Heart rate variability analysis by chaotic global techniques in children with attention deficit hyperactivity disorder

Attention deficit hyperactivity disorder (ADHD) is characterized by decreased attention span, impulsiveness, and hyperactivity. Autonomic nervous system imbalance was previously described in this population. We aim to compare the autonomic function of children with ADHD and controls by analyzing heart rate variability (HRV). Children with ADHD (22 boys, mean age 9.964 years) and 28 controls (15 boys, mean age 9.857 years) rested in supine position with spontaneous breathing for 20 min. Heart rate was recorded beat by beat. HRV analysis was performed by use of chaotic global techniques. ADHD promoted an increase in the chaotic forward parameter. The algorithm which applied all three chaotic global parameters was only the second optimum statistically measured by Kruskal–Wallis (P < 0.0001) and low standard deviations. It was also highly influential by principal component analysis with almost all variation covered by the first two components. The third algorithm which lacked the (high spectral Detrended Fluctuation Analysis) parameter performed best statistically. However, we chose the algorithm which applied all three chaotic globals due to previous studies mentioned in the text—forward and inverse problems. Comparison of the autonomic function by analyzing HRV with chaotic global techniques suggests an increase in chaotic activity in children with ADHD in relation to the control group. © 2015 Wiley Periodicals, Inc. Complexity 21: 412–419, 2016     

Evaluating nonlinear variability of mental fatigue behavioral indices during long‐term attentive task

This study investigates the behavioral indices of attention. A simple repetitive attentive task that resulted in mental fatigue was used consecutively in four trials. In the first step, reaction time and error responses were recorded to evaluate differences among trials. During the task, subjects showed different responses to stimulations. In the second part, to recognize the strategies, multiple clustering methods such as k‐means and fuzzy c‐means were performed in which behavioral indices and nonlinear features were used. In the last section, mental behavior was identified as a result of the chaotic properties of variations in reaction time. Therefore, the Lyapunov exponent of reaction times was evaluated. Results revealed that behavioral indices could distinguish attention from the occurrence of mental fatigue in trials. In addition, the three strategies used by subjects during the test protocol were assessed. Finally, variation of indices extracted from nonlinear analysis, that is, decrease in degree of chaotic behavior determined the transition from attention to mental fatigue. © 2012 Wiley Periodicals, Inc. Complexity, 2012     

Relationships among some chaotic properties of non-autonomous discrete dynamical systems

This paper is concerned with relationships among some chaotic properties of non-autonomous discrete dynamical systems. Some relationships among weak mixing, topologically weak mixing, generic chaos, dense chaos, and sensitivity are investigated. In addition, some equivalent conditions of sensitivity are given and the relationships between sensitivity and Li–Yorke sensitivity are obtained. These results generalize some existing results of autonomous discrete systems, some of which relax the corresponding conditions.     

Special dynamic behaviors of a temporal chaotic system

When dynamic behaviors of temporal chaotic system are analyzed,we find that a temporal chaotic system has not only genetic dynamic behaviors of chaotic reflection,but also has phenomena influencing two chaotic attractors by original values.Along with the system parameters changing to certain value,the system will appear a break in chaotic region,and jump to another orbit of attractors.When it is opposite that the system parameters change direction,the temporal chaotic system appears complicated chaotic behaviors.     

Stabilization of parameters perturbation chaotic system via adaptive backstepping technique

The work of Yassen [M.T. Yassen, Chaos control of chaotic dynamical systems using backstepping design, Chaos Soliton Fract. 27 (2006) 537–548] which mainly investigated the stabilization problem for a class of chaotic systems without the parameters perturbation. This paper is concerned with stabilization problem for a class of parameters perturbation chaotic systems via both backstepping design method and adaptive technique. The proposed controllers can guarantee that the parameters perturbation systems will be stabilized at a fixed bounded point. Furthermore, the paper also proposes controllers to stabilize the uncertain chaotic system at equilibrium point with only backstepping design method. Finally, numerical simulations are given to illustrate the effectiveness of the proposed controllers.     

On chaotic behaviour of the p-adic generalized Ising mapping and its application

In the present paper, by conducting research on the dynamics of the p-adic generalized Ising mapping corresponding to renormalization group associated with the p-adic Ising-Vannemenus model on a Cayley tree, we have determined the existence of the fixed points of a given function. Simultaneously, the attractors of the dynamical system have been found. We have come to a conclusion that the considered mapping is topologically conjugate to the symbolic shift which implies its chaoticity and as an application, we have established the existence of periodic p-adic Gibbs measures for the p-adic Ising-Vannemenus model.     

Projective synchronization of time‒delayed chaotic systems with unknown parameters using adaptive control method

The present article aims to study the projective synchronization between two identical and non?identical time?delayed chaotic systems with fully unknown parameters. Here the asymptotical and global synchronization are achieved by means of adaptive control approach based on Lyapunov–Krasovskii functional theory. The proposed technique is successfully applied to investigate the projective synchronization for the pairs of time?delayed chaotic systems amongst advanced Lorenz system as drive system with multiple delay Rössler system and time?delayed Chua's oscillator as response system. An adaptive controller and parameter update laws for unknown parameters are designed so that the drive system is controlled to be the response system. Numerical simulation results, depicted graphically, are carried out using Runge–Kutta Method for delay?differential equations, showing that the design of controller and the adaptive parameter laws are very effective and reliable and can be applied for synchronization of time?delayed chaotic systems. Copyright © 2014 John Wiley & Sons, Ltd.     

三个耦合的非扩散Lorenz系统的全局混沌同步

以Lyapunov稳定性理论和矩阵论为基础,针对非扩散Lorenz系统,提出了一种三个耦合的恒等系统的全局混沌同步方案.该方案的关键是耦合参数的选取.通过选择适当的耦合参数,使得三个系统的状态变量达到同步,并利用Mathematic软件进行数值仿真.理论分析和仿真结果都表明了该方法的有效性.     

A new adaptive observer design for a class of nonautonomous complex chaotic systems

This article is concerned with designing of a robust adaptive observer for a class of nonautonomous chaotic system with unknown parameters having unknown bounds. The proposed observer is established from the offered output measurement and robust against model uncertainties and external disturbances. Convergence analysis of the observation error dynamics is realized and proved by Lyapunov stabilization theory. Finally, for verification and demonstration, the proposed method is applied to the Chen as an autonomous chaotic system and the electrostatic transducer as a nonautonomous chaotic system. The numerical simulations illustrate the excellent performance of the proposed scheme. © 2014 Wiley Periodicals, Inc. Complexity 21: 145–153, 2015     

SQCF系统的全局时滞混沌同步

考虑到在通信系统中接收端存在时滞问题,针对SQCF系统,以Lyapunov稳定性理论和矩阵论为基础,提出一种系统的全局混沌同步方案.通过选取适当的耦合参数,使误差系统全局渐近稳定,即使得响应系统在t+τ时刻的状态渐进地趋向于驱动系统在t时刻的状态.并用Mathematic软件给出了数值模拟,理论分析和数值仿真表明,该方法可以较快地实现混沌同步.对系统的第三变量z(t)进行扰动,数值仿真表明在扰动系统下仍能很好地保持同步,表明同步系统具有抗干扰性.     

Global synchronization criteria for a class of third-order non-autonomous chaotic systems via linear state error feedback control

This paper investigates the global synchronization of a class of third-order non-autonomous chaotic systems via the master–slave linear state error feedback control. A sufficient global synchronization criterion of linear matrix inequality (LMI) and several algebraic synchronization criteria for single-variable coupling are proven. These LMI and algebraic synchronization criteria are then applied to two classes of well-known third-order chaotic systems, the generalized Lorenz systems and the gyrostat systems, proving that the local synchronization criteria for the chaotic generalized Lorenz systems developed in the existing literature can actually be extended to describe global synchronization and obtaining some easily implemented synchronization criteria for the gyrostat systems.     

Tracking the state of the delay hyperchaotic Lü system using the coullet chaotic system via a single controller

In this article, a partial synchronization scheme is proposed based on Lyapunov stability theory to track the signal of the delay hyperchaotic Lü system using the Coullet system based on only one single controller. The proposed tracking control design has two advantages: only one controller is adopted in our approach and it can allow us to drive the hyperchaotic system to a simple chaotic system even with uncertain parameters. Numerical simulation results are given to demonstrate the effectiveness and robustness of the proposed partial synchronization scheme. © 2014 Wiley Periodicals, Inc. Complexity 21: 125–130, 2016     

Extended Center Manifold, Global Bifurcation and Approximate Solutions of Chen chaotic dynamical system

The study of the chaotic Chen dynamic System (CDS) has been a recent focus in the literature, with numerous works exploring its various chaotic features. However, the majority of these studies have relied primarily on numerical techniques to investigate nonlinear dynamic systems (NLDSs). In this context, our aim is to derive approximate analytical solutions for the CDS by developing an iterative scheme. We have proven the convergence theorem for this scheme, which ensures that our iterative process will converge to the exact solution. Additionally, we introduce a new method for constructing the extended center manifold, a critical component in the analysis of dynamical systems. The characteristics of the global bifurcation of the system components within the parameter space are explored. The error analysis of the iterated solutions demonstrates the efficiency of the present technique. We present both three-dimensional (3D) and two-dimensional (2D) phase portraits of the system. The 3D portrait reveals a feedback loop pattern, while the 2D portrait, which represents the interaction of the system components, exhibits multiple pools and cross pools. Furthermore, we illustrate the global bifurcation by visualizing the components of the CDS against the space parameters. The sensitivity of CDS to innitesimal variations in the initial conditions (ICs) are tested. It is found that even minor changes can lead to signicant alterations in the system.     

Synchronization and antisynchronization of N‐coupled fractional‐order complex chaotic systems with ring connection

This paper is devoted to investigate synchronization and antisynchronization of N‐coupled general fractional‐order complex chaotic systems described by a unified mathematical expression with ring connection. By means of the direct design method, the appropriate controllers are designed to transform the fractional‐order error dynamical system into a nonlinear system with antisymmetric structure. Thus, by using the recently established result for the Caputo fractional derivative of a quadratic function and a fractional‐order extension of the Lyapunov direct method, several stability criteria are derived to ensure the occurrence of synchronization and antisynchronization among N‐coupled fractional‐order complex chaotic systems. Moreover, numerical simulations are performed to illustrate the effectiveness of the proposed design.     

A general scheme for Q-S synchronization of chaotic systems with unknown parameters and scaling functions

This work investigates Q-S synchronization of non-identical chaotic systems with unknown parameters and scaling function. The sufficient conditions for achieving Q-S synchronization with a double-desired scaling function of two different chaotic systems (including different dimensional systems) are derived based on the Lyapunov stability theory. By the adaptive control technique, the corresponding parameter update laws are proposed such that the Q-S synchronization of non-identical chaotic systems is to be obtained. Two illustrative numerical simulations are also given to demonstrate the effectiveness of the proposed scheme.     

Synchronization of fractional‐order chaotic systems with disturbances via novel fractional‐integer integral sliding mode control and application to neuron models

In this paper, a novel fractional‐integer integral type sliding mode technique for control and generalized function projective synchronization of different fractional‐order chaotic systems with different dimensions in the presence of disturbances is presented. When the upper bounds of the disturbances are known, a sliding mode control rule is proposed to insure the existence of the sliding motion in finite time. Furthermore, an adaptive sliding mode control is designed when the upper bounds of the disturbances are unknown. The stability analysis of sliding mode surface is given using the Lyapunov stability theory. Finally, the results performed for synchronization of three‐dimensional fractional‐order chaotic Hindmarsh‐Rose (HR) neuron model and two‐dimensional fractional‐order chaotic FitzHugh‐Nagumo (FHN) neuron model.     

The control of chaotic systems with unknown parameters and external disturbance via backstepping‐like scheme

This article addresses the adaptive control of chaotic systems with unknown parameters, model uncertainties, and external disturbance. We first investigate the control of a class of chaotic systems and then discuss the control of general chaotic systems. Based on the backstepping‐like procedure, some novel criteria are proposed via adaptive control scheme. As an example to illustrate the application of the proposed method, the control and synchronization of the modified Chua's chaotic system is also investigated via a single input. Some numerical simulations are given to demonstrate the robustness and efficiency of the proposed approach. © 2016 Wiley Periodicals, Inc. Complexity 21: 573–583, 2016     

Adaptive impulsive synchronization of uncertain delayed chaotic system with full unknown parameters via discrete‐time drive signals

This article investigates the adaptive impulsive synchronization of delayed chaotic system with full unknown parameters. Aiming at this problem, we propose a new adaptive strategy, in which both the adaptive–impulsive controller and the parameters adaptive laws are designed via the discrete‐time signals from the drive system. The corresponding theoretical proof is given to guarantee the effectiveness of the proposed strategy. Moreover, the concrete adaptive strategies are achieved for delayed Hopfield neural network, optical Ikeda system and the well‐known delayed Lü chaotic system. As expected, numerical simulations show the effectiveness of the proposed strategy. This method has potential applications in parameters estimation, secure communication, and cryptanalysis when only discrete signals are transmitted in communication channel. © 2014 Wiley Periodicals, Inc. Complexity 21: 43–51, 2016     

Multi‐switching combination–combination synchronization of non‐identical fractional‐order chaotic systems

In this paper, multi‐switching combination–combination synchronization scheme has been investigated between a class of four non‐identical fractional‐order chaotic systems. The fractional‐order Lorenz and Chen's systems are taken as drive systems. The combination–combination of multi drive systems is then synchronized with the combination of fractional‐order Lü and Rössler chaotic systems. In multi‐switching combination–combination synchronization, the state variables of two drive systems synchronize with different state variables of two response systems simultaneously. Based on the stability of fractional‐order chaotic systems, the multi‐switching combination–combination synchronization of four fractional‐order non‐identical systems has been investigated. For the synchronization of four non‐identical fractional‐order chaotic systems, suitable controllers have been designed. Theoretical analysis and numerical results are presented to demonstrate the validity and feasibility of the applied method. Copyright © 2017 John Wiley & Sons, Ltd.