In this paper, time delay Lienard's equations are considered, by using the theory of concidence degree, a sufficient condition of existence of at least one 2π-periodic solution is obtained.
In this paper, we define some non-elementary amplitude functions that are giving solutions to some well-known second-order nonlinear ODEs and the Lorenz equations, but not the chaos case. We are giving the solutions a...In this paper, we define some non-elementary amplitude functions that are giving solutions to some well-known second-order nonlinear ODEs and the Lorenz equations, but not the chaos case. We are giving the solutions a name, a symbol and putting them into a group of functions and into the context of other functions. These solutions are equal to the amplitude, or upper limit of integration in a non-elementary integral that can be arbitrary. In order to define solutions to some short second-order nonlinear ODEs, we will make an extension to the general amplitude function. The only disadvantage is that the first derivative to these solutions contains an integral that disappear at the second derivation. We will also do a second extension: the two-integral amplitude function. With this extension we have the solution to a system of ODEs having a very strange behavior. Using the extended amplitude functions, we can define solutions to many short second-order nonlinear ODEs.展开更多
In this paper, we study the number of limit cycles appeared in Hopf bifurcations of a Lienard system with multiple parameters. As an application to some polynomial Lienard systems of the form x= y, y= -gin（x） - fn...In this paper, we study the number of limit cycles appeared in Hopf bifurcations of a Lienard system with multiple parameters. As an application to some polynomial Lienard systems of the form x= y, y= -gin（x） - fn（X）y, we obtain a new lower bound of maximal number of limit cycles which appear in Hopf bifurcation for arbitrary degrees m and n.展开更多
In this paper, we first consider exact solutions for Lienard equation with nonlinear terms of any order. Then,explicit exact bell and kink profile solitary-wave solutions for many nonlinear evolution equations are obt...In this paper, we first consider exact solutions for Lienard equation with nonlinear terms of any order. Then,explicit exact bell and kink profile solitary-wave solutions for many nonlinear evolution equations are obtained by means of results of the Lienard equation and proper deductions, which transform original partial differential equations into the Lienard one. These nonlinear equations include compound KdV, compound KdV-Burgers, generalized Boussinesq,generalized KP and Ginzburg-Landau equation. Some new solitary-wave solutions are found.展开更多
Applying Hopf bifurcation theory and qualitative theory, we give the conditions of the existence and uniqueness of one limit cycle and the existence of two limit cycles for the general cubic Lienard equation. Numerica...Applying Hopf bifurcation theory and qualitative theory, we give the conditions of the existence and uniqueness of one limit cycle and the existence of two limit cycles for the general cubic Lienard equation. Numerical simulation results with one and two limit cycles are given to demonstrate the theoretical results.展开更多
In this paper,some known results concerning the nonexistence of non-trivial periodic solutions for the Lienard system are extended to the more general system
In this note,we study the asymptotic stability in the large of zero solution forLienard equationx″+f(x)x’+g(x)=0.(1)For the previous results please refer to Refs.[1]—[4].The approach often used is Liapunovsecond me...In this note,we study the asymptotic stability in the large of zero solution forLienard equationx″+f(x)x’+g(x)=0.(1)For the previous results please refer to Refs.[1]—[4].The approach often used is Liapunovsecond method,but here we use Filippov transformation for investigation.The results ob-tained are the extension of those in Refs.[1]—[4].We assume henceforth展开更多
In this paper a neuronic equation was investigated. By using qualitative theory of ordinary differential equation, we obtain some conditions for the existence of a unique limit cycle surrounding a singular point or th...In this paper a neuronic equation was investigated. By using qualitative theory of ordinary differential equation, we obtain some conditions for the existence of a unique limit cycle surrounding a singular point or three singular points r espectively.展开更多
Homoclinic solutions were introduced for a prescribed mean curvature Lienard p-Laplacian equation with a deviating argument.It was divided into three parts to discuss the existence of homoclinic solutions.By using an ...Homoclinic solutions were introduced for a prescribed mean curvature Lienard p-Laplacian equation with a deviating argument.It was divided into three parts to discuss the existence of homoclinic solutions.By using an extension of Mawhin's continuation theorem,the existence of a set with 2kT-periodic for a prescribed mean curvature Lienard p-Laplacian equation with a deviating argument was studied.According to a limit on a certain subsequence of 2kT-periodic set,homoclinic solutions were obtained.A numerical example demonstrates the validity of the main results.展开更多
By using topological degree theory and some analysis skill, some sufficient conditions for the existence and uniqueness of periodic solutions for a class of forced Lienard-type equations are obtained.
Some sharp sufficient conditions for generalized Lienard systems to have positive or negative semi-orbits which do not cross the vertical isocline are given. Applications of the main results to some polynomial systems...Some sharp sufficient conditions for generalized Lienard systems to have positive or negative semi-orbits which do not cross the vertical isocline are given. Applications of the main results to some polynomial systems are also presented.展开更多
This paper gives a theorem for the local center of generalized Lienard system; the relative theorems in the references can be deduced from our corollaries.
The main purpose of this paper is to obtain some new sufficient conditions for the continuability of solutions in the future of the lienard system: X'=y-F(x), y'=-g(x).
This paper concerns with global bifurcations of limit cycles for a cubic system.The uniqueness of limit cycles is proved in Hopf and Poincare bifurcations.
Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienard equations with quadratic damping have at most three limit cycles. This implies that the guess in which the system has at ...Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienard equations with quadratic damping have at most three limit cycles. This implies that the guess in which the system has at most two limit cycles is false. We give the sufficient conditions for the system has at most three limit cycles or two limit cycles. We present two examples with three limit cycles or two limit cycles by using numerical simulation.展开更多
Time-delay effects on the dynamics of Li^nard type equation with one fast variable and one slow variable are investigated in the present paper. By using the methods of stability switch and geometric singular perturbat...Time-delay effects on the dynamics of Li^nard type equation with one fast variable and one slow variable are investigated in the present paper. By using the methods of stability switch and geometric singular perturbation, time-delay-induced complex oscillations and bursting are investigated, and in several case studies, the mechanism of the generation of the complex oscillations and bursting is illuminated. Numerical results demonstrate the validity of the theoretical results.展开更多
文摘In this paper, time delay Lienard's equations are considered, by using the theory of concidence degree, a sufficient condition of existence of at least one 2π-periodic solution is obtained.
文摘In this paper, we define some non-elementary amplitude functions that are giving solutions to some well-known second-order nonlinear ODEs and the Lorenz equations, but not the chaos case. We are giving the solutions a name, a symbol and putting them into a group of functions and into the context of other functions. These solutions are equal to the amplitude, or upper limit of integration in a non-elementary integral that can be arbitrary. In order to define solutions to some short second-order nonlinear ODEs, we will make an extension to the general amplitude function. The only disadvantage is that the first derivative to these solutions contains an integral that disappear at the second derivation. We will also do a second extension: the two-integral amplitude function. With this extension we have the solution to a system of ODEs having a very strange behavior. Using the extended amplitude functions, we can define solutions to many short second-order nonlinear ODEs.
基金supported by National Natural Science Foundation of China (Grant No.11271261)
文摘In this paper, we study the number of limit cycles appeared in Hopf bifurcations of a Lienard system with multiple parameters. As an application to some polynomial Lienard systems of the form x= y, y= -gin（x） - fn（X）y, we obtain a new lower bound of maximal number of limit cycles which appear in Hopf bifurcation for arbitrary degrees m and n.
文摘In this paper, we first consider exact solutions for Lienard equation with nonlinear terms of any order. Then,explicit exact bell and kink profile solitary-wave solutions for many nonlinear evolution equations are obtained by means of results of the Lienard equation and proper deductions, which transform original partial differential equations into the Lienard one. These nonlinear equations include compound KdV, compound KdV-Burgers, generalized Boussinesq,generalized KP and Ginzburg-Landau equation. Some new solitary-wave solutions are found.
基金supported by the National Natural Sciences Foundation of China.
文摘Applying Hopf bifurcation theory and qualitative theory, we give the conditions of the existence and uniqueness of one limit cycle and the existence of two limit cycles for the general cubic Lienard equation. Numerical simulation results with one and two limit cycles are given to demonstrate the theoretical results.
文摘In this paper, the uniqueness of limit cycle of a special polynomial Lienard system is discussed and some results under certain conditions are given.
文摘In this paper,some known results concerning the nonexistence of non-trivial periodic solutions for the Lienard system are extended to the more general system
基金Project supported by the National Natural Science Foundation of China.
文摘In this note,we study the asymptotic stability in the large of zero solution forLienard equationx″+f(x)x’+g(x)=0.(1)For the previous results please refer to Refs.[1]—[4].The approach often used is Liapunovsecond method,but here we use Filippov transformation for investigation.The results ob-tained are the extension of those in Refs.[1]—[4].We assume henceforth
文摘In this paper a neuronic equation was investigated. By using qualitative theory of ordinary differential equation, we obtain some conditions for the existence of a unique limit cycle surrounding a singular point or three singular points r espectively.
基金Natural Foundation of Wuyi University,China(No.XQ201305)Young-Middle-Aged Teachers Education Scientific Research Project in Fujian Province,China(No.JA15524)
文摘Homoclinic solutions were introduced for a prescribed mean curvature Lienard p-Laplacian equation with a deviating argument.It was divided into three parts to discuss the existence of homoclinic solutions.By using an extension of Mawhin's continuation theorem,the existence of a set with 2kT-periodic for a prescribed mean curvature Lienard p-Laplacian equation with a deviating argument was studied.According to a limit on a certain subsequence of 2kT-periodic set,homoclinic solutions were obtained.A numerical example demonstrates the validity of the main results.
基金Scientific Research Fund of Zhejiang Provincial Education Department (20070605)
文摘By using topological degree theory and some analysis skill, some sufficient conditions for the existence and uniqueness of periodic solutions for a class of forced Lienard-type equations are obtained.
文摘Some sharp sufficient conditions for generalized Lienard systems to have positive or negative semi-orbits which do not cross the vertical isocline are given. Applications of the main results to some polynomial systems are also presented.
文摘This paper gives a theorem for the local center of generalized Lienard system; the relative theorems in the references can be deduced from our corollaries.
文摘The main purpose of this paper is to obtain some new sufficient conditions for the continuability of solutions in the future of the lienard system: X'=y-F(x), y'=-g(x).
文摘This paper concerns with global bifurcations of limit cycles for a cubic system.The uniqueness of limit cycles is proved in Hopf and Poincare bifurcations.
基金Supported by the National Natural Science Foundation of ChinaNational Key Basic Research Special Found (No. G1998020307).
文摘Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienard equations with quadratic damping have at most three limit cycles. This implies that the guess in which the system has at most two limit cycles is false. We give the sufficient conditions for the system has at most three limit cycles or two limit cycles. We present two examples with three limit cycles or two limit cycles by using numerical simulation.
基金supported by the National Natural Science Foundation of China(11102078 and 11032009)Foundation of Jiangxi Education Committee of China(GJJ1169)
文摘Time-delay effects on the dynamics of Li^nard type equation with one fast variable and one slow variable are investigated in the present paper. By using the methods of stability switch and geometric singular perturbation, time-delay-induced complex oscillations and bursting are investigated, and in several case studies, the mechanism of the generation of the complex oscillations and bursting is illuminated. Numerical results demonstrate the validity of the theoretical results.
