Proceedings of the Japan Academy, Series A, Mathematical Sciences

A computer-assisted proof of existence of a periodic solution

Tomoyuki Miyaji and Hisashi Okamoto

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We consider a three-dimensional dynamical system proposed in \textit{Physica D}, \textbf{164}, (2002), 168–186. It is a conservative system and is unusual in that most of the solutions are unbounded. The paper presented a conjecture that an unstable periodic orbit determines directions of unbounded orbits of helical form. In the present paper we prove existence and local uniqueness of the conjectured periodic orbit by a method of numerical verification.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 90, Number 10 (2014), 139-144.

First available in Project Euclid: 4 December 2014

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Zentralblatt MATH identifier

Primary: 37C27: Periodic orbits of vector fields and flows 65G20: Algorithms with automatic result verification

Three-dimensional dynamical system periodic orbit numerical verification


Miyaji, Tomoyuki; Okamoto, Hisashi. A computer-assisted proof of existence of a periodic solution. Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 10, 139--144. doi:10.3792/pjaa.90.139.

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  • O. Caprani and K. Madsen, Iterative methods for interval inclusion of fixed points, BIT 18 (1978), no. 1, 42–51.
  • A. D. D. Craik and H. Okamoto, A three-dimensional autonomous system with unbounded “bending” solutions, Phys. D 164 (2002), no. 3–4, 168–186.
  • Z. Galias, Investigations of periodic orbits in electronic circuits with interval Newton method, in Symposium on Circuits and Systems, ISCAS'98 (Monterey, 1998), 370–373, Proc. IEEE Int. vol. 3, IEEE, Piscataway, NJ, 1998.
  • Z. Galias, Counting low-period cycles for flows, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 16 (2006), no. 10, 2873–2886.
  • T. Kapela and C. Simó, Computer assisted proofs for nonsymmetric planar choreographies and for stability of the Eight, Nonlinearity 20 (2007), no. 5, 1241–1255.
  • T. Kapela and P. Zgliczyński, The existence of simple choreographies for the $N$-body problem–-a computer-assisted proof, Nonlinearity 16 (2003), no. 6, 1899–1918.
  • G. Mayer, Epsilon-inflation in verification algorithms, J. Comput. Appl. Math. 60 (1995), no. 1-2, 147–169.
  • T. Miyaji, H. Okamoto and A. D. D. Craik, A four-leaf chaotic attractor of a three-dimensional dynamical system, Internat. J. Bifur. Chaos. (to appear).
  • R. E. Moore, Interval analysis, Prentice Hall, Englewood Cliffs, NJ, 1966.
  • R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to interval analysis, SIAM, Philadelphia, PA, 2009.
  • \.I. Pehlivan, Four-scroll stellate new chaotic system, Optoelect. Adva. Mater. 5 (2011), no. 9, 1003–1006.
  • S. M. Rump, Verification methods: rigorous results using floating-point arithmetic, Acta Numer. 19 (2010), 287–449.
  • M. Capinski, J. Cyranka, Z. Galias, T. Kapela, M. Mrozek, P. Pilarczyk, D. Wilczak, P. Zgliczyński and M. Żelawski, Computer Assisted Proofs in Dynamics, Jagiellonian University,
  • P. Zgliczynski, $C^{1}$ Lohner algorithm, Found. Comput. Math. 2 (2002), no. 4, 429–465.