A new lemma for the Caputo fractional derivatives, when , is proposed in this paper. This result has proved to be useful in order to apply the fractional-order extension of Lyapunov direct method, to demonstrate the stability of many fractional order systems, which can be nonlinear and time varying.

► A new bio-inspired algorithm, namely krill herd (KH) is proposed for global optimization. ► The time-dependent position of the krill individuals is formulated by three main factors. ► The KH algorithm has a better performance than well-known methods in the literature. In this paper, a novel biologically-inspired algorithm, namely krill herd (KH) is proposed for solving optimization tasks. The KH algorithm is based on the simulation of the herding behavior of krill individuals. The minimum distances of each individual krill from food and from highest density of the herd are considered as the objective function for the krill movement. The time-dependent position of the krill individuals is formulated by three main factors: (i) movement induced by the presence of other individuals (ii) foraging activity, and (iii) random diffusion. For more precise modeling of the krill behavior, two adaptive genetic operators are added to the algorithm. The proposed method is verified using several benchmark problems commonly used in the area of optimization. Further, the KH algorithm is compared with eight well-known methods in the literature. The KH algorithm is capable of efficiently solving a wide range of benchmark optimization problems and outperforms the exciting algorithms.

This paper presents two new lemmas related to the Caputo fractional derivatives, when , for the case of general quadratic forms and for the case where the trace of the product of a rectangular matrix and its transpose appear. Those two lemmas allow using general quadratic Lyapunov functions and the trace of a matrix inside a Lyapunov function respectively, in order to apply the fractional-order extension of Lyapunov direct method, to analyze the stability of fractional order systems (FOS). Besides, the paper presents a theorem for proving uniform stability in the sense of Lyapunov for fractional order systems. The theorem can be seen as a complement of other methods already available in the literature. The two lemmas and the theorem are applied to the stability analysis of two Fractional Order Model Reference Adaptive Control (FOMRAC) schemes, in order to prove the usefulness of the results.

Fractional calculus is at this stage an arena where many models are still to be introduced, discussed and applied to real world applications in many branches of science and engineering where nonlocality plays a crucial role. Although researchers have already reported many excellent results in several seminal monographs and review articles, there are still a large number of non-local phenomena unexplored and waiting to be discovered. Therefore, year by year, we can discover new aspects of the fractional modeling and applications. This review article aims to present some short summaries written by distinguished researchers in the field of fractional calculus. We believe this incomplete, but important, information will guide young researchers and help newcomers to see some of the main real-world applications and gain an understanding of this powerful mathematical tool. We expect this collection will also benefit our community.

This survey intends to report some of the major documents and events in the area of fractional calculus that took place since 1974 up to the present date.

► Novel Chaotic Improved Firefly Algorithms (CFAs) are presented for global optimization. ► Twelve different chaotic maps are utilized to improve the attraction term of the algorithm. ► Comparing the new chaotic algorithms with the standard FA demonstrates superiority of the CFAs for the benchmark functions. A recently developed metaheuristic optimization algorithm, firefly algorithm (FA), mimics the social behavior of fireflies based on the flashing and attraction characteristics of fireflies. In the present study, we will introduce chaos into FA so as to increase its global search mobility for robust global optimization. Detailed studies are carried out on benchmark problems with different chaotic maps. Here, 12 different chaotic maps are utilized to tune the attractive movement of the fireflies in the algorithm. The results show that some chaotic FAs can clearly outperform the standard FA.

In this paper, an optimal homotopy-analysis approach is described by means of the nonlinear Blasius equation as an example. This optimal approach contains at most three convergence-control parameters and is computationally rather efficient. A new kind of averaged residual error is defined, which can be used to find the optimal convergence-control parameters much more efficiently. It is found that all optimal homotopy-analysis approaches greatly accelerate the convergence of series solution. And the optimal approaches with one or two unknown convergence-control parameters are strongly suggested. This optimal approach has general meanings and can be used to get fast convergent series solutions of different types of equations with strong nonlinearity.

In this paper a Lorenz-like system, describing convective fluid motion in rotating cavity, is considered. It is shown numerically that this system, like the classical Lorenz system, possesses a homoclinic trajectory and a chaotic self-excited attractor. However, for the considered system, unlike the classical Lorenz system, along with self-excited attractor a hidden attractor can be localized. Analytical-numerical localization of hidden attractor is demonstrated.

In this paper, a combination of stripe soliton and lump soliton is discussed to a reduced (3+1)-dimensional Jimbo–Miwa equation, in which such solution gives rise to two different excitation phenomena: fusion and fission. Particularly, a new combination of positive quadratic functions and hyperbolic functions is considered, and then a novel nonlinear phenomenon is explored. Via this method, a pair of resonance kink stripe solitons and rogue wave is studied. Rogue wave is triggered by the interaction between lump soliton and a pair of resonance kink stripe solitons. It is exciting that rogue wave must be attached to the stripe solitons from its appearing to disappearing. The whole progress is completely symmetry, the rogue wave starts itself from one stripe soliton and lose itself in another stripe soliton. The dynamic properties of the interaction between one stripe soliton and lump soliton, rogue wave are discussed by choosing appropriate parameters.

This paper aims to reveal the nonlinear dynamic mechanism of asymmetric tristable energy harvesters and enhance the energy harvesting performance for different excitations. The general harmonic balance solutions of asymmetric tristable energy harvesters and the corresponding Jacobian matrix for estimating the stability of these analytical solutions are deduced. It is found that asymmetric tristable energy harvesters have seven analytical solutions (four stable solutions) under the appropriate excitation frequency and amplitude, which are verified by using basins of attraction. The influence mechanism of asymmetry of potential wells on energy harvesting performance is analyzed. The results show that the potential barrier is a key factor to determine the orbit height of high-energy interwell oscillations, which influences the amplitude of the output voltage. The influence essence of asymmetry on tristable energy harvesters is to change their potential wells and adjust the distribution of their potential energy. By changing the unstable equilibrium positions, a series of tristable energy harvesters with different dynamic characteristics can be obtained, which will benefit energy harvesting under various excitation conditions.

This paper investigates the problem of finite-time synchronization for complex networks with time-varying delays and semi-Markov jump topology. The network topologies are assumed to switch from one to another at different instants. Such a switching is governed by a semi-Markov process which are time-varying and dependent on the sojourn-time . Attention is focused on proposing some synchronization criteria guaranteeing the underlying network is stochastically finite-time synchronized. By using the properties of Kronecker product combined with the Lyapunov–Krasovskii method, the solutions to the finite-time synchronization problem are formulated in the form of low-dimensional linear matrix inequalities. Finally, a numerical example is given to demonstrate the effectiveness of our proposed approach.

► Flow and heat transfer of a nanofluid over a stretching sheet are studied. ► Nanofluid flow with Brownian motion and Thermophoresis effects is investigated. ► FEM with linear/quadratic basis functions has been implemented. ► Excellent correlation has been achieved with FDM and earlier published results. ► Effects of various parameters on heat and mass transfer rate are shown. Steady, laminar boundary fluid flow which results from the non-linear stretching of a flat surface in a nanofluid has been investigated numerically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The resulting non-linear governing equations with associated boundary conditions are solved using variational finite element method (FEM) with a local non-similar transformation. The influence of Brownian motion number ( ), thermophoresis number ( ), stretching parameter ( ) and Lewis number ( ) on the temperature and nanoparticle concentration profiles are shown graphically. The impact of physical parameters on rate of heat transfer (− ′(0)) and mass transfer (− ′(0)) is shown in tabulated form. Some of results have also been compared with explicit finite difference method (FDM). Excellent validation of the present numerical results has been achieved with the earlier nonlinearly stretching sheet problem of Cortell for local Nusselt number without taking the effect of Brownian motion and thermophoresis.

This paper investigates the problem of finite-time H-infinity synchronization for complex networks with time-varying delays and semi-Markov jump topology. The network topologies are assumed to switch from one to another at different instants. Such a switching is governed by a semi-Markov process which are time-varying and dependent on the sojourn-time h. Attention is focused on proposing some synchronization criteria guaranteeing the underlying network is stochastically finite-time H-infinity synchronized. By using the properties of Kronecker product combined with the Lyapunov-Krasovskii method, the solutions to the finite-time H-infinity synchronization problem are formulated in the form of low-dimensional linear matrix inequalities. Finally, a numerical example is given to demonstrate the effectiveness of our proposed approach. (C) 2014 Elsevier B.V. All rights reserved.

We describe, very briefly, the basic ideas and current developments of the homotopy analysis method, an analytic approach to get convergent series solutions of strongly nonlinear problems, which recently attracts interests of more and more researchers. Definitions of some new concepts such as the homotopy-derivative, the convergence-control parameter and so on, are given to redescribe the method more rigorously. Some lemmas and theorems about the homotopy-derivative and the deformation equation are proved. Besides, a few open questions are discussed, and a hypothesis is put forward for future studies.

► Chaos-enhanced accelerated particle swarm optimization algorithms are presented. ► Twelve different chaotic maps are utilized to tune the main parameter of the accelerated particle swarm optimization. ► Chaotic accelerated particle swarm optimization algorithm outperforms the non-chaotic one. ► It has very good performance in comparison with other chaotic algorithms. ► To validate, a complex engineering problem is also solved. There are more than two dozen variants of particle swarm optimization (PSO) algorithms in the literature. Recently, a new variant, called accelerated PSO (APSO), shows some extra advantages in convergence for global search. In the present study, we will introduce chaos into the APSO in order to further enhance its global search ability. Firstly, detailed studies are carried out on benchmark problems with twelve different chaotic maps to find out the most efficient one. Then the chaotic APSO (CAPSO) will be compared with some other chaotic PSO algorithms presented in the literature. The performance of the CAPSO algorithm is also validated using three engineering problems. The results show that the CAPSO with an appropriate chaotic map can clearly outperform standard APSO, with very good performance in comparison with other algorithms and in application to a complex problem.

This review provides the latest developments and trends in the application of fractional calculus (FC) in biomedicine and biology. Nature has often showed to follow rather simple rules that lead to the emergence of complex phenomena as a result. Of these, the paper addresses the properties in respiratory lung tissue, whose natural solutions arise from the midst of FC in the form of non-integer differ-integral solutions and non-integer parametric models. Diffusion of substances in human body, e.g. drug diffusion, is also a phenomena well known to be captured with such mathematical models. FC has been employed in neuroscience to characterize the generation of action potentials and spiking patters but also in characterizing bio-systems (e.g. vegetable tissues). Despite the natural complexity, biological systems belong as well to this class of systems, where FC has offered parsimonious yet accurate models. This review paper is a collection of results and literature reports who are essential to any versed engineer with multidisciplinary applications and bio-medical in particular.

► Modification of the one of old method for finding exact solutions of nonlinear differential equations is considered. ► Examples of application of method are given. ► Merits and demerits of method are discussed. One of old methods for finding exact solutions of nonlinear differential equations is considered. Modifications of the method are discussed. Application of the method is illustrated for finding exact solutions of the Fisher equation and nonlinear ordinary differential equation of the seven order. It is shown that the method is one of the most effective approaches for finding exact solutions of nonlinear differential equations. Merits and demerits of the method are discussed.

► This paper reports a surprising discovery of a simple chaotic system with only one stable equilibrium point. ► The Ši’lnikov homoclinic criterion is not applicable for the new system. ► The attracting basin of the stable equilibrium expands gradually as the parameter increases. If you are given a simple three-dimensional autonomous quadratic system that has only one stable equilibrium, what would you predict its dynamics to be, stable or periodic? Will it be surprising if you are shown that such a system is actually chaotic? Although chaos theory for three-dimensional autonomous systems has been intensively and extensively studied since the time of Lorenz in the 1960s, and the theory has become quite mature today, it seems that no one would anticipate a possibility of finding a three-dimensional autonomous quadratic chaotic system with only one stable equilibrium. The discovery of the new system, to be reported in this Letter, is indeed striking because for a three-dimensional autonomous quadratic system with a single stable node-focus equilibrium, one typically would anticipate non-chaotic and even asymptotically converging behaviors. Although the equilibrium is changed from an unstable saddle-focus to a stable node-focus, therefore the familiar Ši’lnikov homoclinic criterion is not applicable, it is demonstrated to be chaotic in the sense of having a positive largest Lyapunov exponent, a fractional dimension, a continuous broad frequency spectrum, and a period-doubling route to chaos.

► Error is shown in former solutions for impulsive fractional differential equations. ► A correct solution is found for those impulsive Caputo fractional Cauchy problems. ► Some sufficient conditions are established for the existence of solutions. This paper is motivated from some recent papers treating the problem of the existence of a solution for impulsive differential equations with fractional derivative. We firstly show that the formula of solutions in cited papers are incorrect. Secondly, we reconsider a class of impulsive fractional differential equations and introduce a correct formula of solutions for a impulsive Cauchy problem with Caputo fractional derivative. Further, some sufficient conditions for existence of the solutions are established by applying fixed point methods. Some examples are given to illustrate the results.

Chaotic oscillators have been realized using field-programmable gate arrays (FPGAs) showing good results. However, only 2-scrolls have been observed experimentally, and all reported works use commercially-available software tools for FPGA synthesis. In this manner, as a first contribution we show the FPGA realization of two multi-scroll chaotic oscillators that are characterized by their maximum Lyapunov exponent (MLE) for generating from 2- to 6-scrolls. The first multi-scroll chaotic oscillator is based on saturated function series and the second on Chua’s circuit. As a second contribution, we show their hardware realization by applying two numerical methods: Forward Euler (FE) and Runge Kutta (RK). The advantage of realizing those multi-scroll chaotic oscillators is that one can avoid the use of multiplier entities, thus optimizing FPGA resources and increasing the processing speed, as we show by realizing single constant multiplication (SCM) blocks. The experiments are verified by performing co-simulation for an FPGA Spartan 3 of Xilinx. Finally, experimental results are shown for different values of MLE (already optimized) for both multi-scroll chaotic oscillators, and the FPGA used resources are listed for generating 6-scrolls when applying FE and RK.