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

Abstract

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.

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

“…The first three papers [1][2][3] give a detailed description of the specific dynamics of the oscillations arising, directly or indirectly, from the main tongues of parametric resonance. In the present paper we take a more general point of view and concentrate on the symmetries of the system rather than on (the specific form of) the equations of motion.…”

“…These are not surprisingly, just the familiar normal coordinates for our pendulum system [1][2][3]. The matrix representation of the RE symmetry in the new coordinates is found as follows: 15) and this is the diagonal matrix we were looking for.…”

“…Nevertheless, this interpretation must be taken with a grain of salt, since in a nonlinear system the linear superposition of two solutions does not generate a third one; or, put otherwise, the mixed motions are not (linearly) decomposable into constituent motions. Moreover, such an interpretation would suggest that a mixed motion can only be stable when both of its mothers are stable, but nothing could be further from the truth [2,3,16]. Fig.…”

“…In a Hamiltonian system these two moments (the birth of the quadruplet and the Hopf bifurcation) coincide, but in the presence of dissipation the birth precedes the bifurcation. Now, the eigenvalues corresponding to a motion confined to a plane in stroboscopic phase space can never form a quadruplet, since for such motions the eigenvalues always come in pairs and lie either on the real axis or on the unit circle (the latter being replaced by the reduced circle [2] if dissipation is present); these motions can therefore not undergo Hopf bifurcations. In this subsection we will show that such motions can be recognized by the presence of a pure spatial symmetry.…”

“…Other mode interactions which according to the isotropy would be equally possible, such as 2e --+ A ~ 1~, are not observed in practice; this is to say, the bifurcation from 1:~ to A is quite common (and important) but a bifurcation from 2:~ to A has never been witnessed in the parameter ranges investigated in Refs. [1][2][3]. It should be noted that one vertex may correspond with two or more different (i.e., physically distinguishable) types of motion.…”