From the Sharkovskii theorem to periodic orbits for the Rössler system. (English) Zbl 1490.37031
Summary: We extend Sharkovskii’s theorem to the cases of \(N\)-dimensional maps which are close to 1D maps, with an attracting \(n\)-periodic orbit. We prove that, with relatively weak topological assumptions, there exist also \(m\)-periodic orbits for all \(m \triangleright n\) in Sharkovskii’s order, in the nearby.
We also show, as an example of application, how to obtain such a result for the Rössler system with an attracting periodic orbit, for four sets of parameter values. The proofs are computer-assisted.
We also show, as an example of application, how to obtain such a result for the Rössler system with an attracting periodic orbit, for four sets of parameter values. The proofs are computer-assisted.
MSC:
| 37C27 | Periodic orbits of vector fields and flows |
| 37J46 | Periodic, homoclinic and heteroclinic orbits of finite-dimensional Hamiltonian systems |
| 37J39 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.) |
Keywords:
Sharkovskii theorem; Roessler system; periodic orbit; covering relation; computer-assisted proofSoftware:
CAPD DynSysReferences:
| [1] | Block, L.; Guckenheimer, J.; Misiurewicz, M.; Young, L., Periodic points and topological entropy of one dimensional maps, (Lecture Notes Math., vol. 819 (11 2006)), 18 · Zbl 0447.58028 |
| [2] | Burns, Keith; Hasselblatt, Boris, The Sharkovsky theorem: a natural direct proof, Am. Math. Mon., 118, 3, 229-244 (2011) · Zbl 1218.37045 |
| [3] | Computer assisted proofs in dynamics C++ library |
| [4] | Gierzkiewicz, A.; Zgliczyński, P., C++ source code, Available online on |
| [5] | Gierzkiewicz, A.; Zgliczyński, P., Periodic orbits in the Rössler system, Commun. Nonlinear Sci. Numer. Simul., 101, Article 105891 pp. (2021) · Zbl 1472.34060 |
| [6] | Kapela, T.; Mrozek, M.; Wilczak, D.; Zgliczyński, P., CAPD::DynSys: a flexible C++ toolbox for rigorous numerical analysis of dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 101, Article 105578 pp. (2021) · Zbl 1473.37004 |
| [7] | Neumaier, A., Interval Methods for Systems of Equations, Encyclopedia of Mathematics and Its Applications (1991), Cambridge University Press |
| [8] | Rössler, O. E., An equation for continuous chaos, Phys. Lett. A, 57, 5, 397-398 (1976) · Zbl 1371.37062 |
| [9] | Sharkovskii, A. N., Co-existence of cycles of a continuous mapping of the line into itself, Ukr. Math. J.. Ukr. Math. J., Int. J. Bifurc. Chaos Appl. Sci. Eng., 5, 1263-1273 (1995), (in Russian), English translation in · Zbl 0890.58012 |
| [10] | Štefan, P., A theorem of Sarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line, Commun. Math. Phys., 54, 3, 237-248 (1977) · Zbl 0354.54027 |
| [11] | Zgliczyński, P., Multidimensional perturbations of one-dimensional maps and stability of Šarkovskĭ ordering, Int. J. Bifurc. Chaos, 09, 09, 1867-1876 (1999) · Zbl 1089.37502 |
| [12] | Zgliczyński, P., Sharkovskii’s theorem for multidimensional perturbations of one-dimensional maps, Ergod. Theory Dyn. Syst., 19, 6, 1655-1684 (1999) · Zbl 0949.37020 |
| [13] | Zgliczyński, P.; Gidea, M., Covering relations for multidimensional dynamical systems, J. Differ. Equ., 202, 1, 32-58 (2004) · Zbl 1061.37013 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
