## Research

My main research focuses on the theory of dynamical systems and its applications. I am particularly interested in the dynamics of the famous Lotka-Volterra system with constant terms. The dynamics of such systems are surprisingly rich [P1,P2]. The bifurcation turns out to be the unfolding of the degenerate Bogdanov-Takens bifurcation. Keyword: Codimension-three Bogdanov-Takens bifurcation, transcritical bifurcation, first integrals.

While I was working as a post-doc in Institut Teknologi Bandung (ITB), I also work on another applications of dynamical systems which are predator-prey systems with periodic forcing and coupled oscillators with widely spaced frequencies. Together with JM Tuwankotta, we had two students; Livia Owen who successfully completed her master degree with “Dynamics and bifurcations of three coupled oscillators” as her topic [Un9] and Eric with “Predator prey model with non-monotonic functional response and seasonal perturbation” as his topic [Un8]. We also managed to publish our paper about coupled oscillators in Journal of Physics A: Mathematical and Theoretical [P3]. One thing that I would like to do is to investigate more on the swallow tail bifurcation. To our knowledge, this bifurcation rarely occurs in applications. Keyword: coupled oscillators, seasonal perturbation, swallowtail bifurcation.

In 2011, I went to Italy for a summer workshop and met Carles Simo. He taught us a new ODE integration method, which is called the Taylor method with automatic differentiation. He showed that this is one of the accurate integrators. I would like to do more research and to develop a software in Python about this method [Un24]. Keyword: numerical integration, Taylor method, automatic differentiation.

In 2014, I got a Fulbright award and had a chance to go to Athens, Ohio for a four month research visit. My supervisor at that time was Prof. Winfried Just. I thought that why did I not try to continue my research on the bifurcation of Lotka Volterra equation. I analyzed a generic system that has the same property as the Lotka Volterra system in [P4]. Keyword: codimension-one invariant manifold, codimension-three Bogdanov-Takens bifurcation, Hopf-transcritical bifurcation.

Finally, I like mathematics and I am eager to know more how mathematics is used in the real world applications. When I was a student, I worked on how to simulate gas flow inside a pipe [Pro2]. After I became a full time lecturer, I have became a supervisor for our bachelor students since then. There are quite variety of topics here. Together with my bachelor students, we try to solve some logical puzzles using mathematics modeling [Pro1, Un1, Un2, Un3]. Also, we learned to find a numerical solution of a three-dimensional diffusion equation using finite element methods [Un4]. We also learned the application of mathematics in insurance, the topics are pension funding, insurance design, Markov model in multiple decrements insurance and critical illness insurance [Un5, Un15, Un21, Un23]. Other topics of our bachelor student’s final projects are application of mathematics in finance. We learned about credit default swap, option pricing, fractal market hypothesis, portfolio optimization, default probability, volatility computation, herding behavior in stock markets, short selling and many more [Un6, Un7, Un 10, Un11, Un12, Un16, Un18-20, Un26-31, Pro3]. We also learned other topics such as artificial immune system [Un13], mathematical model of dengue fever and cancer [Un17, Un22], multi phase queuing system [Un32], small world network and its applications [Un14, Un25] and car detection model [Un33]. Keyword: mathematical modeling, financial mathematics, logical puzzles, insurance, numerical method, time series.

**Journals**

[P4] **K V I Saputra**, Dynamical systems with a codimension one invariant manifold: the unfoldings and its bifurcations, Int. J. Bifurcation Chaos, 25, 1550091 (2015). Preprint link, journal link.

[P3] J.M. Tuwankotta, F Adi-Kusumo, **K V I Saputra**, A three dimensional singularly perturbed conservative system with symmetry breaking. J. Phys. A: Math. Theor. 46 (2013) 305101. Preprint link, journal link.

[P2] **K V I Saputra**, L van Veen and G R W Quispel, The saddle-node–transcritical bifurcation in a population model with constant rate harvesting, Disc. Cont. Dyn. B 14 no. 1 (2010) pp. 233-250. Preprint link, journal link.

[P1] **K V I Saputra**, L van Veen and G R W Quispel, First Integrals of Lotka-Volterra systems with constant terms, J. Phys. A: Math. Theor. 43 (2010) 225207. Preprint link, journal link.

**Proceedings**

[Pro3] Yunita Wijaya, **K V I Saputra, **S S Kim, Estimasi Volatilitas dan Value at Risk Indeks LQ45 dengan Generalized Pareto Distribution, Prosiding Seminar Nasional Matematika 10 (2015). (In Indonesian)

[Pro2] Susanti, **K V I Saputra**, S Lukas, Solving logical puzzles using mathematical model, Proceeding of the 5th International Symposium on Computational Sciences (2012). (Preprint)

[Pro1] H B Sulistyarso, S Trihandaru, L Mucharam, S Siregar, **K V I Saputra** and S Canggih, Solusi Model Aliran Gas Dalam Pipa pada Kondisi Line Packing Menggunakan Skema Richtmyer, Proceedings ITB 36A 2, 2004. (In Indonesian)

**Preprints**

[Pr1] **K V I Saputra**, Torus bifurcation in the HIV-1 with Cytotoxic T-Lymphocites responses model. In preparation

**Students’ final projects**

[Un1] Dina Stefani (2011), Solving pixel puzzles using mathematical model and best first search, Successfully completed bachelor thesis.

[Un2] Olivia (2011), Mathematical model of KenKen and genetic algorithm. Successfully completed bachelor thesis.

[Un3] Andrew (2011), Mathematical model and genetic algorithm to solve logical puzzle Hashiwokakero. Successfully completed bachelor thesis.

[Un4] Ferry Tan (2011), Numerical simulation of non-linear time-dependent, three dimensional heat equations using finite element method, Successfully completed bachelor thesis.

[Un5] Jessica P. Oetomo, Marline Cecilia (2012), Mathematical model of insurance funds, Successfully completed bachelor thesis.

[Un6] Jessie Carolyn (2012), Valuation of Credit Default Swap (CDS), Successfully completed bachelor thesis.

[Un7] Christian Xaverius (2012), Valuation of American option using binomial tree method, Successfully completed bachelor thesis.

[Un8] Eric (2009) Predator prey model with non-monotonic functional response and seasonal perturbation. Successfully completed Master thesis. (co-supervised with J.M. Tuwankotta)

[Un9] Livia Owen (2010) Dynamics and bifurcations of three coupled oscillators. Successfully Completed Master thesis. (co-supervised with J.M. Tuwankotta)

[Un10] Gillian Tampi (2013) Characteristics of Indonesian Stock Markets under Fractal Market Hypothesis. Successfully completed bachelor thesis.

[Un11] Cantika Puteri Utama (2013) Analysis of Foreign Exchange Markets under Hyperbolic Setting in Option Evaluation. Successfully completed bachelor thesis.

[Un12] Andre Jonathan Cahyadi (2013) Time Series Forecasting using Hybrid Neural Network and ARIMA. Successfully completed bachelor thesis.

[Un13] Ellen Pramana (2013) Hybrid Genetic and Artificial Immune System. Successfully completed bachelor thesis.

[Un14] Michael Kam (2013) Analysis and Simulation of Small-World and Scale-Free Network. Successfully completed bachelor thesis.

[Un15] Irene Azalia Christanto (2013) Optimal Insurance Design under Rank Dependent Expected Utility. Successfully completed bachelor thesis.

[Un16] Illaria Hana, Inke Maria Olivia, Margaretha (2014) Application of Time Series Analysis in Stock Market. Successfully completed bachelor theses.

[Un17] Indra Winarta (2014) Modeling and Optimization of Cancer Medicine. Successfully completed bachelor thesis.

[Un18] Adeline Putri Lestari (2015) Estimation of default probability using replicated Merton model. Successfully completed bachelor theses.

[Un19] Shelly (2015) Effect of American stock market to Indonesian stock market using MVMQ-Caviar. Successfully completed bachelor theses.

[Un20] Andrew Maxell (2015) Herding behaviour in Indonesian stock market. Successfully completed bachelor theses.

[Un21] Alvinie Gracia Marchella (2016) Pricing an accident insurance with a Markov model. Successfully completed bachelor theses.

[Un22] Andika Putra (2016) Transcritical bifurcation in mathematical model of dengue fever. Successfully completed bachelor theses.

[Un23] Alvin Kurnia Halim (2016) Mathematical modelling of claim delay for critical illness insurance using generalized linear model. Successfully completed bachelor theses.

[Un24] Ignatius Albert Rahardjo (2016) ODE Integration using Taylor series method via automatic differentiation, implemented in Python library. On progress

[Un25] Yordan Saputra (2016) Examining Christakis’ theory regarding contagion effect in a social network in XYZ University. On progress

[Un26] Ryan Natagawa (2016) Mathematical modelling of stock liquidity after the change of lot size. On progress

[Un27] Jaclyn Ariandra (2016) Mathematical model competitiveness in developing countries using a panel regression model. On progress

[Un28] Felicia Handayani (2016) Analysis of Keynes model using two stage least square methods. On progress

[Un29] Glenn Gerren Piettojo (2016) Volatility estimation using Garch model, EWMA model and implied method. On progress

[Un30] Rheza Dwiputra (2016) Portofolio optimizations using MAD model, Markowitz model and single index model. On progress

[Un31] Eric Aldiano (2016) Study of short selling in American stock market and its effects on volatility and liquidity. On progress

[Un32] Rheinhard Brain (2016) Analysis of multi phase queuing system in Sentul drag race registration. On progress

[Un33] Roberto Julianto Kondorura (2016) CCTV Vehicle detection using a modified Viola Jones Algorithm. On progress

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