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

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

The double-kissing nano-crush for bifurcation lesions : development, bioengineering, fluid dynamics and initial clinical testing

Background When possible, a single stent technique to treat coronary bifurcation disease is preferable. However, when two stents are required, there is scope to improve upon existing techniques. The crush technique has already been improved with the introduction of double kissing (DK) and mini-crush. We sought to refine and simplify the mini-crush technique, retaining its advantages whilst avoiding its disadvantages, by developing a DK nano-crush technique. Methods The DK nano-crush method allows complete lesion coverage of a bifurcation lesion without excessive metal layers. This is achieved by positioning the SB stent with minimal protrusion into the main branch (MB), implantation of the SB stent with an undeployed balloon in the MB, immediate kissing balloon inflation with formation of a minimal neocarina, stenting the MB, re-crossing the proximal part of the SB without crossing double metal layer, and final kissing. We demonstrate this technique with benchtop implantation, micro computed tomography reconstruction, computational fluid dynamics (CFD) modelling and clinically with angiographic and intravascular imaging. Results The DK nano-crush was practically feasible and resulted in full ostial coverage. CFD analysis demonstrated minimally disturbed blood flow. The technique was successfully utilised in nine patients with bifurcation lesions with excellent angiographic outcomes and no adverse events over twelve months. Conclusions The DK nano-crush technique may represent the ultimate refinement of the original ‘crush’ with a number of practical and theoretical advantages. It remains to be tested against other bifurcation techniques in prospective trials

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: 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