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 f2 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, f2)-plane where all four motion types are stable, as well as regions where all four are unstable. As a third result we mentio~a 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, surprisingly enough, not unlike the effect of dissipation.

Summary Several recently published studies discuss the concept of inductive resistivity-logging devices with oblique transmitting and/or receiving coils. Both wireline induction and logging-while-drilling (LWD) propagation resistivity-tool concepts have been considered. Directional resistivity measurements and improved anisotropy measurements are among the benefits promised by this type of device. Analyses based on point-magnetic dipole antennas were used to illustrate these potential benefits. The effects of a metallic mandrel, borehole, and invasion were not considered because of the absence of a suitable forward model. This paper characterizes mandrel, borehole, and invasion effects for a variety of candidate tilt-coil devices with antenna array parameters similar to those of the previous studies. The characterization is based on calculations from a new forward model that includes tilted transmitting and receiving coils of finite diameter embedded in a concentric cylindrical structure. Important details of the forward model used in the calculations are also provided. Introduction Conventional propagation resistivity devices are routinely used for geosteering applications. Because data from these devices have essentially no azimuthal sensitivity, the LWD engineer is greatly aided by a priori information regarding the proximity of the target bed relative to other geologic features such as shales and water-bearing zones. Suitable a priori information is often available from offset logs. In cases in which offset logs are not fully useful because of changing depositional environments or different tectonic settings, azimuthally sensitive resistivity data would improve the quality of the geosteering effort. One way to achieve azimuthal sensitivity to benefit geo-steering (and to use it for imaging) is to construct a tool similar to a conventional propagation resistivity device, but with the transmitters and/or receivers tilted with respect to the axis of the drill collar. In fact, directional resistivity tools(DRTs) have been proposed in the literature for this pur-pose.1–3 To the knowledge of the authors, DRTs have only been analyzed with point-dipolemodels, which ignore both the drill collar and the finite size of the antennas. For apparent lack of a suitable forward model, mandrel, borehole, and invasion effects have not been considered in the literature. A model has been developed that accounts for tilted transmitters and receivers embedded in arbitrary layers of a concentric cylindrical structure. Many important details of this model are discussed in Appendix A. The term mandrel effect is used here to denote the difference between values calculated with a point-dipole model and the model that accounts for the mandrel encompassed by the antennas. Mandrel effects on DRT measurements will be grouped into three categories:Absolute effects where the mandrel primarily attenuates the signals because of a reduction in the magnetic moment of the antennas.Residual effects that remain after an air-hang calibration is applied to the data.Perturbations to the azimuthal sensitivity of the measurement caused by the finite size of the antennas and the drill collar. Algorithms that transform raw tool measurements to resistivity values can be based on computationally simple point-dipole solutions without significantly degrading the accuracy of the results if mandrel effects associated with categories 1 and 2 can be suppressed. For conventional LWD propagation resistivity measurements, mandrel effects of type 1 are addressed by air-hang calibration. Algorithms that suppress type 2 mandrel effects are discussed in the literature.4 Type 3 mandrel effects are not discussed here.

In this paper we study the dynamics of a system of two linearly coupled, parametrically driven pendulums, subject to viscous dissipation. It is a continuation of the previous paper (E.J. Banning and J.P. van der Weele (1995)), in which we treated the Hamiltonian case. The damping has several important consequences. For instance, the driving amplitude now has to exceed a threshold value in order to excite non-trivial motion in the system. Furthermore, dissipative systems (can) exhibit attraction in phase space, making limit cycles, Arnol'd tongues and chaotic attractors a distinct possibility. We discuss these features in detail. Another consequence of the dissipation is that it breaks the time-reversal symmetry of the system. This means that several, formerly distinct motions now fall within the same symmetry class and may for instance annihilate each other in a saddle-node bifurcation. Implications of this are encountered throughout the paper, and we shall pay special attention to its effect on the interaction between two of the normal modes of the system.

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. (t997) 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 lc~ and 2~q (two of the normal motions of the system) is less degenerate, causing the intermediary mixed motion known as MP to manifest itself more strongly.

This paper is about mode interaction in systems of coupled nonlinear oscillators. The main ideas are demonstrated by means of a model consisting of two coupled, parametrically driven pendulums. On the basis of this we also discuss mode interaction in the Faraday experiment ͑as observed by Ciliberto and Gollub͒ and in running animals. In all these systems the interaction between two modes is seen to take place via a third mode: This interaction mode is a common daughter, born by means of a symmetry breaking bifurcation, of the two interacting modes. Thus, not just any two modes can interact with each other, but only those that are linked ͑in the system's group-theoretical hierarchy͒ by a common daughter mode. This is the quintessence of mode interaction. In many cases of interest, the interaction mode is seen to undergo further bifurcations, and this can eventually lead to chaos. These stages correspond to lower and lower levels of symmetry, and the constraints imposed by group theory become less and less restrictive. Indeed, the precise sequence of events during these later stages is determined not so much by group-theoretical stipulations as by the accidental values of the nonlinear terms in the equations of motion.