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

The two-hole ground state of the Hubbard-Anderson model, approximated by a variational RVB-type wave function

In this paper the Hubbard-Anderson model on a square lattice with two holes is studied. The ground state (GS) is approximated by a variational RVB-type wave function. The holes interact by exchange of a localized spin excitation (SE), which is created or absorbed if a hole moves to a nearest-neighbour site. An SE can move over the sublattice on which it is created. A variational calculation of the GS and the GS-energy is performed for an open-ended 4 × 4 lattice with two holes with the restriction that the SE is neighbouring both holes and does not move over its sublattice. It is found that the two holes prefer a bound state in which their mutual distance is 1 or V2 (with lattice spacing 1)

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

Mode competition in a system of two coupled, 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 ¿ 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, ¿)-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

Mode competition in a system of two parametrically driven pendulums: te 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

Mode interaction in horses, tea, and other nonlinear oscillators: the universal role of symmetry

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 Gollub) 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 hierarchy) 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

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

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