Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridyna...Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.展开更多
For any given positive integer n ≥ 1, the Euler function φ（n） is defined to be the number of positive integers not exceeding n which are relatively prime to n. w（n） is defined to be the number of different prime...For any given positive integer n ≥ 1, the Euler function φ（n） is defined to be the number of positive integers not exceeding n which are relatively prime to n. w（n） is defined to be the number of different prime divisors of n. Some kind of equations involving Euler＇s function is studied in the paper.展开更多
Dynamical variables of coupled nonlinear oscillators can exhibit different synchronization patterns depending on the designed coupling scheme. In this paper, a non-fragile linear feedback control strategy with multipl...Dynamical variables of coupled nonlinear oscillators can exhibit different synchronization patterns depending on the designed coupling scheme. In this paper, a non-fragile linear feedback control strategy with multiplicative controller gain uncertainties is proposed for realizing the mixed-synchronization of Chua＇s circuits connected in a drive-response configuration. In particular, in the mixed-synchronization regime, different state variables of the response system can evolve into complete synchronization, anti-synchronization and even amplitude death simultaneously with the drive variables for an appropriate choice of scaling matrix. Using Lyapunov stability theory, we derive some sufficient criteria for achieving global mixed-synchronization. It is shown that the desired non-fragile state feedback controller can be constructed by solving a set of linear matrix inequalities （LMIs）. Numerical simulations are also provided to demonstrate the effectiveness of the proposed control approach.展开更多
We formulate a “Moore’s law” for photonic integrated circuits (PICs) and their spatial integration density using two methods. One is decomposing the integrated photonics devices of diverse types into equivalent bas...We formulate a “Moore’s law” for photonic integrated circuits (PICs) and their spatial integration density using two methods. One is decomposing the integrated photonics devices of diverse types into equivalent basic elements, which makes a comparison with the generic elements of electronic integrated circuits more meaningful. The other is making a complex compo- nent equivalent to a series of basic elements of the same functionality, which is used to calculate the integration density for func- tional components realized with different structures. The results serve as a benchmark of the evolution of PICs and we can con- clude that the density of integration measured in this way roughly increases by a factor of 2 per year. The prospects for a continued increase of spatial integration density are discussed.展开更多
This paper considers the chaos synchronization of the modified Chua 's circuit with x|x| function. We firstly show that a couple of the modified Chua systems with different parameters and initial conditions can be...This paper considers the chaos synchronization of the modified Chua 's circuit with x|x| function. We firstly show that a couple of the modified Chua systems with different parameters and initial conditions can be synchronized using active control when the values of parameters both in drive system and response system are known aforehand.Furthermore, based on Lyapunov stability theory we propose an adaptive active control approach to make the states of two identical Chua systems with unknown constant parameters asymptotically synchronized. Moreover the designed controller is independent of those unknown parameters. Numerical simulations are given to validate the proposed synchronization approach.展开更多
In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and dedu...In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and deduce the corresponding special cases.展开更多
The through silicon via （TSV） technology has proven to be the critical enabler to realize a three-dimensional （3D） gigscale system with higher performance but shorter interconnect length. However, the received dig...The through silicon via （TSV） technology has proven to be the critical enabler to realize a three-dimensional （3D） gigscale system with higher performance but shorter interconnect length. However, the received digital signal after trans- mission through a TSV channel, composed of redistribution layers （RDLs）, TSVs, and bumps, is degraded at a high data-rate due to the non-idealities of the channel. We propose the Chebyshev multisection transformers to reduce the signal reflec- tion of TSV channel when operating frequency goes up to 20 GHz, by which signal reflection coefficient （$11） and signal transmission coefficient （$21） are improved remarkably by 150% and 73.3%, respectively. Both the time delay and power dissipation are also reduced by 4% and 13.3%, respectively. The resistance-inductance-conductance-capacitance （RLGC） elements of the TSV channel are iterated from scattering （S）-parameters, and the proposed method of weakening the signal reflection is verified using high frequency simulator structure （HFSS） simulation software by Ansoft.展开更多
The ultimate proof of our understanding of nature and engineering systems is reflected in our ability to control them.Since fractional calculus is more universal, we bring attention to the controllability of fractiona...The ultimate proof of our understanding of nature and engineering systems is reflected in our ability to control them.Since fractional calculus is more universal, we bring attention to the controllability of fractional order systems. First,we extend the conventional controllability theorem to the fractional domain. Strictly mathematical analysis and proof are presented. Because Chua＇s circuit is a typical representative of nonlinear circuits, we study the controllability of the fractional order Chua＇s circuit in detail using the presented theorem. Numerical simulations and theoretical analysis are both presented, which are in agreement with each other.展开更多
Based on the α-β bifurcation curves and the special characteristics of chaotic spectrum of chua’s circuit, this paper presents here a method for designing a Chua’s circuit which approximately satisfy specified spe...Based on the α-β bifurcation curves and the special characteristics of chaotic spectrum of chua’s circuit, this paper presents here a method for designing a Chua’s circuit which approximately satisfy specified spectrum distribution range.展开更多
The fast-slow effect can be observed in a typical non-smooth electric circuit with order gap between the natural frequency and the excitation frequency. Numerical simulations are employed to show complicated behaviour...The fast-slow effect can be observed in a typical non-smooth electric circuit with order gap between the natural frequency and the excitation frequency. Numerical simulations are employed to show complicated behaviours, especially different types of busting phenomena. The bifurcation mechanism for the bursting solutions is analysed by assuming the forms of the solutions and introducing the generalized Jacobian matrix at the non-smooth boundaries, which can also be used to account for the evolution of the complicated structures of the phase portraits with the variation of the parameter. Period-adding bifurcation has been explored through the computation of the eigenvalues related to the solutions. At the non-smooth boundaries the so-called ＇single crossing bifurcation＇ can occur, corresponding to the case where the eigenvalues jump only once across the imaginary axis, which leads the periodic burster to have a quasi-periodic oscillation.展开更多
Chua＇s circuit is a well-known nonlinear electronic model, having complicated nonsmooth dynamic behaviors. The stability and boundary equilibrium bifurcations for a modified Chua＇s circuit system with the smooth deg...Chua＇s circuit is a well-known nonlinear electronic model, having complicated nonsmooth dynamic behaviors. The stability and boundary equilibrium bifurcations for a modified Chua＇s circuit system with the smooth degree of 3 are studied. The parametric areas of stability are specified in detail. It is found that the bifurcation graphs of the su- percritical and irregular pitchfork bifurcations are similar to those of the piecewise-smooth continuous （PWSC） systems caused by piecewise smoothness. However, the bifurcation graph of the supercritical Hopf bifurcation is similar to those of smooth systems. There- fore, the boundary equilibrium bifurcations of the non-smooth systems with the smooth degree of 3 should receive more attention due to their special features.展开更多
The eigenvalue space of the canonical four-dimensional Chua＇s circuit which can realize every eigenvalue for fourdimensional system is studied in this paper. First, the analytical relations between the circuit parame...The eigenvalue space of the canonical four-dimensional Chua＇s circuit which can realize every eigenvalue for fourdimensional system is studied in this paper. First, the analytical relations between the circuit parameters and the eigenvalues of the system are established, and therefore all the circuit parameters can be determined explicitly by any given set of eigenvalues. Then, the eigenvalue space of the circuit is investigated in two cases by the nonlinear elements used. According to the types of the eigenvalues, some novel hyperchaotic attractors are presented. Further, the dynamic behaviours of the circuit are studied by the bifurcation diagrams and the Lyapunov spectra of the eigenvalues.展开更多
文摘Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.
基金Foundation item： Supported by the National Natural Science Foundation of China（10671056）
文摘For any given positive integer n ≥ 1, the Euler function φ（n） is defined to be the number of positive integers not exceeding n which are relatively prime to n. w（n） is defined to be the number of different prime divisors of n. Some kind of equations involving Euler＇s function is studied in the paper.
基金Project supported by the Foundation for Distinguished Young Talents in Higher Education of Guangdong Province of China(Grant No. LYM10074)the Natural Science Foundation of Guangdong Province,China (Grant No. 9451042001004076)
文摘Dynamical variables of coupled nonlinear oscillators can exhibit different synchronization patterns depending on the designed coupling scheme. In this paper, a non-fragile linear feedback control strategy with multiplicative controller gain uncertainties is proposed for realizing the mixed-synchronization of Chua＇s circuits connected in a drive-response configuration. In particular, in the mixed-synchronization regime, different state variables of the response system can evolve into complete synchronization, anti-synchronization and even amplitude death simultaneously with the drive variables for an appropriate choice of scaling matrix. Using Lyapunov stability theory, we derive some sufficient criteria for achieving global mixed-synchronization. It is shown that the desired non-fragile state feedback controller can be constructed by solving a set of linear matrix inequalities （LMIs）. Numerical simulations are also provided to demonstrate the effectiveness of the proposed control approach.
文摘We formulate a “Moore’s law” for photonic integrated circuits (PICs) and their spatial integration density using two methods. One is decomposing the integrated photonics devices of diverse types into equivalent basic elements, which makes a comparison with the generic elements of electronic integrated circuits more meaningful. The other is making a complex compo- nent equivalent to a series of basic elements of the same functionality, which is used to calculate the integration density for func- tional components realized with different structures. The results serve as a benchmark of the evolution of PICs and we can con- clude that the density of integration measured in this way roughly increases by a factor of 2 per year. The prospects for a continued increase of spatial integration density are discussed.
文摘This paper considers the chaos synchronization of the modified Chua 's circuit with x|x| function. We firstly show that a couple of the modified Chua systems with different parameters and initial conditions can be synchronized using active control when the values of parameters both in drive system and response system are known aforehand.Furthermore, based on Lyapunov stability theory we propose an adaptive active control approach to make the states of two identical Chua systems with unknown constant parameters asymptotically synchronized. Moreover the designed controller is independent of those unknown parameters. Numerical simulations are given to validate the proposed synchronization approach.
基金Supported by the PCSIRT of Education of China(IRT0621)Supported by the Innovation Program of Shanghai Municipal Education Committee of China(08ZZ24)Supported by the Henan Innovation Project for University Prominent Research Talents of China(2007KYCX0021)
文摘In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and deduce the corresponding special cases.
基金Project supported by the National Natural Science Foundation of China(Grant No.61204044)
文摘The through silicon via （TSV） technology has proven to be the critical enabler to realize a three-dimensional （3D） gigscale system with higher performance but shorter interconnect length. However, the received digital signal after trans- mission through a TSV channel, composed of redistribution layers （RDLs）, TSVs, and bumps, is degraded at a high data-rate due to the non-idealities of the channel. We propose the Chebyshev multisection transformers to reduce the signal reflec- tion of TSV channel when operating frequency goes up to 20 GHz, by which signal reflection coefficient （$11） and signal transmission coefficient （$21） are improved remarkably by 150% and 73.3%, respectively. Both the time delay and power dissipation are also reduced by 4% and 13.3%, respectively. The resistance-inductance-conductance-capacitance （RLGC） elements of the TSV channel are iterated from scattering （S）-parameters, and the proposed method of weakening the signal reflection is verified using high frequency simulator structure （HFSS） simulation software by Ansoft.
基金supported by the National Natural Science Foundation of China(Grant Nos.51109180 and 51479173)the Fundamental Research Funds for the Central Universities,China(Grant No.201304030577)+1 种基金the Northwest A&F University Foundation,China(Grant No.2013BSJJ095)the Scientific Research Foundation on Water Engineering of Shaanxi Province,China(Grant No.2013slkj-12)
文摘The ultimate proof of our understanding of nature and engineering systems is reflected in our ability to control them.Since fractional calculus is more universal, we bring attention to the controllability of fractional order systems. First,we extend the conventional controllability theorem to the fractional domain. Strictly mathematical analysis and proof are presented. Because Chua＇s circuit is a typical representative of nonlinear circuits, we study the controllability of the fractional order Chua＇s circuit in detail using the presented theorem. Numerical simulations and theoretical analysis are both presented, which are in agreement with each other.
文摘Based on the α-β bifurcation curves and the special characteristics of chaotic spectrum of chua’s circuit, this paper presents here a method for designing a Chua’s circuit which approximately satisfy specified spectrum distribution range.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10972091 and 10872080)
文摘The fast-slow effect can be observed in a typical non-smooth electric circuit with order gap between the natural frequency and the excitation frequency. Numerical simulations are employed to show complicated behaviours, especially different types of busting phenomena. The bifurcation mechanism for the bursting solutions is analysed by assuming the forms of the solutions and introducing the generalized Jacobian matrix at the non-smooth boundaries, which can also be used to account for the evolution of the complicated structures of the phase portraits with the variation of the parameter. Period-adding bifurcation has been explored through the computation of the eigenvalues related to the solutions. At the non-smooth boundaries the so-called ＇single crossing bifurcation＇ can occur, corresponding to the case where the eigenvalues jump only once across the imaginary axis, which leads the periodic burster to have a quasi-periodic oscillation.
基金supported by the National Natural Science Foundation of China(Nos.U1204106,11372282,11272024,and 11371046)the National Basic Research Program of China(973 Program)(Nos.2012CB821200 and 2012CB821202)
文摘Chua＇s circuit is a well-known nonlinear electronic model, having complicated nonsmooth dynamic behaviors. The stability and boundary equilibrium bifurcations for a modified Chua＇s circuit system with the smooth degree of 3 are studied. The parametric areas of stability are specified in detail. It is found that the bifurcation graphs of the su- percritical and irregular pitchfork bifurcations are similar to those of the piecewise-smooth continuous （PWSC） systems caused by piecewise smoothness. However, the bifurcation graph of the supercritical Hopf bifurcation is similar to those of smooth systems. There- fore, the boundary equilibrium bifurcations of the non-smooth systems with the smooth degree of 3 should receive more attention due to their special features.
基金Project supported by the National Natural Science Foundation of China (Grant No. 50877007)
文摘The eigenvalue space of the canonical four-dimensional Chua＇s circuit which can realize every eigenvalue for fourdimensional system is studied in this paper. First, the analytical relations between the circuit parameters and the eigenvalues of the system are established, and therefore all the circuit parameters can be determined explicitly by any given set of eigenvalues. Then, the eigenvalue space of the circuit is investigated in two cases by the nonlinear elements used. According to the types of the eigenvalues, some novel hyperchaotic attractors are presented. Further, the dynamic behaviours of the circuit are studied by the bifurcation diagrams and the Lyapunov spectra of the eigenvalues.
