Tomasz Stachowiak 氏
Many dynamical systems, like Lotka-Volterra equations in biology or oligopoly models in economy, are constructed without Hamiltonian structure but directly by specifying the right-hand sides of dx/dt = f(x). Similar systems appear in reductions of more difficult problems, like the famous Lorenz system, or truncated normal forms. The question of their integrability has to be approached differently than in the Liouville-Arnold context, and in particular, two-dimensional systems cannot be analysed with the Morales-Ramis theory. Fortunately, there exists a classical approach for such systems, with polynomial f(x), which relies on their invariant algebraic varieties given by the so called Darboux polynomials. The search for rational first integrals (or the proof of non-existence) is usually the primary focus, but in modern times the analysis has been extended also to Liouvillian functions. I will present the fundamentals of this approach and show how it can be used for the Duffing oscillator, among other examples, to obtain non-meromorphic (hypergeometric) first integrals.