Web page of Zin Arai

Professor
Department of Mathematical and Computing Science, School of Computing
Institute of Science Tokyo (formerly Tokyo Institute of Technology)

Research Supervisor
JST PRESTO "Mathematical Sciences for the Future"

email: zinc.titech.ac.jp
CV

Research Interests

I have been studying dynamical systems using topological and computational methods. My particular interest lies in the real and complex Hénon maps, especially in the structure of their parameter spaces. I am also working on the application of topological and computational methods to several areas, including chemistry, engineering, and biology.

Keywords: Conley index, Computational Topology, Hyperbolic Dynamics, Validated Numerics, Computer Assisted Proofs, Monodromy, Hénon map, etc.

Publications

Research Papers

• Aru Toyoda, Tamaki Maruhashi, Suchinda Malaivijitnond, Kazunari Matsudaira, Zin Arai, Ikki Matsuda, Hiroki Koda, Macaque progressions: passing order during single‑file movements reflects the social structure of a wild stump‑tailed macaque group, Primates, January 29, 2023, https://doi.org/10.1007/s10329-023-01055-y
• Hiroki Koda, Zin Arai, and Ikki Matsuda, Agent-based simulation for reconstructing social structure by observing collective movements with special reference to single-file movement, PLOS ONE, December 3, 2020, https://doi.org/10.1371/journal.pone.0243173
• Takuro Tsutsumi, Yuriko Ono, Zin Arai, and Tetsuya Taketsugu, Visualization of the Dynamics Effect: Projection of on-the-Fly Trajectories to the Subspace Spanned by the Static Reaction Path Network, J. Chem. Theory Comput. 2020, 16, 7, 4029–4037, https://doi.org/10.1021/acs.jctc.0c00018
• Zin Arai, Yutaka Ishii and Hiroki Takahasi, Boundary of the horseshoe locus for the Hénon family, SIAM Journal on Applied Dynamical Systems 17-3 (2018), 2234-2248, DOI: 10.1137/18M1174684, arXiv version: arXiv:1803.09381.
• Takuro Tsutsumi, Yuriko Ono, Zin Arai and Tetsuya Taketsugu, Visualization of the intrinsic reaction coordinate and global reaction route map by classical multidimensional scaling, Journal of Chemical Theory and Computation 14(8) (2018) 4263-4270, DOI: 10.1021/acs.jctc.8b00176.
• Zin Arai and Yutaka Ishii, On parameter loci of the Hénon family, Communications in Mathematical Physics 361(2) (2018), 343-414, DOI: 10.1007/s00220-018-3174-0. arXiv version: arXiv:1501.01368.
• Zin Arai, On Loops in the Hyperbolic Locus of the Complex Hénon Map and Their Monodromies, Physica D 334 (2016), 133-140. PDF of the preprint version, 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 B31 (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.

Proceedings and survey articles

• 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, 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
• Zin Arai, On Applications of Rigorous Computing to Dynamical Systems (Japanese), RIMS Kokyuroku. PDF
• 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. PDF

Programs

Programs for the characterization of the maximal entropy locus of complex Hénon family

Here are Programs and Data for the rigorous verifications used in "On parameter loci of the Hénon family" by ZA and Yutaka Ishii. See the instructions on how to comiple and execute the program. (Last update: Feb 22, 2016, some minor bugs are fixed)

Program for proving hyperbolicity of the complex Hénon map

Here are Programs and Data for the computation of monodromies associated to loops in the hyperbolic horseshoe locus of the complex Hénon Map.

Program for proving hyperbolicity of the real Hénon map

Here is a program to prove the uniform hyperbolicity of the chain recurrent set of the real Hénon maps. The algorithm of the program is described in "On Hyperbolic Plateaus of the Hénon Maps". To run the program you need MATLAB and GAIO. You also need CAPD if you want to compile the program by yourself.

README

realhenon.tar.gz

Programs for rigorous verification of tangencies

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.

Patches for BIAS and Profil on PowerPC G5

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.

Miscellaneous

GAIO memo (Japanese)

A Japanese introduction of GAIO (obsolate).

Iruka Hotel

I'm the drummer of Iruka Hotel, a three-piece rock band formed in Kyoto.