and **WORKSHOP on INTEGRAL and DIFFERENTIAL EQUATIONS (WIDE) 2015.**

Date : 3 of August 2015

Venue : Lab Tek III, Institut Teknologi Bandung, R Seminar 1.2.

Dynamics and Bifurcations Day 2015 is the second small conference on Dynamical Systems at ITB (the first one was in 2014). This conference serves as an opening ceremony for the Summer Course on Dynamical System VI (2015).

**Lecture 1: On the mathematical analysis of vibrations of axially moving strings.**

W.T. van Horssen, Department of Applied Mathematics, Technical University Delft, THE NETHERLANDS.

Abstract

In this paper the transversal vibrations of an axially moving string with constant or time-varying length, time-varying velocity, and/or time-varying tension are studied. By using a multiple timescales perturbation method, asymptotic approximations of the solutions of the formulated initial-boundary value problems are constructed. The applicability of Galerkin’s truncation method and the applicability of the method of characteristic coordinates for these types of problems are discussed. The presence of internal resonances and autoresonances are described in detail.

For conveyor belt problems it will be shown how the two timescales perturbation method in combination with the method of characteristic coordinates can be used to construct asymptotic approximations of the solutions on long timescales. Also for these conveyor belt problems it turned out that Galerkin’s truncation method was not applicable to obtain asymptotic results on long timescales.

**Lecture 2: Bifurcation analysis on the Interaction between Cervical Cancer Cells, Effector Cells, and IL-2 Compound Model.**

Fajar Adi Kusumo, Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Gadjah Mada, Yogyakarta – INDONESIA

Abstract :

We consider the bifurcation analysis of a mathematical model describing the interaction between cervical cancer cells, effector cells, and IL-2 compounds with a perturbation terms. The interactions between the effector cells and the IL-2 compounds are assumed has non constant rates. It is from the fact that the interaction rates between these components depend on many factors, such as enzyme reaction process, protein reactions, medical factors, etc. In this paper we introduce a perturbation term as a function of time. Cervical cancer is a malignant tumour caused by Human Papillomavirus (HPV). The HPV proteins E6 and E7 are respectively inactivate p53 and pRb genes in the human body. The genes play an important role in regulating normal cell division and apoptosis due to uncontrolled divisions of the infected cells. The other important components are the immune system and the effector cells. The immune system is designed to detect the presence of antigens. While, the effector cells is needed to destroy the HPV-infected cells by stimulation of IL-2. A treatment by using a part of tumour tissue to enhance the immune responses by in-vitro fertilisation so that cervical cancer can be cured is called immunotherapy. We assumed that the interaction of the three components follow biochemical reactions which is modelled by Michaelis-Menten kinetics function. In this paper, we study the dynamics of the system and the bifurcation related to the variation of its parameter values.

**Lecture 3: On the Dynamics and Bifurcations on A Periodically Perturbed Predator-Prey type of Dynamical Systems **

J.M. Tuwankotta, Analysis and Geometry, FMIPA-Institut Teknologi Bandung

Abstract:

A class of dynamical systems of the Predator-Prey type is analyzed where a response function of

the Holling type 4 is included. This response function is chosen to include the group defense mechanism of the prey, as the density of the prey become larger. The numerical bifurcations analysis for the equilibrium of the unperturbed system is done using AUTO. We have found three codimension two bifurcations, i.e. cusp bifurcation, Bogdanov-Takens bifurcation and Bautin bifurcation.

We proceed with introducing the time periodic perturbation by taking a periodic variation in the carrying capacity as a model of seasonal behavior. The system then becomes nonautonomous. We are going back into the autonomous setting by coupling the system with a two dimensional oscillators system. For this system, we present a proof for the existence of a periodic solution in the neighborhood of an isolated equilibrium of the unperturbed system, and also an

alternative proof for a classical result on the period of periodic solution for periodic vector field. The numerical bifurcations analysis for the periodic solution is presented; where we have found the occurrence of Swallowtail bifurcation. We conclude the presentation with describing the occurrence of a strange attractor in the system, and a discussion on future works.