Heteroclinic Trajectories Research Articles

Switching homoclinic networks

A heteroclinic network for an equivariant ordinary differential equation is called switching if each sequence of heteroclinic trajectories in it is shadowed by a nearby trajectory. It is called forward switching if this holds for positive trajectories. We provide an elementary example of a switching robust homoclinic network and a related example of a forward switching asymptotically stable robust homoclinic network. The examples are for five-dimensional equivariant ordinary differential equations.

Solitary Waves in Massive Nonlinear SN-Sigma Models

The solitary waves of massive (1+1)-dimensional nonlinear S^N-sigma models are unveiled. It is shown that the solitary waves in these systems are in one-to-one correspondence with the separatrix trajectories in the repulsive N-dimensional Neumann mechanical problem. There are topological (heteroclinic trajectories) and non-topological (homoclinic trajectories) kinks. The stability of some embedded sine-Gordon kinks is discussed by means of the direct estimation of the spectra of the second-order fluctuation operators around them, whereas the instability of other topological and non-topological kinks is established applying the Morse index theorem.

Essentially asymptotically stable homoclinic networks

Melbourne [An example of a nonasymptotically stable attractor, Nonlinearity 4(3) (1991), pp. 835–844] discusses an example of a robust heteroclinic network that is not asymptotically stable but which has the strong attracting property called essential asymptotic stability. We establish that this phenomenon is possible for homoclinic networks, where all heteroclinic trajectories are symmetry related. Moreover, we study a transverse bifurcation from an asymptotically stable to an essentially asymptotically stable homoclinic network. The essentially asymptotically stable homoclinic network turns out to attract all nearby points except those on codimension-one stable manifolds of equilibria outside the homoclinic network.

Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos

We study the Hamiltonian system of two point vortices, embedded inexternal strain and rotation. This external deformation field mimicsthe influence of neighboring vortices or currents in complex flows.When the external field is stationary, the equilibria of the twovortices, symmetric with respect to the center of the plane, aredetermined. The stability analysis indicates that two saddle pointslie at the crossing of separatrices, which bound streamfunctionlobes having neutral centers.   When the external field varies periodically with time, resonancebecomes possible between the forcing and the oscillation of vorticesaround the neutral centers. A multiple time-scale expansion providesthe slow-time evolution equation for these vortices, which, for weakperiodic deformation, oscillate within their original (steady)trajectory. These analytical results accurately compare withnumerical integration of the complete equations of motion. As theperiodic deformation field increases, this vortex oscillationmigrates out of the original trajectories, towards the location ofthe separatrices. With a periodic external field, these separatriceshave given way to heteroclinic trajectories with multipleself-intersections, as shown by the calculation of the Melnikovfunction.    Chaos appears in vortex trajectories as they enter theaperiodic domain around the heteroclinic curves. In fact, thischaotic domain progressively fills out the plane, replacing KAM toriand cantori, as the periodic deformation field reaches finiteamplitude. The appearance of windows of periodicity is illustrated.

Generalized MSTB models: Structure and kink varieties

In this paper, we describe the structure of a class of two-component scalar field models in a (1+1) Minkowskian space–time which generalize the well-known Montonen–Sarker–Trullinger–Bishop — hence MSTB-model. This class includes all the field models whose static field equations are equivalent to the Newton equations of two-dimensional type I Liouville mechanical systems, with a discrete set of instability points. We offer a systematic procedure to characterize these models and to identify the solitary wave or kink solutions as homoclinic or heteroclinic trajectories in the analogous mechanical system. This procedure is applied to a one-parametric family of generalized MSTB models with a degree-eight polynomial as potential energy density.

Drift of slow variables in slow-fast Hamiltonian systems

We study the drift of slow variables in a slow-fast Hamiltonian system with several fast and slow degrees of freedom. Keeping the slow variables frozen, for any periodic trajectory of the fast subsystem we define an action. For a family of periodic orbits, the action is a scalar function of the slow variables and can be considered as a Hamiltonian function which generates some slow dynamics. These dynamics depend on the family of periodic orbits. Assuming that for the frozen slow variables the fast system has a pair of hyperbolic periodic orbits connected by two transversal heteroclinic trajectories, we prove that for any path composed of a finite sequence of slow trajectories generated by action Hamiltonians, there is a trajectory of the full system whose slow component shadows the path.

Hydraulic-jump behavior of a thin film flowing down an inclined plane under an electrostatic field

The hydraulic-jump phenomenon of a thin fluid layer flowing down an inclined plane under an electrostatic field is explored by using a global bifurcation theory. First, the existence of hydraulic-hump wave has been found from heteroclinic trajectories of an associated ordinary differential equation. Then, the jump behavior has been characterized by introducing an intensity function on the variations of Reynolds number and surfave-wave speed. Finally, we have investigated the nonlinear stability of traveling shock waves triggered from a hydraulic jump by integrating the initial-value problem directly. At a given wave speed there exists a certain value of Reynolds number beyond which a time-dependent buckling of the free surface appears. Like the other wave motions such as periodic and pulse-like solitary waves, the hydraulic-jump waves are also found to become more unstable as the electrostatic field is getting stronger.

Graphs of NMS Flows on S 3 with Knotted Saddle Orbits and No Heteroclinic Trajectories

We consider NMS systems on S 3 without heteroclinic trajectories connecting two saddles orbits and we build the dual graphs associated with this kind of .ows. We prove that flows coming from essential attachments of orientable round 1-handles can be reproduced from the corresponding dual graph.

Shock wave solutions of the compound Burgers–Korteweg–de Vries equation

In this paper, shock wave solutions of the compound Burgers–KdV equation is studied by using qualitative theory of differential equation and ansatz method. The existence and uniqueness of different types of travelling waves of the system are obtained by analyzing qualitatively the existence and uniqueness of the heteroclinic trajectories of the ordinary differential equation. We construct some travelling wave solutions and indicate their stability and bifurcation.

Homoclinic and heteroclinic transfer trajectories between planar Lyapunov orbits in the sun-earth and earth-moon systems

In this paper a method for finding homoclinic and heteroclinic connections between Lyapunov orbits using invariant manifolds in a given energy surface of the planar restricted circular three body problem is developed. Moreover, the systematic application of this method to a range of Jacobi constants provides a classification of the connections in bifurcation families. The models used correspond to the Sun-Earth+Moon and the Earth-Moon cases.

ENERGY-BASED CONTROL OF THE BUTTERFLY ROBOT

We address the control problem for the Butterfly, an interesting example of 2-dof underactuated mechanical system. This robot consists of a butterfly-shaped rotational link on whose rim a ball rolls freely. The control objective is to stabilize the robot at a certain unstable equilibrium. To this end, exploiting the existence of heteroclinic trajectories, we extend a previously proposed energy-based technique. Simulation results show the effectiveness of the presented method.

Time series analysis of homoclinic nonlinear systems using a wavelet transform method

Homoclinic (and heteroclinic) trajectories are closed paths in phase space that connect one or more saddle points. They play an important role in the study of dynamical systems and are associated with the creation/destruction of limit cycles as a parameter is varied. Often, this creation/destruction process involves complicated sequences of bifurcations in small regions of parameter space and there is now an established theoretical framework for the study of such systems.

Time series analysis of homoclinic nonlinear systems using a wavelet transform method

Homoclinic (and heteroclinic) trajectories are closed paths in phase space that connect one or more saddle points. They play an important role in the study of dynamical systems and are associated with the creation/destruction of limit cycles as a parameter is varied. Often, this creation/destruction process involves complicated sequences of bifurcations in small regions of parameter space and there is now an established theoretical framework for the study of such systems. The eigenvalues of saddle points in the phase space determine the behaviour of the system. In this article we present a new eigenvalue estimation technique based on a wavelet transformation of a time series under study and compare it with an existing method based on phase space reconstruction. We find that the two methods give good agreement with theory using clean model data, but where noisy data are analysed the wavelet technique is both more robust and easier to implement.

On admissibility for parabolic equations in Rn

We consider the parabolic equation (P) utu = F(x,u), (t,x) 2 R+ × R n and the corresponding semiflowin the phase space H 1 . We give conditions on the nonlinearity F(x,u), ensuring that all bounded sets of H 1 are �-admissibile in the sense of Rybakowski. If F(x,u) is asymptotically linear, under appropriate non- resonance conditions, we use Conley's index theory to prove the existence of nontrivial equilibria of (P) and of heteroclinic trajectories joining some of these equilibria. The results obtained in this paper extend earlier results of Rybakowski concerning parabolic equations on bounded open subsets of R n .

Closed trajectories in planar relay feedback systems

A planar system of piecewise linear differential equations with a line of discontinuity, ℤ2–symmetry and a linear part having negative determinant is investigated. Using the theory of differential inclusions and an appropriate Poincaré map a complete analysis is provided. This analysis applies to uniqueness and non-uniqueness for the initial value problem, stability of stationary points, sliding motion solutions, number and stability of periodic trajectories and existence of pairs of heteroclinic trajectories connecting two saddle points forming heteroclinic cycles. A complete bifurcation diagram is given.

Construction of homoclinic and heteroclinic trajectories in mechanical systems with several equilibrium positions

A new approach for a construction of homo and heteroclinic trajectories of some principal non-linear dynamical systems is utilized here, namely the non-linear Schrodinger equation, non-autonomous Duffing equation and the equation of a parametrically excited damped pendulum are considered. Pade’ and quasi-Pade’ approximants and a convergence condition used earlier in the theory of non-linear normal vibration modes made possible to solve a boundary-value problems formulated for the orbits and to determine initial amplitude values of the trajectories with admissible precision. The approach proposed here is more exact than the generally accepted one because it is not necessary to use here separatrix trajectories of the corresponding autonomous equations.

Solitary Waves of a Perturbed sine-Gordon Equation**The project supported by the China Postdoctoral Science Foundation (Grant No. 28) and the Shanghai Scientic and Technological Development Foundation (Grant No. 98JC14032)

Based on the bifurcation and the idea that the solitary waves and shock waves of partial differential equations correspond respectively to the homoclinic and heteroclinic trajectories of nonlinear ordinary differential equations satisfied by the travelling waves, different conditions for the existence of solitary waves of a perturbed sine-Gordon equation are obtained. All of the corresponding approximate solitary wave solutions are given by integrating the derived approximate equations directly.

Abundance ofhomoclinic and heteroclinic orbits and rigorous bounds for the topologicalentropy for the Hénon map

We show how to link topological tools with a localhyperbolic behaviour to prove the existence ofhomoclinic and heteroclinic trajectories for a map. Weapply this technique for the Hénon map h withclassical parameter values (a = 1.4,b = 0.3). For thismap we give a computer-assisted proof of the existence ofan infinite number of homoclinic and heteroclinictrajectories. We also introduce the method forcomputation of the lower bound of the topological entropyof a map based on the covering relations involvingdifferent iterations of the map and we prove that thetopological entropy of h is larger than 0.3381.

Shock wave profiles in the burnett approximation

This paper is devoted to a discussion of the profiles of shock waves using the full nonlinear Burnett equations of hydrodynamics as they appear from the Chapman-Enskog solution to the Boltzmann equation. The system considered is a dilute gas composed of rigid spheres. The numerical analysis is carried out by transforming the hydrodynamic equations into a set of four first-order equations in four dimensions. We compare the numerical solutions of the Burnett equations, obtained using Adam's method, with the well known direct simulation Monte Carlo method for different Mach numbers. An exhaustive mathematical analysis of the results offered here has been done mainly in connection with the existence of heteroclinic trajectories between the two stationary points located upflow and downflow. The main result of this study is that such a trajectory exists for the Burnett equations for Mach numbers greater than 1. Our numerical calculations suggest that heteroclinic trajectories exist up to a critical Mach number ( approximately 2.69) where local mathematical analysis and numerical computations reveal a saddle-node-Hopf bifurcation. This upper limit for the existence of heteroclinic trajectories deserves further clarification.

NUMERICAL CHARACTERISTICS ON THE SET OF HETEROCLINIC POINTS OF MORSE - SMALE DIFFEOMORPHISMS ON SURFACES

For Morse–Smale diffeomorphisms on closed surfaces, we investigate the properties of numerical characteristics of heteroclinic trajectories with respect to the local structure of direct product in a small neighborhood of a saddle periodic point.