On the maximum number of period annuli for second order conservative equations
Abstract
We consider a second order scalar conservative differential equation whose potential function is a Morse function with a finite number of critical points and is unbounded at infinity. We give an upper bound for the number of nonglobal nontrivial period annuli of the equation and prove that the upper bound obtained is sharp. We use tree theory in our considerations.
Keyword : conservative equation, Morse function, period annulus, binary tree
How to Cite
Gritsans, A., & Yermachenko, I. (2021). On the maximum number of period annuli for second order conservative equations. Mathematical Modelling and Analysis, 26(4), 612-630. https://doi.org/10.3846/mma.2021.13979
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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S. Atslega and F. Sadyrbaev. On periodic solutions of Liénard type equations. Mathematical Modelling and Analysis, 18(5):708–716, 2013. https://doi.org/10.3846/13926292.2013.871651
C. Berge. Topological Spaces: Including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity. Dover Publications, Inc., Mineola, NY, 1997.
S. Biasotti, D. Giorgi, M. Spagnuolo and B. Falcidieno. Reeb graphs for shape analysis and applications. Theoretical Computer Science, 392(1):5–22, 2008. https://doi.org/10.1016/j.tcs.2007.10.018
C. Christopher and C. Li. Limit Cycles of Differential Equations. Advanced Courses in Mathematics. CRM Barcelona, Birkha¨user Verlag, Basel, 2007.
F. Dumortier. Limit cycles of differential equations (book review of MR2325099). Bulletin of the American Mathematical Society, 46(4):697–701, 2009. https://doi.org/10.1090/S0273-0979-09-01267-1
I. Gelbukh. Loops in Reeb graphs of n-manifolds. Discrete & Computational Geometry, 59(4):843–863, 2018. https://doi.org/10.1007/s00454-017-9957-9
Y. Kozmina. On the polynomials of optimal shape generating maximum number of period annuli. Innovative Infotechnologies for Science, Business and Education, 2(13):13–19, 2012. Available from Internet: http://old.kolegija.lt/dokumentai_img/IITSBE2012213.pdf
Y. Kozmina and F. Sadyrbaev. On a maximal number of period annuli. Abstract and Applied Analysis, 2011:8, 2011. https://doi.org/10.1155/2011/393875
F. Mańosas and J. Villadelprat. A note on the critical periods of potential systems. International Journal of Bifurcation and Chaos, 16(3):765–774, 2006. https://doi.org/10.1142/S0218127406015155
P. Mardešíc, D. Marín and J. Villadelprat. The period function of reversible quadratic centers. Journal of Differential Equations, 224(1):120–171, 2006. https://doi.org/10.1016/j.jde.2005.07.024
W. Marzantowicz, M. Kaluba and N. Silva. On representation of the Reeb graph as a sub-complex of manifold. Topological Methods in Nonlinear Analysis, 45(1):287–307, 2015. https://doi.org/10.12775/TMNA.2015.015
L .P. Michalak. Realization of a graph as the Reeb graph of a Morse function on a manifold. Topological Methods in Nonlinear Analysis, 52(2):749–762, 2018. https://doi.org/10.12775/TMNA.2018.029
L. Nicolaescu. An Invitation to Morse Theory. Universitext, Springer, New York, 2007. https://doi.org/10.1007/978-0-387-49510-1
M. Potter. Set Theory and its Philosophy: A Critical Introduction. Oxford University Press, New York, 2004. https://doi.org/10.1093/acprof:oso/9780199269730.001.0001
D. Rojas. On the upper bound of the criticality of potential systems at the outer boundary using the Roussarie-Ecalle compensator. Journal of Differential Equations, 267(6):3922–3951, 2019. https://doi.org/10.1016/j.jde.2019.04.021
M. Sabatini. Liénard limit cycles enclosing period annuli, or enclosed by period annuli. Rocky Mountain Journal of Mathematics, 35(1):253–266, 2005. https://doi.org/10.1216/rmjm/1181069780
G. Valiente. Algorithms on Trees and Graphs. Springer-Verlag, Berlin, 2002. https://doi.org/10.1007/978-3-662-04921-1
D. Vrajitoru and W. Knight. Practical Analysis of Algorithms. Springer, Cham, 2014. https://doi.org/10.1007/978-3-319-09888-3
D.B. West. Introduction to Graph Theory. Prentice Hall, Inc., Upper Saddle River, NJ, 1996.
L. Yang and X. Zeng. The period function of potential systems of polynomials with real zeros. Bulletin des Sciences Mathématiques, 133(6):555–577, 2009. https://doi.org/10.1016/j.bulsci.2009.05.002
P. Yu. Chapter 1 bifurcation, limit cycle and chaos of nonlinear dynamical systems. In J.Q. Sun and A.C.J. Luo (Eds.), Bifurcation and Chaos in Complex Systems, volume 1 of Edited Series on Advances in Nonlinear Science and Complexity, pp. 1–125. Elsevier Science, 2006. https://doi.org/10.1016/S1574-6909(06)01001-X