3bedd40d-b9bd-425d-bd47-1fbe068919f020221209101814263wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON SYSTEMS AND CONTROL2224-28561991-876310.37394/23203http://wseas.org/wseas/cms.action?id=40731520221520221710.37394/23203.2022.17https://wseas.com/journals/sac/2022.phpAndrés GabrielGarcíaDepartamento de Ingeniería Eléctrica Universidad Tecnológica Nacional-FRBB 11 de Abril 461, Bahía Blanca, Buenos Aires ARGENTINAThis paper provides an upper bound for the number of periodic orbits in planar systems. The research results in, [7], and, [8], allows one to produce a bound on the number of periodic orbits/limit cycles. Introducing the concept of Maximal Grade and Maximal Number of Periodic Orbits, a simple algebraic calculation leads to an upper bound on the number of periodic trajectories for general second order systems. In particular, it also applies to polynomial ODE’s. As far as the author is aware, such a powerful result is not available in the literature. Instead, the methods in this paper provide a tool to determine an upper bound on the periodic orbits/limit cycles for a wide range of dynamical systems.1292022129202249850355https://wseas.com/journals/sac/2022/b125103-033(2022).pdf10.37394/23203.2022.17.55https://wseas.com/journals/sac/2022/b125103-033(2022).pdfA. Lins Neto, W. de Melo, C.C. Pugh, On Liénard Equations, Proc. Symp. Geom. and Topol., Springer Lectures Notes in Math. Number 597, pp. 335–357, 1977. Linear System Theory. Wilson Rugh, Pearson, 2nd Edition. 1995. Oscillations in Planar Dynamic Systems (SERIES ON ADVANCES IN MATHEMATICS FOR APPLIED SCIENCES). R. E. Mickens. World Scientific. 1996. 10.1090/s0002-9939-07-08688-1F. Dumortier, D. Panazzolo and R. Roussarie. More Limit Cycles than Expected i n Liénard Equations. Proceedings of the American Mathematical Society, Vol. 135, 6, 1895 1904, 2007. Asymptotic Methods in Analysis. N. G. De Brujin. Dover Publications.2010. 10.1016/j.exmath.2016.12.001Jaume Llibre and Xiang Zhang. Limit cycles of the classical Liènard differential systems: A survey on the Lins Neto, de Melo and Pugh’s conjecture, Exposition. Math., vol.35, 386–299, 2017. DOI: [10.1016/j.exmath.2016.12. García, Andrés Gabriel (2019). First Integrals vs Limit Cycles. https://arxiv.org/abs/1909.07845, doi = 10.48550/ARXIV.1909.07845. 10.31181/rme200102143hHe , J.-H. and García , A. (2021). The simplest amplitude-period formula for non-conservative oscillators. Reports in Mechanical Engineering, 2(1), 143-148. https://doi.org/10.31181/rme200102143h 10.1088/1361-6544/ac7691Rodrigo D Euzébio and Jaume Llibre and Durval J Tonon. (2022). Lower bounds for the number of limit cycles in a generalised Rayleigh–Liénard oscillator. IOP Publishing: Nonlinearity, 35(8), 3883. https://dx.doi.org/10.1088/1361- 6544/ac7691 10.1016/j.jsc.2016.02.004Xianbo Suna, Wentao Huang (2017). Bounding the number of limit cycles for a polynomial Liénard system by using regular chains. Journal of Symbolic Computation, 79, 197-210. 10.1016/j.cnsns.2021.105716Yifan Hu, Wei Niu, Bo Huang (2021). Bounding the number of limit cycles for parametric Liénard systems using symbolic computation methods. Communications in Nonlinear Science and Numerical Simulation, 96, 105716. https://doi.org/10.1016/j.cnsns.2021.105716.