Integer–Fractional Order of Identical Eigenvalues Chaotic Oscillator: Analysis and Shadow Economy Application

The rich dynamical analysis and predefined-time synchronization of simple memristive chaotic systems are of great significance in fully understanding the mechanism of dynamics formation and expanding the potential applications of chaotic systems. A four-dimensional memristive chaotic system with only a single nonlinear term is proposed to reveal various dynamic behaviors under the change of parameters and initial conditions, and to realize effective synchronization control. Based on dissipativity analysis and Lyapunov exponent computation, and combined with bifurcation analysis and multi steady state exploration, it is shown that the system possesses an infinite number of unstable equilibrium points and exhibits homogeneous and heterogeneous multistability, including point attractors, periodic attractors, and chaotic attractors. Moreover, it is found that amplitude modulation of the output signals of the system can be precisely achieved by adjusting internal parameters of the memristor, thus providing a theoretical basis for achieving effective amplitude modulation of periodic and chaotic signals. A predefined-time sliding mode surface with linear and bidirectional power-law nonlinear decay terms is constructed to address synchronization. Sufficient conditions for predefined-time convergence of synchronization errors are derived using Lyapunov stability theory, and a double-stage sliding mode controller with an adjustable upper bound on synchronization time is designed. The resulting control law ensures an adjustable synchronization time bound and rapid error suppression under arbitrary disturbances. Numerical simulations further confirm the effectiveness and robustness of the proposed control scheme, indicating that even under external disturbances or significant variations in initial conditions, the error variables can still converge precisely within the predefined time.

Adaptive Synchronization of Memristor - based Chaotic Neural Systems

To explore the impact of unknown terms and parameters on chaotic characteristics in chaotic systems, this paper examines the effects of state variables and unknown parameters. The study focuses on different combinations of linear, nonlinear, and constant terms It primarily investigates the role of multi-order state variables and their application to chaotic system models of varying dimensions. Firstly, by simulating a three-dimensional chaotic system, the paper analyzes how different combinations of nonlinear terms and initial conditions affect the system's chaotic behavior. Secondly, it evaluates the chaotic characteristics of a four-dimensional system, combining nonlinear terms with unknown parameters, using tools such as Lyapunov index diagrams, sample entropy, and dynamic trajectory plots. Finally, the paper integrates the constructed chaotic system with chaotic mapping to develop a two-level key chaotic image encryption system, thoroughly assessing its security and resistance to interference.

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In this work, we report a new chaotic population biology system with one prey and two predators. Our new chaotic population model is derived by introducing two nonlinear interaction terms between the prey and predator-2 to the Samardzija-Greller population biology system (1988).We show that the new chaotic population biology system has a greater value of Maximal Lyapunov Exponent (MLE) than the Maximal Lyapunov Exponent (MLE) of the Samardzija- Greller population biology system (1988).We carry out a detailed bifurcation analysis of the new chaotic population biology system with one prey and two predators. We also show that the new chaotic population biology model exhibits multistability with coexisting chaotic attractors. Next, we use the integral sliding mode control (ISMC) for the complete synchronization of the new chaotic population biology system with itself, taken as the master and slave chaotic population biology systems. Finally, for practical use of the new chaotic population biology system, we design an electronic circuit design using Multisim (Version 14.0).

This research aims to investigate the influence of model parameters and fractional order on a novel mathematical model with tangent hyperbolic memristor. This investigation conducted by applying Lyapunov exponents and bifurcation analysis. We utilize the Lyapunov exponent theory to understand and characterize these chaotic behaviors under fractional indices. The Lyapunov exponent, bifurcation, and phase diagrams have been depicted to explore the intricate dynamics of the chaotic system governed by the chaotic equation. A novel approach termed Atangana-Baleanu-Caputo (ABC) fractional derivative (FD) to generate phase portraits and gain insights into the system’s behavior. The random numbers generated by the chaotic system are employed to distort the image through an amalgamated image encryption (AIE) algorithm. Subsequently, the integrity of the scrambled image has been assessed using various image security evaluation methods to reinforce the notion that combining the chaotic system and image can constitute a valuable encryption key. Finally, the chaotic model circuit realization uses active and passive components, and the outcomes are compared with the numerical simulations.

Large Amplitude Oscillatory Shear Rheology of Living Fibroblasts: Path-Dependent Steady States

We present and discuss philosophy and methodology of chaotic evolution that is theoretically supported by chaos theory. We introduce four chaotic systems, that is, logistic map, tent map, Gaussian map, and Hénon map, in a well-designed chaotic evolution algorithm framework to implement several chaotic evolution (CE) algorithms. By comparing our previous proposed CE algorithm with logistic map and two canonical differential evolution (DE) algorithms, we analyse and discuss optimization performance of CE algorithm. An investigation on the relationship between optimization capability of CE algorithm and distribution characteristic of chaotic system is conducted and analysed. From evaluation result, we find that distribution of chaotic system is an essential factor to influence optimization performance of CE algorithm. We propose a new interactive EC (IEC) algorithm, interactive chaotic evolution (ICE) that replaces fitness function with a real human in CE algorithm framework. There is a paired comparison-based mechanism behind CE search scheme in nature. A simulation experimental evaluation is conducted with a pseudo-IEC user to evaluate our proposed ICE algorithm. The evaluation result indicates that ICE algorithm can obtain a significant better performance than or the same performance as interactive DE. Some open topics on CE, ICE, fusion of these optimization techniques, algorithmic notation, and others are presented and discussed.

This chapter deals with the statistical description of chaotic signals and systems. Although chaotic systems are purely deterministic they can be modelled, analysed and designed by using probability measures and statistical characteristics, such as probability density functions and correlation functions, widely used in signal theory and in engineering applications. The chapter describes the problem of statistical analysis, the performance evaluation of chaotic signal processing schemes. This requires the introduction of an extended calculus because random processes interact with chaotic signals. The chapter examines the solution of inverse problem, the synthesis and design of chaotic systems from prescribed statistical characteristics of signals to be generated. Chaos communication schemes are analysed and the results are presented in terms of performance criteria commonly used in communication engineering. The fundamentals of statistical analysis of chaotic systems are explained systematically. The theory is mainly developed for multidimensional systems and illustrated with examples for one-dimensional case. The chapter presents and develops tools for the statistical analysis.

A new approach to eliminating of chaotic ferroresonant oscillations in power transformer

4D chaotic system-based secure data hiding method to improve robustness and embedding capacity of videos

We propose a novel population-based optimization algorithm, Chaotic Evolution (CE), which uses ergodic property of chaos to implement exploration and exploitation functions of an evolutionary algorithm. CE introduces a mathematical mechanism into an iterative process of evolution and simulates ergodic motion in a search space with a simple principle. A control parameter, direction factor rate, is proposed to guide search direction in CE. It is easy to extend its search capability by using different chaotic system in CE algorithm framework. The scalability of CE is higher than that of some other evolutionary computation algorithms. A series of comparative evaluations and investigations is conducted to analyse characteristics of the proposal. Our proposal can obtain better optimization performance by comparing with differential evolution and some of its variants. We point out that the chaos theory is used not only to describe and explain a non-linear system, but also to implement a variety of optimization algorithms based on its ergodic property.

Elimination of chaotic ferroresonance in power transformer by ISFCL

In this study, the K feedback gain vector parameters that are used for the control of three degree of freedom four rotor quadcopter system (3 DOF Hover) are optimized with the Enhanced Equilibrium Optimization Algorithm (E2O). The E2O algorithm is proposed with using parameters obtained from fractional order chaotic oscillator models instead of random variables. The Basic EO algorithm is inspired by volume-mass balance. In EO algorithm, each particle is called a motion that searches a parameter vector space. However, random coefficients derived from uniform distribution are used in the parameters updating process or in the generation of the initial population. The E2O algorithm was proposed by using vectors obtained from fractional order chaotic oscillators instead of stochastic coefficients in the basic Equilibrium optimization algorithm. Genesio Tesi, Rössler, Lotka Volterra fractional-order chaotic oscillator models were used in the E2O algorithm to optimize K feedback gain vector of 3 DOF Hover. The order and initial conditions the fractional chaotic oscillator models were experimentally adjusted for the control of 3 DOF problem. Thus, suitable fractional-order chaotic models for the problem were obtained. The E2O algorithm results are compared with the Stochastic Multi Parameter Optimization (SMDO) and Discreet Stochastic Optimization (DSO) algorithms for the system’s pitch, roll and yaw angles.

A solution to the problem of control of nonlinear chaotic dynamical systems, is proposed with the use of differential flatness theory and of adaptive fuzzy control theory. Considering that the dynamical model of chaotic systems is unknown, an adaptive fuzzy controller is designed. By applying differential flatness theory the chaotic system's model is written in a linear form, and the resulting control inputs are shown to contain nonlinear elements which depend on the system's parameters. The nonlinear terms which appear in the control inputs of the transformed dynamical model are approximated with the use of neuro-fuzzy networks. It is proven that a suitable learning law can be defined for the aforementioned neuro-fuzzy approximators so as to preserve the closed-loop system stability. Moreover, with the use of Lyapunov stability analysis it is proven that the proposed adaptive fuzzy control scheme results in H ∞ tracking performance, which means that the influence of the modeling errors and the external disturbances on the tracking error is attenuated to an arbitrary desirable level. Simulation experiments confirm the efficiency of the proposed adaptive fuzzy control method, using as a case study the model of the Lorenz chaotic oscillator.