Darboux integral

A first integral of a polynomial vector field on the plane, which has a specific form, the product of (non-integer) powers and exponentials of rational functions.

For upper and lower integral sums of a Riemann integrable function, see Darboux sums.

Invariant curves, factors and cofactors

Let $v=P(x,y)\partial_x+Q(x,y)\partial_y$ be a polynomial vector field on the plane, with $P,Q\in\R[x,y]$, having only isolated singularities (i.e., $\gcd(P,Q)=1$); denote by $\omega$ the polynomial $1$-form $-Q(x,y)\rd x+P(x,y)\rd y$ annulating $v$, so that $\omega\cdot v\equiv0$.

An reduced (square-free) algebraic curve $\Gamma=\{R(x,y)=0\}$ is called an invariant curve, or particular integral of the field $v$ (resp., the form $\omega)$, if $v$ is tangent to the curve at all smooth points of the latter. This means that the polynomial $\rd R\cdot v$ vanishes on $\Gamma$, hence is divisible by $R$: $$ \rd R\cdot v=KR,\qquad K=K(x,y)\in\R[x,y]. $$ The polynomial $K$ is called the cofactor of the curve $\Gamma$.

If the polynomial $R$ is reducible, $R=R_1\cdots R_k$ (without repetitions), then each of the irreducible invariant curves $\Gamma_i=\{R_i=0\}$ is a particular integral with the corresponding cofactors $K_i$ defined by the equations $\rd R_i\cdot v = R_i K_i$, $i=1,\dots,k$. Obviously, $K=K_1+\cdots+K_k$.

Integrability theorems

Let $n=\deg v=\max(\deg P,\deg Q)$ be the degree of the vector field. In general, it is not possible to place an upper bound on the degree $m=\deg \Gamma=\deg R$ of its integral curves, the degree of the cofactor is always $n+(m-1)-m=n-1$. Thus the space of possible cofactors is finite-dimensional and its dimension is $\frac12 n(n+1)$. This number is the natural bound for the number of "nontrivial" invariant curves.

Theorem (G. Darboux). If a polynomial vector field has $k\ge \frac12 n(n+1)+1$ different irreducible algebraic invariant curves $\Gamma_1=\{R_1=0\},\dots\Gamma_k=\{R_k=0\}$, then the field admits a first integral $H$, such that $\rd H\cdot v\equiv0$, of the form $$ H(x,y)=R_1^{\lambda_1}(x,y)\cdots R_k^{\lambda_k}(x,y),\qquad \lambda_1,\dots,\lambda_k\in\C, \tag DI $$ with the complex exponents $\lambda_1,\dots,\lambda_k$ not all equal to zero.

Definition. The function (DI) is referred to as the Darboux integral of the vector field.

Theorem (J.-P. Jouanolou, 1979). If the number $k$ of the different irreducible algebraic invariant curves is greater or equal to $\frac12 n(n+1)+2$ (one more than before), then $v$ has a rational first integral $$ H(x,y)=\frac{F(x,y)}{G(x,y)},\qquad F,G\in\C[x,y],\ G\not\equiv0. $$

Vector fields with a Darbouxian (or rational) integrals are called integrable: they cannot exhibit limit cycles, one of the most elusive objects defined by ordinary differential equations on the plane.

Darbouxian integrability in the Pfaffian form

A Pfaffian form $\omega$ determines the distribution of lines tangent to the plane; this distribution does not change (except for the formal domain) after multiplication of $\omega$ by a rational function. Thus without loss of generality instead of polynomial, one can consider the class of rational Pfaffian forms tangent to the same vector field.

Definition. A Pfaffian differential equation $\omega=0$ with a rational 1-form $\omega$ is called Darboux integrable, if $\omega$ is a closed rational 1-form, $\rd \omega=0$.

Denote by $\varSigma\subset\C^2$ the algebraic curve which is the polar locus of the rational 1-form $\omega$, $\varSigma=\Gamma_1\cup\cdots\cup\Gamma_k$ is its representation as the union of the irreducible components defined by polynomial equations $\Gamma_i=\{R_i=0\}$, $i=1,\dots,k$.

Lemma. A closed rational 1-form with the polar locus $\varSigma$ has the form $$ \omega=\sum_{i=1}^k \lambda_i\frac{\rd R_i}{R_i}+\rd \biggl(\frac{F}{G}\biggr),\qquad \lambda_i\in\C,\ R_i,F,G\in\C[x,y]. $$

A vector field which is tangent to a closed rational 1-form admits a generalized Darboux first integral, $$ H(x,y)=\prod_{i=1}^k R_i^{\lambda_i}\cdot\exp(F/G), $$ which also excludes the appearance of limit cycles.

Appearance of the exponential factors can be explained by confluence of distinct but close algebraic invariant curves.