Twisted and nontwisted bifurcations induced by diffusion

We discuss a diffusively perturbed predator-prey system. Freedman and Wolkowicz showed that the corresponding ODE can have a periodic solution that bifurcates from a homoclinic loop. When the diffusion coefficients are large, this solution represents a stable, spatially homogeneous time-periodic solution of the PDE. We show that when the diffusion coefficients become small, the spatially homogeneous periodic solution becomes unstable and bifurcates into spatially nonhomogeneous periodic solutions. The nature of the bifurcation is determined by the twistedness of an equilibrium/homoclinic bifurcation that occurs as the diffusion coefficients decrease. In the nontwisted case two spatially nonhomogeneous simple periodic solutions of equal period are generated, while in the twisted case a unique spatially nonhomogeneous double periodic solution is generated through period-doubling.     

Non-existence of Shilnikov chaos in continuous-time systems

In this paper,a non-existence condition for homoclinic and heteroclinic orbits in n-dimensional,continuous-time,and smooth systems is obtained.Based on this result and an elementary example,it can be conjectured that there is a fourth kind of chaos in polynomial ordinary differential equation(ODE) systems characterized by the nonexistence of homoclinic and heteroclinic orbits.     

On the Chaotic Dynamics of a Spherical Pendulum with a Harmonically Vibrating Suspension

The equations of motion for a lightly damped spherical pendulum are considered. The suspension point is harmonically excited in both vertical and horizontal directions. The equations are approximated in the neighborhood of resonance by including the third order terms in the amplitude. The stability of equilibrium points of the modulation equations in a four-dimensional space is studied. The periodic orbits of the spherical pendulum without base excitations are revisited via the Jacobian elliptic integral to highlight the role played by homoclinic orbits. The homoclinic intersections of the stable and unstable manifolds of the perturbed spherical pendulum are investigated. The physical parameters leading to chaotic solutions in terms of the spherical angles are derived from the vanishing Melnikov–Holmes–Marsden (MHM) integral. The existence of real zeros of the MHM integral implies the possible chaotic motion of the harmonically forced spherical pendulum as a result from the transverse intersection between the stable and unstable manifolds of the weakly disturbed spherical pendulum within the regions of investigated parameters. The chaotic motion of the modulation equations is simulated via the 4th-order Runge–Kutta algorithms for certain cases to verify the analysis.     

Sine-Gordon方程的截断系统的同宿轨道

研究Ｓｉｎｅ Ｇｏｒｄｏｎ方程的广义渐近惯性流形上的常微分方程组，证实了在一定参数条件下存在Ｗｉｇｇｉｎｓ［１］意义下的同宿轨道．计算表明，与Ｂｉｓｈｏｐ［２］用数值计算得到的Ｓｉｎｅ Ｇｏｒｄｏｎ方程产生混沌的参数值尚有差别，考虑到同宿出现参数值往往低于混沌出现参数值，故结果在定性上正确，而且改进了文［１］中的结果．     

Multi-pulse homoclinic orbits and chaotic dynamics for an axially moving viscoelastic beam

The multi-pulse homoclinic orbits and chaotic dynamics for an axially moving viscoelastic beam are investigated in the case of 1:2 internal resonance. On the basis of the modulation equations derived by the method of multiple scales, the theory of normal form is utilized to find the explicit formulas of normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method is employed to analyze the global bifurcations for the axially moving viscoelastic beam. The results obtained here indicate that there exist the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the resonant case, leading to chaos in the system. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found. To illustrate the theoretical predictions, we present visualizations of these complicated structures.     

Travelling Breathers in Klein-Gordon Lattices as Homoclinic Orbits to <Emphasis Type="BoldItalic">p</Emphasis>-Tori

Klein-Gordon chains are one-dimensional lattices of nonlinear oscillators in an anharmonic on-site potential, linearly coupled with their first neighbors. In this paper, we study the existence in such networks of spatially localized solutions, which appear time periodic in a referential in translation at constant velocity. These solutions are called travelling breathers. In the case of travelling wave solutions, the existence of exact solutions has been obtained by Iooss and Kirchgässner. Formal multiscale expansions have been used by Remoissenet to derive approximate solutions of travelling breathers in the form of modulated plane waves. James and Sire have studied the existence of specific travelling breather solutions, consisting in pulsating travelling waves which are exactly translated of 2 lattice sites after a fixed propagation time T. In this paper, we generalize this approach to pulsating travelling waves which are exactly translated of p≥ 3 sites after a given time T p being arbitrary. By formulating the problem as a dynamical system, one is able to reduce the system locally to a finite dimensional set of ordinary differential equations (ODE), whose dimension depends on the parameter values of the problem. We prove that the principal part of this system of ODE admits homoclinic connections to p-tori under general conditions on the potential. One can obtain leading order approximations of these homoclinic connections and these orbits should correspond, for the oscillator chain, to small amplitude travelling breather solutions superposed on an exponentially small quasi-periodic tail.     

Multi-pulse chaotic motions of functionally graded truncated conical shell under complex loads

The global bifurcations and multi-pulse orbits of an aero-thermo-elastic functionally graded material (FGM) truncated conical shell under complex loads are investigated with the case of 1:2 internal resonance and primary parametric resonance. The method of multiple scales is utilized to obtain the averaged equations. Based on the averaged equations obtained, the normal form theory is employed to find the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method developed by Haller and Wiggins is used to analyze the multi-pulse homoclinic bifurcations and chaotic dynamics of the FGM truncated conical shell. The analytical results obtained here indicate that there exist the multi-pulse Shilnikov-type homoclinic orbits for the resonant case which may result in chaos in the system. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found. The diagrams show a gradual breakup of the homoclinic tree in the system as the dissipation factor is increased. Numerical simulations are presented to illustrate that for the FGM truncated conical shell, the multi-pulse Shilnikov-type chaotic motions can occur. The influence of the structural-damping, the aerodynamic-damping, and the in-plane and transverse excitations on the system dynamic behaviors is also discussed by numerical simulations. The results obtained here mean the existence of chaos in the sense of the Smale horseshoes for the FGM truncated conical shell.     

Bifurcations of heteroclinic loop accompanied by pitchfork bifurcation

In this paper, using the local coordinate moving frame approach, we investigate bifurcations of generic heteroclinic loop with a hyperbolic equilibrium and a nonhyperbolic equilibrium which undergoes a pitchfork bifurcation. Under some generic hypotheses, the existence of homoclinic loop, heteroclinic loop, periodic orbit and three or four heteroclinic orbits is obtained. In addition, the non-coexistence conditions for homoclinic loop and periodic orbit are also given. Note that the results achieved here can be extended to higher dimensional systems.     

Weak Solutions of a Generalized Boussinesq System

We study the convergence of homoclinic orbits and heteroclinic orbits in the dynamical system governing traveling wave solutions of a perturbed Boussinesq systems modeling two-directional propagation of water waves. Nonanalytic weak solutions are found to be limits of these orbits, including compactons, peakons, and rampons, as well as infinitely many mesaons occurring at the same fixed point in the dynamical system. Singularities of solitary wave solutions in the system are also studied to understand the important impact of both linear and nonlinear dispersion terms on the regularity of these solutions.     

Singular perturbation theory for homoclinic orbits in a class of near-integrable Hamiltonian systems

This paper describes a new type of orbits homoclinic to resonance bands in a class of near-integrable Hamiltonian systems. It presents a constructive method for establishing whether small conservative perturbations of a family of heteroclinic orbits that connect pairs of points on a circle of equilibria will yield transverse homoclinic connections between periodic orbits in the resonance band resulting from the perturbation. In any given example, this method may be used to prove the existence of such transverse homoclinic orbits, as well as to determine their precise shape, their asymptotic behavior, and their possible bifurcations. The method is a combination of the Melnikov method and geometric singular perturbation theory for ordinary differential equations.     

Sliding homoclinic orbits and bifurcations of three-dimensional piecewise affine systems

Sliding dynamics is a peculiar phenomenon to discontinuous dynamical systems, while homoclinic orbits play a role in studying the global dynamics of dynamical systems. This paper provides a method to ensure the existence of sliding homoclinic orbits of three-dimensional piecewise affine systems. In addition, sliding cycles are obtained by bifurcations of the systems with sliding homoclinic orbits to saddles. Two examples with simulations of sliding homoclinic orbits and sliding cycles are provided to illustrate the effectiveness of the results.

     

Three kinds of nonlinear dispersive waves in elastic rods with finite deformation

On the basis of classical linear theory on longitudinal, torsional and flexural waves in thin elastic rods, and taking finite deformation and dispersive effects into consideration, three kinds of nonlinear evolution equations are derived. Qualitative analysis of three kinds of nonlinear equations are presented. It is shown that these equations have homoclinic or heteroclinic orbits on the phase plane, corresponding to solitary wave or shock wave solutions, respectively. Based on the principle of homogeneous balance, these equations are solved with the Jacobi elliptic function expansion method. Results show that existence of solitary wave solution and shock wave solution is possible under certain conditions. These conclusions are consistent with qualitative analysis.     

Global bifurcations in externally excited autoparametric systems

We consider an autoparametric system consisting of an oscillator coupled with an externally excited subsystem. The oscillator and the subsystem are in one-to-one internal resonance. The excited subsystem is in primary resonance. The method of second-order averaging is used to obtain a set of autonomous equations of the second-order approximations to the externally excited system with autoparametric resonance. The Šhilnikov-type homoclinic orbits and chaotic dynamics of the averaged equations are studied in detail. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Šhilnikov-type homoclinic orbits in the averaged equations. The results obtained above mean the existence of the amplitude-modulated chaos for the Smale horseshoe sense in the externally excited system with autoparametric resonance. Furthermore, a detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Nine branches of dynamic solutions are found. Two of these branches emerge from two Hopf bifurcations and the other seven are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, intermittency chaos and homoclinic explosions are also observed.     

Exponential dichotomies and transversal homoclinic orbits in degenerate cases

In this paper, we first give a sufficient condition which assures that a linear differential equation depending on a small parameter admits an exponential dichotomy onR, then we use the result obtained here on exponential dichotomies to investigate the existence of transversal homoclinic orbits of perturbed differential systems in two degenerate cases and obtain a Melnikov-type vector. The results on exponential dichotomies of this paper provide us a tool of proving the transversality of homoclinic orbits in studying degenerate bifurcations.This work is supported by NSF of China.     

Homoclinic orbits of the Davey-Stewartson equations

Itisimportanttostudyintegrablesysteminsolitontheory .Integrablesystemhasmanyproperties,forexample ,infiniteconservationlaws ,Laxpairandsoon .Butrecently ,chaoswasfoundinclassicalintegrablesystemsuchasnonlinearSchr equation (NLS) ,sine_Gordonequation (SG) ,andhomoclinicsolutionsassociatedwithchaoswereobtained[1～ 6 ].Davey_Stewartsonequationsaremodelequationsoftheshallow_waterwaves,whicharecompletelyintegrablesystems .InDavey_Stewartsonequations ( 1 ) ,thefunctionsuandvarerelatedtotheampli…     

New solutions for surface tension driven spreading of a thin film

The standard fourth-order non-linear PDE modelling the flow of thin fluid film subject to surface tension is studied. The Lie group method is used to reduce the model equation from a fourth-order PDE to a fourth-order ODE. Analytical solutions are obtained for certain cases. Where analytical progress cannot be made, we determine numerical solutions.     

梁中非线性弯曲波传播特性的研究

研究了梁中的非线性弯曲波的传播特性,同时考虑了梁的大挠度引起的几何非线性效应和 梁的转动惯性导致的弥散效应,利用Hamilton变分法建立了梁中非线性弯曲波的波动方程. 对该方程进行了定性分析,在不同的条件下,该方程在相平面上存在同宿轨道或异宿轨道, 分别对应于方程的孤波解或冲击波解. 利用Jacobi椭圆函数展开法,对该非线性方程进行 求解,得到了非线性波动方程的准确周期解及相对应的孤波解和冲击波解,讨论了这些解存 在的必要条件,这与定性分析的结果完全相同. 利用约化摄动法从非线性弯曲波动方程中导 出了非线性Schr\"{o}dinger方程,从理论上证明了考虑梁的大挠度和转动惯性时梁中存在 包络孤立波.     

Homoclinic flip bifurcation with a nonhyperbolic equilibrium

In this paper, the problem of homoclinic bifurcation accompanied by a transcritical bifurcation is investigated for high-dimensional systems. With the aid of a suitable local coordinate system, the Poincaré map is constructed. Under certain nongeneric conditions (orbit flip and inclination flip homoclinic orbits), the existence, nonexistence, coexistence and uniqueness of homoclinic and periodic orbits are studied. Some known results are extended.