# American Institute of Mathematical Sciences

2003, 2003(Special): 701-708. doi: 10.3934/proc.2003.2003.701

## Topological-numerical approach to the existence of periodic trajectories in ODE's

 1 Jagiellonian University, Institute of Computer Science, ul. Nawojki 11, 30-072 Kraków, Poland

Received  September 2002 Published  April 2003

We discuss an application of a topological-numerical method for proving the existence of a periodic trajectory in a smooth dynamical system in $\mathbb(R)^n$ where a periodic trajectory is numerically observed. The method is based on the Conley index theory and rigorous numerics for ODEs and it is a generalization of the method introduced in [13]. We apply this method to the Rössler equations.
Citation: Paweł Pilarczyk. Topological-numerical approach to the existence of periodic trajectories in ODE's. Conference Publications, 2003, 2003 (Special) : 701-708. doi: 10.3934/proc.2003.2003.701
 [1] Todd Young. A result in global bifurcation theory using the Conley index. Discrete & Continuous Dynamical Systems, 1996, 2 (3) : 387-396. doi: 10.3934/dcds.1996.2.387 [2] M. C. Carbinatto, K. Mischaikow. Horseshoes and the Conley index spectrum - II: the theorem is sharp. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 599-616. doi: 10.3934/dcds.1999.5.599 [3] Marian Gidea. Leray functor and orbital Conley index for non-invariant sets. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 617-630. doi: 10.3934/dcds.1999.5.617 [4] Jintao Wang, Desheng Li, Jinqiao Duan. On the shape Conley index theory of semiflows on complete metric spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1629-1647. doi: 10.3934/dcds.2016.36.1629 [5] Anna Go??biewska, S?awomir Rybicki. Equivariant Conley index versus degree for equivariant gradient maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985 [6] Ketty A. De Rezende, Mariana G. Villapouca. Discrete conley index theory for zero dimensional basic sets. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1359-1387. doi: 10.3934/dcds.2017056 [7] Lingling Lv, Zhe Zhang, Lei Zhang, Weishu Wang. An iterative algorithm for periodic sylvester matrix equations. Journal of Industrial & Management Optimization, 2018, 14 (1) : 413-425. doi: 10.3934/jimo.2017053 [8] Fernando Casas, Cristina Chiralt. A Lie--Deprit perturbation algorithm for linear differential equations with periodic coefficients. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 959-975. doi: 10.3934/dcds.2014.34.959 [9] Peter Giesl. Converse theorem on a global contraction metric for a periodic orbit. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5339-5363. doi: 10.3934/dcds.2019218 [10] Rafael Ortega. Stability and index of periodic solutions of a nonlinear telegraph equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 823-837. doi: 10.3934/cpaa.2005.4.823 [11] Piotr Oprocha, Pawel Wilczynski. Distributional chaos via isolating segments. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 347-356. doi: 10.3934/dcdsb.2007.8.347 [12] Anete S. Cavalcanti. An existence proof of a symmetric periodic orbit in the octahedral six-body problem. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1903-1922. doi: 10.3934/dcds.2017080 [13] Xueting Tian, Shirou Wang, Xiaodong Wang. Intermediate Lyapunov exponents for systems with periodic orbit gluing property. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1019-1032. doi: 10.3934/dcds.2019042 [14] Peter Giesl, James McMichen. Determination of the basin of attraction of a periodic orbit in two dimensions using meshless collocation. Journal of Computational Dynamics, 2016, 3 (2) : 191-210. doi: 10.3934/jcd.2016010 [15] Tatiane C. Batista, Juliano S. Gonschorowski, Fábio A. Tal. Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3315-3326. doi: 10.3934/dcds.2015.35.3315 [16] Peter Giesl. On a matrix-valued PDE characterizing a contraction metric for a periodic orbit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4839-4865. doi: 10.3934/dcdsb.2020315 [17] Evgeniy Timofeev, Alexei Kaltchenko. Nearest-neighbor entropy estimators with weak metrics. Advances in Mathematics of Communications, 2014, 8 (2) : 119-127. doi: 10.3934/amc.2014.8.119 [18] Peter Giesl. Necessary condition for the basin of attraction of a periodic orbit in non-smooth periodic systems. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 355-373. doi: 10.3934/dcds.2007.18.355 [19] Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293 [20] Belgacem Rahal, Cherif Zaidi. On finite Morse index solutions of higher order fractional elliptic equations. Evolution Equations & Control Theory, 2021, 10 (3) : 575-597. doi: 10.3934/eect.2020081

Impact Factor:

## Metrics

• PDF downloads (81)
• HTML views (0)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]