Dynamics and Bifurcations of Three-Coupled Oscillators with Energy-Preserving Nonlinearity

Introductions

Various models in applications of mathematics in science and engineering take the form a differential equation.  Among those models, many of them can be formulated as a system of coupled oscillators.  Starting in 2003, J.M. Tuwankotta has been studying a system of coupled oscillators with a special geometric property of the nonlinearity, namely: the norm-preserving nonlinearity.  This property is not at all artificial, infact there are many examples where nature behaves in this manner.  This property is basically stated that the nonlinearity distribute energy but not adding to nor taking out energy of the system.  There are at least four scientific papers that has been published in a well-known journals, and numerous presentation and conference papers that has been published on this topic.  Moreover, one PhD project and one Master of Science project have been completed on this topic, and one more Master Project is schedulled to be finished in February 2011.


The goal of this research are the following:

  • To analyze the six-dimensional dynamical system by reduction to five-dimensional system using averaging method.

  • To analyze the geometry of the invariant structures in the system.

  • To understand the dynamics and bifurcations in the system.


The method.

We start with writing the system of ordinary differential equations in its Taylor expansion around the origin.  Experts in this field know that this does not restrict ourself too much, instead in the case where the vector field is not real anaytic, we can use the first few terms of the asymptotic expansions.

Our next step is to impose the geometric condition to the nonlinearity, of which we have restricted our self to look only at the quadratic nonlinearity.  This is rather restrictive indeed, but one could argue that putting too many terms in the beginning will add significantly to the complexity of the problem, while the results are only meaningful if we understood what’s going on in the simpler system.  Thus, this is merely a  choice one should made (after all life is finite).

Having the system clearly specified and written down correctly, we proceed with constructing a nonlinear coordinate transformation to bring the system into the so-called Lagrange standard form.  We then take the average of the resulting periodic system, knowing that this is nothing but adding higher order terms in the coordinate transformation. This is done with the goal of pushing all terms that are not in the kernel of the adjoint operator involved, to higher degree of the expanded vector field.  Obviously, this means those terms that can be pushed away are not significant with respect to the dynamics of the system.  This method is known as normalization, where we have added to the literature that this process can be done while preserving the geometric property of the nonlinearity we assumed.

Unlike the low-dimensional case, the reduced, truncated normal form is still too high in dimensional to analyze. Thus the next step of ours is to switch to numerics to describe the dynamics and the bifurcations of the system.

Up to this point, one might wonder why not doing numerics from the very beginning.  One should keep in mind that we are dealing not with a particular system describing a single phenomena regardless its importance.  We are dealing with a generic systems, depending on many unknown parameters, and the results, regardless how trivial it seems, are valid of a wide class of dynamical systems


The results.

Up to this point, we have managed to produce the normal form for the three coupled oscillators system.  The normal form is five dimensional, which shown clearly the interplay between the 1:2-resonance and the 1:0 resonance in the system.  We have add extra assumption to the normal form to simplify the system further.  The assumption is that the couplings between the oscilators are as such as the interaction beetween the first and the third oscillators occurs only through the second oscillators.  Full interaction will be analyzed in the future.

Under this assumption we have found an organizing center for the bifurcation namely the Bogdanov-Takens bifurcation.

This is a nontrivial and quite sophisticated bifurcations.  Infact it is semiglobal, where not only one, but two invariant sturctures are invloved. Those invariant structures are a critical point which undergoes a Saddle-Node bifurcations and Hopf bifurcation.  The Hopf bifurcation throw away a periodic orbit which intertwined with the critical point and produces homoclinic orbit.  One its way to the homoclinic orbit, we see that the periodic orbit undergoes several period-doubling bifurcations which produces more periodic solutions which are mostly unstable.  These are some of the ingredients for chaotic dynamics.

Although period-doublings bifurcations usually followed by the creations of homoclinic orbit and the existence of infinitely many perodic solutions through Shilnikov scenario, we have not yet found that this is the case in our system.  The reason for nonexistence of Shilnikov scenario is still under investigation, however, it falls beyond the scope of this research topic.


Publications: 2012: with Livia Owen, Bogdanov-Takens Bifurcations in Three Coupled Oscillators System with Energy Preserving Nonlinearity, J. Indones. Math. Soc., Vol. 18, No 2, pp. 73-83.

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