On the planar integrable differential systems
Consider a planar dynamical system, i.e., a system of two autonomous ODEs \[ \dot{x} = P(x,y), \quad \dot{y} = Q (x,y). \tag{*} \] Here, \(P,Q : U \to \mathbb R\) are \(C^k\) functions (with possibly \(k=\infty\) or \(k = \omega\), which means we deal with analytic functions), and \(U \subseteq \mathbb R^2\) an open set. This defines a vector field which we denote by \(X\). Denote by \(\Sigma \subset U\) the set of all separatrices (this is a closed and invariant subset), and by \(U_0 = U \backslash \Sigma\) the complement of \(\Sigma\) in \(U\). If system (*), i.e., the vector field \(X\), admits a non-constant function \(H : U_0 \to \mathbb R\) such that \(X(H) = 0\), then \(H\) is a canonical first integral for \(X\), and \(X\) -- or equivalently system (*) -- is said to be integrable. It is well-known that, if \(X\) is the flow generated by a Hamiltonian \(\mathcal{H}\), then the system admits the first integral \(H = \mathcal{H}\) and is hence (recall we are in \(\mathbb R^2\)) integrable. The paper focuses hence on the case when \(X\) is not Hamiltonian. It turns out that non-Hamiltonian integrable systems have a common structure, i.e., can be mapped to a dilation system; it also follows from this that they have a Lie symmetry. More specifically, if \(X\) is non Hamiltonian and assuming \(P\) and \(Q\) are \(C^2\) on \(U\), one looks for an integrating factor, i.e., a \(C^1\)-function \(\rho : V_\rho \to \mathbb{R}\), with \(V_\rho \subseteq U\), such that \[ Y = \rho X = (\rho P) \partial_x + (\rho Q) \partial_y \] is (divergence-free, and hence) Hamiltonian. If this exists, we have that the (\(C^1\)-) Hamiltonian \(H\) (defined on \(V_H \subseteq U\)) for \(Y\) is identified by \(\rho P = - \partial_y H\), \(\rho Q = \partial_x H\). Denote by \(V = V_\rho \cap V_H\) the set in which both \(\rho\) and \(H\) are defined, and by \(Z \subseteq V\) the set where \(\rho (\rho_x H_y - \rho_y H_x) (P_x + Q_y)=0\). If \(Z\) has zero Lebesgue measure in \(V\), then (Theorem 1), passing to variables \[ u=\rho (x,y), \quad v=\rho (x,y) \;H(x,y), \quad ds=-(P_x + Q_y) dt, \] system (*) is (in the open and dense subset \(V \backslash Z\)) mapped onto the dilation system \[ du/ds= u \quad dv/ds=v. \] It is rather clear that (Theorem 2), in this case, system (*) admits a Lie symmetry in \(V \backslash Z\). The authors consider as a nontrivial application the system given by \(P = - y - b (x^2 + y^2)\), \(Q = x\) (with of course \(b \not=0\)). A section is also devoted to the relation of their results with Darboux theory of integrability.
scientific article; zbMATH DE number 6134347
scientific article