Associate Professor

Department of Mathematics

Hokkaido University

I have been studying dynamical systems using **topological** and **computational** methods.
My current interest is on the real and complex **Hénon map**, especially on the structure of its hyperbolic horseshoe locus and the monodromy representation of it.

- 2016 Feb 1-21, ICERM, Brown University.
- 2016 Mar 4-7, Amami-Oshima.
- 2016 Mar 16-19, Tsukuba University.
- 2016 Mar 24-25, Kyoto University.
- 2016 Apr 3-10, ICERM, Brown University.

- Zin Arai and Yutaka Ishii, "On parameter loci of the Hénon family", arXiv:1501.01368, submitted.
- Zin Arai, "On Loops in the Hyperbolic Locus of the Complex Hénon Map and Their Monodromies", to appear in Physica D, PDF, Supplemental Materials (data, programs).
- Zin Arai, "Decomposition and clustering for the visualization of dynamical systems", Mathematical Progress in Expressive Image Synthesis I, Springer (2014), ISBN 978-4-431-55006-8, 13-20.
- Zin Arai, "A rigorous numerical algorithm for computing the linking number of links", Nonlinear Theory and Its Applications, Vol. 4, No. 1 (2013), 104-110, Supplemental Materials (INTLAB source code). There is a typo in the published version (thanks to Dr. Simon Copar). The return value of Algorithm 3 should be doubled. This does not affect the INTLAB code above. PDF of the corrected version
- Zin Arai, Marcio Gameiro, Tomas Gedeon, Hiroshi Kokubu, Konstantin Mischaikow and Hiroe Oka, "Graph-based topological approximation of saddle-node bifurcation in maps", RIMS Kokyuroku Bessatsu, B 31 (2012) 225-241.
- Zin Arai, Kazunori Hayashi and Yasuaki Hiraoka, "Mayer-Vietoris sequences and coverage problems in sensor networks", Japan Journal of Industrial and Applied Mathematics, 28 (2011) 237-250.
- Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka and P. Pilarczyk, "A databases schema for the global dynamics of multi-parameter systems", SIAM Journal on Applied Dynamical Systems, 8 (2009), 757-789. PDF.
- Z. Arai, H. Kokubu and P. Pilarczyk, "Recent development in rigorous computational methods in dynamical systems", Japan Journal of Industrial and Applied Mathematics, 26 (2009) 393-417. PDF.
- Zin Arai, "On Hyperbolic Plateaus of the Hénon Map", Experimental Mathematics, 16:2 (2007), 181--188. PDF
- Zin Arai and Konstantin Mischaikow, "Rigorous Computations of Homoclinic Tangencies", SIAM Journal on Applied Dynamical Systems 5 (2006), 280--292. PDF
- Zin Arai, "Equivalence of Graded Module Braids and Interlocking Sequences", Journal of Mathematics of Kyoto University 43 (2003), 441--449. PDF
- Zin Arai, "Tangencies and the Conley Index", Ergodic Theory and Dynamical Systems 22 (2002), no.4, 973--999.

- Zin Arai, Hiroshi Kokubu and Ippei Obayashi, "Capturing the global behavior of dynamical systems with Conley-Morse graphs", Advances in Cognitive Neurodynamics (III) 2012, Springer, 665--672.
- Zin Arai, "Hyperbolicity, stability and monodromy of dynamical systems", PAMM, Volume 7 (2007), Special Issue: Sixth International Congress on Industrial Applied Mathematics (ICIAM07) and GAMM Annual Meeting.
- Zin Arai, "On applications of computational homology to dynamical systems" (Japanese), Bul. of Jap. Soc. for Ind. and App. Math. 18 (2008).
- Zin Arai, "On computer assisted analysis of discrete dynamical systems" (Japanese), Bul. of Jap. Soc. for Ind. and App. Math. 15 (2005), 20--31.
- Zin Arai, "On Loops in the Hyperbolic Loci of the Complex Hénon Maps" (Japanese), RIMS Kokyuroku. PDF (11 pages, 425KB)
- Zin Arai, "On Applications of Rigorous Computing to Dynamical Systems" (Japanese), RIMS Kokyuroku. PDF (13 pages, 466KB)
- Zin Arai, "The Conley index and transition matrix for discrete dynamical systems and their applications" (Japanese), A survey article based on my master's thesis. Includes an introduction for the Conley index theory and the Morse decomposition. PDF (49 pages, 490KB)

A program which implements the algorithm given in "Rigorous Computations of Homoclinic Tangencies" by ZA and Konstantin Mischaikow is available as tangency.tar.gz. See README for the instruction.

To use BIAS and Profil on PowerPC G5 (PowerPC 970), you have to add some assembler code to the program. Here is an ad-hoc patch for BiasInt.h. Although this patch is for the BIAS package distributed with GAIO, I think you can easily modify it for other versions of BIAS.

- GAIO memo (Japanese)
- A Japanese introduction of GAIO.
- Iruka Hotel
- I'm the drummer of Iruka Hotel, a three-piece rock band formed in Kyoto.