Promoting Dynamical Systems in Indonesia

In 1998, a group of students from Indonesia went to Utrecht University and Delft University of Technology, the Netherlands to do PhD research. This was more or less the beginning for the birth of a research group on Dynamical Systems at ITB (Bandung Institute of Technology), as one of the above mentioned students comes from ITB.

In the year 2001, in collaboration with Utrecht University and Delft University of Technology, ITB organized a Summer Course on Dynamical Systems (SCDS). This was the first activity of this research group. A group of distinguish mathematicians came to give lectures to promote this topic in mathematics. There were Prof. dr F. Verhulst, the late Prof. dr J.J. Duistermaat, and Dr B.W. Rink from Utrecht, and Dr A.H.P. van den Burgh from Delft. From Indonesia side, there were J.M. Tuwankotta (ITB) and S. Fatimah (Universitas Pendidikan Indonesia). There were two series of lectures: Introduction to Dynamical Systems (by Verhulst, van den Burgh, Tuwankotta) and On Geometric Mechanics (by Duistermaat and Rink). During this series of lectures, one of the participants, namely F. Adi-Kusumo, was very much inspired and decided to do PhD research at ITB with J.M. Tuwankotta in the following years. Also another member of the Mathematics Department at ITB, Dr. W. Setya-Budhi, joined the research as a second supervisor.

Left to right: Theo Tuwankotta, Lennaert van Veen, the late Hans Duistermaat and Chris Stolk, currently at the University of Amsterdam.

Three years later, in 2004, the second summer course: SCDS II took place. The topic of this series of lectures was Bifurcation Theory and Numerics. The lectures were delivered by J.M. Tuwankotta (ITB) and Dr. L. van Veen (currently at UOIT, Canada). Our attention at that moment was given especially to the introduction of the powerful numerical continuation software called AUTO in Indonesia. A number of students at that time applied this technique in their own research, mainly in population dynamics. In 2006, the research group organized the Summer Course on Dynamical Systems III. This time we collaborated with Groningen University, the Netherlands and Utrecht University. The first topic that was chosen was rather ambitious, namely Singularity Theory and Catastrophe by Prof. dr. H.W. Broer (Groningen) and Singular Perturbation Theory by Prof. dr. F. Verhulst (Utrecht). This time, there were fewer participants than previously due to the nature of the topic we chose, which requires substantial background in mathematics.

Topic of interest in the group

Resarch-wise, the group concentrates their attention to three topics of research.

• Hamiltonian dynamical Systems. Using normal form theory, we studied the relation between resonances, especially the higher-order ones, with the dynamics of the system. One of our main results is on the size of the resonance domain for two degrees of freedom system. The resonance domain is a domain in phase-space where interaction between the degree of freedom occurs. This is usually represented by energy exchange. We have refined the size by providing sharper estimates from the one in the literature. At the moment, two PhD students are working on this topic related to Hamiltonian Systems.
• Our second interest is on singular perturbation, to be more precise: singularly perturbed conservative systems. This is an ongoing project in which a three-dimensional system of ordinary differential equations is studied. The vector field that defines the system can be written as the perturbation of a conservative system. However, the perturbation removes the conserved quantity. The system is constructed by looking at oscillations with widely separated frequencies which is perturbed nonlinearly using energy-preserving nonlinearity. A few master students were involved in this research, and one PhD student is looking at an extension to three oscillator system. In 2008, we have graduated our first PhD student who is working in this field.
• A more applied topic: dynamics and bifurcations in predator-prey type systems. In this research we study a predator-prey type system which is periodically perturbed. The unperturbed system has a non monotonic response function, which models the influence of group defence among the prey. One of the questions we ask is on the Bodanov-Takens point as the periodic perturbation is turned on. A number of students including one PhD student is working on this topic. There was also a postdoctoral fellow who is now working in a university in Jakarta involved in this topic.

There is also a small topic of research which is not included in the above list, namely discrete dynamical systems: dynamics, bifurcations and integrability, which is a bit too premature to be mentioned here since we have not yet done enough in that area.

Future plan

In the year 2013, the group is planning to organize the Summer Course in Dynamical Systems IV: Workshop on Hamiltonian Mechanics. The list of lectures who have confirmed participation in the series of lectures is the following:

• Prof. dr. H. W. Broer (Rijksuniversiteit Groningen, the Netherlands): Resonance and Fractal Geometry.
• Prof. dr. F. Verhulst (Universiteit Utrecht, the Netherlands): tba.
• Prof. dr. R. Cushman (University of Calgary, Canada): On Integrable Systems.
• Prof. dr. E. van Groessen (Universiteit Twente, the Netherlands): Hamiltonian Systems in Water Waves.
• Prof. dr. G.R.W. Quispel (La Trobe University, Australia): On Geometric Numerical Integration.
• Dr. H. Hansmann (Universiteit Utrecht, the Netherlands): Hamiltonian Quasi-periodic Bifurcation Theory.
• Dr. B.W. Rink (Vrije Universiteit Amsterdam, the Netherlands): Aubry-Mather Theory (Hamiltonian Twist maps and application to crystal model).
  
  Further information on this event will be updated and posted regularly on the workshop home page.