Applied Mathematics and Computation 338 (2018) 55–71 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Reversibility in polynomial systems of ODE’s Maoan Han a,b , Tatjana Petek c,d,e,∗ , Valery G. Romanovski c,e,f a Department of Mathematics, Shanghai Normal University, Shanghai 200234, China b School of Mathematics Sciences, Qufu Normal University, Qufu 273165, China c Faculty of Electrical Engineering and Computer Science, University of Maribor, Maribor SI-2000, Slovenia d Institute of Mathematics, Physics and Mechanics, Ljubljana SI-1000, Slovenia e Center for Applied Mathematics and Theoretical Physics, Maribor SI-2000, Slovenia f Faculty of Natural Science and Mathematics, University of Maribor, Maribor SI-2000, Slovenia a r t i c l e i n f o MSC: 34A34 34C14 13P10 Keywords: Polynomial systems of ODE’s Symmetries Time-reversible system Integrability a b s t r a c t For a given family of real planar polynomial systems of ordinary differential equations de- pending on parameters, we consider the problem of how to find the systems in the family which become time-reversible after some affine transformation. We first propose a general computational approach to solve this problem, and then demonstrate its usage for the case of the family of quadratic systems. © 2018 Elsevier Inc. All rights reserved. 1. Introduction One of important problems arising in the investigation of the qualitative behavior of dynamical systems is determining whether a given system admits some kind of symmetry. In studies of dynamical systems described by autonomous poly- nomial systems of ordinary differential equations, we deal mainly with two kinds of symmetries: The rotational symmetry (Z q -symmetry) and the time-reversible (involutive) symmetry. Z q -invariant systems arise frequently in works related to the second part of Hilbert’s 16th problem, since, due to the existence of such symmetry, it is possible to construct polynomial systems with many limit cycles (see e.g. [6,9,13–15,20] and references therein). The existence of time-reversible symmetry in a polynomial system is related closely to the integrability of the system. For example, consider the real system ˙ u = −v − vU (u, v 2 ), ˙ v = u + V (u, v 2 ), (1.1) where U is an analytic function without constant term, and V is an analytic function whose series expansion at (0,0) starts with terms of order at least two. Clearly, the origin of system (1.1) is either a focus or a center. It is seen easily seen that the phase portrait of system (1.1) remains unchanged after reflection with respect to the Ou axis, and reversion of the sense of every orbit (the reversal of time). Hence, the origin of (1.1) is a center and, by the Poincaré–Lyapunov theorem, the system admits a local analytical integral. In general, it is said that a smooth differential system ˙ x = F (x), x ∈ R n , (1.2) ∗ Corresponding author at: Faculty of Electrical Engineering and Computer Science, University of Maribor, SI-2000 Maribor, Slovenia. E-mail addresses: mahan@shnu.edu.cn (M. Han), tatjana.petek@um.si (T. Petek), valerij.romanovskij@um.si (V.G. Romanovski). https://doi.org/10.1016/j.amc.2018.05.051 0096-3003/© 2018 Elsevier Inc. All rights reserved.