# Singularly Perturbed Conservative systems

**Widely separated frequencies oscillations with energy preserving nonlinearity and constant perturbation**. In this paper, we consider a three dimensional system of ordinary differential equations, which is a normal form for the system of two coupled oscillators with widely spaced frequencies and energy preserving quadratic nonlinearity (see PhysicaD, IntJournalNonlinearMechanics). An elaborate analysis has been devoted in those two papers to describe the dynamics and bifurcations of such system.

Now, we add constant forcing term to the system. The idea behind adding this constant forcing term is the following. We have observed that a sequence of period doubling bifurcations is evident in the system. However, unlike in the literature, this does not seems to lead to chaotic dynamics. The reason for this is that the system exhibit a co-dimension one invariant plane. Thus, we conjecture that by perturbing the system in such a way such that it removes this invariant plane, chaotic dynamics as a consequence of a sequence of period doubling bifurcations.

In addition to the above mentioned, removing the invariant plane also has another interesting consequence. The phase-portrait of the system without constant perturbation is separated into two parts by the invariant plane. Removing the invariant plane implies that the separation is no longer there. We expect to see exciting dynamics such as, coexistence of stable attractor which implies competition in the system.

The result is publish in Journal of Physics A, 2013.

The research is funded by Riset International ITB 2007. This research is in collaboration with Dr. F. Adikusumo and K.V.I. Saputra, PhD.
**Chaos and strange attractor in coupled oscillators with energy-preserving nonlinearity**. In this work we analyze a system of coupled oscillators system. The vector field which defines the flow is basically conservative. However, we include perturbation which removes the conservative nature of the system. This is an ongoing research of JM Tuwankotta since 2003. **The research is funded by Riset KK ITB 2006 and Riset Internasional ITB 2007**. In 2008, Dr Fajar Adi-Kusumo is promoted as PhD. His dissertation: Sistem Dua Oscilator Berpasangan yang Diperturbasi Secara Singular, is based on this research.
**Dynamics and Bifurcations of Three-Coupled Oscillators with Energy-Preserving Nonlinearity****.** Funded by Riset KK ITB 2009. People involved: Livia Owen, MSi (graduated in 2010 with master thesis on this topic).