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1.
Prof. G. Cicogna 《Acta Mechanica》1989,76(3-4):189-202
Summary We shall consider in detail, by means of Melnikov method, the appearance of transverse homoclinic points in 1-dimensional
problems whose potential term is given by an even function perturbed by some parity-breaking term. After having described
the two possible and essentially different situations one can find, we shall introduce two methods in order to carefully analyze
in each case the effects of the symmetry perturbing term on the appearance of homoclinic points (and therefore on the onset
of the typical “Smale horseshoe” chaotic motion). As examples for the various cases, we shall consider a perturbed Duffing
equation the equation for the Josephson junction, and finally an equation for which we can provide the explicit solution,
and which becomes in this way a good test for the results.
With 3 Figures 相似文献
2.
The existence of bounded solutions (including in particular homoclinic and heteroclinic solutions) is studied for non-autonomous perturbed parabolic partial differential equations, without the restriction that the linear variational equation has a unique non-trivial bounded solution. Specifically, an idea applied to ordinary differential equations by Hale (1984) and by Battelli and Laari (1990) is realised in an infinite-dimensional setting. Like other work on related problems, the main technique is Lyapunov?Schmidt reduction; we use that technique here in the context of bounded solutions, rather than the more usual setting of periodic or homoclinic solutions. Moreover, several technical obstacles are circumvented in the infinite-dimensional setting?in particular in the proof of the existence of a solution to the reduced bifurcation equation. Non-uniqueness is shown to occur for the Kuramoto-Sivashinsky equation, demonstrating the need to remove the uniqueness restriction 相似文献
3.
The properties of various numerical methods for the study of the perturbed sine-Gordon (sG) equation with impulsive forcing are investigated. In particular, finite difference and pseudo-spectral methods for discretizing the equation are considered. Different methods of discretizing the Dirac delta are discussed. Various combinations of these methods are then used to model the soliton–defect interaction. A comprehensive study of convergence of all these combinations is presented. Detailed explanations are provided of various numerical issues that should be carefully considered when the sG equation with impulsive forcing is solved numerically. The properties of each method depend heavily on the specific representation chosen for the Dirac delta—and vice versa. Useful comparisons are provided that can be used for the design of the numerical scheme to study the singularly perturbed sG equation. Some interesting results are found. For example, the Gaussian approximation yields the worst results, while the domain decomposition method yields the best results, for both finite difference and spectral methods. These findings are corroborated by extensive numerical simulations. 相似文献
4.
This article studies the boundary element solution of two-dimensional sine-Gordon (SG) equation using continuous linear elements approximation. Non-linear and in-homogenous terms are converted to the boundary by the dual reciprocity method and a predictor–corrector scheme is employed to eliminate the non-linearity. The procedure developed in this paper, is applied to various problems involving line and ring solitons where considered in references [Argyris J, Haase M, Heinrich JC. Finite element approximation to two-dimensional sine-Gordon solitons. Comput Methods Appl Mech Eng 1991;86:1–26; Bratsos AG. An explicit numerical scheme for the sine-Gordon equation in 2+1 dimensions. Appl Numer Anal Comput Math 2005;2(2):189–211, Bratsos AG. A modified predictor–corrector scheme for the two-dimensional sine-Gordon equation. Numer Algorithms 2006;43:295–308; Bratsos AG. The solution of the two-dimensional sine-Gordon equation using the method of lines. J Comput Appl Math 2007;206:251–77; Bratsos AG. A third order numerical scheme for the two-dimensional sine-Gordon equation. Math Comput Simul 2007;76:271–8; Christiansen PL, Lomdahl PS. Numerical solutions of 2+1 dimensional sine-Gordon solitons. Physica D: Nonlinear Phenom 1981;2(3):482–94; Djidjeli K, Price WG, Twizell EH. Numerical solutions of a damped sine-Gordon equation in two space variables. J Eng Math 1995;29:347–69; Dehghan M, Mirzaei D. The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation. Comput Methods Appl Mech Eng 2008;197:476–86]. Using continuous linear elements approximation produces more accurate results than constant ones. By using this approach all cases associated to SG equation, which exist in literature, are investigated. 相似文献
5.
In a well-known science fiction novel by Ray Bradbury, a butterfly in the pre-historic past is accidentally crushed by time
travelers and this seemingly insignificant event radically changes history. We will call this the “Bradbury Butterfly” Effect
(BBE). We know that such an effect occurs in Long Josephson Junctions as described by a time-dependent nonlinear sine-Gordon
equation. This equation states that any alteration within the initial perturbation fundamentally changes the asymptotic state
of the system. The actual manifestation of this effect is proven by a numerical simulation of the time-dependent sine-Gordon
equation. 相似文献
6.
《International Journal of Engineering Science》2007,45(2-8):258-271
In general, this paper deals with nonlinear double sine-Gordon equation with even potential energy which has arisen in many physical phenomena. The nonlinear dispersion problems without a small perturbation parameter are difficult to be solved analytically. Hence, the main concern is focused on solving the traveling wave of the double sine-Gordon equation. As commonly known, the perturbation method is for solving problems with small parameters, and the analytical representation thus derived has, in most cases, a small range of validity. For some nonlinear problems, although an exact analytical solution can be achieved, they often appear in terms of sophisticated implicit functions, and are not convenient for application. Although a variety of transformation methods has been developed for solving the nonlinear dispersion problems, such transformed equations still include nonlinear terms. To overcome these difficulties, a new approach for Newton-harmonic balance (NHB) method with Fourier–Bessel series is presented here. It is applied to solve the higher-order analytical approximations for dispersion relation in double sine-Gordon equation. The Fourier–Bessel series with the NHB method presents excellent improvement from lower-order to higher-order analytical approximations involving the nonlinear terms in the double sine-Gordon equation. Not restricted by the existence of a small perturbation parameter, the method is suitable for small as well as large amplitudes of wavetrains. Excellent agreement with exact solutions is presented in some practical examples. 相似文献
7.
一类非线性振子中有界噪声诱发的混沌运动 总被引:3,自引:0,他引:3
研究谐和外力与有界噪声激励联合作用下的一类非线性振子的混沌运动。利用Melnikov方法,通过计算扰动系统的Melnikov积分,分析了系统在参数发生变化时的同宿分岔,得出系统产生混沌运动的参数阈值,并讨论了有界噪声激励对系统的混沌运动的影响。最后利用数值方法模拟了系统的安全盆的侵蚀状况,并进一步通过计算系统运动的Lyapunov指数,给出了由噪声诱发的混沌运动与噪声激励下非混沌运动之间的差别。 相似文献
8.
In this paper, we derive the dynamic equation of molecular motion for a twisted nematic liquid crystal (NLC) under applied electric and magnetic fields, and show that it takes the form of a double sine-Gordon (DSG) equation. Two kink and anti-kink solitary solutions of the liquid crystal molecules are obtained by using the F-expansion method to solve the DSG equation. Finally, we confirm that the twist of the NLC molecules can propagate in the form of solitary waves. The propagation velocity and amplitude of the kink and anti-kink solitons induced by the electric and magnetic fields are discussed. 相似文献
9.
Jürgen Knobloch Thorsten Rieß Martin Vielitz 《Dynamical Systems: An International Journal》2011,26(3):335-365
Homoclinic snaking refers to the sinusoidal ‘snaking’ continuation curve of homoclinic orbits near a heteroclinic cycle connecting an equilibrium E and a periodic orbit P. Along this curve the homoclinic orbit performs more windings about the periodic orbit. Typically, this behaviour appears in reversible Hamiltonian systems. Here we discuss this phenomenon in systems without any particular structure. We give a rigorous analytical verification of homoclinic snaking under certain assumptions on the behaviour of the stable and unstable manifolds of E and P. We show how the snaking behaviour depends on the signs of the Floquet multipliers of P. Further we present a nonsnaking scenario. Finally, we show numerically that these assumptions are fulfilled in a model equation. 相似文献
10.
Cao Q Wiercigroch M Pavlovskaia EE Thompson JM Grebogi C 《Philosophical transactions. Series A, Mathematical, physical, and engineering sciences》2008,366(1865):635-652
In a recent paper we examined a model of an arch bridge with viscous damping subjected to a sinusoidally varying central load. We showed how this yields a useful archetypal oscillator which can be used to study the transition from smooth to discontinuous dynamics as a parameter, alpha, tends to zero. Decreasing this smoothness parameter (a non-dimensional measure of the span of the arch) changes the smooth load-deflection curve associated with snap-buckling into a discontinuous sawtooth. The smooth snap-buckling curve is not amenable to closed-form theoretical analysis, so we here introduce a piecewise linearization that correctly fits the sawtooth in the limit at alpha=0. Using a Hamiltonian formulation of this linearization, we derive an analytical expression for the unperturbed homoclinic orbit, and make a Melnikov analysis to detect the homoclinic tangling under the perturbation of damping and driving. Finally, a semi-analytical method is used to examine the full nonlinear dynamics of the perturbed piecewise linear system. A chaotic attractor located at alpha=0.2 compares extremely well with that exhibited by the original arch model: the topological structures are the same, and Lyapunov exponents (and dimensions) are in good agreement. 相似文献
11.
非线性偏微分方程的显式解析解,特别是行波解,蕴含了方程的丰富信息,对于描述各种现象的发展规律起着至关重要的作用.本文尝试构造 KdV 方程多种形式的新显式行波解.首先,利用试探函数法和 Matlab计算给出了 Riccat 方程的许多新显式解析解.其次,运用广义 Tanh 函数法以及 Riccati 方程的新解得到了 sine-Gordon 方程的许多新显式解析解.最后,作为新的应用,把三角函数法结合 sine-Gordon 方程的新显式解析解并利用简化的变换形式进一步找到了 KdV 方程的许多新显式行波解.这些结果推广和补充了以往的相关研究成果,特别地,这些方法和新的结果可以用于求解许多非线性偏微分方程的新显式行波解. 相似文献
12.
A. Golmakani 《Dynamical Systems: An International Journal》2011,26(1):61-76
Lorenz-like attractors are known to appear in unfoldings from certain codimension two homoclinic bifurcations for differential equations in ?3 that possess a reflectional symmetry. This includes homoclinic loops under a resonance condition and the inclination-flip homoclinic loops. We show that Lorenz-like attractors also appear in the third possible codimension two homoclinic bifurcation (for homoclinic loops to equilibria with real different eigenvalues); the orbit-flip homoclinic bifurcation. We moreover provide a bifurcation analysis computing the bifurcation curves of bifurcations from periodic orbits and discussing the creation and destruction of the Lorenz-like attractors. Known results for the inclination flip are extended to include a bifurcation analysis. 相似文献
13.
Summary By considering a model equation we are able to derive conditions under which a limit cycle, created (at small amplitude) by a Hopf bifurcation, can be destroyed (at finite amplitude) by an infinite period bifurcation, this latter appearing out of a homoclinic orbit formed by the separatrices of a saddle-point equilibrium state. Further, we are able to extend the methods used for showing the existence of an infinite period bifurcation to calculate the amplitude of the limit cycle over its whole range of existence. These ideas are then applied to an equation arising in the theory of roll waves down an open inclined channel, extending previous work to include the case when the Reynolds number is large with the Froude number close to its critical value for the temporal instability of the uniform flow. Here the governing equation reduces to one similar in form to the model equation.With 3 Figures 相似文献
14.
This study deals with the nonlinear dynamics associated with large deformation of abeam sliding on two-knife edge supports
under external excitation. The beam is referred to as a Gospodnetic-Frisch-Fay beam, after the researchers who reported its
static deformation in closed form. The freedom of the beam to slide on its supports imparts a nonlinear characteristic to
the force-deflection response. The restoring elastic force of the beam possesses characteristics similar to those of the roll-restoring
moment of ships. The Gospodnetic-Frisch-Fay exact solution is given in terms of elliptic functions. A curve fit of the exact
solution up to eleventhorder is constructed to establish the governing equation of motion under external excitation. The dynamic
stability of the unperturbed beam is examined for the damped and undamped cases. The undamped case reveals periodic orbits
and one homoclinic orbit depending on the value of the initial conditions. The response to a sinusoidal excitation at a frequency
below the linear natural frequency is numerically estimated for different excitation amplitude and different values of initial
conditions covered by the area of the homoclinic orbit. The safe basins of attraction are plotted for different values of
excitation amplitude. It is found that the safe region of operation is reduced as the excitation amplitude increases
This paper is dedicated to Prof R N Iyengar of the Indian Institute of Science on the occasion of his formal retirement. 相似文献
15.
研究了S形本构关系的弹性直杆纵振时的混沌行为.用Galerkin原理将杆纵振时的动力控制方程转化为二阶三次非线性微分动力系统;给出了其产生同宿轨道和异宿轨道的条件,得到了同宿轨道的参数方程;借助Melnikov函数给出了系统发生混沌的临界条件;数值计算给出了混沌运动区域随β和γ的变化规律,用分岔图、位移时程曲线、相平面图和Poincaré映射判断了系统的运动行为即定常还是混沌.进一步的研究还表明本构关系中的二次非线性项对系统的动力响应具有很大的影响. 相似文献
16.
Finite-dimensional perturbing operators are constructed using some incomplete information about eigen-solutions of an original and/or adjoint generalized Fredholm operator equation (with zero index). Adding such a perturbing operator to the original one reduces the eigen-space dimension and can, particularly, lead to an unconditionally and uniquely solvable perturbed equation. For the second kind Fredholm operators, the perturbing operators are analyzed such that the spectrum points for an original and the perturbed operators coincide except a spectrum point considered, which can be removed for the perturbed operator. A relation between resolvents of original and perturbed operators is obtained. Effective procedures are described for calculation of the undetermined constants in the right-hand side of an operator equation for the case when these constants must be chosen to satisfy the solvability conditions not written explicitly. Implementation of the methods is illustrated on a boundary integral equation of elasticity. 相似文献
17.
Summary We study the Melnikov criterion for a global homoclinic bifurcation and possible transition to chaos for a single degree of
freedom nonlinear oscillator. This provides a systematic method of treatment for an arbitrary potential expressed as a fourth
order polynomial. The equation of motion has external excitation and a Duffing type nonlinearity with one or two unsymmetric
potential wells. 相似文献
18.
The numerical analysis of the excitation of the Josephson junction (Jj) by an external electromagnetic field has been reported. The fully nonlinear approach requires using the quasiperiodic solutions of the sine-Gordon equation (sGe) which are expressed by the Riemann theta functions. Their local soliton approximation of the nondissipative Jj is presented. Using the self-consistent phenomenological model of a long Jj, the influence of the external electromagnetic field on the volt-ampere characteristics is considered. 相似文献
19.
20.
Lakshmanan M 《Philosophical transactions. Series A, Mathematical, physical, and engineering sciences》2011,369(1939):1280-1300
The Landau-Lifshitz-Gilbert (LLG) equation is a fascinating nonlinear evolution equation both from mathematical and physical points of view. It is related to the dynamics of several important physical systems such as ferromagnets, vortex filaments, moving space curves, etc. and has intimate connections with many of the well-known integrable soliton equations, including nonlinear Schr?dinger and sine-Gordon equations. It can admit very many dynamical structures including spin waves, elliptic function waves, solitons, dromions, vortices, spatio-temporal patterns, chaos, etc. depending on the physical and spin dimensions and the nature of interactions. An exciting recent development is that the spin torque effect in nanoferromagnets is described by a generalization of the LLG equation that forms a basic dynamical equation in the field of spintronics. This article will briefly review these developments as a tribute to Robin Bullough who was a great admirer of the LLG equation. 相似文献