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[en] In this paper, we present a new application of higher order compact finite differences to solve nonlinear initial value problems exhibiting chaotic behaviour. The method involves dividing the domain of the problem into multiple sub-domains, with each sub-domain integrated using higher order compact finite difference schemes. The nonlinearity is dealt using a Gauss–Seidel-like relaxation. The method is, therefore, referred to as the multi-domain compact finite difference relaxation method (MD-CFDRM). In this new application, the MD-CFDRM is used to solve famous chaotic systems and hyperchaotic systems. The main advantage of the new approach is that it offers better accuracy on coarser grids which significantly improves the computational speed of the method. The results are compared with spectral-based multi-domain method.
[en] We have considered a permutation entropy method for analyzing chaotic, noisy, and chaotic noisy series. We have introduced the concept of permutation entropy from a survey of some features of information entropy (Shannon entropy), described the algorithm for its calculation, and indicated the advantages of this approach in the analysis of time series; the application of this method in the analysis of various model systems and experimental data has also been demonstrated.
[en] We define the generalized Bessel–Struve operator in the space of functions analytic in an arbitrary domain. The conditions of equivalence of the generalized Bessel–Struve operator to the operator of second derivative are investigated. We also describe the commutant of the generalized Bessel–Struve operator and establish its hypercyclicity and chaotic nature.
[en] Substitution box is a vital and the only nonlinear component of modern encryption algorithm. S-box is introduced as a confusion component to resist against differential cryptanalysis. Chaos-based encryption is well liked because it exhibits similarity like cryptography. However, chaotic S-boxes possess high maximum differential approximation probability, measured using difference distribution table (DDT) for differential cryptanalysis. Therefore, this paper reports a systematic design methodology to generate chaotic S-box utilizing DDT and that can be used in multimedia encryption algorithms. DDT within the design loop is used to optimize differential approximation probability. The proposed S-box shows very low differential approximation probability as compared to other chaos-based S-box designed recently, while maintaining good cryptographic properties and high value of linear approximation probability. The strength of the proposed cryptographically strong S-box is vetted in the practical implementation of multimedia encryption.
[en] The problem of reconstruction of dynamic systems in the presence of noise using series of interburst intervals is solved. It is shown that the reconstruction procedure can be applied to strongly nonlinear noisy oscillatory processes. The results make it possible to generalize the method for analysis of dynamic systems with respect to recovery time to a wide variety of neuron oscillators.
[en] Reduced-order observers are designed for a class of Lipschitz semilinear descriptor systems. Sufficient conditions for the existence of an observer are characterized in terms of the rank of system operators and solvability of one linear matrix inequality. In application part, the paper considers the issues of secure communication via chaotic systems subject to unknown parameters. Simulations are done on a Lorenz chaotic system to verify the effectiveness of the main result.
[en] Due to the widespread usage of the internet and other wired and wireless communication methods, the security of the transmitted data has become a major requirement. Nuclear knowledge is mainly built upon the exchange of nuclear information which is considered highly sensitive information, so its security has to be enhanced by using high level security mechanisms. Data confidentiality is concerned with the achievement of higher protection For confidential information from unauthorized disclosure or access. Cryptography and steganography are famous and widely used techniques that process information in order to achieve its confidentiality, but sometimes, when used individually, they don’t satisfy a required level of security for highly sensitive data. In this paper, cryptography is accompanied with steganography for constituting a multilayer security techniques that can strengthen the level of security of highly confidential nuclear data that are archived or transmitted through different channel types and noise conditions. (author)
[en] Copula modeling consists in finding a probabilistic distribution, called copula, whereby its coupling with the marginal distributions of a set of random variables produces their joint distribution. The present work aims to use this technique to connect the statistical distributions of weakly chaotic dynamics and deterministic subdiffusion. More precisely, we decompose the jumps distribution of Geisel–Thomae map into a bivariate one and determine the marginal and copula distributions respectively by infinite ergodic theory and statistical inference techniques. We verify therefore that the characteristic tail distribution of subdiffusion is an extreme value copula coupling Mittag–Leffler distributions. We also present a method to calculate the exact copula and joint distributions in the case where weakly chaotic dynamics and deterministic subdiffusion statistical distributions are already known. Numerical simulations and consistency with the dynamical aspects of the map support our results.
[en] We propose a multicircuit oscillator with a common control scheme driving self-sustained oscillations in each circuit, which can exhibit quasi-periodic and chaotic oscillations. The proposed oscillator was investigated by experimental and numerical methods that confirmed the possibility of exciting multifrequency quasi-periodic, chaotic, and hyperchaotic oscillations with several positive Lyapunov exponents.
[en] It is well known that simple and complex dynamics of a nonlinear system are separated by a localizing set that contains all compact invariant sets and corresponds to a function in the phase space of the system. This separation means that, in the complement of the localizing set, the trajectory behavior of the system admits a standard description in the form of several variants, while, in the localizing set, the trajectory behavior of the system can be very complex, for example, chaotic. Domains in the localizing set with a similar standard behavior are indicated, and an external estimate of the domain with complex dynamics is sequentially refined. Two examples are considered.