Darboux integral

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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[1], 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[2]). 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[3], 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[4].

Basic references

  1. See Poincare problem.
  2. Jouanolou, J.-P. Équations de Pfaff algébriques, Lecture Notes in Mathematics, 708. Springer, Berlin, 1979. MR0537038.
  3. Ilyashenko, Yu., Yakovenko, S., Lectures on analytic differential equations, Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008. MR2363178
  4. Christopher, C., Llibre, J., Pereira, J., Multiplicity of invariant algebraic curves in polynomial vector fields, Pacific J. Math. 229 (2007), no. 1, 63--117. MR2276503.
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