A. Goriely, Integrability and Nonintegrability of Dy- namical Systems, ser Advanced Series in Nonlinear Dynamics, World Scientific, 2001.

R. Risch, The problem of integration in finite terms, Transactions of the American Mathematical Society, vol.139, 1968.
DOI : 10.1090/S0002-9947-1969-0237477-8

S. Boyd, L. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, ser, Studies in Applied Mathematics, vol.15, 1994.

B. Tibken, Estimation of the domain of attraction for polynomial systems via LMIs, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187), pp.3860-3864, 2000.
DOI : 10.1109/CDC.2000.912314

D. Henrion and J. B. Lasserre, Inner Approximations for Polynomial Matrix Inequalities and Robust Stability Regions, IEEE Transactions on Automatic Control, vol.57, issue.6, pp.1456-1467, 2012.
DOI : 10.1109/TAC.2011.2178717

URL : https://hal.archives-ouvertes.fr/hal-00588754

D. Henrion and M. Korda, Convex Computation of the Region of Attraction of Polynomial Control Systems, IEEE Transactions on Automatic Control, vol.59, issue.2, 2013.
DOI : 10.1109/TAC.2013.2283095

URL : https://hal.archives-ouvertes.fr/hal-00723019

V. Zubov and A. Liapunov, Methods of A. M. Lyapunov and Their Application, ser. Translation series, U.S. Atomic Energy Commission, 1961.

A. Vannelli and M. Vidyasagar, Maximal lyapunov functions and domains of attraction for autonomous nonlinear systems, Automatica, vol.21, issue.1, pp.69-80, 1985.
DOI : 10.1016/0005-1098(85)90099-8

P. A. Parillo, Semidefinite programming relaxations for semialgebraic problems, Mathematical Programming, vol.96, issue.2, pp.293-320, 2003.
DOI : 10.1007/s10107-003-0387-5

S. Prajna, A. Papachristodoulou, P. Seiler, P. A. Parrilo, and . Sos-tools, Sum of squares optimization toolbox for MATLAB, Available from http://www.cds.caltech, 2004.

A. Papachristodoulou and S. Prajna, Analysis of Non-polynomial Systems Using the Sum of Squares Decomposition, Positive Polynomials in Control, ser. Lecture Notes in Control and Information Science, pp.23-43, 2005.
DOI : 10.1007/10997703_2

A. Tiwari and G. Khanna, Non-linear systems: Approximating reach sets, HSCC, ser. LNCS, pp.477-492, 2004.
DOI : 10.1007/978-3-540-24743-2_40

URL : http://www.csl.sri.com/~tiwari/papers/hscc04a.ps

A. Platzer, Differential Dynamic Logic for Hybrid Systems, Journal of Automated Reasoning, vol.30, issue.1, pp.143-189, 2008.
DOI : 10.1007/978-1-4612-0601-9

URL : http://www.functologic.com/pub/freedL.pdf

S. Prajna and A. Jadbabaie, Stochastic safety verification using barrier certificates, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601), pp.477-492, 2004.
DOI : 10.1109/CDC.2004.1428804

URL : http://www.seas.upenn.edu/~pappasg/papers/CDC04-Barrier.pdf

M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations and algebraic computation, ser. SYMSAC '81, Proceedings of the fourth ACM symposium on Symbolic, pp.30-35, 1981.
DOI : 10.1145/800206.806368

J. Jouanolou, Equations de Pfaff algebriques, ser, Lecture Notes in Mathematics, 1979.
DOI : 10.1007/bfb0063393

J. M. Ollagnier and A. Nowicki, Derivations of polynomial algebras without Darboux polynomials, Journal of Pure and Applied Algebra, vol.212, issue.7, pp.1626-1631, 2008.
DOI : 10.1016/j.jpaa.2007.10.017

URL : https://doi.org/10.1016/j.jpaa.2007.10.017

A. A. Ahmadi, M. Krstic, and P. A. Parrilo, A globally asymptotically stable polynomial vector field with no polynomial Lyapunov function, IEEE Conference on Decision and Control and European Control Conference, pp.7579-7580, 2011.
DOI : 10.1109/CDC.2011.6161499

A. Bostan, G. Chèze, T. Cluzeau, and J. Weil, Efficient algorithms for computing rational first integrals and Darboux polynomials of planar polynomial vector fields, Mathematics of Computation, vol.85, issue.299, 1310.
DOI : 10.1090/mcom/3007

URL : https://hal.archives-ouvertes.fr/hal-00871663

Y. Man, Computing Closed Form Solutions of First Order ODEs Using the Prelle-Singer Procedure, Journal of Symbolic Computation, vol.16, issue.5, pp.423-443, 1993.
DOI : 10.1006/jsco.1993.1057

Y. Man and M. A. Maccallum, A Rational Approach to the Prelle???Singer Algorithm, Journal of Symbolic Computation, vol.24, issue.1, pp.31-43, 1997.
DOI : 10.1006/jsco.1997.0111

G. Cheze, Computation of Darboux polynomials and rational first integrals with bounded degree in polynomial time, Journal of Complexity, vol.27, issue.2, pp.246-262, 2011.
DOI : 10.1016/j.jco.2010.10.004

URL : https://hal.archives-ouvertes.fr/hal-00517694

J. Lasserre, Moments, Positive Polynomials and Their Applications, ser. Imperial College Press optimization series, 2009.
DOI : 10.1142/p665

A. Vannelli and M. Vidyasagar, Maximal lyapunov functions and domains of attraction for autonomous nonlinear systems, Automatica, vol.21, issue.1, pp.69-80, 1985.
DOI : 10.1016/0005-1098(85)90099-8

A. Korobeinikov and G. C. Wake, Global properties of the three-dimensional predator-prey Lotka-Volterra systems, Journal of Applied Mathematics and Decision Sciences, vol.3, issue.2, pp.155-162, 1999.
DOI : 10.1155/S1173912699000085

Y. Tang, R. Yuan, and Y. Ma, Dynamical behaviors determined by the Lyapunov function in competitive Lotka-Volterra systems, Physical Review E, vol.87, issue.1, 2013.
DOI : 10.1103/PhysRevLett.104.170602