Adaptive Hybrid Projective Synchronization Of Hyper-chaotic Systems

In this paper, we design a procedure to investigate the hybrid projective synchronization (HPS) technique among two identical hyper-chaotic systems. An adaptive control method (ACM) is pro- posed which is based on Lyapunov stability theory (LST). The considered technique globally determines the asymptotical stability and establishes identification of parameter simultaneously via HPS approach. Additionally, numerical simulations are carried out for visualizing the effectiveness and feasibility of discussed scheme by using MATLAB

Exponential stability via aperiodically intermittent control of complex-variable time delayed chaotic systems

summary:This paper focuses on the problem of exponential stability analysis of uncertain complex-variable time delayed chaotic systems, where the parameters perturbation are bounded assumed. The aperiodically intermittent control strategy is proposed to stabilize the complex-variable delayed systems. By taking the advantage of Lyapunov method in complex field and utilizing inequality technology, some sufficient conditions are derived to ensure the stability of uncertain complex-variable delayed systems, where the constrained time delay are considered in the conditions obtained. To protrude the availability of the devised stability scheme, simulation examples are ultimately demonstrated

Multi-objective Active Control Policy Design for Commensurate and Incommensurate Fractional Order Chaotic Financial Systems

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.In this paper, an active control policy design for a fractional order (FO) financial system is attempted, considering multiple conflicting objectives. An active control template as a nonlinear state feedback mechanism is developed and the controller gains are chosen within a multi-objective optimization (MOO) framework to satisfy the conditions of asymptotic stability, derived analytically. The MOO gives a set of solutions on the Pareto optimal front for the multiple conflicting objectives that are considered. It is shown that there is a trade-off between the multiple design objectives and a better performance in one objective can only be obtained at the cost of performance deterioration in the other objectives. The multi-objective controller design has been compared using three different MOO techniques viz. Non Dominated Sorting Genetic Algorithm-II (NSGA-II), epsilon variable Multi-Objective Genetic Algorithm (ev-MOGA), and Multi Objective Evolutionary Algorithm with Decomposition (MOEA/D). The robustness of the same control policy designed with the nominal system settings have been investigated also for gradual decrease in the commensurate and incommensurate fractional orders of the financial system

A Chaotic System with an Infinite Number of Equilibrium Points: Dynamics, Horseshoe, and Synchronization

Discovering systems with hidden attractors is a challenging topic which has received considerable interest of the scientific community recently. This work introduces a new chaotic system having hidden chaotic attractors with an infinite number of equilibrium points. We have studied dynamical properties of such special system via equilibrium analysis, bifurcation diagram, and maximal Lyapunov exponents. In order to confirm the system’s chaotic behavior, the findings of topological horseshoes for the system are presented. In addition, the possibility of synchronization of two new chaotic systems with infinite equilibria is investigated by using adaptive control

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...

Synchronization of complex dynamical networks with fractional order

Complex dynamical networks (CDN) can be applied to many areas in real world, from medicine, biology, Internet to sociology. Study on CDNs has drawn great attention in recent years. Nodes in a CDN can be modelled as systems represented by differential equations. Study has shown that fractional order differential equations (DF) can better represent some real world systems than integer-order DFs. This research work focuses on synchronization in fractional CDNs.  A literature review on CDNs with fractional order has summarized the latest works in this area.  Fractional chaotic systems are studied in our initial investigation.  Fractional calculus is introduced and the relevant fundamentals to model, describe and analyse dynamical networks are presented. It is shown that the structure and topological characteristics of a network can have a big impact on its synchronizability. Synchronizability and its various interpretations in dynamical networks are studied. To synchronize a CDN efficiently, controllers are generally needed. Controller design is one of the main tasks in this research. Our first design is a new sliding mode control to synchronize a dynamical network with two nodes. Its stability has been proven and verified by simulations.  Its convergence speed outperforms Vaidyanathan's scheme, a well-recognized scheme in this area. The design can be generalized to CDNs with more nodes.  As many applications can be modelled as CDNs with node clustering, a different sliding mode control is designed for cluster synchronization of a CDN with fractional order. Its stability is proven by using Lyapunov method. Its convergence and efficiency is shown in a simulation. Besides these nonlinear methods mentioned, linear control is also studied intensively for the synchronization.  A novel linear method for synchronization of fractional CDNs using a new fractional Proportional-Integral (PI) pinning control is proposed.  Its stability is proven and the synchronization criteria are obtained. The criteria have been simplified using two corollaries so the right value for the variables can be easily assigned. The proposed method is compared with the conventional linear method which uses Proportional (P) controller. In the comparison, the mean squared error function is used. The function measures the average of the squared errors and it is an instant indicator of the synchronization efficiency. A numerical simulation is repeated 100 times to obtain the averages over these runs. Each simulation has different random initial values for both controllers. The average of the errors in all the 100 simulations is obtained and the area under the function curve is defined as an overall performance index (OPI), which indicates the controller's overall performance. In control, small overshoot is always desired. In our work, the error variation is also used as a measure.  The maximum variation from the average of 100 simulations is calculated and compared for both methods. With all the statistical comparisons, it is clear that with the same power consumption, the proposed method outperforms the conventional one and achieves faster and smoother synchronization. Communication constraints exist in most real world CDNs. Communication constraints and their impact on control and synchronization of CDNs with fractional order are investigated in our study. A new adaptive method for synchronizing fractional CDN with disturbance and uncertainty is designed. Its stability is proven and its synchronization criteria are obtained for both fractional CDN with known and unknown parameters. Random disturbance is also included in both cases. Our results show that the new method is efficient in synchronizing CDNs with presence of both disturbance and uncertainty

A New 3-D Multistable Chaotic System with Line Equilibrium: Dynamic Analysis and Synchronization

This work introduces a new 3-D chaotic system with a line of equilibrium points. We carry out a detailed dynamic analysis of the proposed chaotic system with five nonlinear terms. We show that the chaotic system exhibits multistability with two coexisting chaotic attractors. We apply integral sliding mode control for the complete synchronization of the new chaotic system with itself as leader-follower systems

Modelagem e sincronização de sistemas caóticos com ênfase em sistemas financeiros

Trabalho de conclusão de curso (graduação)—Universidade de Brasília, Faculdade de Tecnologia, Curso de Graduação em Engenharia de Controle e Automação, 2013.Este trabalho estuda, com base na teoria de estabilidade de Lyapunov, o projeto de controladores robustos para sistemas hipercaóticos financeiros e caóticos unificados na presença de incerteza nos parâmetros e distúrbios limitados. Inicialmente, de modo a explicitar o problema de sincronização e apresentar os conceitos e ferramentas utilizados, uma breve revisão sobre dinâmica não linear, estabilidade, caos e análise tipo Lyapunov-like é apresentada. Na sequência, são estudados os modelos hipercaóticos e caóticos a serem utilizados. Para tanto, são analisados os trabalhos de Yu et al, 2012 e Lu et al, 2001. Estes trabalhos são fundamentais para os resultados apresentados no presente trabalho de graduação. A seguir, um esquema de controle adaptativo para sistemas hipercaóticos financeiros que assegura convergência para zero do erro residual de sincronização é estudado. O esquema é baseado em uma análise tipo Lyapunov-like e assegura convergência assintótica, mesmo na presença de erros de modelagem e distúrbios limitados. Simulações exaustivas são apresentadas com o objetivo de se avaliar a influência dos diversos parâmetros de projeto no desempenho do algoritmo. Finalmente, de modo a ressaltar a aplicabilidade da metodologia de projeto empregada, é considerado o controle adaptativo de um sistema caótico unificado. Como esperado, é mostrado que o erro residual de sincronização converge para zero na presença de erro de modelagem, distúrbios e incerteza paramétrica.The present work studies, based on the Lyapunov Stability Theory, the design of robust controllers for hyperchaotic financial and unified chaotic systems in the presence of uncertainty parameter and bounded disturbances. Initially, in order to explain the problem of synchronization and introduce the concepts and tools used, a brief review of nonlinear dynamics, stability, chaos and a Lyapunov-like analysis type is presented. In the sequel, the hyperchaotic and chaotic models are studied and then used. For this, we analyze the works of Yu et al, 2012 e Lu et al, 2001. These works are critical to the results presented in this graduate work. Following, an adaptive control scheme to a hyperchaotic financial system that assures the convergence of the residual synchronization error to zero is studied. The scheme is based upon a Lyapunov-like analysis type and assures asymptotic convergence, even in the presence of modeling errors and bounded disturbances. Exhaustive simulations are showed aiming to evaluate the influence of various design parameters on the performance of the algorithm. Finally, in order to emphasize the applicability of the design methodology employed, an adaptive control to the unified chaotic system is considered. As expected, is showed that the residual synchronization error converges to zero in the presence of modeling errors, disturbances and parametric uncertainty

Chaos synchronization and its application to secure communication

Chaos theory is well known as one of three revolutions in physical sciences in 20th-century, as one physicist called it: Relativity eliminated the Newtonian illusion of absolute space and time; quantum theory eliminated the Newtonian dream of a controllable measurable process; and chaos eliminates the Laplacian fantasy of deterministic predictability". Specially, when chaos synchronization was found in 1991, chaos theory becomes more and more attractive. Chaos has been widely applied to many scientific disciplines: mathematics, programming, microbiology, biology, computer science, economics, engineering, finance, philosophy, physics, politics, population dynamics, psychology, and robotics. One of most important engineering applications is secure communication because of the properties of random behaviours and sensitivity to initial conditions of chaos systems. Noise-like dynamical behaviours can be used to mask the original information in symmetric cryptography. Sensitivity to initial conditions and unpredictability make chaotic systems very suitable to construct one-way function in public-key cryptography. In chaos-based secure communication schemes, information signals are masked or modulated (encrypted) by chaotic signals at the transmitter and the resulting encrypted signals are sent to the corresponding receiver across a public channel (unsafe channel). Perfect chaos synchronization is usually expected to recover the original information signals. In other words, the recovery of the information signals requires the receiver's own copy of the chaotic signals which are synchronized with the transmitter ones. Thus, chaos synchronization is the key technique throughout this whole process. Due to the difficulties of generating and synchronizing chaotic systems and the limit of digital computer precision, there exist many challenges in chaos-based secure communication. In this thesis, we try to solve chaos generation and chaos synchronization problems. Starting from designing chaotic and hyperchaotic system by first-order delay differential equation, we present a family of novel cell attractors with multiple positive Lyapunov exponents. Compared with previously reported hyperchaos systems with complex mathematic structure (more than 3 dimensions), our system is relatively simple while its dynamical behaviours are very complicated. We present a systemic parameter control method to adjust the number of positive Lyapunov exponents, which is an index of chaos degree. Furthermore, we develop a delay feedback controller and apply it to Chen system to generate multi-scroll attractors. It can be generalized to Chua system, Lorenz system, Jerk equation, etc. Since chaos synchronization is the critical technique in chaos-based secure communication, we present corresponding impulsive synchronization criteria to guarantee that the receiver can generate the same chaotic signals at the receiver when time delay and uncertainty emerge in the transmission process. Aiming at the weakness of general impulsive synchronization scheme, i.e., there always exists an upper boundary to limit impulsive intervals during the synchronization process, we design a novel synchronization scheme, intermittent impulsive synchronization scheme (IISS). IISS can not only be flexibly applied to the scenario where the control window is restricted but also improve the security of chaos-based secure communication via reducing the control window width and decreasing the redundancy of synchronization signals. Finally, we propose chaos-based public-key cryptography algorithms which can be used to encrypt synchronization signals and guarantee their security across the public channel