbit, which can be considered as a criterion for the presence of complex dynamics [4]-[6]. It was certified by Shilniko...
Modular Chaos for Random Processes
Akhmet, Marat (2022-1-01)
In the present paper, an essential generalization of the symbolic dynamics is considered. We apply the notions of abstract self-similar sets and the similarity map for a chaos introduction, which orbits are expanded among infinitely many modules. The dynamics is free of dimensional, metrical and topological assumptions. It unites all the three types of Poincar´e, Li-Yorke and Devaney chaos in a single model, which can be unbounded. The research demonstrates that the dynamics of Poincar´e chaos is of excepti...
Abstract similarity, fractals and chaos
Akhmet, Marat (2021-05-01)
A new mathematical concept of abstract similarity is introduced and is illustrated in the space of infinite strings on a finite number of symbols. The problem of chaos presence for the Sierpinski fractals, Koch curve, as well as Cantor set is solved by considering a natural similarity map. This is accomplished for Poincare, Li-Yorke and Devaney chaos, including multi-dimensional cases. Original numerical simulations illustrating the results are presented.
Citation Formats
M. Akhmet, “Abstract Hyperbolic Chaos,” Discontinuity, Nonlinearity, and Complexity, vol. 11, no. 1, pp. 133–138, 2022, Accessed: 00, 2022. [Online]. Available: https://hdl.handle.net/11511/97714.