This paper aims to investigate the retarded Liénard-type equation+f 1(x)+f 2(x)(t-τ)+f 3(x)2+φ(x)+g(x(t-τ))=0,where τ is a nonnegative constant, f 1,f 2,f 3,φ and g are continuous functions on R. Using Liapu...This paper aims to investigate the retarded Liénard-type equation+f 1(x)+f 2(x)(t-τ)+f 3(x)2+φ(x)+g(x(t-τ))=0,where τ is a nonnegative constant, f 1,f 2,f 3,φ and g are continuous functions on R. Using Liapunov functional method, we establish a sufficient condition on the stability and boundedness of the solutions of above equation. This will generalize the main results of reference [2].展开更多
By coincidence degree,the existence of solution to the boundary value problem of a generalized Liénard equationa(t)x'+F(x,x′)x′+g(x)=e(t), x(0)=x(2π),x′(0)=x′(2π)is proved,where a∈C 1[0,2π],a(t)>...By coincidence degree,the existence of solution to the boundary value problem of a generalized Liénard equationa(t)x'+F(x,x′)x′+g(x)=e(t), x(0)=x(2π),x′(0)=x′(2π)is proved,where a∈C 1[0,2π],a(t)>0(0≤t≤2π),a(0)=a(2π),F(x,y)=f(x)+α|y| β,α>0,β>0 are all constants,f∈C(R,R),e∈C[0,2π]. An example is given as an application.展开更多
In this paper the generalized nonlinear Euler differential equation t^2k（tu＇）u＇＇＋ t（f（u） ＋ k（tu＇））u＇ ＋ g（u） = 0 is considered. Here the functions f（u）, g（u） and k（u） satisfy smoothness conditio...In this paper the generalized nonlinear Euler differential equation t^2k（tu＇）u＇＇＋ t（f（u） ＋ k（tu＇））u＇ ＋ g（u） = 0 is considered. Here the functions f（u）, g（u） and k（u） satisfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub（super） linearity are assumed. W＇e present some necessary and sufficient conditions and some tests for the equivalent planar system to have or fail to have property （X^＋）, which is very important for the existence of periodic solutions and oscillation theory.展开更多
The existence of monotone and non_monotone solutions of boundary value problem on the real line for Liénard equation is studied. Applying the theory of planar dynamical systems and the comparison method of vector...The existence of monotone and non_monotone solutions of boundary value problem on the real line for Liénard equation is studied. Applying the theory of planar dynamical systems and the comparison method of vector fields defined by Liénard system and the system given by symmetric transformation or quasi_symmetric transformation, the invariant regions of the system are constructed. The existence of connecting orbits can be proved. A lot of sufficient conditions to guarantee the existence of solutions of the boundary value problem are obtained. Especially, when the source function is bi_stable, the existence of infinitely many monotone solusion is obtained.展开更多
In this paper, we deal with the existence of unbounded orbits of the mapping $$\left\{ \begin{gathered} \theta _1 = \theta + 2n\pi + \frac{1}{\rho }\mu (\theta ) + o(\rho ^{ - 1} ), \hfill \\ \rho _1 = \rho + c - \mu ...In this paper, we deal with the existence of unbounded orbits of the mapping $$\left\{ \begin{gathered} \theta _1 = \theta + 2n\pi + \frac{1}{\rho }\mu (\theta ) + o(\rho ^{ - 1} ), \hfill \\ \rho _1 = \rho + c - \mu '(\theta ) + o(1), \rho \to \infty \hfill \\ \end{gathered} \right.$$ , where n is a positive integer, c is a constant and μ(θ) is a 2π-periodic function. We prove that if c > 0 and μ(θ) ≠ 0, θ, ∈ [0, 2?], then every orbit of the given mapping goes to infinity in the future for ρ large enough; if c < 0 and μ(θ) ≠ 0, θ ∈ [0, 2π], then every orbit of the given mapping goes to infinity in the past for ρ large enough. By using this result, we prove that the equation x″+f(x)x′+ax +?bx ?+?(x)=p(t) has unbounded solutions provided that a, b satisfy $1/\sqrt a + 1/\sqrt b = 2/n$ and ?(x) satisfies some limit conditions. At the same time, we obtain the existence of 2π-periodic solutions of this equation.展开更多
Using the theroy of topological degree, the paper studies the periodic solutions to a type of neutral Liénard equation with state-dependent deviation variable. A sufficient condition for the existence of periodic...Using the theroy of topological degree, the paper studies the periodic solutions to a type of neutral Liénard equation with state-dependent deviation variable. A sufficient condition for the existence of periodic solution is obtained.展开更多
We are concerned with the existence of quasi-periodic solutions for the following equation x" + Fx (x, t)x' + ω2x + φ(x,t) = 0,where F and φ are smooth functions and 2π-periodic in t, ω＞ 0 is a constant...We are concerned with the existence of quasi-periodic solutions for the following equation x" + Fx (x, t)x' + ω2x + φ(x,t) = 0,where F and φ are smooth functions and 2π-periodic in t, ω＞ 0 is a constant. Under some assumptions on the parities of F and φ, we show that the Dancer's function, which is used to study the existence of periodic solutions, also plays a role for the existence of quasi-periodic solutions and the Lagrangian stability (i.e. all solutions are bounded).展开更多
We present some exact integrability cases of the extended Liénard equation y′′+ f(y)(y′)n +k(y)(y′)m + g(y)y′+ h(y) = 0, with n > 0 and m > 0 arbitrary constants, while f(y), k(y), g(y), and h(y) are a...We present some exact integrability cases of the extended Liénard equation y′′+ f(y)(y′)n +k(y)(y′)m + g(y)y′+ h(y) = 0, with n > 0 and m > 0 arbitrary constants, while f(y), k(y), g(y), and h(y) are arbitrary functions. The solutions are obtained by transforming the equation Liénard equation to an equivalent first kind first order Abel type equation given bydv/dy= f(y)v3-n+ k(y)v3-m+ g(y)v2+ h(y)v3, with v = 1/y′.As a first step in our study we obtain three integrability cases of the extended quadratic-cubic Liénard equation,corresponding to n = 2 and m = 3, by assuming that particular solutions of the associated Abel equation are known. Under this assumption the general solutions of the Abel and Liénard equations with coefficients satisfying some differential conditions can be obtained in an exact closed form. With the use of the Chiellini integrability condition, we show that if a particular solution of the Abel equation is known, the general solution of the extended quadratic cubic Liénard equation can be obtained by quadratures. The Chiellini integrability condition is extended to generalized Abel equations with g(y) ≡ 0 and h(y) ≡ 0, and arbitrary n and m, thus allowing to obtain the general solution of the corresponding Liénard equation. The application of the generalized Chiellini condition to the case of the reduced Riccati equation is also considered.展开更多
This work focuses on stochastic Lienard equations with state-dependent switching. First, the existence and uniqueness of a strong solution are obtained by successive construction method. Next, strong Feller property i...This work focuses on stochastic Lienard equations with state-dependent switching. First, the existence and uniqueness of a strong solution are obtained by successive construction method. Next, strong Feller property is proved by introducing certain auxiliary processes and using the Radon-Nikodym derivatives and truncation arguments. Based on these results, positive Harris recurrence and exponential ergodicity are obtained under the Foster-Lyapunov drift conditions. Finally, examples using van der Pol equations are presented for illustrations, and the corresponding Foster-Lyapunov functions for the examples are constructed explicitly.展开更多
Liénard’s equation is a kind of important ordinary differential equations frequently appearing in engineering and technology, and hence receives great attention of many mathematicians. In 1949, H. J. Eckweiler c...Liénard’s equation is a kind of important ordinary differential equations frequently appearing in engineering and technology, and hence receives great attention of many mathematicians. In 1949, H. J. Eckweiler conjectured that the equation +μsin+x=0 has infinite number of limit cycles. Then H. S. Hochstadt and B. Stephan, R. N. D’Heedene and others proved that this equation has at least n limit cycles in the interval |x|<(n+1)π for specified parameter μ. In 1980, Professor Zhang Zhifen proved that this equation has exact n limit cycles in the interval |x|<(n+1)π for any nonzero parameter μ, and thus pushed the related work forward greatly. In this paper, we shall prove that the Liénard’s equation has exact n limit cycles in a finite interval under a class of very general condition.展开更多
In this paper, a time delay Liénard’s equation is considered, by the coincidence degree theory. Sufficient conditions for the existence of at least one T-periodic solution are obtained.
In this paper, we provide a Hopf bifurcation diagram of Lienard equation with a discrete delay, by using the (?) - D decomposition, one can determine the stability domain of the equilibrium and Hopf bifurcation curves...In this paper, we provide a Hopf bifurcation diagram of Lienard equation with a discrete delay, by using the (?) - D decomposition, one can determine the stability domain of the equilibrium and Hopf bifurcation curves in the parameter space.展开更多
In this paper, a Liénard equation with a deviating argument has been studied by means of Mawhin’s continuation theorem. A new result guaranteeing the existence of periodic solutions is obtained.
We analytically derived the complex Ginzburg-Landau equation from the Liénard form of the discrete FitzHugh Nagumo model by employing the multiple scale expansions in the semidiscrete approximation. The complex G...We analytically derived the complex Ginzburg-Landau equation from the Liénard form of the discrete FitzHugh Nagumo model by employing the multiple scale expansions in the semidiscrete approximation. The complex Ginzburg-Landau equation now governs the dynamics of a pulse propagation along a myelinated nerve fiber where the wave dispersion relation is used to explain the famous phenomena of propagation failure and saltatory conduction. Stability analysis of the pulse soliton solution that mimics the action potential fulfills the Benjamin-Feir criteria for plane wave solutions. Finally, results from our numerical simulations show that as the dissipation along the myelinated axon increases, the nerve impulse broadens and finally degenerates to front solutions.展开更多
文摘This paper aims to investigate the retarded Liénard-type equation+f 1(x)+f 2(x)(t-τ)+f 3(x)2+φ(x)+g(x(t-τ))=0,where τ is a nonnegative constant, f 1,f 2,f 3,φ and g are continuous functions on R. Using Liapunov functional method, we establish a sufficient condition on the stability and boundedness of the solutions of above equation. This will generalize the main results of reference [2].
文摘By coincidence degree,the existence of solution to the boundary value problem of a generalized Liénard equationa(t)x'+F(x,x′)x′+g(x)=e(t), x(0)=x(2π),x′(0)=x′(2π)is proved,where a∈C 1[0,2π],a(t)>0(0≤t≤2π),a(0)=a(2π),F(x,y)=f(x)+α|y| β,α>0,β>0 are all constants,f∈C(R,R),e∈C[0,2π]. An example is given as an application.
文摘In this paper the generalized nonlinear Euler differential equation t^2k（tu＇）u＇＇＋ t（f（u） ＋ k（tu＇））u＇ ＋ g（u） = 0 is considered. Here the functions f（u）, g（u） and k（u） satisfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub（super） linearity are assumed. W＇e present some necessary and sufficient conditions and some tests for the equivalent planar system to have or fail to have property （X^＋）, which is very important for the existence of periodic solutions and oscillation theory.
文摘The existence of monotone and non_monotone solutions of boundary value problem on the real line for Liénard equation is studied. Applying the theory of planar dynamical systems and the comparison method of vector fields defined by Liénard system and the system given by symmetric transformation or quasi_symmetric transformation, the invariant regions of the system are constructed. The existence of connecting orbits can be proved. A lot of sufficient conditions to guarantee the existence of solutions of the boundary value problem are obtained. Especially, when the source function is bi_stable, the existence of infinitely many monotone solusion is obtained.
基金the National Natural Science Foundation of China(Grant No.10471099)the Fund of Beijing Education Committee(Grant No.KM200410028003)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,Ministry of Education of China
文摘In this paper, we deal with the existence of unbounded orbits of the mapping $$\left\{ \begin{gathered} \theta _1 = \theta + 2n\pi + \frac{1}{\rho }\mu (\theta ) + o(\rho ^{ - 1} ), \hfill \\ \rho _1 = \rho + c - \mu '(\theta ) + o(1), \rho \to \infty \hfill \\ \end{gathered} \right.$$ , where n is a positive integer, c is a constant and μ(θ) is a 2π-periodic function. We prove that if c > 0 and μ(θ) ≠ 0, θ, ∈ [0, 2?], then every orbit of the given mapping goes to infinity in the future for ρ large enough; if c < 0 and μ(θ) ≠ 0, θ ∈ [0, 2π], then every orbit of the given mapping goes to infinity in the past for ρ large enough. By using this result, we prove that the equation x″+f(x)x′+ax +?bx ?+?(x)=p(t) has unbounded solutions provided that a, b satisfy $1/\sqrt a + 1/\sqrt b = 2/n$ and ?(x) satisfies some limit conditions. At the same time, we obtain the existence of 2π-periodic solutions of this equation.
基金Supported by the Education Department of Shanghai and the Youth Projecet Foundation of USST.
文摘Using the theroy of topological degree, the paper studies the periodic solutions to a type of neutral Liénard equation with state-dependent deviation variable. A sufficient condition for the existence of periodic solution is obtained.
基金supported by the Special Funds for Major State Basic Research Projects(973 Projects)NSFC(Grant No.10325103)TRAPOYT.
文摘We are concerned with the existence of quasi-periodic solutions for the following equation x" + Fx (x, t)x' + ω2x + φ(x,t) = 0,where F and φ are smooth functions and 2π-periodic in t, ω＞ 0 is a constant. Under some assumptions on the parities of F and φ, we show that the Dancer's function, which is used to study the existence of periodic solutions, also plays a role for the existence of quasi-periodic solutions and the Lagrangian stability (i.e. all solutions are bounded).
文摘We present some exact integrability cases of the extended Liénard equation y′′+ f(y)(y′)n +k(y)(y′)m + g(y)y′+ h(y) = 0, with n > 0 and m > 0 arbitrary constants, while f(y), k(y), g(y), and h(y) are arbitrary functions. The solutions are obtained by transforming the equation Liénard equation to an equivalent first kind first order Abel type equation given bydv/dy= f(y)v3-n+ k(y)v3-m+ g(y)v2+ h(y)v3, with v = 1/y′.As a first step in our study we obtain three integrability cases of the extended quadratic-cubic Liénard equation,corresponding to n = 2 and m = 3, by assuming that particular solutions of the associated Abel equation are known. Under this assumption the general solutions of the Abel and Liénard equations with coefficients satisfying some differential conditions can be obtained in an exact closed form. With the use of the Chiellini integrability condition, we show that if a particular solution of the Abel equation is known, the general solution of the extended quadratic cubic Liénard equation can be obtained by quadratures. The Chiellini integrability condition is extended to generalized Abel equations with g(y) ≡ 0 and h(y) ≡ 0, and arbitrary n and m, thus allowing to obtain the general solution of the corresponding Liénard equation. The application of the generalized Chiellini condition to the case of the reduced Riccati equation is also considered.
基金Supported by the National Natural Science Foundation of China(No.11171024)the National Science Foundation,United States(No.DMS-0907753)
文摘This work focuses on stochastic Lienard equations with state-dependent switching. First, the existence and uniqueness of a strong solution are obtained by successive construction method. Next, strong Feller property is proved by introducing certain auxiliary processes and using the Radon-Nikodym derivatives and truncation arguments. Based on these results, positive Harris recurrence and exponential ergodicity are obtained under the Foster-Lyapunov drift conditions. Finally, examples using van der Pol equations are presented for illustrations, and the corresponding Foster-Lyapunov functions for the examples are constructed explicitly.
文摘Liénard’s equation is a kind of important ordinary differential equations frequently appearing in engineering and technology, and hence receives great attention of many mathematicians. In 1949, H. J. Eckweiler conjectured that the equation +μsin+x=0 has infinite number of limit cycles. Then H. S. Hochstadt and B. Stephan, R. N. D’Heedene and others proved that this equation has at least n limit cycles in the interval |x|<(n+1)π for specified parameter μ. In 1980, Professor Zhang Zhifen proved that this equation has exact n limit cycles in the interval |x|<(n+1)π for any nonzero parameter μ, and thus pushed the related work forward greatly. In this paper, we shall prove that the Liénard’s equation has exact n limit cycles in a finite interval under a class of very general condition.
文摘In this paper, a time delay Liénard’s equation is considered, by the coincidence degree theory. Sufficient conditions for the existence of at least one T-periodic solution are obtained.
文摘In this paper, we provide a Hopf bifurcation diagram of Lienard equation with a discrete delay, by using the (?) - D decomposition, one can determine the stability domain of the equilibrium and Hopf bifurcation curves in the parameter space.
基金supported by NSF of the Educational Bureau of Anhui Province(No.KJ2009B103Z, 2009SQRZ083, KJ2008B235)the Key Young Item of Anhui University of Finance and Economics (No.ACKYQ0811ZD)
文摘In this paper, a Liénard equation with a deviating argument has been studied by means of Mawhin’s continuation theorem. A new result guaranteeing the existence of periodic solutions is obtained.
文摘We analytically derived the complex Ginzburg-Landau equation from the Liénard form of the discrete FitzHugh Nagumo model by employing the multiple scale expansions in the semidiscrete approximation. The complex Ginzburg-Landau equation now governs the dynamics of a pulse propagation along a myelinated nerve fiber where the wave dispersion relation is used to explain the famous phenomena of propagation failure and saltatory conduction. Stability analysis of the pulse soliton solution that mimics the action potential fulfills the Benjamin-Feir criteria for plane wave solutions. Finally, results from our numerical simulations show that as the dissipation along the myelinated axon increases, the nerve impulse broadens and finally degenerates to front solutions.
