In this paper we construct, by using the theory of exponential dichotomies, a Melnikov-type function by which we can detect the existence of homoclinic orbits for the perturbed systems x = g(x) + epsilon h(t, x, epsil...In this paper we construct, by using the theory of exponential dichotomies, a Melnikov-type function by which we can detect the existence of homoclinic orbits for the perturbed systems x = g(x) + epsilon h(t, x, epsilon). Our result of this paper may be complementary to that of K.J.Palmer([3]).展开更多
Based on the theory of generalized exponeatial dichotomies, some typical nonautonomouslinear systems with time-translation parameters or with time-scale parameters are discussed.Conditions of parameters are given to d...Based on the theory of generalized exponeatial dichotomies, some typical nonautonomouslinear systems with time-translation parameters or with time-scale parameters are discussed.Conditions of parameters are given to determine by spectral gaps the generalized exponentialdichotomies of those systems and their local L'-limit systexns. These conclusions are applied tosrvcalled SVC systems and HFO systems for their stability.展开更多
In this paper, we study the relationship between exponential dichotomy and quadratic Lyapunov function for the linear equation x△=A(t)x on time scales. Moreover, for the nonlinear perturbed equation x△= A(t)x ...In this paper, we study the relationship between exponential dichotomy and quadratic Lyapunov function for the linear equation x△=A(t)x on time scales. Moreover, for the nonlinear perturbed equation x△= A(t)x + f(t, x) we give the instability of the zero solution when f is sufficiently small.展开更多
In this paper, we consider the existence and uniqueness of the solutions which are pseudo almost automorphic in distribution for a class of non-autonomous stochastic differential equations in a Hilbert space. In concl...In this paper, we consider the existence and uniqueness of the solutions which are pseudo almost automorphic in distribution for a class of non-autonomous stochastic differential equations in a Hilbert space. In conclusion, we use the Banach contraction mapping principle and exponential dichotomy property to obtain our main results.展开更多
The inch purpose of this paper is to study homocliaic bifurcation in a degenerate case of differential equations of form i = f(x, ε) and i = g(x) + h(t, x. ε). By using theory of exponential dichotomies and Liapunov...The inch purpose of this paper is to study homocliaic bifurcation in a degenerate case of differential equations of form i = f(x, ε) and i = g(x) + h(t, x. ε). By using theory of exponential dichotomies and Liapunov-Schmidt method, the authors obtain the Melnikov-like vectors by which the existence of homoclinic orbits can be detected.展开更多
It is well known that if x' = A(t)x admits an exponential dichotomy and f(t, x) is a bounded function and has a small Lipschitz constant, then system X' = A(x)x + f(t, x) has a unique bounded solution. In this...It is well known that if x' = A(t)x admits an exponential dichotomy and f(t, x) is a bounded function and has a small Lipschitz constant, then system X' = A(x)x + f(t, x) has a unique bounded solution. In this paper, we shall generalize this theorem to the case when f(t, x) is an unbounded function.展开更多
The main purpose of this paper is to investigate the existence of almost periodic solutions for the Duffing differential equation.By combining the theory of exponential dichotomies with Liapunov functions,we obtain an...The main purpose of this paper is to investigate the existence of almost periodic solutions for the Duffing differential equation.By combining the theory of exponential dichotomies with Liapunov functions,we obtain an intersting result on the existence of almost periodic solutions.展开更多
The aim of this paper is to investigate the existence and uniqueness of almost periodic solutions for the forced Rayleigh equation. By combining the theory of exponential dichotomies with Liapunov functions, we obtain...The aim of this paper is to investigate the existence and uniqueness of almost periodic solutions for the forced Rayleigh equation. By combining the theory of exponential dichotomies with Liapunov functions, we obtain an interesting result on the existence of almost periodic solutions.展开更多
By applying Horn fixed point theorem and the method of exponential dichotomy,we investigate the existence of almost periodic solution to nonlinear ordinarydifferential equation.
In this paper, we deal with the problem on the existence of periodic solution of higher periodic system. Using the exponential dichotomy and the Schauder's fixed point theorem,we establish the sufficient condition...In this paper, we deal with the problem on the existence of periodic solution of higher periodic system. Using the exponential dichotomy and the Schauder's fixed point theorem,we establish the sufficient conditions which guarantee the existence and uniqueness and stability of periodic solution.展开更多
By using the theory of exponential dichotomies and the method of Liapunov-Schmidt,in this paper we investigate the existence of homoclinic solutions for autonomous differential equations and obtain a Melnikov-type vec...By using the theory of exponential dichotomies and the method of Liapunov-Schmidt,in this paper we investigate the existence of homoclinic solutions for autonomous differential equations and obtain a Melnikov-type vector. We show that if the Melnikov-type vector satisfies some conditions then autonomous differential equations have homoclinic solutions. Moreover, our result holds only for autonomous differential equations.展开更多
In this paper, a suitable local coordinate system is constructed by using exponential dichotomies and generalizing the Floquet method from periodic systems to nonperiodic systems. Then the Poincare map is established ...In this paper, a suitable local coordinate system is constructed by using exponential dichotomies and generalizing the Floquet method from periodic systems to nonperiodic systems. Then the Poincare map is established to solve various problems in homoclinic bifurcations with codimension one or two. Bifurcation diagrams and bifurcation curves are given.展开更多
Using the theory of invariant manifolds, we give local expressions of the stable and unstable manifolds for normally hyperbolic invariant tori, and study the existence of transverse orbits heteroclinic to hyperbolic i...Using the theory of invariant manifolds, we give local expressions of the stable and unstable manifolds for normally hyperbolic invariant tori, and study the existence of transverse orbits heteroclinic to hyperbolic invariant tori. These extend and improve the corresponding results obtained in [3-5].展开更多
In this paper,we establish the existence of local stable manifolds for a semi-linear differential equation,where the linear part is a Hille-Yosida operator on a Banach space and the nonlinear forcing term f satisfies ...In this paper,we establish the existence of local stable manifolds for a semi-linear differential equation,where the linear part is a Hille-Yosida operator on a Banach space and the nonlinear forcing term f satisfies the ψ-Lipschitz conditions,where ψ belongs to certain classes of admissible function spaces.The approach being used is the fixed point arguments and the characterization of the exponential dichotomy of evolution equations in admissible spaces of functions defined on the positive half-line.展开更多
We consider non-autonomous ordinary differential equations in two cases.One is the one-dimensional case that admits a condition of hyperbolicity,and the other one is the higher-dimensional case that admits an exponent...We consider non-autonomous ordinary differential equations in two cases.One is the one-dimensional case that admits a condition of hyperbolicity,and the other one is the higher-dimensional case that admits an exponential dichotomy.For differential equations of this kind,we give a rigorous treatment of the Harmonic Balance Method to look for almost periodic solutions.展开更多
The main purpose of this paper is to investigate existence of transversal homoclinic orbits of nonlinear system of two small parameters in a degenerate case. By making use ofthe theory of exponential dichotomy and Lia...The main purpose of this paper is to investigate existence of transversal homoclinic orbits of nonlinear system of two small parameters in a degenerate case. By making use ofthe theory of exponential dichotomy and Liapunov-Schmidt method, we construct Melnikov-like functions, and show that if these functions satisfy some conditions thentransversal homoclinic orbits of nonlinear system exists.展开更多
A generalized method combining the exponential dichotomy and the theory of transversality was used to give conditions for the persistence and transversality of homoclinic orbits under small perturbation for the diffeo...A generalized method combining the exponential dichotomy and the theory of transversality was used to give conditions for the persistence and transversality of homoclinic orbits under small perturbation for the diffeomorphisms.展开更多
By means of the fixed point theorem and exponential dichotomy, in this paper we investigate the existence of almost periodic solutions to a class of ndimensional almost periodic systems, which is more general than the...By means of the fixed point theorem and exponential dichotomy, in this paper we investigate the existence of almost periodic solutions to a class of ndimensional almost periodic systems, which is more general than the systems in [1-2]. We generalize and improve some results in [3].展开更多
By using the geometrical method, the higher order Melnikov vector and the associatedcriteria for the persistence, transversality and tangency of homoclinic and heteroclinic orbits are established. Examples of applicat...By using the geometrical method, the higher order Melnikov vector and the associatedcriteria for the persistence, transversality and tangency of homoclinic and heteroclinic orbits are established. Examples of application are also given.展开更多
文摘In this paper we construct, by using the theory of exponential dichotomies, a Melnikov-type function by which we can detect the existence of homoclinic orbits for the perturbed systems x = g(x) + epsilon h(t, x, epsilon). Our result of this paper may be complementary to that of K.J.Palmer([3]).
文摘Based on the theory of generalized exponeatial dichotomies, some typical nonautonomouslinear systems with time-translation parameters or with time-scale parameters are discussed.Conditions of parameters are given to determine by spectral gaps the generalized exponentialdichotomies of those systems and their local L'-limit systexns. These conclusions are applied tosrvcalled SVC systems and HFO systems for their stability.
文摘In this paper, we study the relationship between exponential dichotomy and quadratic Lyapunov function for the linear equation x△=A(t)x on time scales. Moreover, for the nonlinear perturbed equation x△= A(t)x + f(t, x) we give the instability of the zero solution when f is sufficiently small.
基金The Undergraduate Research Training Program Grant(J1030101)the NSF(11271151)of China
文摘In this paper, we consider the existence and uniqueness of the solutions which are pseudo almost automorphic in distribution for a class of non-autonomous stochastic differential equations in a Hilbert space. In conclusion, we use the Banach contraction mapping principle and exponential dichotomy property to obtain our main results.
文摘The inch purpose of this paper is to study homocliaic bifurcation in a degenerate case of differential equations of form i = f(x, ε) and i = g(x) + h(t, x. ε). By using theory of exponential dichotomies and Liapunov-Schmidt method, the authors obtain the Melnikov-like vectors by which the existence of homoclinic orbits can be detected.
基金supported by NSFC(19671017)and by NSF(A970)of Fujian.
文摘It is well known that if x' = A(t)x admits an exponential dichotomy and f(t, x) is a bounded function and has a small Lipschitz constant, then system X' = A(x)x + f(t, x) has a unique bounded solution. In this paper, we shall generalize this theorem to the case when f(t, x) is an unbounded function.
基金This work is supported by NSF of China,No.19401013
文摘The main purpose of this paper is to investigate the existence of almost periodic solutions for the Duffing differential equation.By combining the theory of exponential dichotomies with Liapunov functions,we obtain an intersting result on the existence of almost periodic solutions.
文摘The aim of this paper is to investigate the existence and uniqueness of almost periodic solutions for the forced Rayleigh equation. By combining the theory of exponential dichotomies with Liapunov functions, we obtain an interesting result on the existence of almost periodic solutions.
文摘By applying Horn fixed point theorem and the method of exponential dichotomy,we investigate the existence of almost periodic solution to nonlinear ordinarydifferential equation.
文摘In this paper, we deal with the problem on the existence of periodic solution of higher periodic system. Using the exponential dichotomy and the Schauder's fixed point theorem,we establish the sufficient conditions which guarantee the existence and uniqueness and stability of periodic solution.
文摘By using the theory of exponential dichotomies and the method of Liapunov-Schmidt,in this paper we investigate the existence of homoclinic solutions for autonomous differential equations and obtain a Melnikov-type vector. We show that if the Melnikov-type vector satisfies some conditions then autonomous differential equations have homoclinic solutions. Moreover, our result holds only for autonomous differential equations.
文摘In this paper, a suitable local coordinate system is constructed by using exponential dichotomies and generalizing the Floquet method from periodic systems to nonperiodic systems. Then the Poincare map is established to solve various problems in homoclinic bifurcations with codimension one or two. Bifurcation diagrams and bifurcation curves are given.
基金Supported by the National Natural Science Foundation of China Shanghai Natural Science Foundation.
文摘Using the theory of invariant manifolds, we give local expressions of the stable and unstable manifolds for normally hyperbolic invariant tori, and study the existence of transverse orbits heteroclinic to hyperbolic invariant tori. These extend and improve the corresponding results obtained in [3-5].
基金Deanship of Scientific Research at Majmaah University for supporting this work under Project Number No.R-1441-27.
文摘In this paper,we establish the existence of local stable manifolds for a semi-linear differential equation,where the linear part is a Hille-Yosida operator on a Banach space and the nonlinear forcing term f satisfies the ψ-Lipschitz conditions,where ψ belongs to certain classes of admissible function spaces.The approach being used is the fixed point arguments and the characterization of the exponential dichotomy of evolution equations in admissible spaces of functions defined on the positive half-line.
基金supported by the National Natural Science Foundation of China(Grants No.12071296 and No.11871273)partially supported by the National Natural Science Foundation of China(Grants Nos.12090014,12031020 and 12271509)。
文摘We consider non-autonomous ordinary differential equations in two cases.One is the one-dimensional case that admits a condition of hyperbolicity,and the other one is the higher-dimensional case that admits an exponential dichotomy.For differential equations of this kind,we give a rigorous treatment of the Harmonic Balance Method to look for almost periodic solutions.
文摘The main purpose of this paper is to investigate existence of transversal homoclinic orbits of nonlinear system of two small parameters in a degenerate case. By making use ofthe theory of exponential dichotomy and Liapunov-Schmidt method, we construct Melnikov-like functions, and show that if these functions satisfy some conditions thentransversal homoclinic orbits of nonlinear system exists.
文摘A generalized method combining the exponential dichotomy and the theory of transversality was used to give conditions for the persistence and transversality of homoclinic orbits under small perturbation for the diffeomorphisms.
基金The work is supported by the Natural Sciences Foundation of Hunan Province under Grant 03JJY3014 and the Item of the Government of Science and Technology of Yonezhou.
文摘By means of the fixed point theorem and exponential dichotomy, in this paper we investigate the existence of almost periodic solutions to a class of ndimensional almost periodic systems, which is more general than the systems in [1-2]. We generalize and improve some results in [3].
文摘By using the geometrical method, the higher order Melnikov vector and the associatedcriteria for the persistence, transversality and tangency of homoclinic and heteroclinic orbits are established. Examples of application are also given.