Dynamics of the Fractional-Order Lorenz System Based on Adomian Decomposition Method and Its DSP Implementation

Dear Editor,Dynamics and digital circuit implementation of the fractional-order Lorenz system are investigated by employing Adomian decomposition method(ADM).Dynamics of the fractional-order Lorenz system with derivative order and parameter varying is analyzed by means of Lyapunov exponents(LEs),bifurcation diagram.

A novel variable-order fractional chaotic map and its dynamics

In recent years,fractional-order chaotic maps have been paid more attention in publications because of the memory effect.This paper presents a novel variable-order fractional sine map(VFSM)based on the discrete fractional calculus.Specially,the order is defined as an iterative function that incorporates the current state of the system.By analyzing phase diagrams,time sequences,bifurcations,Lyapunov exponents and fuzzy entropy complexity,the dynamics of the proposed map are investigated comparing with the constant-order fractional sine map.The results reveal that the variable order has a good effect on improving the chaotic performance,and it enlarges the range of available parameter values as well as reduces non-chaotic windows.Multiple coexisting attractors also enrich the dynamics of VFSM and prove its sensitivity to initial values.Moreover,the sequence generated by the proposed map passes the statistical test for pseudorandom number and shows strong robustness to parameter estimation,which proves the potential applications in the field of information security.

Numerical analysis of a simplest fractional-order hyperchaotic system

In this paper,a simplest fractional-order hyperchaotic(SFOH)system is obtained when the fractional calculus is applied to the piecewise-linear hyperchaotic system,which possesses seven terms without any quadratic or higher-order polynomials.The numerical solution of the SFOH system is investigated based on the Adomian decomposition method(ADM).The methods of segmentation and replacement function are proposed to solve this system and analyze the dynamics.Dynamics of this system are demonstrated by means of phase portraits,bifurcation diagrams,Lyapunov exponent spectrum(LEs)and Poincarésection.The results show that the system has a wide chaotic range with order change,and large Lyapunov exponent when the order is very small,which indicates that the system has a good application prospect.Besides,the parameter a is a partial amplitude controller for the SFOH system.Finally,the system is successfully implemented by digital signal processor(DSP).It lays a foundation for the application of the SFOH system.

A memristive map with coexisting chaos and hyperchaos

By introducing a discrete memristor and periodic sinusoidal functions,a two-dimensional map with coexisting chaos and hyperchaos is constructed.Various coexisting chaotic and hyperchaotic attractors under different Lyapunov exponents are firstly found in this discrete map,along with which other regimes of coexistence such as coexisting chaos,quasiperiodic oscillation,and discrete periodic points are also captured.The hyperchaotic attractors can be flexibly controlled to be unipolar or bipolar by newly embedded constants meanwhile the amplitude can also be controlled in combination with those coexisting attractors.Based on the nonlinear auto-regressive model with exogenous inputs(NARX)for neural network,the dynamics of the memristive map is well predicted,which provides a potential passage in artificial intelligencebased applications.

Solutions and memory effect of fractional-order chaotic system:A review

Fractional calculus is a 300 years topic,which has been introduced to real physics systems modeling and engineering applications.In the last few decades,fractional-order nonlinear chaotic systems have been widely investigated.Firstly,the most used methods to solve fractional-order chaotic systems are reviewed.Characteristics and memory effect in those method are summarized.Then we discuss the memory effect in the fractional-order chaotic systems through the fractionalorder calculus and numerical solution algorithms.It shows that the integer-order derivative has full memory effect,while the fractional-order derivative has nonideal memory effect due to the kernel function.Memory loss and short memory are discussed.Finally,applications of the fractional-order chaotic systems regarding the memory effects are investigated.The work summarized in this manuscript provides reference value for the applied scientists and engineering community of fractional-order nonlinear chaotic systems.

A novel hyperchaotic map with sine chaotification and discrete memristor

Discrete memristor has become a hotspot since it was proposed recently.However,the design of chaotic maps based on discrete memristor is in its early research stage.In this paper,a memristive seed chaotic map is proposed by combining a quadratic discrete memristor with the sine function.Furthermore,by applying the chaotification method,we obtain a high-dimensional chaotic map.Numerical analysis shows that it can generate hyperchaos.With the increase of cascade times,the generated map has more positive Lyapunov exponents and larger hyperchaotic range.The National Institute of Standards and Technology(NIST)test results show that the chaotic pseudo-random sequence generated by cascading two seed maps has good unpredictability,and it indicates the potential in practical application.

Effect of AlCl_3 concentration on nanoparticle removal by coagulation

In recent years,engineered nanoparticles,as a new group of contaminants emerging in natural water,have been given more attention AlCl3 In order to understand the behavior of nanoparticles in the conventional water treatment process,three kinds of nanoparticle suspensions,namely multi-walled carbon nanotube-humic acid（MWCNT-HA）,multiwalled carbon nanotube-N,N-dimethylformamide（MWCNT-DMF） and nano TiO2-humic acid（TiO2-HA） were employed to investigate their coagulation removal efficiencies with varying aluminum chloride（AlCl3） concentrations AlCl3 Results showed that nanoparticle removal rate curves had a reverse ＂U＂ shape with increasing concentration of aluminum ion（Al^（3＋） ）AlCl3 More than 90% of nanoparticles could be effectively removed by an appropriate Al^（3＋） concentration AlCl3 At higher Al^（3＋） concentration,nanoparticles would be restabilized AlCl3 The hydrodynamic particle size of nanoparticles was found to be the crucial factor influencing the effective concentration range（ECR） of Al^（3＋） for nanoparticle removal AlCl3 The ECR of Al^（3＋） followed the order MWCNT-DMF 〉 MWCNT-HA 〉 TiO2-HA,which is the reverse of the nanoparticle size trend AlCl3 At a given concentration,smaller nanoparticles carry more surface charges,and thus consume more coagulants for neutralization AlCl3 Therefore,over-saturation occurred at relatively higher Al^（3＋） concentration and a wider ECR was obtained AlCl3 The ECR became broader with increasing p H because of the smaller hydrodynamic particle size of nanoparticles at higher p H values AlCl3 A high ionic strength of Na Cl can also widen the ECR due to its strong potential to compress the electric double layer AlCl3 It was concluded that it is important to adjust the dose of Al^（3＋） in the ECR for nanoparticle removal in water treatment.