Bounds for bounded motion around a perturbed fixed point

We consider a dissipative map of the plane with a bounded perturbation term. This perturbation represents e.g. an extra time dependent term, a coupling to another system or noise. The unperturbed map has a spiral attracting fixed point. We derive an analytical/numerical method to determine the effect of the additional term on the phase portrait of the original map, as a function of the δ bound on the perturbation. This method yields a value δ c such that for δδ c the orbits about the attractor are certainly bounded. In that case we obtain a largest region in which all orbits remain bounded and a smallest region in which these bounded orbits are captured after some time (the analogue of 'basin' and 'attractor respectively')

Transient periodic behaviour related to a saddle-node bifurcation

The authors investigate transient periodic orbits of dissipative invertible maps of R2. Such orbits exist just before, in parameter space, a saddle-node pair is formed. They obtain numerically and analytically simple scaling laws for the duration of the transient, and for the region of initial conditions which evolve into transient periodic orbits. An estimate of this region is then obtained by the construction-after extension of the map to C2-of the stable manifolds of the two complex saddles in C2 that bifurcate ino the real saddle-node pai

Interacting solitary waves in a damped driven Lennard-Jones chain

It is shown analytically that pulse solitary waves in a chain with Lennard-Jones type nearest neighbor interaction are strongly localized and marginally stable in the high energy limit.\ud \ud In a damped and periodically driven chain we obtain numerically families of states whose behavior is similar to that of equally many oscillators. We observe a period doubling sequence in a one-solitary wave family and bifurcation to (quasi-) periodic motion in a family of two solitary waves. We conclude that the damped and driven chain admits asymptotically stable states living on a low-dimensional manifold in phase space. These results depend sensitively on the shape of the driving term

Period-doubling density waves in a chain

The authors consider a one-dimensional chain of N+2 identical particles with nearest-neighbour Lennard-Jones interaction and uniform friction. The chain is driven by a prescribed periodic motion of one end particle, with frequency v and 'strength' parameter alpha . The other end particle is held fixed. They demonstrate numerically that there is a region in the alpha -v plane where the chain has a stable state in which a density wave runs to and fro between the two ends of the chain, similarly to a ball bouncing between two walls. More importantly, they observe a period-doubling transition to chaos, for fixed v and increasing alpha , while the localised (solitary wave) character of the motion is preserve

The shape of the basin of attraction and transients via transformation to normal form

A method is derived to approximate the basin and investigate the transients of an attracting fixed point with complex multipliers for an analytic invertible mapping of the real plane. It consists of constructing approximate but explicit expressions for the normal transformation about the fixed point. This transformation has the whole basin as domain, and transforms the nonlinear mapping restricted to the basin, to a linear one.\ud \ud With this transformation a non-negative functional for the basin is defined whose value decreases along an orbit. Explicit expressions are derived for a sequence of contours approximating a level line of this functional. At large values of the functional a level line approximates the basin boundary.\ud \ud Calculation of a sequence of level lines yields estimates for that part of the basin where regular (monotonic) convergence to the attracting fixed point is to be expected. The shape of the level lines demonstrates the occurrence of transient-periodic behavior. The way in which the unstable manifold of the saddle, born together with the attractor from a tangent bifurcation, intersects the level lines yields a criterion for a saddle-sink connection. In that case the stable manifold as part of the basin boundary has a simple structure, in contrast to the fractal structure that occurs when there is a homoclinic orbit, which causes final state sensitivity. The calculations are carried out for the Hénon mapping

Ultra-sharp soliton switching in a directional coupler

By a numerical investigation it is shown that a directional coupler, described by two linearly coupled non-linear Schro¨dinger equations, can be used to construct a soliton switch with an extremely narrow transition region

Substrate-induced pairing of Si ad-dimers on the Si(100)surface

The interaction between Si ad-dimers on the Si(100) surface has been studied by total-energy calculations with a three-particle Stillinger-Weber potential. We have found a strong attractive interaction between neighboring Si ad-dimers located in neighboring on-top and deep-channel positions in adjacent substrate dimer rows. This should result in a four-atomic block consisting of two dimers as an important elementary object of the Si(100) kinetics