HOMOCLINIC ORBITS FOR A CLASS OF THE SECOND ORDER HAMILTONIAN SYSTEMS

The existence of homoclinic orbits is obtained by the variational approach for a class of second order Hamiltonian systems q（t） ＋ ↓△V（t, q（t）） = 0, where V（t, x） = -K（t, x） ＋ W（t, x）, K（t, x） is neither a quadratic form in x nor periodic in t and W（t, x） is superquadratic in x.

EXPONENTIAL DICHOTOMIES,HOMOCLINIC ORBITS AND METHODS OF MELNIKOV

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]).

HOMOCLINIC ORBITS OF THE DAVEY- STEWARTSON EQUATIONS

The homoclinic solutions problem of the Davey-Stewartson ( DS) Equations were studied. By using the Hirota's bilinear method, the homoclinic orbits of the Davey-Stewartson Equations were obtained through the dependent variable transformation. The homoclinic orbits of the Davey-Stewartson Equations were discussed.

POSITIVE HOMOCLINIC ORBITS FOR A CLASS OF ASYMPTOTICALLY PERIODIC SECOND ORDER DIFFERENTIAL EQUATIONS

This note studies the existence of positive homoclinic orbits of the second order equation-u″+α(x)u=β(x)u q+γ(x)u p, x∈R,where 1<q<p.Assume that the coefficient functions α(x),β(x) and γ(x) are asymptotically periodic and satisfy0<a≤α(x), 0<γ(x)≤B, -M≤β(x)≤M.A positive homoclinic orbit of the equation is obtained by means of variational methods.

Limit cycles and homoclinic orbits and their bifurcation of Bogdanov-Takens system

A quantitative analysis of limit cycles and homoclinic orbits, and the bifurcation curve for the Bogdanov-Takens system are discussed. The parameter incremental method for approximate analytical-expressions of these problems is given. These analytical-expressions of the limit cycle and homoclinic orbit are shown as the generalized harmonic functions by employing a time transformation. Curves of the parameters and the stability characteristic exponent of the limit cycle versus amplitude are drawn. Some of the limit cycles and homoclinic orbits phase portraits are plotted. The relationship curves of parameters μ and A with amplitude a and the bifurcation diagrams about the parameter are also given. The numerical accuracy of the calculation results is good.

ON THE HOMOCLINIC ORBITS IN A CLASS OF TWO-DEGREE-OF-FREEDOM SYSTEMS UNDER THE RESONANCE CONDITIONS

A class of two-degree-of-freedom systems in resonance with an external, parametric excitation is investigated, the existence of the periodic solutions locked to Omega is proved by the use of the method of multiple scales. This systems can be transformed into the systems of Wiggins under some conditions. A calculating formula which determines the existence of homoclinic orbits of the systems is given.

Homoclinic orbits in three-dimensional Shilnikov-type chaotic systems

In this paper, the Pad6 approximant and analytic solution in the neighborhood of the initial value are introduced into the process of constructing the Shilnikov type homoclinic trajectories in three-dimensional nonlinear dynamical systems. The P1D controller system with quadratic and cubic nonlinearities, the simplified solar-wind-driven-magnetosphere-ionosphere system, and the human DNA sequence system are considered. With the aid of presenting a new condition, the solutions of solving the boundary-value problems which are formulated for the trajectory and evaluating the initial amplitude values become available. At the same time, the value of the bifurcation parameter is obtained directly, which is almost consistent with the numerical result.

Homoclinic orbits for some(2+1)-dimensional nonlinear Schrdinger-like equations

Chaos is closely associated with homoclinic orbits in deterministic nonlinear dynamics. In this paper, analytic expressions of homoclinic orbits for some （2＋1）- dimensional nonlinear Schrodinger-like equations are constructed based on Hirota＇s bilinear method, including long wave-short wave resonance interaction equation, generalization of the Zakharov equation, Mel＇nikov equation, and g-Schrodinger equation are constructed based on Hirota＇s bilinear method.

HOMOCLINIC ORBITS FOR SECOND ORDER HAMILTONIAN SYSTEM WITH QUADRATIC GROWTH

Some existence and multiplicity of homoclinic orbit for second order Hamiltonian system x-a(t)x + Wx(t, x)=0 are given by means of variational methods, where the potential V(t, x)=-a(t)|s|2 + W(t, s) is quadratic in s at infinity and subquadratic in s at zero, and the function a(t) satisfies the growth condition lim→∞∫_t￣(t+l) a(t)dt=+∞,l∈R￣1.

Homoclinic orbits of first order discrete Hamiltonian systems with super linear terms

In this paper we consider the first order discrete Hamiltonian systems {x1（n＋1）-x1（n）=Hx2（n,x（n））,x2（n）-x2（n-1）=Hx1（n,x（n））,where x（n） = （x2（n）x1（n））∑ R^2N, H（n,z） = 1/2S（n）z. z ＋ R（n,z） is periodic in n and superlinear as {z} →4 ∞. We prove the existence and infinitely many （geometrically distinct） homoclonic orbits of the system by critical point theorems for strongly indefinite functionals.

Bifurcation of Degenerate Homoclinic Orbits toSaddle-Center in Reversible Systems

The authors study the bifurcation of homoclinic orbits from a degenerate homoclinic orbit in reversible system. The unperturbed system is assumed to have saddle-center type equilibrium whose stable and unstable manifolds intersect in two-dimensional manifolds. A perturbation technique for the detection of symmetric and nonsymmetric homoctinic orbits near the primary homoclinic orbits is developed. Some known results are extended.

HOMOCLINIC ORBITS FOR A CLASS OF HAMILTONIAN SYSTEMS WITH POTENTIALS CHANGING SIGN

In this paper, we prove the existence of nontrivial homoclinic orbits for a class of Hamiltonian systems with potential changing sign. We use Mountain Pass Lemma.

Singular Homoclinic Orbits and Limit Cycles in Positive Second Order Slow-Fast Systems

In the paper we consider a wide class of slow-fast second order systems and give sufficient conditions for the existence of a singular limit cycle related to a homoclinic orbit.

Bifurcations of double homoclinic flip orbits with resonant eigenvalues

Concerns double homoclinic loops with orbit flips and two resonant eigenvalues in a four-dimensional system. We use the solution of a normal form system to construct a singular map in some neighborhood of the equilibrium, and the solution of a linear variational system to construct a regular map in some neighborhood of the double homoclinic loops, then compose them to get the important Poincaré map. A simple calculation gives explicitly an expression of the associated successor function. By a delicate analYSiS of the bifurcation equation, we obtain the condition that the original double homoclinic loops are kept, and prove the existence and the existence regions of the large 1-homoclinic orbit bifurcation surface, 2-fold large 1-periodic orbit bifurcation surface, large 2-homoclinic orbit bifurcation surface and their approximate expressions. We also locate the large periodic orbits and large homoclinic orbits and their number.

BIFURCATION OF DEGENERATE HOMOCLINIC ORBITS

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.

INFINITELY MANY HOMOCLINIC ORBITS FOR A CLASS OF HAMILTONIAN SYSTEMS WITH SYMMETRY

This paper deals via variational methods with the existence of infinitely many homoclinic orbits for a class of first order time dependent Hamiltonian systems=JH z(t,z)without any periodicity assumption on H, providing that H(t,z) iseven with respect to z∈R 2N , superquadratic or subquadratic as |z|→∞, and satisfies some additional assumptions.

CODIMENSION 3 BIFURCATIONS OFHOMOCLINIC ORBITS WITH ORBITFLIPS AND INCLINATION FLIPS

The homoclinic bifurcations in four dimensional vector fields are investigated by setting up a local coordinates near the homoclinic orbit. This homoclinic orbit is nonprincipal in the meanings that its positive semi-orbit takes orbit flip and its unstable foliation takes inclination flip. The existence, nonexistence, uniqueness and coexistence of the 1-homoclinic orbit and the 1-periodic orbit are studied. The existence of the twofold periodic orbit and three-fold periodic orbit are also obtained.

THE EXISTENCE OF HOMOCLINIC ORBITS FOR HAMILTONIAN SYSTEMS WITH THE POTENTIALS CHANGING SIGN

This paper studies the existence of nontrival homoclinic orbits of the Hamiltonian systems -L(t)q+V′(t,q)=0 by using the critical point theory, where the potential V(t,q)=b(t)W(q) can change sign. Under a new kind of "superquadratic" condition on W, some new results are obtained.

HOMOCLINIC ORBITS IN PERTURBED GENERALIZED HAMILTONIAN SYSTEMS

It this paper we obtain existence and bifurcation theorems for homoclinic orbits in three-dimeensional,time dependent and independent,perturbations of generalized Hamiltonian differential equations defined on three-dimensional Poisson manifolds.Thed we apply them to a truncated spectral model of the quasi-geostrophic flow on a cyclic β-plane.

A generalized Padé approximation method of solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators

An intrinsic extension of Pad′e approximation method, called the generalized Pad′e approximation method, is proposed based on the classic Pad′e approximation theorem. According to the proposed method, the numerator and denominator of Pad′e approximant are extended from polynomial functions to a series composed of any kind of function, which means that the generalized Pad′e approximant is not limited to some forms, but can be constructed in different forms in solving different problems. Thus, many existing modifications of Pad′e approximation method can be considered to be the special cases of the proposed method. For solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators, two novel kinds of generalized Pad′e approximants are constructed. Then, some examples are given to show the validity of the present method. To show the accuracy of the method, all solutions obtained in this paper are compared with those of the Runge–Kutta method.