Euler Buckling in a Potential Field

Summary. We consider elastic buckling of an inextensible rod with free ends, confined to the plane, and in the presence of distributed body forces derived from a potential. We formulate the geometrically nonlinear (Euler) problem with nonzero preferred curvature, and show that it may be written as a three-degree-of-freedom Hamiltonian system. We focus on the special case of an initially straight rod subject to body forces derived from a quadratic potential uniform in one direction; in this case the system reduces to two degrees of freedom. We find two classes of trivial (straight) solutions and study the primary non-trivial branches bifurcating from one of these classes, as a load parameter, or the rod's length, increases. We show that the primary branches may be followed to large loads (lengths) and that segments derived from primary solutions may be concatenated to create secondary solutions, including closed loops, implying the existence of disconnected branches. At large loads all finite energy solutions approach homoclinic and heteroclinic orbits to the other class of straight states, and we prove the existence of an infinite set of such `spatially chaotic' solutions, corresponding to arbitrary concatenations of `simple' homoclinic and heteroclinic orbits. We illustrate our results with numerically computed equilibria and global bifurcation diagrams. Received July 9, 1999; accepted March 16, 2000 Online publication May 31, 2000     

Lyapunov-Schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies

This article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits. In addition, tangencies in the intersection of un/stable manifolds are allowed. The main result is a reduction to algebraic equations of the problem to find all solutions that are close to the heteroclinic network for all time, and their parameter values. A leading order expansion is given in terms of the time spent near vertices and, if applicable, the location on the non-trivial tangent directions. The only difference between a periodic orbit and an equilibrium is that the time parameter is discrete for a periodic orbit. The essential assumptions are hyperbolicity of the vertices and transversality of parameters. Using the result, conjugacy to shift dynamics for a generic homoclinic orbit to a periodic orbit is proven. Finally, equilibrium-to-periodic orbit heteroclinic cycles of various types are considered.     

Exact front, soliton and hole solutions for a modified complex Ginzburg-Landau equation from the harmonic balance method

A novel approach of using harmonic balance (HB) method is presented to find front, soliton and hole solutions of a modified complex Ginzburg-Landau equation. Three families of exact solutions are obtained, one of which contains two parameters while the others one parameter. The HB method is an efficient technique in finding limit cycles of dynamical systems. In this paper, the method is extended to obtain homoclinic/heteroclinic orbits and then coherent structures. It provides a systematic approach as various methods may be needed to obtain these families of solutions. As limit cycles with arbitrary value of bifurcation parameter can be found through parametric continuation, this approach can be extended further to find analytic solution of complex quintic Ginzburg-Landau equation in terms of Fourier series.     

Stationary layered solutions in {\Bbb R}^2 for an Allen–Cahn system with multiple well potential

We study entire solutions on of the elliptic system where is a multiple-well potential. We seek solutions which are “heteroclinic,” in two senses: for each fixed they connect (at ) a pair of constant global minima of , and they connect a pair of distinct one dimensional stationary wave solutions when . These solutions describe the local structure of solutions to a reaction-diffusion system near a smooth phase boundary curve. The existence of these heteroclinic solutions demonstrates an unexpected difference between the scalar and vector valued Allen–Cahn equations, namely that in the vectorial case the transition profiles may vary tangentially along the interface. We also consider entire stationary solutions with a “saddle” geometry, which describe the structure of solutions near a crossing point of smooth interfaces. Received April 15, 1996 / Accepted: November 11, 1996     

Heteroclinic Ratchets in Networks of Coupled Oscillators

We analyze an example system of four coupled phase oscillators and discover a novel phenomenon that we call a “heteroclinic ratchet”; a particular type of robust heteroclinic network on a torus where connections wind in only one direction. The coupling structure has only one symmetry, but there are a number of invariant subspaces and degenerate bifurcations forced by the coupling structure, and we investigate these. We show that the system can have a robust attracting heteroclinic network that responds to a specific detuning Δ between certain pairs of oscillators by a breaking of phase locking for arbitrary Δ>0 but not for Δ≤0. Similarly, arbitrary small noise results in asymmetric desynchronization of certain pairs of oscillators, where particular oscillators have always larger frequency after the loss of synchronization. We call this heteroclinic network a heteroclinic ratchet because of its resemblance to a mechanical ratchet in terms of its dynamical consequences. We show that the existence of heteroclinic ratchets does not depend on symmetry or number of oscillators but depends on the specific connection structure of the coupled system.     

一类二阶微分方程的异宿解

利用变分法获得了一类二阶微分方程异宿解存在的一个充分条件.将连续情况下的二阶微分方程异宿解研究推广到离散情况下来研究.     

A parameter study of epitaxial crystal growth

A third order autonomous ordinary differential equation is studied that is derived from a mathematical model of epitaxial crystal growth on misoriented crystal substrates. The solutions of the ODE correspond to the traveling wave solutions of a nonlinear partial differential equation which is related to the Kuramoto–Sivashinsky equation. The fixed points, the periodic solutions, and the heteroclinic orbits of the ODE are analysed, and stability results are given. A variety of nonlinear phenomena are observed, including Gavrilov–Guckenheimer bifurcations, homoclinic bifurcations, and a cascade of period doublings.     

The Conley index along heteroclinic trajectories of reaction–diffusion equations

It is well known that hyperbolic equilibria of reaction–diffusion equations have the homotopy Conley index of a pointed sphere, the dimension of which is the Morse index of the equilibrium. A similar result concerning the homotopy Conley index along heteroclinic solutions of ordinary differential equations under the assumption that the respective stable and unstable manifolds intersect transversally, is due to McCord. This result has recently been generalized by Dancer to some reaction–diffusion equations by using finite-dimensional approximations. We extend McCord?s result to reaction–diffusion equations. Additionally, an error in the original proof is corrected.     

Heteroclinic and Periodic Cycles in a Perturbed Convection Model

Vivancos and Minzoni (New Choatic behaviour in a singularly perturbed model, preprint) proposed a singularly perturbed rotating convection system to model the Earth's dynamo process. Numerical simulation shows that the perturbed system is rich in chaotic and periodic solutions. In this paper, we show that if the perturbation is sufficiently small, the system can only have simple heteroclinic solutions and two types of periodic solutions near the simple heteroclinic solutions. One looks like a figure “Delta” and the other looks like a figure “Eight”. Due to the fast - slow characteristic of the system, the reduced slow system has a relay nonlinearity (“Asymptotic Method in Singularly Perturbed Systems,” Consultants Bureau, New York and London, 1994) - solutions to the slow system are continuous but their derivative changes abruptly at certain junction surfaces. We develop new types of Melnikov integral and Lyapunov-Schmidt reduction methods which are suitable to study heteroclinic and periodic solutions for systems with relay nonlinearity.     

Exact solutions of nonlocal nonlinear field equations in cosmology

We consider a method for seeking exact solutions of the equation of a nonlocal scalar field in a nonflat metric. In the Friedmann-Robertson-Walker metric, the proposed method can be used in the case of an arbitrary potential except linear and quadratic potentials, and it allows obtaining solutions in quadratures depending on two arbitrary parameters. We find exact solutions for an arbitrary cubic potential, which consideration is motivated by string field theory, and also for exponential, logarithmic, and power potentials. We show that the k-essence field can be added to the model to obtain exact solutions satisfying all the Einstein equations.     

Mountain pass solutions to a generalized Frenkel-Kontorova model

Science China Mathematics - We study a generalized Frenkel-Kontorova model and obtain periodic and heteroclinic mountain pass solutions. The heteroclinic mountain pass solution in the second...     

Peaked singular wave solutions associated with singular curves

We present new types of singular wave solutions with peaks in this paper. When a heteroclinic orbit connecting two saddle points intersects with the singular curve on the topological phase plane for a generalized KdV equation, it may be divided into segments. Different combinations of these segments may lead to different singular wave solutions, while at the intersection points, peaks on the waves can be observed. It is shown for the first time that there coexist different types of singular waves corresponding to one heteroclinic orbit.     

Homoclinic and heteroclinic flows in global attractor for Davey-stewartson II equation

By the variable transformation and generalized Hirota method, exact homoclinic and heteroclinic solutions for Davey-Stewartson II （DSII） equation are obtained. For perturbed DSII equation, the existence of a global attractor is proved. The persistence of homoclinic and heteroclinic flows is investigated, and the special homoclinic and heteroclinic structure in attractors is shown.     

Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity

Entire solutions for monostable reaction-diffusion equations with nonlocal delay in one-dimensional spatial domain are considered. A comparison argument is employed to prove the existence of entire solutions which behave as two traveling wave solutions coming from both directions. Some new entire solutions are also constructed by mixing traveling wave solutions with heteroclinic orbits of the spatially averaged ordinary differential equations, and the existence of such a heteroclinic orbit is established using the monotone dynamical systems theory. Key techniques include the characterization of the asymptotic behaviors of solutions as t→−∞ in term of appropriate subsolutions and supersolutions. Two models of reaction-diffusion equations with nonlocal delay arising from mathematical biology are given to illustrate main results.     

Traveling fronts of a real supercritical quintic Ginzburg-Landau equation coupled by a slow diffusion mode

In this paper, we investigate the existence of traveling front solutions for a class of quintic Ginzburg-Landau equations coupled with a slow diffusion mode. By employing the theory of geometric singular perturbations, we turn the problem into a geometric perturbation problem. We demonstrate the intersection property of the critical manifold and further validate the existence of heteroclinic orbits by computing the zeros of the Melnikov function on the critical manifold. The results demonstrate that under certain parameters, there is 1 or 2 heteroclinic solutions, confirming the existence of traveling front solutions for the considered quintic Ginzburg-Landau equation coupled with a slow diffusion mode.     

Existence of traveling wave solutions for diffusive predator–prey type systems

In this work we investigate the existence of traveling wave solutions for a class of diffusive predator–prey type systems whose each nonlinear term can be separated as a product of suitable smooth functions satisfying some monotonic conditions. The profile equations for the above system can be reduced as a four-dimensional ODE system, and the traveling wave solutions which connect two different equilibria or the small amplitude traveling wave train solutions are equivalent to the heteroclinic orbits or small amplitude periodic solutions of the reduced system. Applying the methods of Wazewski Theorem, LaSalle?s Invariance Principle and Hopf bifurcation theory, we obtain the existence results. Our results can apply to various kinds of ecological models.     

Persistent switching near a heteroclinic model for the geodynamo problem

Modelling chaotic and intermittent behaviour, namely the excursions and reversals of the geomagnetic field, is a big problem far from being solved. Armbruster et al. [5] considered that structurally stable heteroclinic networks associated to invariant saddles may be the mathematical object responsible for the aperiodic reversals in spherical dynamos. In this paper, invoking the notion of heteroclinic switching near a network of rotating nodes, we present analytical evidences that the mathematical model given by Melbourne et al. [19] contributes to the study of the georeversals. We also present numerical plots of solutions of the model, showing the intermittent behaviour of trajectories near the heteroclinic network under consideration.     

Traveling waves in the complex Ginzburg-Landau equation

Summary In this paper we consider a modulation (or amplitude) equation that appears in the nonlinear stability analysis of reversible or nearly reversible systems. This equation is the complex Ginzburg-Landau equation with coefficients with small imaginary parts. We regard this equation as a perturbation of the real Ginzburg-Landau equation and study the persistence of the properties of the stationary solutions of the real equation under this perturbation. First we show that it is necessary to consider a two-parameter family of traveling solutions with wave speedυ and (temporal) frequencyθ; these solutions are the natural continuations of the stationary solutions of the real equation. We show that there exists a two-parameter family of traveling quasiperiodic solutions that can be regarded as a direct continuation of the two-parameter family of spatially quasi-periodic solutions of the integrable stationary real Ginzburg-Landau equation. We explicitly determine a region in the (wave speedυ, frequencyθ)-parameter space in which the weakly complex Ginzburg-Landau equation has traveling quasi-periodic solutions. There are two different one-parameter families of heteroclinic solutions in the weakly complex case. One of them consists of slowly varying plane waves; the other is directly related to the analytical solutions due to Bekki & Nozaki [3]. These solutions correspond to traveling localized structures that connect two different periodic patterns. The connections correspond to a one-parameter family of heteroclinic cycles in an o.d.e. reduction. This family of cycles is obtained by determining the limit behaviour of the traveling quasi-periodic solutions as the period of the amplitude goes to ∞. Therefore, the heteroclinic cycles merge into the stationary homoclinic solution of the real Ginzburg-Landau equation in the limit in which the imaginary terms disappear.     

Traveling wave solutions of diffusive Hindmarsh–Rose-type equations with recurrent neural feedback

From the perspective of bifurcation theory, this study investigates the existence of traveling wave solutions for diffusive Hindmarsh–Rose-type (dHR-type) equations with recurrent neural feedback (RNF). The applied model comprises two additional terms: 1) a diffusion term for the conduction process of action potentials and 2) a delay term. The delay term is introduced because if a neuron excites a second neuron, the second neuron, in turn, excites or inhibits the first neuron. To probe the existence of traveling wave solutions, this study applies center manifold reduction and a normal form method, and the results demonstrate the existence of a heteroclinic orbit of a three-dimensional vector for dHR-type equations with RNF near a fold–Hopf bifurcation. Finally, numerical simulations are presented.