To continue the discussion in （Ⅰ ） and （ Ⅱ ）,and finish the study of the limit cycle problem for quadratic system （ Ⅲ ）m=0 in this paper. Since there is at most one limit cycle that may be created from critic...To continue the discussion in （Ⅰ ） and （ Ⅱ ）,and finish the study of the limit cycle problem for quadratic system （ Ⅲ ）m=0 in this paper. Since there is at most one limit cycle that may be created from critical point O by Hopf bifurcation,the number of limit cycles depends on the different situations of separatrix cycle to be formed around O. If it is a homoclinic cycle passing through saddle S1 on 1 ＋ax-y = 0,which has the same stability with the limit cycle created by Hopf bifurcation,then the uniqueness of limit cycles in such cases can be proved. If it is a homoclinic cycle passing through saddle N on x= 0,which has the different stability from the limit cycle created by Hopf bifurcation,then it will be a case of two limit cycles. For the case when the separatrix cycle is a heteroclinic cycle passing through two saddles at infinity,the discussion of the paper shows that the number of limit cycles will change from one to two depending on the different values of parameters of system.展开更多
In this paper, (a) we rerise Theorem 2 of Ref [1] omit the condition V_7>0 .(b) we discuss the relative positions of six curves M(s￣2, r)=0, J( s￣2, r)=0, L(s￣2,r)=0, T(s￣2,r)=0, Under the condition of the (1.3...In this paper, (a) we rerise Theorem 2 of Ref [1] omit the condition V_7>0 .(b) we discuss the relative positions of six curves M(s￣2, r)=0, J( s￣2, r)=0, L(s￣2,r)=0, T(s￣2,r)=0, Under the condition of the (1.3) distri-butions of limit cycles, we expand the variable regions of parameters ( s , r) and clearly. show them in figure, (c) we study the (1, 3) distributions of limit cycles of one kind quadratic systems with two singular points at the infinite: and (d) we give a generalmethod to discuss the ( 1 ,3) distibutions`of limit cycles of system (1.1) whatever there isone, two or three singular points at the infinite.展开更多
In a previous paper, we have proved that a planar quadratic system with invariant parabola r has at most one limit cycle. In this paper, we use geometric characteristics to give necessary and sufficient conditions un&...In a previous paper, we have proved that a planar quadratic system with invariant parabola r has at most one limit cycle. In this paper, we use geometric characteristics to give necessary and sufficient conditions un'der which a PQSp with three non-degenerate singular points can be transformed into twO different definite forms. In this wayl we obtain all the bifurcations of such a system.展开更多
Based on the linear quantum transformation theory,we present a new approach to obtain the explicit expressions of energy spectrum and simplify the derivations of partition functions for general multi-mode boson and fe...Based on the linear quantum transformation theory,we present a new approach to obtain the explicit expressions of energy spectrum and simplify the derivations of partition functions for general multi-mode boson and fermion quadratic systems.展开更多
In this paper,the authors revisit decentralized control of linear quadratic(LQ)systems.Instead of imposing an assumption that the process and observation noises are Gaussian,the authors assume that the controllers are...In this paper,the authors revisit decentralized control of linear quadratic(LQ)systems.Instead of imposing an assumption that the process and observation noises are Gaussian,the authors assume that the controllers are restricted to be linear.The authors show that the multiple decentralized control models,the form of the best linear controllers is identical to the optimal controllers obtained under the Gaussian noise assumption.The main contribution of the paper is the solution technique.Traditionally,optimal controllers for decentralized LQ systems are identified using dynamic programming,maximum principle,or spectral decomposition.The authors present an alternative approach which is based by combining elementary building blocks from linear systems,namely,completion of squares,state splitting,static reduction,orthogonal projection,(conditional)independence of state processes,and decentralized estimation.展开更多
In this study,a dynamic model for an inverted pendulum system(IPS)attached to a car is created,and two different control methods are applied to control the system.The designed control algorithms aim to stabilize the p...In this study,a dynamic model for an inverted pendulum system(IPS)attached to a car is created,and two different control methods are applied to control the system.The designed control algorithms aim to stabilize the pendulum arms in the upright position and the car to reach the equilibrium position.Grey Wolf Optimization-based Linear Quadratic Regulator(GWO-LQR)and GWO-based Fuzzy LQR(FLQR)control algorithms are used in the control process.To improve the performance of the LQR and FLQR methods,the optimum values of the coefficients corresponding to the foot points of the membership functions are determined by the GWO algorithm.Both a graphic and a numerical analysis of the outcomes are provided.In the comparative analysis,it is observed that the GWO-based FLQR method reduces the settling time by 22.58% and the maximum peak value by 18.2% when evaluated in terms of the angular response of the pendulum arm.Furthermore,this approach outperformed comparable research in the literature with a settling time of 2.4 s.These findings demonstrate that the suggested GWO-based FLQR controlmethod outperforms existing literature in terms of the time required for the pendulum arm to reach equilibrium.展开更多
This research considers the tracking problem of a moving target in distributed sensor networks with a limited sensing range(LSR)affected by non-Gaussian noise.In such sensor networks,observation loss due to LSR is a p...This research considers the tracking problem of a moving target in distributed sensor networks with a limited sensing range(LSR)affected by non-Gaussian noise.In such sensor networks,observation loss due to LSR is a prevalent issue that has received insufficient attention.We introduce a time-varying random variable to describe whether the sensor observes a moving target at each moment.When a single sensor node is unable to receive information from other nodes,it cannot update its state estimation of the moving target once the target moves beyond this node’s observation range.We propose an information flow topology within distributed sensor networks to facilitate the reception of prior state estimation data transmitted by neighboring nodes.Based on this information,a quadratic distributed estimator is designed for each sensor,and an output injection term is introduced to handle unstable systems.Finally,a numerical example is provided to illustrate the effectiveness of the proposed control scheme.展开更多
Making full use of the operator ordering method and the integration within ordered products,we obtain the analytical evolution law of a general quadratic state in the amplitude decay channel,and find that it is determ...Making full use of the operator ordering method and the integration within ordered products,we obtain the analytical evolution law of a general quadratic state in the amplitude decay channel,and find that it is determined not only by the decay rate of the amplitude decay channel but also by the coefficients of the initial quadratic state.Further,the quantum statistical properties of the initial quadratic state for amplitude decay are investigated via its average photon number and photon-counting distribution,and its Wigner distribution function evolution is discussed in detail.展开更多
We transform the quadratic system into the special system of Type (Ⅲ)a=0' and hence a string sufficient conditions are established to ensure that the considered system has at most one limit cycle.
The main results of this paper are as follows: （ⅰ） The important formulas, given by Bautin, of three focal quantities for the specific form of quadratle system （E2） have been generalized to the general form of ...The main results of this paper are as follows: （ⅰ） The important formulas, given by Bautin, of three focal quantities for the specific form of quadratle system （E2） have been generalized to the general form of （E2）. （ⅱ） By using the method in [13], a kind of （E2） possessing at least four limit cycles is given. Theorem 2 herein contains the results in [11--13] on （1,3）-distribution of limit cycles of （E2）.展开更多
In this paper, we discuss the Poincaré bifurcation for a class of quadratic systems with an unbounded triangular region and a center region. It is proved, by Poincaré bifurcation, that inside the center regi...In this paper, we discuss the Poincaré bifurcation for a class of quadratic systems with an unbounded triangular region and a center region. It is proved, by Poincaré bifurcation, that inside the center region quadratic system perturbed by quadratic polynomial perturbation may generate three limit cycles.展开更多
In this paper, we prove that a planar quadratic systems with a 3rd-order weak focus has at most one limit cycle, and a planar quadratic system with a 2nd-order weak focus has at most two limit cycles.
The maximal number of limit cycles for a particular type Ⅲ system x = -y + lx2 + mxy, y =x(1 + ax + by) is studied and some errors that appeared in the paper by Suo Mingxia and Yue Xiting (Annals of Differential Equa...The maximal number of limit cycles for a particular type Ⅲ system x = -y + lx2 + mxy, y =x(1 + ax + by) is studied and some errors that appeared in the paper by Suo Mingxia and Yue Xiting (Annals of Differential Equations, 2003,19(3):397-401) are corrected. By translating the system to be considered into the Lienard type and by using some related properties, we obtain several theorems with suitable conditions coefficients of the system, under which we prove that the system has at most two limit cycles. The conclusions improve the results given in Suo and Yue's paper mentioned above.展开更多
Some properties such as oscillation, stability, existence of periodic solutions and quadratic integrability of solutions based on a class of second order nonlinear delayed systems are analyzed by using the V-function,...Some properties such as oscillation, stability, existence of periodic solutions and quadratic integrability of solutions based on a class of second order nonlinear delayed systems are analyzed by using the V-function, the Lyapunov functional or the Beuman-Bihari inequality, and some sufficient conditions based on those properties are given. Finally, the conclusions are applied to over-voltage models based on three-phase nonsynchronous closing of switches appearing in the power systems, the results in accord with the background physical meaning are obtained. And all the conditions of the conclusions are easy to validate, so the conclusions have definite theoretical meaning and are easy to apply in practice.展开更多
In this paper, we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex e...In this paper, we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x^2 ＋ y^2 ＋ 1 = 0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincare disc.展开更多
It is proved that the quadratic system with a weak saddle has at most one limit cycle,and that if this system has a separatrix cycle passing through the weak saddle,then the stability of the separatrix cycle is contra...It is proved that the quadratic system with a weak saddle has at most one limit cycle,and that if this system has a separatrix cycle passing through the weak saddle,then the stability of the separatrix cycle is contrary to that of the singular point surrounded by it.展开更多
In this paper foe bifurcation of critical points for the quadratic systems of type(II)and (III) is investigated. and an answer to the problem given in[1] is given.
We study the number and distribution of critical points as we Ⅱ as algebraic solutions of a cubic system close related to the general quadratic system.
This paper is devoted to discussing the topological classification of the quartic invariant algebraic curves for a quadratic system.We obtain sufficient and necessary conditions which ensure that the homoclinic cycle ...This paper is devoted to discussing the topological classification of the quartic invariant algebraic curves for a quadratic system.We obtain sufficient and necessary conditions which ensure that the homoclinic cycle of the system is defined by the quartic invariant algebraic curve.Finally,the corresponding global phase diagrams are drawn.展开更多
In [2-5], cubic, quartic or quintic homoclinic cycles are found. In this paper, we present a quadratic system with homoclinic cycle which is described by a sextic curve.
基金Project supported by the National Natural Science Foundation of China （10471066）.
文摘To continue the discussion in （Ⅰ ） and （ Ⅱ ）,and finish the study of the limit cycle problem for quadratic system （ Ⅲ ）m=0 in this paper. Since there is at most one limit cycle that may be created from critical point O by Hopf bifurcation,the number of limit cycles depends on the different situations of separatrix cycle to be formed around O. If it is a homoclinic cycle passing through saddle S1 on 1 ＋ax-y = 0,which has the same stability with the limit cycle created by Hopf bifurcation,then the uniqueness of limit cycles in such cases can be proved. If it is a homoclinic cycle passing through saddle N on x= 0,which has the different stability from the limit cycle created by Hopf bifurcation,then it will be a case of two limit cycles. For the case when the separatrix cycle is a heteroclinic cycle passing through two saddles at infinity,the discussion of the paper shows that the number of limit cycles will change from one to two depending on the different values of parameters of system.
文摘In this paper, (a) we rerise Theorem 2 of Ref [1] omit the condition V_7>0 .(b) we discuss the relative positions of six curves M(s￣2, r)=0, J( s￣2, r)=0, L(s￣2,r)=0, T(s￣2,r)=0, Under the condition of the (1.3) distri-butions of limit cycles, we expand the variable regions of parameters ( s , r) and clearly. show them in figure, (c) we study the (1, 3) distributions of limit cycles of one kind quadratic systems with two singular points at the infinite: and (d) we give a generalmethod to discuss the ( 1 ,3) distibutions`of limit cycles of system (1.1) whatever there isone, two or three singular points at the infinite.
文摘In a previous paper, we have proved that a planar quadratic system with invariant parabola r has at most one limit cycle. In this paper, we use geometric characteristics to give necessary and sufficient conditions un'der which a PQSp with three non-degenerate singular points can be transformed into twO different definite forms. In this wayl we obtain all the bifurcations of such a system.
基金Supported by the National Natural Science Foundation of China under Grant No.19575044.
文摘Based on the linear quantum transformation theory,we present a new approach to obtain the explicit expressions of energy spectrum and simplify the derivations of partition functions for general multi-mode boson and fermion quadratic systems.
基金supported in part by the Natural Sciences and Engineering Research Council of Canada(NSERC)Discovery Grant under Grant No.RGPIN-2021-0351.
文摘In this paper,the authors revisit decentralized control of linear quadratic(LQ)systems.Instead of imposing an assumption that the process and observation noises are Gaussian,the authors assume that the controllers are restricted to be linear.The authors show that the multiple decentralized control models,the form of the best linear controllers is identical to the optimal controllers obtained under the Gaussian noise assumption.The main contribution of the paper is the solution technique.Traditionally,optimal controllers for decentralized LQ systems are identified using dynamic programming,maximum principle,or spectral decomposition.The authors present an alternative approach which is based by combining elementary building blocks from linear systems,namely,completion of squares,state splitting,static reduction,orthogonal projection,(conditional)independence of state processes,and decentralized estimation.
文摘In this study,a dynamic model for an inverted pendulum system(IPS)attached to a car is created,and two different control methods are applied to control the system.The designed control algorithms aim to stabilize the pendulum arms in the upright position and the car to reach the equilibrium position.Grey Wolf Optimization-based Linear Quadratic Regulator(GWO-LQR)and GWO-based Fuzzy LQR(FLQR)control algorithms are used in the control process.To improve the performance of the LQR and FLQR methods,the optimum values of the coefficients corresponding to the foot points of the membership functions are determined by the GWO algorithm.Both a graphic and a numerical analysis of the outcomes are provided.In the comparative analysis,it is observed that the GWO-based FLQR method reduces the settling time by 22.58% and the maximum peak value by 18.2% when evaluated in terms of the angular response of the pendulum arm.Furthermore,this approach outperformed comparable research in the literature with a settling time of 2.4 s.These findings demonstrate that the suggested GWO-based FLQR controlmethod outperforms existing literature in terms of the time required for the pendulum arm to reach equilibrium.
基金National Natural Science Foundation of China(No.61803081)。
文摘This research considers the tracking problem of a moving target in distributed sensor networks with a limited sensing range(LSR)affected by non-Gaussian noise.In such sensor networks,observation loss due to LSR is a prevalent issue that has received insufficient attention.We introduce a time-varying random variable to describe whether the sensor observes a moving target at each moment.When a single sensor node is unable to receive information from other nodes,it cannot update its state estimation of the moving target once the target moves beyond this node’s observation range.We propose an information flow topology within distributed sensor networks to facilitate the reception of prior state estimation data transmitted by neighboring nodes.Based on this information,a quadratic distributed estimator is designed for each sensor,and an output injection term is introduced to handle unstable systems.Finally,a numerical example is provided to illustrate the effectiveness of the proposed control scheme.
文摘Making full use of the operator ordering method and the integration within ordered products,we obtain the analytical evolution law of a general quadratic state in the amplitude decay channel,and find that it is determined not only by the decay rate of the amplitude decay channel but also by the coefficients of the initial quadratic state.Further,the quantum statistical properties of the initial quadratic state for amplitude decay are investigated via its average photon number and photon-counting distribution,and its Wigner distribution function evolution is discussed in detail.
文摘We transform the quadratic system into the special system of Type (Ⅲ)a=0' and hence a string sufficient conditions are established to ensure that the considered system has at most one limit cycle.
文摘The main results of this paper are as follows: （ⅰ） The important formulas, given by Bautin, of three focal quantities for the specific form of quadratle system （E2） have been generalized to the general form of （E2）. （ⅱ） By using the method in [13], a kind of （E2） possessing at least four limit cycles is given. Theorem 2 herein contains the results in [11--13] on （1,3）-distribution of limit cycles of （E2）.
基金Supported by NSF and RFDP of China and China Postdoctoral Science Foundation (No.10471014).
文摘In this paper, we discuss the Poincaré bifurcation for a class of quadratic systems with an unbounded triangular region and a center region. It is proved, by Poincaré bifurcation, that inside the center region quadratic system perturbed by quadratic polynomial perturbation may generate three limit cycles.
基金Supported by the National Natural Science Foundation of China (19671071).
文摘In this paper, we prove that a planar quadratic systems with a 3rd-order weak focus has at most one limit cycle, and a planar quadratic system with a 2nd-order weak focus has at most two limit cycles.
文摘The maximal number of limit cycles for a particular type Ⅲ system x = -y + lx2 + mxy, y =x(1 + ax + by) is studied and some errors that appeared in the paper by Suo Mingxia and Yue Xiting (Annals of Differential Equations, 2003,19(3):397-401) are corrected. By translating the system to be considered into the Lienard type and by using some related properties, we obtain several theorems with suitable conditions coefficients of the system, under which we prove that the system has at most two limit cycles. The conclusions improve the results given in Suo and Yue's paper mentioned above.
文摘Some properties such as oscillation, stability, existence of periodic solutions and quadratic integrability of solutions based on a class of second order nonlinear delayed systems are analyzed by using the V-function, the Lyapunov functional or the Beuman-Bihari inequality, and some sufficient conditions based on those properties are given. Finally, the conclusions are applied to over-voltage models based on three-phase nonsynchronous closing of switches appearing in the power systems, the results in accord with the background physical meaning are obtained. And all the conditions of the conclusions are easy to validate, so the conclusions have definite theoretical meaning and are easy to apply in practice.
基金partially supported by a MINECO/FEDER grant MTM2013-40998-Pan AGAUR grant number 2014 SGR568+2 种基金the grants FP7-PEOPLE-2012-IRSES 318999 and 316338the MINECO/FEDER grant UNAB13-4E-1604partially supported by FCT/Portugal through UID/MAT/04459/2013
文摘In this paper, we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x^2 ＋ y^2 ＋ 1 = 0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincare disc.
文摘It is proved that the quadratic system with a weak saddle has at most one limit cycle,and that if this system has a separatrix cycle passing through the weak saddle,then the stability of the separatrix cycle is contrary to that of the singular point surrounded by it.
文摘In this paper foe bifurcation of critical points for the quadratic systems of type(II)and (III) is investigated. and an answer to the problem given in[1] is given.
文摘We study the number and distribution of critical points as we Ⅱ as algebraic solutions of a cubic system close related to the general quadratic system.
文摘This paper is devoted to discussing the topological classification of the quartic invariant algebraic curves for a quadratic system.We obtain sufficient and necessary conditions which ensure that the homoclinic cycle of the system is defined by the quartic invariant algebraic curve.Finally,the corresponding global phase diagrams are drawn.
文摘In [2-5], cubic, quartic or quintic homoclinic cycles are found. In this paper, we present a quadratic system with homoclinic cycle which is described by a sextic curve.
