Web page of Zin ARAI

Associate Professor
Department of Mathematics
Hokkaido University

Research Interest

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.

Publications

Research Papers

• Zin Arai, "On Loops in the Hyperbolic Locus of the Complex Hénon Map and Their Monodromies",
  preprint, revised on 2008 Nov 13, PDF, Supplemental Materials (data, programs).
• 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" to appear in "Advances in Cognitive Neurodynamics (III)" 2012, Springer.
• 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)

Travel Schedule

• 2012 Jan 13-14, Kyoto University.
• 2012 Jan 20-22, Kyoto.
• 2012 Jan 28-29, Nara.
• 2012 Feb 13-17, Yamaguchi University.
• 2012 Feb 20-22, Kyushu University.
• 2012 Feb 23-24, Aihara "First" Project, Tokyo University.
• 2012 Mar 02-03, Waseda University.
• 2012 Mar 13-16, Aisin AW CO., LTD, Aichi.
• 2012 Mar 19-23, Hawaii, Workshop on CMG.
• 2012 Mar 26-28, MSJ meeting, Tokyo.
• 2012 Mar 29-30, "Kanzume" workshop, Kyoto.

Programs

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 Computation 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.

List of My Talks in International Conferences

"Capturing the Global Behavior of Dynamical Systems with Topological and Graph Theoretical Methods"

Emerging Topics in Nonlinear Science, Goldrain, Italy (16 Sep, 2010).

"Hyperbolicity, monodromy and pruning fronts"

Various Aspects of Dynamical Systems, Vadodara, India (29 Aug, 2010).

"Rigorous verification of global properties of dynamical systems"

8th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Dresden, Germany (May 28, 2010).

"Rigorous verification of uniform hyperbolicity, subshifts of finite type and the pruning front"

Workshop on Computational Differential Geometry, Topology, and Dynamics (a part of FoCM2009), Toronto, Canada (November 19, 2009).

"Rigorous Computation of the Monodromy and the Pruning Front of Dynamical Systems"

2009 International Symposium on Nonlinear Theory and its Applications (NOLTA 2009), Sapporo, Japan (October 19, 2009).

"A Database Schema for the Analysis of Global Dynamics"

7th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2009), Crete, Greece (Sep 21, 2009).

"Hyperbolicity, Stability and Monodromy of Dynamical Systems"

Topology and Computer 2009, Tokyo Institute of Technology, Tokyo (Sep 2, 2009).

"Development and Applications of an Algorithm for Proving Structural Stability"

Dynamics, Topology and Computations (DyToComp2009), Bedlweo, Poland (June 6, 2009). presentation file (PDF, 27MB)

"Rigorous Monodromy Computation and its Application to the Pruning Front Theory"

SIAM Conference on Applications of Dynamical Systems 2009, Snowbird, USA (May 17, 2009).

"Towards the verification of qualitative properties of dynamical systems"

International Workshop on Numerical Verification and its Applications 2009, Okinawa (March 28, 2009).

"On rigorous computational methods for proving uniform hyperbolicity"

Computational Topology and Dynamics Workshop, Bozeman, Montana, USA (Aug 11, 2008).

"Rigorous Verification of the hyperbolicity of dynamical systems and its applications"

Foundations of Computational Mathematics "FoCM08", City University of Hong Kong, Hong Kong (Jun 17, 2008).

"A hyperbolicity verification algorithm and its application to the real Henon map"

Computer Assisted Nonlinear Dynamics "Candy08", Imperial College, London, UK (Mar 17, 2008).

"On Rigorous Verification of Stability and Bifurcation of Dynamical Systems"

2008 International Workshop on Numerical Verification and its Applications, Okinawa, Japan (Mar 1, 2008).

"Hyperbolicity and Monodromy of Real and Complex Henon Maps"

International Conference on Topology and its Applications 2007, Kyoto, Japan (Dec 7, 2007).

"Hyperbolicity, Stability and Monodromy of Dynamics Systems"

6th International Congress on Industrial and Applied Mathematics, Zurich, Swiss (Jul 2007). presentation PDF

"An Algorithm for Hyperbolicity Verification and its Application"

NOLTA 2006, Bologna, Italy (Sep 2006).

"A Hyperbolicity Verification Algorithm and its Application"

Dynamics, Topology, and Computation 2006, Bedlewo, Poland (Jun 2006). presentation (gzipped Keynote archive, 4.1MB), presentation (PDF 5.2MB)

"An Algorithm for Hyperbolicity Verification and its Application"

Computational and Topological Aspects of Dynamics, Lorentz Center, Netherlands (May 2006). presentation (PDF 532KB).

"Verification of Hyperbolicity and Non-Hyperbolicity via Topological Methods"

SIAM Conference on Applications of Dynamical Systems, Snowbird, USA (May 2005). abstract (PDF 302KB).

"Computational Study of Bifurcations and Stability"

NCTS 2005 Japan-Taiwan Conference on Dynamical Systems, National Center for Theoretical Sciences, Taiwan (Mar 2005). abstract (PDF 315KB).

"Rigorous Computations of Homoclinic Tangencies"

US-Japan Workshop on Dynamics and Computations, Shonan Village Center, Japan (Mar 2004).
abstract (PDF 2.1MB), abstract (Keynote files 3.5MB)

"Numerical Detection of Structurally Unstable Connecting Orbits via the Conley Index Theory"

NSC Winter Seminar, Hokkaido University, Japan (Feb 2003). abstract (PDF 524KB).

"The Conley index for the bundle maps"

Second Workshop on the Conley Index and Related Topics, Serbrooke University, Canada (Aug 2001).

"Homoclinic Bifurcations and the Conley Index"

SIAM Pacific Rim Dynamical Systems Conference, Hawaii, USA (Aug 2000).

"Homoclinic/Heteroclinic Tangencies and the Conley Index"

International Workshop on New Trends in Dynamical Systems, Ryukoku 2000, Seta, Japan (Jun 2000).

Miscellaneous

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.