Generalized synchronization and control for incommensurate fractional unified chaotic system and applications in secure communication

summary:A fractional differential controller for incommensurate fractional unified chaotic system is described and proved by using the Gershgorin circle theorem in this paper. Also, based on the idea of a nonlinear observer, a new method for generalized synchronization (GS) of this system is proposed. Furthermore, the GS technique is applied in secure communication (SC), and a chaotic masking system is designed. Finally, the proposed fractional differential controller, GS and chaotic masking scheme are showed by using numerical and experimental simulations

Modified Projective Synchronization between Different Fractional-Order Systems Based on Open-Plus-Closed-Loop Control and Its Application in Image Encryption

A new general and systematic coupling scheme is developed to achieve the modified projective synchronization (MPS) of different fractional-order systems under parameter mismatch via the Open-Plus-Closed-Loop (OPCL) control. Based on the stability theorem of linear fractional-order systems, some sufficient conditions for MPS are proposed. Two groups of numerical simulations on the incommensurate fraction-order system and commensurate fraction-order system are presented to justify the theoretical analysis. Due to the unpredictability of the scale factors and the use of fractional-order systems, the chaotic data from the MPS is selected to encrypt a plain image to obtain higher security. Simulation results show that our method is efficient with a large key space, high sensitivity to encryption keys, resistance to attack of differential attacks, and statistical analysis

A new buffering theory of social support and psychological stress

A dynamical model linking stress, social support, and health has been recently proposed and numerically analyzed from a classical point of view of integer-order calculus. Although interesting observations have been obtained in this way, the present work conducts a fractional-order analysis of that model. Under a periodic forcing of an environmental stress variable, the perceived stress has been analyzed through bifurcation diagrams and two well-known metrics of entropy and complexity, such as spectral entropy and C0 complexity. The results obtained by numerical simulations have shown novel insights into how stress evolves with frequency and amplitude of the perturbation, as well as with initial conditions for the system variables. More precisely, it has been observed that stress can alternate between chaos, periodic oscillations, and stable behaviors as the fractional order varies. Moreover, the perturbation frequency has revealed a narrow interval for the chaotic oscillations, while its amplitude may present different values indicating a low sensitivity regarding chaos generation. Also, the perceived stress has been noted to be highly sensitive to initial conditions for the symptoms of stress-related ill-health and for the social support received from family and friends. This work opens new directions of research whereby fractional calculus might offer more insight into psychology, life sciences, mental disorders, and stress-free well-being

Symmetry in Chaotic Systems and Circuits

Symmetry can play an important role in the field of nonlinear systems and especially in the design of nonlinear circuits that produce chaos. Therefore, this Special Issue, titled “Symmetry in Chaotic Systems and Circuits”, presents the latest scientific advances in nonlinear chaotic systems and circuits that introduce various kinds of symmetries. Applications of chaotic systems and circuits with symmetries, or with a deliberate lack of symmetry, are also presented in this Special Issue. The volume contains 14 published papers from authors around the world. This reflects the high impact of this Special Issue

Fractional derivative models for the spread of diseases

This thesis considers the mathematical modelling of disease, using fractional differential equations in order to provide a tool for the description of memory effects. In Chapter 3 we illustrate a commensurate fractional order tumor model, and we find a critical value of the fractional derivative dependent on the parameter values of the model. For fractional derivatives of orders less than the critical value an unstable equilibrium point of the system becomes stable. In order to show changes in the observed areas of attraction of two stable points in the system, we then consider a fractional order SIR epidemic model and investigate the change from a monostable to a bistable system.;Chapter 4 considers a model for virus dynamics where the fractional orders for populations are different, called an incommensurate system. An approximate analytical solution for the characteristic equation of the incommensurate model is found when the different fractional orders are similar and close to the critical value of the fractional order of the commensurate system. In addition, the instability boundary is found as a function of both parameters. A comparison between analytical and numerical results shows the high accuracy of this approximation.;Chapter 5 consists of two parts, in the first part we generalise the integer Fisher's equation to be a space-time fractional differential equation and consider travelling wave solutions. In the second part we generalise an integer SIR model with spatial heterogeneity, which was studied by Murray [117], to a space-time fractional derivative model. We apply the (G0/G)-expansion method and find travelling wave solutions, although in this case we must consider the Jumarie's modified Riemann-Liouville fractional derivative. Finally, we consider the effect of changing the orders of time and space fractional derivatives on the location and speed of the travelling wave solution.This thesis considers the mathematical modelling of disease, using fractional differential equations in order to provide a tool for the description of memory effects. In Chapter 3 we illustrate a commensurate fractional order tumor model, and we find a critical value of the fractional derivative dependent on the parameter values of the model. For fractional derivatives of orders less than the critical value an unstable equilibrium point of the system becomes stable. In order to show changes in the observed areas of attraction of two stable points in the system, we then consider a fractional order SIR epidemic model and investigate the change from a monostable to a bistable system.;Chapter 4 considers a model for virus dynamics where the fractional orders for populations are different, called an incommensurate system. An approximate analytical solution for the characteristic equation of the incommensurate model is found when the different fractional orders are similar and close to the critical value of the fractional order of the commensurate system. In addition, the instability boundary is found as a function of both parameters. A comparison between analytical and numerical results shows the high accuracy of this approximation.;Chapter 5 consists of two parts, in the first part we generalise the integer Fisher's equation to be a space-time fractional differential equation and consider travelling wave solutions. In the second part we generalise an integer SIR model with spatial heterogeneity, which was studied by Murray [117], to a space-time fractional derivative model. We apply the (G0/G)-expansion method and find travelling wave solutions, although in this case we must consider the Jumarie's modified Riemann-Liouville fractional derivative. Finally, we consider the effect of changing the orders of time and space fractional derivatives on the location and speed of the travelling wave solution

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

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. [...