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In this study, the higher order interactions, which are complex interactions involving more than two neurons, are studied in two real-world networks. These interactions play a crucial role in the brain's functioning, as they can influence the network's synchronization and dynamics. This paper analyzes the application of directed simplicial complexes, the mathematical realization of the higher order interactions, to study the synchronization of two Macaque rhesus brain networks. Interestingly, the directionally allows for the consideration of different simplicial motifs. The Hindmarsh–Rose chaotic system has been used to model the node dynamics. Also, electrical and chemical couplings are considered to be the interaction mechanism. The findings from this study provide valuable insights into the complex dynamics of brain networks. [ABSTRACT FROM AUTHOR]

The Lozi map is well-known and has been studied in various researches. By combining three research trends (discrete map, memristor and fractional calculus) we investigate a fractional memristive Lozi map in this work. Firstly the Grunwald–Letnikov fractional difference operator is used to introduce the new fractional map with no equilibrium point. Then, the coexistence of several chaotic hidden attractors is shown, along with the coexistence of a number of bifurcations, depending on the values of the initial conditions. We found attractive dynamics and characteristics of this fractional Lozi map. The realization with hardware platform illustrates the map's feasibility. [ABSTRACT FROM AUTHOR]

An editorial focuses on the recent advancements in fractional calculus and its applications in physical systems. It emphasizes the utility of fractional calculus in describing complex systems with nonlocal characteristics. The versatility of fractional calculus across various scientific disciplines, including physics, engineering, and biology, is highlighted.

The dynamics of coupled chaotic circuits have particular importance in many applications. The Chua circuit is a well-known chaotic circuit that has been applied in various studies. In this paper, we investigate the network of Chua circuits in different coupling schemes. We consider three cases where the circuits are coupled through resistors, memristors, and inductors. The synchronization behavior of the network is examined for each case. It is observed that the nonstationary chimera is formed in coupled circuits in all coupling schemes. Complete synchronization can be achieved only in resistive coupling. In memristive coupled circuits, increasing the coupling strength leads to the instability of the network, while in the inductor coupling, the synchronization appears only among two variables, and the third variable cannot be synchronized. [ABSTRACT FROM AUTHOR]

Synchronization in coupled oscillators is of high importance in secure communication and information processing. Due to this reason, a significant number of studies have been performed to investigate the synchronization state in coupled circuits. Diffusive coupling is the simplest connection between the oscillators, which can be implemented through a variable resistor between two variables of two circuits. The Chua's circuit is the most famous chaotic circuit whose dynamics have been investigated in many studies. However, Wang–Zhang–Bao (WZB) is another chaotic circuit that can exhibit exciting behaviors such as bistability. Thus, this study aims to investigate the cooperative dynamics of the WZB circuit in its elegant parameter values. To this issue, first, we explored the dynamic behavior of the elegant WZB circuit using the bifurcation diagrams, the Lyapunov exponents, and the basins of attraction. Based on the results, we found the range of the bifurcation parameter and the initial conditions wherein the system is bistable. Subsequently, setting the parameters in the monostable region, we studied the synchronization state of two diffusively coupled WZB circuits analytically and numerically. Consequently, we used master stability functions and temporally averaged synchronization error as the analytical and numerical tools to explore the synchronization state. Then we numerically examined the synchronization state in a network of 100 nonlocally coupled WZB oscillators. As a result, we found imperfect chimera and phase synchronization in the studied network before getting synchronized. [ABSTRACT FROM AUTHOR]

A novel memristive oscillator with infinite equilibria is proposed in this paper. The oscillator has six lines of equilibria, which make it different from known ones. Interestingly, the oscillator displays chaos and attractive features. Implementation of the oscillator via analog components is presented to verify its feasibility. [ABSTRACT FROM AUTHOR]

Here, a 2D megastable oscillator with a square-wave function is proposed. The oscillator shows an infinite number of coexisting jagged-shape limit cycles. It has an unstable equilibrium point. Based on this equilibrium, the innermost attractor is self-excited, while the other ones are hidden. The number of sharp edges of the jagged-shape limit cycles increases by moving the attractor from the innermost one toward the outside. The basin of attraction of the limit cycles is also studied. Then, a sinusoidal force is applied to the oscillator, and its chaotic dynamics are discussed. Various dynamics of the forced oscillator by changing the amplitude and frequency of the forced term are discussed. The basin of attraction of the forced system is investigated. To the best of our knowledge, the jagged-shape attractors in multi-stable oscillators have not been studied before. [ABSTRACT FROM AUTHOR]

Chaotic dynamics and synchronization in fractional systems described by non-integer order difference operators have attracted increasing attention in recent years. However, very few papers have been published regarding the chaotic behaviors of fractional discrete-time neural networks and their synchronization properties. A novel chaotic h-fractional discrete neural network model is presented in this paper. This model is described using the h-fractional Caputo difference operator of order ν . The paper also presents a novel theorem, which assures that two chaotic fractional discrete neural networks achieve synchronized dynamics using very simple linear control laws. Additionally, the chaotic dynamics of the network are stabilized at the origin via a suitable controller. Finally, simulation results are carried out to demonstrate the feasibility of the theoretical results developed herein. [ABSTRACT FROM AUTHOR]

The deadly outbreak of the second wave of Covid-19, especially in worst hit lower-middle-income countries like India, and the drastic rise of another growing epidemic of Mucormycosis, call for an efficient mathematical tool to model pandemics, analyse their course of outbreak and help in adopting quicker control strategies to converge to an infection-free equilibrium. This review paper on prominent pandemics reveals that their dispersion is chaotic in nature having long-range memory effects and features which the existing integer-order models fail to capture. This paper thus puts forward the use of fractional-order (FO) chaos theory that has memory capacity and hereditary properties, as a potential tool to model the pandemics with more accuracy and closeness to their real physical dynamics. We investigate eight FO models of Bombay plague, Cancer and Covid-19 pandemics through phase portraits, time series, Lyapunov exponents and bifurcation analysis. FO controllers (FOCs) on the concepts of fuzzy logic, adaptive sliding mode and active backstepping control are designed to stabilise chaos. Also, FOCs based on adaptive sliding mode and active backstepping synchronisation are designed to synchronise a chaotic epidemic with a non-chaotic one, to mitigate the unpredictability due to chaos during transmission. It is found that severity and complexity of the models increase as the memory fades, indicating that FO can be used as a crucial parameter to analyse the progression of a pandemic. To sum it up, this paper will help researchers to have an overview of using fractional calculus in modelling pandemics more precisely and also to approximate, choose, stabilise and synchronise the chaos control parameter that will eliminate the extreme sensitivity and irregularity of the models. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear dynamics of the biological system modeled by the fractional incommensurate order Van der Pol equations are investigated. The stability of the proposed fractional non-autonomous system is analyzed by varying both the fractional order derivative and system parameters. Moreover, very interesting phenomena such as symmetry, multi-stability and coexistence of attractors are discovered in the considered biological system. Numerical simulations are performed by considering the Caputo fractional derivative and results are reported by means of bifurcation diagrams, computation of the largest Lyapunov exponent, phase portraits in 2D and 3D projections. [ABSTRACT FROM AUTHOR]

It is almost more than a year that earth has faced a severe worldwide problem called COVID-19. In December 2019, the origin of the epidemic was found in China. After that, this contagious virus was reported almost all over the world with different variants. Besides all the healthcare system attempts, quarantine, and vaccination, it is needed to study the dynamical behavior of this disease specifically. One of the practical tools that may help scientists analyze the dynamical behavior of epidemic disease is mathematical models. Accordingly, here, a novel mathematical system is introduced. Also, the complex behavior of this model is investigated considering different dynamical analyses. The results represent that some range of parameters may lead the model to chaotic behavior. Moreover, comparing the two same bifurcation diagrams with different initial conditions reveals that the model has multi-stability. [ABSTRACT FROM AUTHOR]

Chaotic maps, memristive maps, and maps with special fixed points have attracted significant researches recently. We introduce a simple methodology to construct a memristor-based map without fixed points in this work. In our methodology, a tiny perturbation is included into a memristor-based map to change its fixed points. Therefore, the presence and disappearance of fixed points can be controlled. By considering two general structures of discrete maps, seven memristive non-fixed-point maps have been reported. We have found that the methodology is useful for designing not only non-fixed-point maps with memristor but also non-fixed-point maps without memristor. To confirm the physical realization of the proposed maps, a hardware implementation of one of such maps is performed using a microcontroller. [ABSTRACT FROM AUTHOR]

Understanding extreme events attracts scientists due to substantial impacts. In this work, we study the emergence of extreme events in a fractional system derived from a Liénard-type oscillator. The effect of fractional-order derivative on the extreme events has been investigated for both commensurate and incommensurate fractional orders. Especially, such a system displays multistability and coexistence of multiple extreme events. [ABSTRACT FROM AUTHOR]

This paper considers the distributed consensus control of nth-order uncertain nonlinear multi-agent systems (MASs) with fuzzy dead-zone (DZ). Unlike the existing works which describes the DZ constraint by the conventional deterministic models, this work describes the DZ by the fuzzy and uncertain value. Therefore, some imposed restricting assumptions on the DZ parameters such as known bounds of the DZ slopes, or linear and deterministic model for DZ description which exist in the previous works are removed in this work. In the proposed scheme, the neural network (NN) is invoked to model the uncertain nonlinearities of the follower agents and uncertain interaction of their neighbors. To decrease the number of adaptive parameters and computational load, maximum norm of the weight coefficients of the NN is considered as an adjustable parameter and tuned online. Also, to decrease the controller complexity and eliminate the repeated derivatives of nonlinear functions in the controller, the dynamic surface control (DSC) method is used to develop the proposed distributed consensus scheme. Stability analysis based on the Lyapunov direct method guarantees that all of the signals in the closed-loop MAS are semi-globally uniformly ultimately bounded and the consensus error converges to a bound dependent on the design parameters. Therefore, appropriate choice of the design parameters makes the bound small. Simulation results on a well-known example demonstrate the effectiveness of the developed consensus algorithm. [ABSTRACT FROM AUTHOR]

Chimera state is a peculiar spatiotemporal pattern, wherein the coherence and incoherence coexist in the network of coupled identical oscillators. In this paper, we study the chimera states in a network of impact oscillators with nonlocal coupling. We investigate the effects of the coupling strength and the coupling range on the network behavior. The results reveal the emergence of the chimera state for significantly small values of coupling strength, and higher coupling strength values lead to unbounded motions in the oscillators. We also study the network in the case of excitation failure. We observe that the coupling helps in the maintenance of an oscillatory motion with a lower amplitude in the failed oscillator. [ABSTRACT FROM AUTHOR]

The paper investigates control and synchronization of fractional-order maps described by the Caputo h-difference operator. At first, two new fractional maps are introduced, i.e., the Two-Dimensional Fractional-order Lorenz Discrete System (2D-FoLDS) and Three-Dimensional Fractional-order Wang Discrete System (3D-FoWDS). Then, some novel theorems based on the Lyapunov approach are proved, with the aim of controlling and synchronizing the map dynamics. In particular, a new hybrid scheme is proposed, which enables synchronization to be achieved between a master system based on a 2D-FoLDS and a slave system based on a 3D-FoWDS. Simulation results are reported to highlight the effectiveness of the conceived approach. [ABSTRACT FROM AUTHOR]

In this paper, we are particularly interested in the fractional form of the Hénon-Lozi type map. Using discrete fractional calculus, we show that the general behavior of the proposed fractional order map depends on the fractional order. The dynamical properties of the new generalized map are investigated by applying numerical tools such as: phase portrait, bifurcation diagram, largest Lyapunov exponent, and 0-1 test. It shows that the fractional order Hénon-Lozi map exhibits a range of different dynamical behaviors including chaos and coexisting attractors. Furthermore, a one-dimensional control law is proposed to stabilize the states of the fractional order map. Numerical results are presented to illustrate the findings. [ABSTRACT FROM AUTHOR]

Coronavirus disease 2019 is a recent strong challenge for the world. In this paper, an epidemiology model is investigated as a model for the development of COVID-19. The propagation of COVID-19 through various sub-groups of society is studied. Some critical parameters, such as the background of mortality without considering the disease state and the speed of moving people from infected to resistance, affect the conditions of society. In this paper, early warning indicators are used to predict the bifurcation points in the system. In the interaction of various sub-groups of society, each sub-group can have various parameters. Six cases of the sub-groups interactions are studied. By coupling these sub-groups, various dynamics of the whole society are investigated. [ABSTRACT FROM AUTHOR]

This paper proposes three fractional discrete chaotic systems based on the Rulkov, Chang, and Zeraoulia-Sprott rational maps. The dynamics of the proposed maps are investigated by means of phase plots and bifurcations diagrams. Adaptive stabilization schemes are proposed for each of the three maps and the convergence of the states is established by using the Lyapunov method. Furthermore, a combination synchronization scheme is proposed whereby a combination of the fractional Rulkov and Chang maps is synchronized to the fractional Zeraoulia-Sprott map. Numerical results are used to confirm the findings of the paper. [ABSTRACT FROM AUTHOR]

Investigating new chaotic flows has been a hot topic for many years. Studying the chaotic attractors of systems with various properties illuminates a lamp to reveal the vague of the generation of chaotic attractors. A new chaotic system in the spherical coordinates is proposed in this paper. The system's solution is inside a predefined sphere, and its attractor cannot cross the sphere. Investigation of equilibrium points of the system shows that the system has eight equilibria, and all of them are saddle. Bifurcation analysis of the system depicts the period-doubling route to chaos with changing the bifurcation parameter. Also, Lyapunov exponents in the studied interval of the bifurcation parameter are discussed. The basin of attraction of the system is investigated to show the sensitivity of the system to initial conditions. [ABSTRACT FROM AUTHOR]

This paper studies the dynamics of a new fractional-order map with no fixed points. Through phase plots, bifurcation diagrams, largest Lyapunov exponent, it is shown that the proposed fractional map exhibit chaotic and periodic behavior. New Hidden chaotic attractors are observed, and transient state is found to exist. Complexity of the new map is also analyzed by employing approximate entropy. Results, show that the fractional map without fixed point have high complexity for certain fractional order. In addition, a control scheme is introduced. The controllers stabilize the states of the fractional map and ensure their convergence to zero asymptotically. Numerical results are used to verify the findings. [ABSTRACT FROM AUTHOR]

In this paper, a new hyperchaotic memristor oscillator is proposed. Different dynamical properties of the proposed system such as dissipativity, equilibrium points, and their stabilities, Lyapunov exponents and Kaplan-Yorke dimension are investigated. The system has a line of equilibria. Thus it belongs to the category of systems with hidden attractors. Investigation of the stability of the line of equilibria shows that the line is stable in some intervals and unstable in some others. Bifurcation analysis of the system reveals several coexisting attractors in some range of parameters which indicates multistability. Various coexisting attractors of the proposed system are studied. [ABSTRACT FROM AUTHOR]

The article presents the correction for a article titled "Recent advancement of fractional calculus and its applications in physical systems" published online on December 8, 2023.

Multi-layer networks are quite fundamental to study structural and functional properties of various biological systems. In this study, a two-layer neuronal network is considered and the interactions between the spiral pattern in one layer and a homogeneous state in the other layer, under the effect of inter-layer bidirectional connection are investigated. Spiral wave has been confirmed to play a significant role in many complex systems. In this regard, in laminar structured systems in particular, it is crucial to study the dynamics of the spiral wave affected by the inter-layer interactions. Here, for each layer (sub-network), a regular network with eight-neighbor connection is designed. For the local dynamics of each neuron, the magnetic Fitzhugh–Nagumo (FN) neuronal model is introduced. The results show that depending on the level of interactions between the two layers, four different types of collective electrical activity can occur. When the inter-layer connection is weak, the layer with spiral pattern does not change while the homogeneous state of the other layer is broken by a blurred spiral pattern. As the inter-layer connection is strengthened, the dynamics of the spiral wave changes significantly, leading to unstable spiral wave and spiral turbulence. However, by a further increase in the inter-layer coupling strength, the spiral wave does not exist at all. [ABSTRACT FROM AUTHOR]

Multilayer neuronal networks provide the opportunity of considering multiple types of synaptic connections. In this paper, a two-layer neuronal network with chemical intralayer connections and electrical interlayer connections is investigated, where two layers have different spiking and bursting behaviours. The effects of synaptic time delay and also the self-feedback time delay on the synchronization are studied by calculating a statistical synchronization factor for intralayer synchronization and defining an interlayer synchronization error. It is observed that the interlayer coupling strength changes the spiking behaviour of first-layer neurons toward the chaotic behaviour of the second layer. The intralayer coupling is effective on the collective behaviour of the layers. Computing the synchronization measures for the time delays separately, shows the effects of each time delay on the network's synchronized behaviour. It also reveals that the lowest interlayer synchronization error and the highest intralayer synchronization factors are attained by using the self-feedback delay. [ABSTRACT FROM AUTHOR]

Parameter identification (PI) is very important in system analysis and controller design and has wide applications in industry. PI is an essential step in designing mathematical models of the dynamical systems based on the measured data. Identification of chaotic systems is very challenging due to the butterfly effect. Recently, a state-space-based cost function based on Gaussian mixture model (GMM) has been proposed for the PI of the chaotic systems. In there, a GMM as a statistical model of the measured data modeled static features of the chaotic attractors in the state space. In this paper, we propose a new method which incorporates static and dynamic features in the GMM modeling in order to achieve a better cost function for the PI problem. The proposed method can extract suitable information from the trajectory of the chaotic attractor. We conduct some experiments for one-dimensional PI. Using measured data from 4D and 3D chaotic systems, empirical results indicate success of the proposed method. [ABSTRACT FROM AUTHOR]

We propose a modified Fitzhugh-Nagumo neuron (MFNN) model. Based on this model, an integer-order MFNN system (case A) and a fractional-order MFNN system (case B) were investigated. In the presence of electromagnetic induction and radiation, memductance and induction can show a variety of distributions. Fractional-order magnetic flux can then be considered. Indeed, a fractional-order setting can be acceptable for non-uniform diffusion. In the case of an MFNN system with integer-order discontinuous magnetic flux, the system has chaotic and non-chaotic attractors. Dynamical analysis of the system shows the birth and death of period doubling, which is a sign of antimonotonicity. Such a behavior has not been studied previously in the dynamics of neurons. In an MFNN system with fractional-order discontinuous magnetic flux, different attractors such as chaotic and periodic attractors can be observed. However, there is no sign of antimonotonicity. [ABSTRACT FROM AUTHOR]

In the field of genetics, it is well known that a specific genetic behavior may be influenced by more than one gene. There is a similar concept in genetic algorithms (GAs), called epistasis, which is the interaction between genes. This study demonstrates that, in spite of what is generally assumed, GAs are not an efficient optimization tool. This is because the main operator, mating (crossover), cannot function properly in epistatic optimization problems. In non-epistatic problems, although a GA can possibly provide a correct solution, it is an inefficient and time-consuming algorithm. As proof, we used conventional test functions and introduced new ones and confirmed our claim with simulation results. [ABSTRACT FROM AUTHOR]

The last two decades have seen many literatures on the mathematical and computational analysis of neuronal activities resulting in many mathematical models to describe neuron. Many of those models have described the membrane potential of a neuron in terms of the leakage current and the synaptic inputs. Only recently researchers have proposed a new neuron model based on the electromagnetic induction theorem, which considers inner magnetic fluctuation and external electromagnetic radiation as a significant missing part that can participate in neural activity. While the flux coupling of the membrane is considered equivalent to a memductance function of a memristor, standard memductance model of α+3βϕ2 has been used in the literatures, but in this paper we propose a new memductance function based on discontinuous flux coupling. Various dynamical properties of the neuron model with discontinuous flux coupling are studied and interestingly the proposed model shows hyperchaotic behavior which was not identified in the literatures. Furthermore, we consider a ring network of the proposed model and investigate whether the chimera state can emerge. The chimera state relates to the state with simultaneously coherence and incoherence in oscillatory networks and has received much attention in recent years. [ABSTRACT FROM AUTHOR]

The present study investigates the dynamical properties of a fractional order coronary artery system and its control with uncertainties in the parameters. The fractional order model of the coronary artery system (FOCA) is derived using the Grünwald-Letnikov method and the properties of the FOCA system are discussed. Bifurcation plots of the system in the parameter space are derived and presented. The novelty of the paper is the identification of the multistable feature shown by the FOCA system, which has not been discussed in the research literature. Various coexisting attractors are also presented to show the multistability. An adaptive sliding mode controller is designed to stabilize the chaotic oscillations in the FOCA system. Numerical simulations are conducted to indicate the effectiveness of the controller. [ABSTRACT FROM AUTHOR]

Neural network is important for a wide range of applications. Especially, a small neural network can display various complex behaviors. In this work, the investigations of a Hopfield neural network and its field programmable gate array (FPGA) implementation have been reported. The considered Hopfield neural network is simple because it includes only three neurons. It is interesting that we observed chaos and numerous coexisting attractors in such a network. In addition, the network has been implemented via an FPGA platform to verify its feasibility. [ABSTRACT FROM AUTHOR]

This letter gives a general review on the monostability, bistability, periodicity and chaos in gene regulatory network. Some simple motifs that generate monostability, bistability, periodicity and chaos are analytically and numerically reported. Further research directions of the nonlinear dynamics of gene regulatory network are discussed. [ABSTRACT FROM AUTHOR]

This paper presents new results related to the coexistence of function-based hybrid synchronization types between non-identical incommensurate fractional-order systems characterized by different dimensions and orders. Specifically, a new theorem is illustrated, which ensures the coexistence of the full-state hybrid function projective synchronization (FSHFPS) and the inverse full-state hybrid function projective synchronization (IFSHFPS) between wide classes of three-dimensional master systems and four-dimensional slave systems. In order to show the capability of the approach, a numerical example is reported, which illustrates the coexistence of FSHFPS and IFSHFPS between the incommensurate chaotic fractional-order unified system and the incommensurate hyperchaotic fractional-order Lorenz system. [ABSTRACT FROM AUTHOR]

Iron pyrite (FeS2) thin films were fabricated by spin coating the solution of FeS2 nanocrystals of ∼40 nm in size on glass substrates, followed by annealing in a sulfur environment at different temperatures. The effect of sulfurization temperature on the morphology, structural, optical and electrical properties was investigated. With increase of the sulfurization temperature, the grain size and crystallinity of the films was improved, although some cracks and voids were observed on the surface of thin films. The band gap of the FeS2 films was decreased at higher sulfurization temperature. The electrical properties were also changed, including the increasing in resistivity and the decrease in Hall mobility, with increase of sulfurization temperature. The change in the optical and electrical properties of the FeS2 thin films was explained based on the changes of phase, morphology, surface, and grain boundary property. [ABSTRACT FROM AUTHOR]

Designing and analyzing new chaotic systems with desired properties has gotten much interest recently. Through applying modification in a recent rare chaotic flow which has adjustable Kaplan-Yorke dimension, a flexible chaotic system is introduced in this work. In the mentioned modification, two parameters are added which can control amplitude of the attractor and largest Lyapunov exponent. Statistical and dynamical properties of such system could be discovered by applying stability analysis, estimating Lyapunov exponents, local Kaplan-Yorke dimension, and plots of state space. Interestingly, this system is in group of systems which have coexisting attractors, so it is a multistable chaotic system. By considering a practical application of this rare system, meaningful relation between the value of local Kaplan-Yorke dimension and the ability of the system as a random number generator can be seen in statistical tests. [ABSTRACT FROM AUTHOR]

For the past 4 years, there has been a rapid rise in the study of chaotic systems with curves of equilibria which are categorized as systems with hidden attractors. There is still significant controversy surrounding the shapes of equilibrium points. This paper presents a new three-dimensional autonomous chaotic system with hyperbolic sine equilibrium. Fundamental dynamical properties and complex dynamics of the system have been discovered by using equilibrium analysis, phase portrait, Poincaré map, bifurcation diagram and Lyapunov spectrum. It is crucial to note that there are bistable hidden chaotic attractors in the introduced system. Furthermore, in order to show the feasibility of the new system with hyperbolic sine equilibrium, its electronic circuit has been implemented. [ABSTRACT FROM AUTHOR]

Hyperjerk systems have received considerable interest in the literature because of their simplicity and complex dynamical properties. In this work, we introduce a novel hyperjerk system with an absolute nonlinearity and a quintic term. Interestingly, the hyperjerk system exhibits hyperchaotic behavior. Dynamics and the feasibility of the hyperjerk system are discovered by using numerical analysis and circuit implementation. Moreover, adaptive controllers have been designed for stabilization and synchronization of the new hyperjerk system. The control results have been established by using Lyapunov stability theory, and numerical simulations with MATLAB have been shown to illustrate the validity of the constructed adaptive controllers. [ABSTRACT FROM AUTHOR]

The topic related to the coexistence of different synchronization types between fractional-order chaotic systems is almost unexplored in the literature. Referring to commensurate and incommensurate fractional systems, this paper presents a new approach to rigorously study the coexistence of some synchronization types between nonidentical systems characterized by different dimensions and different orders. In particular, the paper shows that identical synchronization ( IS), antiphase synchronization ( AS), and inverse full state hybrid projective synchronization ( IFSHPS) coexist when synchronizing a three-dimensional master system with a fourth-dimensional slave system. The approach, which can be applied to a wide class of chaotic/hyperchaotic fractional-order systems in the master-slave configuration, is based on two new theorems involving the fractional Lyapunov method and stability theory of linear fractional systems. Two examples are provided to highlight the capability of the conceived method. In particular, referring to commensurate systems, the coexistence of IS, AS, and IFSHPS is successfully achieved between the chaotic three-dimensional Rössler system of order 2.7 and the hyperchaotic four-dimensional Chen system of order 3.84. Finally, referring to incommensurate systems, the coexistence of IS, AS, and IFSHPS is successfully achieved between the chaotic three-dimensional Lü system of order 2.955 and the hyperchaotic four-dimensional Lorenz system of order 3.86. [ABSTRACT FROM AUTHOR]

There has been an increasing interest in studying fractional-order chaotic systems and their synchronization. In this paper, the fractional-order form of a system with stable equilibrium is introduced. It is interesting that such a three-dimensional fractional system can exhibit chaotic attractors. Full-state hybrid projective synchronization scheme and inverse full-state hybrid projective synchronization scheme have been designed to synchronize the three-dimensional fractional system with different four-dimensional fractional systems. Numerical examples have verified the proposed synchronization schemes. [ABSTRACT FROM AUTHOR]

Recently, the time delay has considerable attention in the existence of chaos in the nonlinear dynamical systems. In this paper, we therefore develop a new cascade-coupled chaotic synchronization model based on the first-order nonlinear time-delayed chaotic system using the simple Lur'e system to provide more robust secure system. Adaptive observer-based master systems are designed to synchronize with the given chaotic slave systems at the receiver where the dynamical model is subjected to certain time delay values at the slave systems. Two cases of being system parameters match and mismatch are considered. The adaptive observer laws are carefully designed to confirm the convergence of error dynamics. Numerical simulations reveal the effectiveness of the time delay on the recovered message security using the chaotic masking technique. As a result, the information message is able to be recovered correctly with sufficient precision over the noiseless and noisy channels and under predefined time-delay values compared to the existing systems. [ABSTRACT FROM AUTHOR]

Although different hyperjerk systems have been discovered, a few hyperjerk systems can exhibit hyperchaotic behavior. In this work, we introduce a new hyperjerk system with hyperchaotic attractors. By investigating dynamics of the system, we have observed the different coexisting attractors such as coexistence of period-2 attractors, or coexistence of period-2 attractor and quasiperiodic attractor. It is worth noting that this striking phenomenon is rarely reported in a hyperjerk system. The proposed system has been realized with electronic components. The agreement between the simulation and experimental results indicates the feasibility of the hyperjerk system. Moreover, chaos control and synchronization of such hyperjerk system have been also reported. [ABSTRACT FROM AUTHOR]

Novel chaotic system designs and their engineering applications have received considerable critical attention. In this paper, a new three-dimensional chaotic system and its application are introduced. The interesting aspects of this chaotic system are the absence of equilibrium points and the coexisting of limit cycle and torus. Basic dynamics of the no-equilibrium system have been executed by means of phase portraits, bifurcation diagram, continuation, and Lyapunov exponents. Experimental results of the electronic circuit realizing the no-equilibrium system have been reported to show system's feasibility. By using the chaoticity of the new system without equilibrium, we have developed a random bit generator for practical signal encryption application. Numerical results illustrate the usefulness of the random bit generator. [ABSTRACT FROM AUTHOR]