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 Poincaré-Birkhoff chains.\ud \ud 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.\ud 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

Mode competition in a system of two parametrically driven pendulums: the role of symmetry

This paper is the final part in a series of four on the dynamics of two coupled, parametrically driven pendulums. In the previous three parts (Banning and van der Weele, Mode competition in a system of two parametrically driven pendulums; the Hamiltonian case, Physica A 220 (1995) 485¿533; Banning et al., Mode competition in a system of two parametrically driven pendulums; the dissipative case, Physica A 245 (1997) 11¿48; Banning et al., Mode competition in a system of two parametrically driven pendulums with nonlinear coupling, Physica A 245 (1997) 49¿98) we have given a detailed survey of the different oscillations in the system, with particular emphasis on mode interaction. In the present paper we use group theory to highlight the role of symmetry. It is shown how certain symmetries can obstruct period doubling and Hopf bifurcations; the associated routes to chaos cannot proceed until these symmetries have been broken. The symmetry approach also reveals the general mechanism of mode interaction and enables a useful comparison with other systems

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

The squeeze effect in non-integrable Hamiltonian systems

In non-integrable Hamiltonian systems (represented by mappings of the plane) the stable island around an elliptic fixed point is generally squeezed into the fixed point by three saddle points, when the rotation number ρ of the motion at the fixed point approaches 1/3. At ρ=1/3 the island is reduced to one single point.\ud A detailed investigation of this squeeze effect, and some of its global implications, is presented by means of a typical two-dimensional area-preserving map. In particular, it turns out that the squeeze effect occurs in any mapping for which the Taylor expansion around the fixed point contains a quadratic term, whereas it does not occur if the first non-linear term is cubic. We illustrate this with two physical examples: a compass needle in an oscillating field, showing the squeeze effect, and a ball which bounces on a vibrating plane, for which the squeeze effect does not occur

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

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.\ud \ud 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 2ß (two of the normal motions of the system) is less degenerate, causing the intermediary mixed motion known as MP to manifest itself more strongly

Phase-length distributions in intermittent band switching

The distribution of phase lengths t for intermittent band switching is investigated. Its form is observed to deviate from a exponential function; a minimal phase length is seen to exist and the probabilities for the first few occuring phase lengths are often strongly enhanced or suppressed. These deviations are analyzed and described explicitly in terms of the parameters of a model map

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

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 Poincaré-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.\ud \ud Our results can be regarded as a supplement to the Poincaré-Birkhoff theorem, for the case that the twist condition is not satisfied