Abstract
In this article we investigated the problem of detecting the chaotic behavior of the dynamical system with a Hamiltonian structure using pattern recognition methods. We carried out a numerical constructed of the phase space structure of these dynamical system which represented on 2D Poincare’ sections of a special points cloud in chaotic cases. This chaotic regions are characterized by many various bad formalizing analytically forms. We are classified these forms on 2d sections in simply and multiply connected, inside and outside located. We adapted a deformable active contours method for closed curves to automatic detecting these chaotic regions of the dynamical system.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Ruchkin, C.: The general conception of the intellectual investigation of the regular and chaotic behavior of the dynamical system hamiltonian structure. Applied Non-Linear Dynamical Systems. Springer Proceedings in Mathematics and Statistics, vol. 93. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08266-0_17
Ruchkin, K.A.: Development of computer system for analysis of Poincare’ sections / K.A. Ruchkin. “Artificial intelligence”. - Donetsk: NAS Ukraine. - 2009. - No̱ 1., pp. 83–87 (2009)
Neumark, Yu.I.: The multidimensional geometry and image recognition. Soros Educational Magazine Number 7, pp. 119–123 (1996)
Nejmark Yu.I.: New the approach to numerical research of specific dynamic systems by pattern recognition and statistical modeling methods / Yu.I. Nejmark, I.V. Kotelnikov, L.G. Teklina. News High Schools “Application-oriented nonlinear dynamics”. - 2010. - 18. - No̱2, pp. 3–15 (2010)
Neimark, Yu.I., Teklina, L.G.: On possibilities of using pattern recognition methods to study mathematical models. Pattern Recognit. Image Anal. 22(1), 144–149 (2012). https://doi.org/10.1134/S1054661812010282
Neimark, Yu.I., Kotelnikov, I.V., Teklina, L.G.: Investigation of the structure of the phase space of a dynamical system as a pattern recognition problem. In: Conference on MMRO-12, pp. 177–180. M. MaksPress (2005)
Gottwald, G.A., Melbourne, I.: A new test for chaos in deterministic systems. Proc. Roy. Soc. A 460, 603–611 (2004)
Gottwald, G.A., Melbourne, I.: Testing for chaos in deterministic systems with noise. Physica D 212(1–2), 100–110 (2005)
Gottwald, G.A., Melbourne, I.: Comment on “Reliability of the 0–1 test for chaos”. Phys. Rev. E 77, 028201 (2008)
Gottwald, G.A., Melbourne, I.: On the implementation of the 0–1test for chaos. SIAM J. Appl. Dyn. 8, 129–145 (2009)
Gottwald, G.A., Melbourne, I.: On the validity of the 0–1 test for chaos. Nonlinearity 22, 1367–1382 (2009)
Gottwald, G.A., Melbourne, I.: A Huygens principle for diffusion and anomalous diffusion in spatially extended systems. Proc. Natl. Acad. Sci. USA 110, 8411–8416 (2013)
Gottwald, G.A., Melbourne, I.: Central limit theorems and suppression of anomalous diffusion for systems with symmetry. submitted. (2013). https://arxiv.org/abs/1404.0770
Gottwald, G.A., Melbourne, I.: A test for a conjecture on the nature of attractors for smooth dynamical systems. Chaos 24, 024403 (2014). https://doi.org/10.1063/1.4868984
Zachilas, L., Psarianos I.: Examining the chaotic behavior in dynamical systems by means of the 0-1 Test. J. Appl. Math. 2012, 681296 (2012). https://doi.org/10.1155/2012/681296
Kyriakopoulos, N., Koukouloyannis, V., Skokos, Ch., Kevrekidis, P.: Chaotic behavior of three interacting vortices in a confined bose-einstein condensate. Chaos 24, 024410 (2013). https://doi.org/10.1063/1.4882169
Manos, Th., Skokos, Ch., Athanassoula, E., Bountis T.: Studying the global dynamics of conservative dynamical systems using the SALI chaos detection method. Nonlinear Phenomena Complex Syst. 11, 171–176 (2008)
Skokos, C., Parsopoulos, K.E., Patsis, P.A., Vrahatis, M.N.: Particle swarm optimization: an efficient method for tracing periodic orbits in three-dimensional galactic potentials. Monthly Notices Roy. Astron. Soc. 359(1), 251–260 (2005). https://doi.org/10.1111/j.1365-2966.2005.08892
Petalas, Y., Parsopoulos, K., Vrahatis, M.: Stochastic optimization for detecting periodic orbits of nonlinear map**s. Nonlinear Phenom. Complex Syst. 11, 285–291 (2008)
Petalas, Y., Antonopoulos, Ch., Bountis, T., Vrahatis, M.: Detecting Resonances using Evolutionary Algorithms. In: roceedings of the International Conference of “Computational Methods in Sciences and Engineering”(ICCMSE 2006). CRC Press, Boca Raton (2019). https://doi.org/10.1201/9780429070655-106
Strijov, V., Shakin, V.: An algorithm for clustering of the phase trajectory of a dynamic system. Math. Commun. Suppl. 1, 159–165 (2001)
Kass, M., Witkin, A., Terzopoulos, D.: snakes: active contour models. Int. J. Comput. Vis. 1(4), 321–331 (1988)
Cootes, T.F., Taylor, C.J., Cooper, D.H., Graham, J.: Active shape models - their training and application. Comput. Vis. Image Understand. 61(1), 38–59 (1995)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Ruchkin, A., Ruchkin, C. (2021). Detection of Chaotic Behavior in Dynamical Systems Using a Method of Deformable Active Contours. In: Awrejcewicz, J. (eds) Perspectives in Dynamical Systems II: Mathematical and Numerical Approaches. DSTA 2019. Springer Proceedings in Mathematics & Statistics, vol 363. Springer, Cham. https://doi.org/10.1007/978-3-030-77310-6_13
Download citation
DOI: https://doi.org/10.1007/978-3-030-77310-6_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-77309-0
Online ISBN: 978-3-030-77310-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)