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The Number of Orbits of Periodic Box-Ball Systems [chapter]

Akihiro Mikoda, Shuichi Inokuchi, Yoshihiro Mizoguchi, Mitsuhiko Fujio
2006 Lecture Notes in Computer Science  
We showed some relational equations of soliton sizes, a box size and the number of orbits.  ...  Varying a box size, we also found out some simulation results of the periodicity of orbits of pBBS with same kinds of solitons.  ...  Acknowledgments The author thanks the anonymous reviewers for their valuable comments.  ... 
doi:10.1007/11839132_15 fatcat:ay5rmdvzjzedve4m7khc2qbzti

Periodic orbit analysis for the deterministic path-preference traffic flow cellular automaton [article]

Yoichi Nakata, Yoshihiro Ohta, Sigeo Ihara
2017 arXiv   pre-print
By considering the behavior of periodic orbits, we also determine the point where such gaps in the flow appear and the exact value of the flow.  ...  We find that the number of particles is dominant to the dynamics of this cellular automaton and observed that there are not only expected phase shift but also several gaps as the number of particles increases  ...  This research is supported by Platform for Dynamic Approaches to Living System (the Platform Project for Supporting in Drug Discovery and Life Science Research) from the Ministry of Education, Culture,  ... 
arXiv:1409.3311v3 fatcat:uuk422ffnjglbhzypnusjdjnga

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Lan Wen, Zhihong Xia
2018 Transactions of the American Mathematical Society  
Like the closing lemma, the connecting lemma is of fundamental importance in dynamical systems.  ...  Hayashi recently proved the C 1 connecting lemma for stable and unstable manifolds of a hyperbolic invariant set. In this paper, we prove several very general C 1 connecting lemmas.  ...  Since z is not periodic of f , all terms in the negative orbit of z are hence distinct.  ... 
doi:10.1090/s0002-9947-00-02553-8 fatcat:wejdlc32kfgtlbmscslv2pjwwi

How to detect Wada basins

Alexandre Wagemakers, Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain, Alvar Daza, Miguel A. F. Sanjuán, Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA, Department of Applied Informatics, Kaunas University of Technology, Studentu 50-415, Kaunas LT-51368, Lithuania
2017 Discrete and continuous dynamical systems. Series B  
We present a review of the different techniques available to study a special kind of fractal basins of attraction known as Wada basins, which have the intriguing property of having a single boundary separating  ...  We expose several approaches to identify this topological property that rely on different, but not exclusive, definitions of the Wada property.  ...  This work has been financially supported by the Spanish State Research Agency (AEI) and the European Regional Development Fund (ERDF, EU) under Projects No. FIS2016-76883-P and No.  ... 
doi:10.3934/dcdsb.2020330 fatcat:kfe65zl5pfah7mmwbqdfsozrvu

How to detect Wada Basins [article]

Alexandre Wagemakers, Alvar Daza, Miguel A.F. Sanjuan
2020 arXiv   pre-print
We present a review of the different techniques available to study a special kind of fractal basins of attraction known as Wada basins, which have the intriguing property of having a single boundary separating  ...  We expose several approaches to identify this topological property that rely on different, but not exclusive, definitions of the Wada property.  ...  unstable periodic orbits shows a single period 3 orbit on the boundary of the basins.  ... 
arXiv:2005.05923v2 fatcat:lsi6ni5wbney5e4deg2gybx3gm

Chaotic dynamics in a three-dimensional superconducting microwave billiard

H. Alt, H.-D. Gräf, R. Hofferbert, C. Rangacharyulu, H. Rehfeld, A. Richter, P. Schardt, A. Wirzba
1996 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
Concerning the periodic orbits of the classical system we present a comparison with the length spectrum of the resonator and introduce a new bouncing ball formula for electromagnetic billiards.  ...  We analyze the statistical properties of the measured spectrum in terms of several methods originally derived for quantum systems like eigenvalue statistics and periodic orbits and obtain according to  ...  ensemble of resonances, but above all by the lengths of the shortest periodic orbits of the classical system.  ... 
doi:10.1103/physreve.54.2303 pmid:9965335 fatcat:tfbw2ntburfsnoai5vxric2mye

FRACTAL AND WADA EXIT BASIN BOUNDARIES IN TOKAMAKS

JEFFERSON S. E. PORTELA, IBERÊ L. CALDAS, RICARDO L. VIANA, MIGUEL A. F. SANJUÁN
2007 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering  
Moreover, identifying the flux surfaces as KAM tori of the corresponding Hamiltonian system, the existence of a chaotic region implies the breaking of flux surfaces.  ...  The creation of an outer layer of chaotic magnetic field lines in a tokamak is useful to control plasma-wall interactions.  ...  Acknowledgments This work was made possible through partial financial support from the following Brazil-  ... 
doi:10.1142/s021812740701986x fatcat:r7ragzegsfdqzhepxs2uxmxpfe

Page 8003 of Mathematical Reviews Vol. , Issue 2001K [page]

2001 Mathematical Reviews  
A sequence of the symbols {1,2,3} is called a generator if it is the code of a stable billiard periodic orbit in a triangle with sides numbered as 1, 2, 3 (a periodic trajectory in a polygon Q is called  ...  The author gives a comprehensive survey of the state of the art of proving ergodic properties of hard ball systems (HBS).  ... 

Constraint Based Computation of Periodic Orbits of Chaotic Dynamical Systems [chapter]

Alexandre Goldsztejn, Laurent Granvilliers, Christophe Jermann
2013 Lecture Notes in Computer Science  
Periodic orbits play a key role in understanding chaotic systems.  ...  In this paper, we show that the (numerical) constraint programming framework provides a very convenient and efficient method for computing periodic orbits of chaotic dynamical systems: Indeed, the flexibility  ...  Unsafe boxes must be properly handled so as to allow rigorously counting the number of real periodic orbits.  ... 
doi:10.1007/978-3-642-40627-0_57 fatcat:o22hfc27djb4dbgntwfqu3e5kq

GALTON'S QUINCUNX: RANDOM WALK OR CHAOS?

KEVIN JUDD
2007 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering  
In 1873 Francis Galton had constructed a simple mechanical device where a ball is dropped vertically through a harrow of pins that deflect the ball sideways as it falls.  ...  This paper presents evidence that this assumption is almost certainly not valid and that the quincunx has the richer, more predictable qualities of a low-dimensional deterministic dynamical system.  ...  In (a) the periodic orbit passes through the side of the box and in (b) the periodic orbit that involves a stick-and-roll motion.  ... 
doi:10.1142/s0218127407020129 fatcat:sj6rf6vhjraxhpsy4lr7m25upa

Nanodiamond Collective Electron States and their Localization [article]

Ivan A. Denisov, Andrey A. Zimin, Leslie A. Bursill, Peter I. Belobrov
2013 arXiv   pre-print
Three distinct classes of collective electron states have been found: collective bonding orbitals resembling the morphology of 3D-modulated particle in a box solutions; surface-localized non-bonding conductive  ...  Intrinsic spin existence is supposed to result from the collective and spread nature of subsurface orbitals, allowing spin-density fluctuation effects to become significant on this length scale.  ...  Generally, the observed orbital morphologies arise from near-spherical symmetry of the system and similarity with the particle-in-a-box case.  ... 
arXiv:1307.4633v2 fatcat:epzuef5n2fbdlknch4ywbmyeya

Riddling Crisis

Satoshi Hayama
1999 Progress of theoretical physics  
In this case, as a parameter is varied, the destruction of the chaotic attractor occurs quite gradually, and we can observe chaotic transients for a sufficient number of iterations in a quite wide range  ...  Furthermore, windows of periodic attractors appear many times after this change occurs.  ...  Acknowledgements The author would like to thank Emeritus Professor M. Yamaguchi who deceased in December, 1998, Univ. of Kyoto for continuous encouragement. The author is grateful to Professor H.  ... 
doi:10.1143/ptp.101.519 fatcat:cvzat5orczf2loqvqe5js6pw4y

Finite-size effects on active chaotic advection

Takashi Nishikawa, Zoltán Toroczkai, Celso Grebogi, Tamás Tél
2002 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
The dynamics of the advected particle can be chaotic even with a flow field that is simply time periodic.  ...  Using the example of an autocatalytic reaction, AϩB→2B, we show that the balance between dissipation in the particle dynamics and production due to reaction leads to a steady state distribution of the  ...  With this analogy in mind, we consider the following model. Take the same number of balls as the number of particles in the system.  ... 
doi:10.1103/physreve.65.026216 pmid:11863641 fatcat:7sg2cpbwpndkhf7vmms43a2x6y

Bethe ansatz atq= 0 and periodic box–ball systems

Atsuo Kuniba, Akira Takenouchi
2006 Journal of Physics A: Mathematical and General  
Based on the Bethe ansatz at q=0, we propose explicit formulas for the dynamical period and the size of certain orbits under the time evolution in A^(1)_n case.  ...  A class of periodic soliton cellular automata is introduced associated with crystals of non-exceptional quantum affine algebras.  ...  Acknowledgments A.K. thanks Mo-Lin Ge, Chengming Bai and the organizing committee of DGMTP XXIII for the hospitality at Nankai Institute of Mathematics, where a part of this work was presented.  ... 
doi:10.1088/0305-4470/39/11/002 fatcat:ttjkkmzzzjf4ve5kwhyvxn3ide

New corner stones in dissipative granular gases [article]

P. Evesque
2005 arXiv   pre-print
dimension of this billiard-like system from 13-d to 1-d, since the motion is 1d and quasi-periodic for large enough forcing; the result remains valid with 2 balls.  ...  , but decreases when the number of balls N increases.  ...  very regular in the case of the 1-ball system under periodic excitation.  ... 
arXiv:cond-mat/0507267v1 fatcat:mlgiypaunvdadgdzpsftmft7dq
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