A concrete numerical example of Z6-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 having at least 24 limit cycles and the configurations of compound eyes are given by using the bifurcati...A concrete numerical example of Z6-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 having at least 24 limit cycles and the configurations of compound eyes are given by using the bifurcation theory of planar dynamical systems and the method of detection functions. There is reason to conjecture that the Hilbert number H(2k + 1) ? (2k + I)2 - 1 for the perturbed Hamiltonian systems.展开更多
Based on the homotopy function, a new iterative algorithm for solving nonlinear polynomial systems called homotopy iteration method has been developed. All solutions can be found easily without the need of choosing pr...Based on the homotopy function, a new iterative algorithm for solving nonlinear polynomial systems called homotopy iteration method has been developed. All solutions can be found easily without the need of choosing proper initial values. Compared to the homotopy continuation method, the solution process of the present method is much simplified, and the computation efficiency as well as the reliability for obtaining all solutions are also improved. Applicating the method to a number of problems coming from engineering, the results are satisfactory.展开更多
The paper proposes an approach to transmit electric power system dynamics in the SCADA. With the prevalent application of digital substation automation system, it is feasible for the remote terminal units (RTUs) to co...The paper proposes an approach to transmit electric power system dynamics in the SCADA. With the prevalent application of digital substation automation system, it is feasible for the remote terminal units (RTUs) to collect phasors within a substation. However, limited communication capacity remains the bottleneck that prevents SCADA from transmitting system dynamics. This paper proposes to compress dynamics data with curve fitting in the RTUs and reconstruct the dynamics in the SCADA server for reducing communication demand. Dispatchers in the control center can thus get system dynamics with a delay of several seconds. Simulation result shows that for a power system under disturbance with short-circuit that once occurred and was cleared, the SCADA can approximate the original dynamics with satisfying precision using limited degree polynomial fitting. The approach is highly scalable and adaptable, and can be implemented on existing communication infrastructure with a few software modifications. The approach has extensive application potential.展开更多
Consider the polynomial differential system of degree m of the form x=-y(1+μ(a_(2)x-a_(1)y))+x(v(a_(1)x+a_(2)y)+Ω_(m-1)(x,y)),y=x(1+μ(a_(2)x-a_(1)y))+y(v(a_(1)x+a_(2)y)+Ω_(m-1)(x,y)),whereμandνare real numbers s...Consider the polynomial differential system of degree m of the form x=-y(1+μ(a_(2)x-a_(1)y))+x(v(a_(1)x+a_(2)y)+Ω_(m-1)(x,y)),y=x(1+μ(a_(2)x-a_(1)y))+y(v(a_(1)x+a_(2)y)+Ω_(m-1)(x,y)),whereμandνare real numbers such that(μ^(2)+v^(2))(μ+v(m-2))(a_(1)^(2)+a_(2)^(2))≠m>2 andΩ_(m−1)(x,y)is a homogenous polynomial of degree m−1.A conjecture,stated in J.Differential Equations 2019,suggests that whenν=1,this differential system has a weak center at the origin if and only if after a convenient linear change of variable(x,y)→(X,Y)the system is invariant under the transformation(X,Y,t)→(−X,Y,−t).For every degree m we prove the extension of this conjecture to any value ofνexcept for a finite set of values ofμ.展开更多
In this article, we study the expansion of the first order Melnikov function in a near-Hamiltonian system on the plane near a double homoclinic loop. We obtain an explicit formula to compute the first four coeffcients...In this article, we study the expansion of the first order Melnikov function in a near-Hamiltonian system on the plane near a double homoclinic loop. We obtain an explicit formula to compute the first four coeffcients, and then identify the method of finding at least 7 limit cycles near the double homoclinic loop using these coefficients. Finally, we present some interesting applications.展开更多
In this paper the development of reactive software is transformed into a control problem, and the supervisory control theory for discrete event dynamic systems is suggested to solve this control problem. The operating...In this paper the development of reactive software is transformed into a control problem, and the supervisory control theory for discrete event dynamic systems is suggested to solve this control problem. The operating environment under consideration is viewed as a controlled plant, the software under development as the corresponding controller, and the software requirements as the corresponding control objective. This idea leads to a constructive approach of software design, which ensures properties required a priori of the software under development. In this way the validation of the software under development is reduced to the validation of properties independent of implementation process. We reveal the inconsistence in using the concept of reachability to specify software requirements and clarify six different definitions of reachability. Two different definitions of invariance for specifying software requirements are also clarified. We then show how to synthesize the required controller or obtain software design solutions if the underlying software requirements are specified by several new combinations of reachability and invariance. The topic of this paper falls into the scope of software cybernetics that explores the interplay between software and control.展开更多
This paper presents a quantum algorithm to decide whether a Boolean equation system F has a solution and to compute one if F does have solutions with any given success probability.The runtime complexity of the algorit...This paper presents a quantum algorithm to decide whether a Boolean equation system F has a solution and to compute one if F does have solutions with any given success probability.The runtime complexity of the algorithm is polynomial in the size of F and the condition number of certain Macaulay matrix associated with F.As a consequence,the authors give a polynomial-time quantum algorithm for solving Boolean equation systems if their condition numbers are polynomial in the size of F.The authors apply the proposed quantum algorithm to the cryptanalysis of several important cryptosystems:The stream cipher Trivum,the block cipher AES,the hash function SHA-3/Keccak,the multivariate public key cryptosystems,and show that they are secure under quantum algebraic attack only if the corresponding condition numbers are large.This leads to a new criterion for designing such cryptosystems which are safe against the attack of quantum computers:The corresponding condition number.展开更多
研究了饱和输入下一类多项式系统的数据驱动控制问题.针对一类受噪声影响的未知多项式系统,设计了一种直接基于数据驱动的饱和控制器实现系统镇定,其中系统噪声具体形式未知但存在二次型上界.首先,在随机输入作用下离线采集系统的输入...研究了饱和输入下一类多项式系统的数据驱动控制问题.针对一类受噪声影响的未知多项式系统,设计了一种直接基于数据驱动的饱和控制器实现系统镇定,其中系统噪声具体形式未知但存在二次型上界.首先,在随机输入作用下离线采集系统的输入与输出数据.然后,通过采集到的离线数据以及平方和方法确定系统状态反馈控制器增益,引入的饱和约束机制可以对控制输入进行限幅.最后,通过在van der Pol振荡器系统中的数值仿真和电路实验验证了所提出控制策略的有效性.展开更多
Let m, n and k be positive integers, m>1 and k≤n . Let Z m =Z/ m Z denote the residue class ring modulo m . Let Z denote the group of additive characters of ring Z m , x 0 denote the identity of Z . A necessary an...Let m, n and k be positive integers, m>1 and k≤n . Let Z m =Z/ m Z denote the residue class ring modulo m . Let Z denote the group of additive characters of ring Z m , x 0 denote the identity of Z . A necessary and sufficient condition for the system f\-1, ..., f\-k∈ Z m [ x\-1, ..., x\-n ] to be orthogonal is given: for all additive characters ψ 1, ..., ψ k ∈Z with ψ i≠x 0 for some i, 1≤i≤k, ∑(c\-1, ..., c\-n)∈ Z n m ψ(f\-1(c\-1, ..., c\-n))...ψ k(f\-k(c\-1, ..., c\-n))=0. As corollaries, the result of Shiue_Sun_Zhang and the result of Sun are obtained.展开更多
In this paper,we present a semi-Lagrangian(SL)method based on a non-polynomial function space for solving the Vlasov equation.We fnd that a non-polynomial function based scheme is suitable to the specifcs of the targe...In this paper,we present a semi-Lagrangian(SL)method based on a non-polynomial function space for solving the Vlasov equation.We fnd that a non-polynomial function based scheme is suitable to the specifcs of the target problems.To address issues that arise in phase space models of plasma problems,we develop a weighted essentially non-oscillatory(WENO)scheme using trigonometric polynomials.In particular,the non-polynomial WENO method is able to achieve improved accuracy near sharp gradients or discontinuities.Moreover,to obtain a high-order of accuracy in not only space but also time,it is proposed to apply a high-order splitting scheme in time.We aim to introduce the entire SL algorithm with high-order splitting in time and high-order WENO reconstruction in space to solve the Vlasov-Poisson system.Some numerical experiments are presented to demonstrate robustness of the proposed method in having a high-order of convergence and in capturing non-smooth solutions.A key observation is that the method can capture phase structure that require twice the resolution with a polynomial based method.In 6D,this would represent a signifcant savings.展开更多
This paper presents a subspace identification method for closed-loop systems with unknown deterministic disturbances.To deal with the unknown deterministic disturbances,two strategies are implemented to construct the ...This paper presents a subspace identification method for closed-loop systems with unknown deterministic disturbances.To deal with the unknown deterministic disturbances,two strategies are implemented to construct the row space that can be used to approximately represent the unknown deterministic disturbances using the trigonometric functions or Bernstein polynomials depending on whether the disturbance frequencies are known.For closed-loop identification,CCF-N4SID is extended to the case with unknown deterministic disturbances using the oblique projection.In addition,a proper Bernstein polynomial order can be determined using the Akaike information criterion(AIC)or the Bayesian information criterion(BIC).Numerical simulation results demonstrate the effectiveness of the proposed identification method for both periodic and aperiodic deterministic disturbances.展开更多
基金This work was supported by the Strategic Research (Grant No. 7000934) from the City University of Hong Kong.
文摘A concrete numerical example of Z6-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 5 having at least 24 limit cycles and the configurations of compound eyes are given by using the bifurcation theory of planar dynamical systems and the method of detection functions. There is reason to conjecture that the Hilbert number H(2k + 1) ? (2k + I)2 - 1 for the perturbed Hamiltonian systems.
文摘Based on the homotopy function, a new iterative algorithm for solving nonlinear polynomial systems called homotopy iteration method has been developed. All solutions can be found easily without the need of choosing proper initial values. Compared to the homotopy continuation method, the solution process of the present method is much simplified, and the computation efficiency as well as the reliability for obtaining all solutions are also improved. Applicating the method to a number of problems coming from engineering, the results are satisfactory.
基金Supported by the State Key Development Program for Basic Research of China (Grant No. 2009CB219701)National Natural Science Foundation of China (Grant No. 50595414)Youth Scientific and Technological Innovation Project of CSEE
文摘The paper proposes an approach to transmit electric power system dynamics in the SCADA. With the prevalent application of digital substation automation system, it is feasible for the remote terminal units (RTUs) to collect phasors within a substation. However, limited communication capacity remains the bottleneck that prevents SCADA from transmitting system dynamics. This paper proposes to compress dynamics data with curve fitting in the RTUs and reconstruct the dynamics in the SCADA server for reducing communication demand. Dispatchers in the control center can thus get system dynamics with a delay of several seconds. Simulation result shows that for a power system under disturbance with short-circuit that once occurred and was cleared, the SCADA can approximate the original dynamics with satisfying precision using limited degree polynomial fitting. The approach is highly scalable and adaptable, and can be implemented on existing communication infrastructure with a few software modifications. The approach has extensive application potential.
基金Supported by Grant NNSF of China(Grant No.12171491)the Ministerio de Ciencia,Innovación y Universidades,Agencia Estatal de Investigación grants MTM2016-77278-P(FEDER)and PID2019-104658GB-I00(FEDER)+1 种基金the Agència de Gestiód’Ajuts Universitaris i de Recerca grant 2017SGR1617the H2020 European Research Council grant MSCA-RISE-2017-777911。
文摘Consider the polynomial differential system of degree m of the form x=-y(1+μ(a_(2)x-a_(1)y))+x(v(a_(1)x+a_(2)y)+Ω_(m-1)(x,y)),y=x(1+μ(a_(2)x-a_(1)y))+y(v(a_(1)x+a_(2)y)+Ω_(m-1)(x,y)),whereμandνare real numbers such that(μ^(2)+v^(2))(μ+v(m-2))(a_(1)^(2)+a_(2)^(2))≠m>2 andΩ_(m−1)(x,y)is a homogenous polynomial of degree m−1.A conjecture,stated in J.Differential Equations 2019,suggests that whenν=1,this differential system has a weak center at the origin if and only if after a convenient linear change of variable(x,y)→(X,Y)the system is invariant under the transformation(X,Y,t)→(−X,Y,−t).For every degree m we prove the extension of this conjecture to any value ofνexcept for a finite set of values ofμ.
基金the National Natural Science Foundation of China (10671127)
文摘In this article, we study the expansion of the first order Melnikov function in a near-Hamiltonian system on the plane near a double homoclinic loop. We obtain an explicit formula to compute the first four coeffcients, and then identify the method of finding at least 7 limit cycles near the double homoclinic loop using these coefficients. Finally, we present some interesting applications.
文摘In this paper the development of reactive software is transformed into a control problem, and the supervisory control theory for discrete event dynamic systems is suggested to solve this control problem. The operating environment under consideration is viewed as a controlled plant, the software under development as the corresponding controller, and the software requirements as the corresponding control objective. This idea leads to a constructive approach of software design, which ensures properties required a priori of the software under development. In this way the validation of the software under development is reduced to the validation of properties independent of implementation process. We reveal the inconsistence in using the concept of reachability to specify software requirements and clarify six different definitions of reachability. Two different definitions of invariance for specifying software requirements are also clarified. We then show how to synthesize the required controller or obtain software design solutions if the underlying software requirements are specified by several new combinations of reachability and invariance. The topic of this paper falls into the scope of software cybernetics that explores the interplay between software and control.
基金supported by the National Natural Science Foundation of China under Grant No.11688101and NKRDP 2018YFA0704705。
文摘This paper presents a quantum algorithm to decide whether a Boolean equation system F has a solution and to compute one if F does have solutions with any given success probability.The runtime complexity of the algorithm is polynomial in the size of F and the condition number of certain Macaulay matrix associated with F.As a consequence,the authors give a polynomial-time quantum algorithm for solving Boolean equation systems if their condition numbers are polynomial in the size of F.The authors apply the proposed quantum algorithm to the cryptanalysis of several important cryptosystems:The stream cipher Trivum,the block cipher AES,the hash function SHA-3/Keccak,the multivariate public key cryptosystems,and show that they are secure under quantum algebraic attack only if the corresponding condition numbers are large.This leads to a new criterion for designing such cryptosystems which are safe against the attack of quantum computers:The corresponding condition number.
文摘研究了饱和输入下一类多项式系统的数据驱动控制问题.针对一类受噪声影响的未知多项式系统,设计了一种直接基于数据驱动的饱和控制器实现系统镇定,其中系统噪声具体形式未知但存在二次型上界.首先,在随机输入作用下离线采集系统的输入与输出数据.然后,通过采集到的离线数据以及平方和方法确定系统状态反馈控制器增益,引入的饱和约束机制可以对控制输入进行限幅.最后,通过在van der Pol振荡器系统中的数值仿真和电路实验验证了所提出控制策略的有效性.
文摘Let m, n and k be positive integers, m>1 and k≤n . Let Z m =Z/ m Z denote the residue class ring modulo m . Let Z denote the group of additive characters of ring Z m , x 0 denote the identity of Z . A necessary and sufficient condition for the system f\-1, ..., f\-k∈ Z m [ x\-1, ..., x\-n ] to be orthogonal is given: for all additive characters ψ 1, ..., ψ k ∈Z with ψ i≠x 0 for some i, 1≤i≤k, ∑(c\-1, ..., c\-n)∈ Z n m ψ(f\-1(c\-1, ..., c\-n))...ψ k(f\-k(c\-1, ..., c\-n))=0. As corollaries, the result of Shiue_Sun_Zhang and the result of Sun are obtained.
基金AFOSR and NSF for their support of this work under grants FA9550-19-1-0281 and FA9550-17-1-0394 and NSF grant DMS 191218。
文摘In this paper,we present a semi-Lagrangian(SL)method based on a non-polynomial function space for solving the Vlasov equation.We fnd that a non-polynomial function based scheme is suitable to the specifcs of the target problems.To address issues that arise in phase space models of plasma problems,we develop a weighted essentially non-oscillatory(WENO)scheme using trigonometric polynomials.In particular,the non-polynomial WENO method is able to achieve improved accuracy near sharp gradients or discontinuities.Moreover,to obtain a high-order of accuracy in not only space but also time,it is proposed to apply a high-order splitting scheme in time.We aim to introduce the entire SL algorithm with high-order splitting in time and high-order WENO reconstruction in space to solve the Vlasov-Poisson system.Some numerical experiments are presented to demonstrate robustness of the proposed method in having a high-order of convergence and in capturing non-smooth solutions.A key observation is that the method can capture phase structure that require twice the resolution with a polynomial based method.In 6D,this would represent a signifcant savings.
基金partially supported by National Key Research and Development Program of China(2019YFC1510902)National Natural Science Foundation of China(62073104)+1 种基金Natural Science Foundation of Heilongjiang Province(LH2022F024)China Postdoctoral Science Foundation(2022M710965)。
文摘This paper presents a subspace identification method for closed-loop systems with unknown deterministic disturbances.To deal with the unknown deterministic disturbances,two strategies are implemented to construct the row space that can be used to approximately represent the unknown deterministic disturbances using the trigonometric functions or Bernstein polynomials depending on whether the disturbance frequencies are known.For closed-loop identification,CCF-N4SID is extended to the case with unknown deterministic disturbances using the oblique projection.In addition,a proper Bernstein polynomial order can be determined using the Akaike information criterion(AIC)or the Bayesian information criterion(BIC).Numerical simulation results demonstrate the effectiveness of the proposed identification method for both periodic and aperiodic deterministic disturbances.
