J.P. van der Weele

Multibreathers and homoclinic orbits in 1-dimensional nonlinear lattices

Spatially localized, time-periodic excitations, known as discrete breathers, have been found to occur in a wide variety of 1-dimensional (1-D) lattices of nonlinear oscillators with nearest-neighbour coupling. Eliminating the time-dependence from the differential-difference equations of motion, and taking into account only the N largest Fourier modes, we view these solutions as orbits of a (non-integrable) 2N-D map. For breathers to occur, the trivial rest state of the lattice must be a hyperbolic fixed point of the map, with an N-D stable and an N-D unstable manifold. The breathers and multibreathers (with one and more spatial oscillations respectively) are then directly related to the intersections of these manifolds, and hence to homoclinic orbits of the 2N-D map. This is explicitly shown here on a discretized nonlinear Schrodinger equation with only one Fourier mode (N=1), represented by a 2-D map. We then construct the 2N-D map for an array of nonlinear oscillators, with nearest neighbour coupling and a quartic on-site potential, and demonstrate how a one-Fourier-mode representation (via a 2-D map) can be used to provide remarkably accurate initial conditions for the breather and multibreather solutions of the system.

The birth of twin Poincaré-Birkhoff chains near 1:3 resonance

For a typical area-preserving map we describe the birth process of two twin Poincare-Birkhoff chains, i.e. two rings consisting of center points alternated by saddles, wound around an elliptic fixed point. These twin chains are not born out of the elliptic fixed point, but in the plane, from an annular region where the rotation number has a rational extremum. This situation generically occurs near a 1:3 resonance. We find that the birth of two twin PB chains in such an annular region requires first the birth of two “dimerized” chains of saddle-center pairs, by a tangent bifurcation. The transition from two dimerized chains to two PB chains involves the breakup of homoclinic saddle connections and the formation of heteroclinic connections; it amounts to the reconnection phenomenon of Howard and Hohs. Our results can be regarded as a supplement to the Poincare-Birkhoff theorem, for the case that the twist condition is not satisfied.

The birth process of periodic orbits in non-twist maps

We study the birth process of periodic orbits in non-twist systems, by means of a model map which contains all the typical features of such a system. The most common form of the birth process, or standard scenario, is described in detail. This scenario involves several steps: first one “dimerized” chain of saddle-center pairs is born, then a second, and eventually these two chains are reconnected into two Poincare-Birkhoff chains. We also discuss several variations on this standard scenario. These variations can give rise to arbitrarily many chains, intertwined in a complex fashion, and the reconnection of these chains can be highly non-trivial. Finally we study the effect of dissipation on the birth process. For sufficiently small dissipation one can still recognize the birth and reconnection processes, but with several new features. In the first place, the chains do not consist anymore of conservative saddles and centers, but rather of dissipative saddles and nodes. Furthermore, the dissipation destrtoys the symmetry between the inner and outer chains, and as a result the reconnection does not take place in one single step anymore, but in three.

BIFURCATIONS IN TWO-DIMENSIONAL REVERSIBLE MAPS

We give a treatment of the non-resonant bifurcations involving asymmetric fixed points with Jacobian J≠1 in reversible mappings of the plane. These bifurcations include the saddle-node bifurcation not in the neighbourhood of a fixed point with J≠1, as well as the so-called transcritical bifurcations and generalized Rimmer bifurcations taking place at a fixed point with Jacobian J≠1. The bifurcations are illustrated by some simple examples of model maps. The Rimmer type of bifurcation, with e.g. a center point with J≠1 changing into a saddle with Jacobian J≠1, an attractor and a repeller, occurs under more general conditions, i.e. also in non-reversible mappings if only a certain order of local reversibility is satisfied. These Rimmer bifurcations are important in connection with the emergence of dissipative features in non-measure-preserving reversible dynamical systems.

The order—chaos—order sequence in the spring pendulum

We study the motions of a spring pendulum as a function of its two control parameters (the ratio of the spring and pendulum frequencies, and the energy). It is shown that in the limits for very small and very large parameter values the dynamics of the spring pendulum is predominantly regular, while at intermediate parameter values the majority of initial conditions lead to chaotic trajectories. Thus, upon varying the parameters from small to large values one typically witnesses a transition from order to chaos and back to order again. Similar order?chaos?order sequences are observed in many other dynamical systems, and the spring pendulum is a representative example. In this context, we also discuss the phenomenon for which the spring pendulum is famous, namely the to-and-fro transfer between spring- and pendulum-like behaviour when the spring frequency is (approximately) twice the pendulum frequency. This turns out to play an important role in the order-chaos-order sequence.

On the scaling factors α(z) and δ(z)

Abstract We introduce a new method to determine the scaling factors α ( z ) and δ ( z ) for the period-doubling route to chaos in dissipative systems, exemplified by the one-dimensional mapping x n +1 =1− λ ∣ x n ∣ z . With the help of the Feigenbaum universal functions g ( x ) and h ( x ) we derive the inequality α z − α δ ( z ) α z , implying in particular that δ ( z ) remains finite (≲30) in the limit z → ∞.

Window scaling in one-dimensional maps

We describe both the internal structure and the width of the periodic windows in one-dimensional maps, by considering a universal local submap. Both features are found to depend only on the order of the extremum of this submap. Moreover, we discuss how the windows are grouped in accumulating families, and we calculate the scaling of the widths within these families.

Short-phase anomalies in intermittent band switching

The distribution of phase lengths t for intermittent band switching is investigated for small t. Some typical deviations from exponential behaviour are reported, in particular the occurrence of a minimal phase length with enhanced probability.

Mode competition in a system of two parametrically driven pendulums with nonlinear coupling

This paper is part three in a series on the dynamics of two coupled, parametrically driven pendulums. In the previous parts Banning and van der Weele (1995) and Banning et al. (1997) studied the case of linear coupling; the present paper deals with the changes brought on by the inclusion of a nonlinear (third-order) term in the coupling. Special attention will be given to the phenomenon of mode competition. The nonlinear coupling is seen to introduce a new kind of threshold into the system, namely a lower limit to the frequency at which certain motions can exist. Another consequence is that the mode interaction between 1? and 2s (two of the normal motions of the system) is less degenerate, causing the intermediary mixed motion known as MP to manifest itself more strongly.

Mode competition in a system of two parametrically driven pendulums; the Hamiltonian case*

We study the mode competition in a Hamiltonian system of two parametrically driven pendulums, linearly coupled by a torsion spring. First we make a classification of all the periodic motions in four main types: the trivial motion, two ‘normal modes’, and a mixed motion. Next we determine the stability regions of these motions, i.e., we calculate for which choices of the driving parameters (angular frequency ω and amplitude A) the respective types of motion are stable. To this end we take the (relatively simple) uncoupled case as our starting point and treat the coupling K as a control parameter. Thus we are able to predict the behaviour of the pendulums for small coupling, and find that increasing the coupling does not qualitatively change the situation anymore. One interesting result is that we find stable (and also Hopf bifurcated) mixed motions outside the stability regions of the other motions. Another remarkable feature is that there are regions in the (A, ω)-plane where all four motion types are stable, as well as regions where all four are unstable. As a third result we mention the fact that the coupling (i.e. the torsion spring) tends to destabilize the normal mode in which the pendulums swing in parallel fashion. The effects of the torsion spring on the stability region of this mode is, suprisingly enough, not unlike the effect of dissipation.