高间Melnikov方法

本文把原有Melnikov方法推广到高阶情况.找到了二阶次谐Melnikov函数表达式,并且证明了在一定条件下可以用二阶次谐Melnikov函数来判定系统的次谐或超次谐的存在.     

Homoclinic orbits for a perturbed quintic-cubic nonlinear Schrödinger equation

The existence of homoclinic orbits for a perturbed cubic-quintic nonlinear Schrödinger equation with even periodic boundary conditions under the generalized parameters conditions is established. We combined geometric singular perturbation theory, Melnikov analysis, and integrable theory to prove the persistence of homoclinic orbits.     

准对称流的混沌和共振流线

采用计算Melnikov函数的方法 ,研究了描述qth(q=3或 6 )准对称流流体粒子运动的动力系统 .文中在分析未扰动系统轨道解析表示的基础上 ,深入考察了扰动系统的分岔情况 .结果表明 ,扰动系统在一定条件下能够分支出混沌和共振流线 .     

Chaotic dynamics of the vibro-impact system under bounded noise perturbation

In this paper, chaotic dynamics of the vibro-impact system under bounded noise excitation is investigated by an extended Melnikov method. Firstly, the Melnikov method in the deterministic vibro-impact system is extended to the stochastic case. Then, a typical stochastic Duffing vibro-impact system is given to application. The analytic conditions for occurrence of chaos are derived by using the random Melnikov process in the mean-square-value sense. In addition, the numerical simulations confirm the validity of analytic results. Also, the influences of interesting system parameters on the chaotic dynamics are discussed.     

周期扰动下卫星运动系统的动力学性质

本文运用Melnikov方法对平面卫星运动系统在周期扰动下所表现出来的动力学性质进行了探讨.首先运用次谐Melnikov方法给出了卫星轨道在周期扰动下存在次谐周期轨道的条件,并进一步运用同宿.Melnikov方法证实了该系统存在Smale马蹄意义下的混沌性质.     

Complete Sets of Mutually Orthogonal Hypercubes and Their Connections to Affine Resolvable Designs

. Recently, Laywine and Mullen proved several generalizations of Bose's equivalence between the existence of complete sets of mutually orthogonal Latin squares of order n and the existence of affine planes of order n. Laywine further investigated the relationship between sets of orthogonal frequency squares and affine resolvable balanced incomplete block designs. In this paper we generalize several of Laywine's results that were derived for frequency squares. We provide sufficient conditions for construction of an affine resolvable design from a complete set of mutually orthogonal Youden frequency hypercubes; we also show that, starting with a complete set of mutually equiorthogonal frequency hypercubes, an analogous construction can always be done. In addition, we give conditions under which an affine resolvable design can be converted to a complete set of mutually orthogonal Youden frequency hypercubes or a complete set of mutually equiorthogonal frequency hypercubes.     

Transversal homoclinic points and hyperbolic sets for non-autonomous maps II

The method of Melnikov is generalized to non-autonomous maps. If the Melnikov function has infinitely many zeros with derivatives bounded away from zero then the system admits a generalized hyperbolic set as it was introduced in part I. The developed theory is applied to almost periodically perturbed differential equations.

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Zusammenfassung Die Methode von Melnikov wird verallgemeinert für nichtautonome Abbildungen. Falls die Melnikov-Funktion unendlich viele Nullstellen mit von Null weg beschränkten Ableitungen hat, dann enthält das System eine verallgemeinerte hyberbolische Menge, wie sie in Teil I eingeführt wurde. Die entwickelte Theorie wird auf fast periodisch gestörte Systeme angewandt.

     

Melnikov method for discontinuous planar systems

This paper treats the occurrence of homoclinic solutions in planar systems with discontinuous right-hand side. More precisely, we deal with a T$T$-periodic perturbed system such that the unperturbed system is an autonomous possessing homoclinic orbit. By means of the so-called “non-smooth” Melnikov function there is shown the existence of a homoclinic solution for a perturbed system. The non-smooth Melnikov function is derived, and the method of how to find it in concrete problems is also introduced.     

Homoclinic waves in a contaminant transport model

A contaminant transport model is studied with nonlinear sorption. Using the Melnikov method, it is shown that homoclinic concentration waves exist under certain conditions. We obtain the analytical form of such a concentration wave.     

Effect of bounded noise on the chaotic motion of a Duffing Van der pol oscillator in a ϕ6 potential

This paper investigates the chaotic behavior of an extended Duffing Van der pol oscillator in a ϕ⁶ potential under additive harmonic and bounded noise excitations for a specific parameter choice. From Melnikov theorem, we obtain the conditions for the existence of homoclinic or heteroclinic bifurcation in the case of the ϕ⁶ potential is bounded, which are complemented by the numerical simulations from which we illustrate the bifurcation surfaces and the fractality of the basins of attraction. The results show that the threshold amplitude of bounded noise for onset of chaos will move upwards as the noise intensity increases, which is further validated by the top Lyapunov exponents of the original system. Thus the larger the noise intensity results in the less possible chaotic domain in parameter space. The effect of bounded noise on Poincare maps is also investigated.     

Splitting Potential and the Poincaré-Melnikov Method for Whiskered Tori in Hamiltonian Systems

Summary. We deal with a perturbation of a hyperbolic integrable Hamiltonian system with n+1 degrees of freedom. The integrable system is assumed to have n -dimensional hyperbolic invariant tori with coincident whiskers (separatrices). Following Eliasson, we use a geometric approach closely related to the Lagrangian properties of the whiskers, to show that the splitting distance between the perturbed stable and unstable whiskers is the gradient of a periodic scalar function of n phases, which we call splitting potential. This geometric approach works for both the singular (or weakly hyperbolic) case and the regular (or strongly hyperbolic) case, and provides the existence of at least n+1 homoclinic intersections between the perturbed whiskers. In the regular case, we also obtain a first-order approximation for the splitting potential, that we call Melnikov potential. Its gradient, the (vector) Melnikov function, provides a first-order approximation for the splitting distance. Then the nondegenerate critical points of the Melnikov potential give rise to transverse homoclinic intersections between the whiskers. Generically, when the Melnikov potential is a Morse function, there exist at least 2 ⁿ critical points. The first-order approximation relies on the n -dimensional Poincaré-Melnikov method, to which an important part of the paper is devoted. We develop the method in a general setting, giving the Melnikov potential and the Melnikov function in terms of absolutely convergent integrals, which take into account the phase drift along the separatrix and the first-order deformation of the perturbed hyperbolic tori. We provide formulas useful in several cases, and carry out explicit computations that show that the Melnikov potential is a Morse function, in different kinds of examples. Received January 18, 1999; final revision received October 25, 1999; accepted December 12, 1999     

Effect of random noise on chaotic motion of a particle in a ϕ6 potential

The chaotic behaviors of a particle in a triple well ϕ⁶ potential possessing both homoclinic and heteroclinic orbits under harmonic and Gaussian white noise excitations are discussed in detail. Following Melnikov theory, conditions for the existence of transverse intersection on the surface of homoclinic or heteroclinic orbits for triple potential well case are derived, which are complemented by the numerical simulations from which we show the bifurcation surfaces and the fractality of the basins of attraction. The results reveal that the threshold amplitude of harmonic excitation for onset of chaos will move downwards as the noise intensity increases, which is further verified by the top Lyapunov exponents of the original system. Thus the larger the noise intensity results in the more possible chaotic domain in parameter space. The effect of noise on Poincare maps is also investigated.     

Qualitative analysis of the anisotropic two-body problem under Seeliger's potential

In this paper we study the two-body problem that describes the motion of two-point masses in an anisotropic space under the influence of the Seeliger potential. We will show that the set of initial conditions leading to collisions and ejections has positive measure and study the capture and escape solutions in the positive-energy case using the infinity manifold. We will also apply the Melnikov method to show that the flow on the zero-energy manifold of another potential which is the sum of the classical Keplerian potential and the anisotropic Seeliger's potential perturbation is chaotic.     

Melnikov method and detection of chaos for non-smooth systems

We extend the Melnikov method to non-smooth dynamical systems to study the global behavior near a non-smooth homoclinic orbit under small time-periodic perturbations. The definition and an explicit expression for the extended Melnikov function are given and applied to determine the appearance of transversal homoclinic orbits and chaos. In addition to the standard integral part, the extended Melnikov function contains an extra term which reflects the change of the vector field at the discontinuity. An example is discussed to illustrate the results.     

Exponential asymptotics of oscillatory integrals. Application to the problem of persistence of homoclinic connections for the 02iω resonance

We explain in this Note how to obtain an exponentially small equivalent of an oscillatory integral when it involves solutions of nonlinear differential equation. The method proposed in this Note enables us to study the problem of existence of homoclinic connections to 0 for vector fields admitting a 0²iω resonance at the origin. This problem could not be solved by a direct application of the classical Melnikov method since the Melnikov function is given in this case by an exponentially small oscillatory integral.     

Irreducible function bases for simple fluids and liquid crystal films

We present a rigorous derivation of the canonical representation of a class of constitutive functions for liquid crystal films which has been widely used in various special forms in the fields of emulsion chemistry and cell-membrane biology. The representation yields the largest class of functions compatible with an appropriate definition of fluidity. The method used also furnishes established representation formulas in the classical theories of capillarity and three-dimensional compressible fluids.     

Exponential asymptotics of bi-oscillatory integrals. Application to the problem of persistence of homoclinic connections for the (iω0)2iω1 resonance

We explain in this Note how to obtain an exponentially small equivalent of a bioscillatory integral when it involves solutions of a nonlinear differential equation. The method proposed in this Note enables us to study the problem of existence of homoclinic connections for vector fields admitting a (iω₀)²iω₁ resonance at the origin. This problem could not be solved by a direct application of the classical Melnikov method since the Melnikov function is given in this case by an exponentially small bi-oscillatory integral.     

Asymmetric type II periodic motions for nonlinear impact oscillators

In this paper a general class of nonlinear impact oscillators is considered for Type II periodic motions. This system can be used to model an inverted pendulum impacting on rigid walls under external periodic excitation. The unperturbed system possesses a pair of homoclinic cycles and three separate families of periodic orbits inside and outside the homoclinic cycles via the identification given by the impact law. By approximating the Poincaré map to O(ε)$O\left(\epsilon \right)$ directly, a general method of Melnikov type for detecting the existence of asymmetric Type II subharmonic orbits outside the homoclinic cycles is presented.     

Heteroclinic bifurcation in a class of planar piecewise smooth systems with multiple zones

We discuss heteroclinic bifurcation in a class of periodically excited planar piecewise smooth systems with discontinuities on finitely many smooth curves intersecting at the origin. Assume that the unperturbed system has a hyperbolic saddle in each subregion, and those saddles are connected by a heteroclinic cycle that crosses every switching curve transversally exactly once. We present a method of Melnikov type to derive sufficient conditions under which the perturbed stable and unstable manifolds intersect transversally. Such transversal intersections imply that the corresponding Poincaré map has a transverse heteroclinic cycle. As applications, we present examples with 2 and 4 switching curves respectively. Our numerical simulations suggest that such transversal intersections result in the appearance of chaotic motions in those example systems.