JournalsrmiVol. 14, No. 3pp. 481–517

Inverse problems in the theory of analytic planar vector

  • Natalia Sadovskaia

    Universidat Politècnica de Catalunya, Barcelona, Spain
  • Rafael O. Ramírez

    Universitat Rovira i Virgili, Tarragona, Spain
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In this communication we state and analyze the new inverse problems in the theory of differential equations related to the construction of an analytic planar vector field from a given, finite number of solutions, trajectories or partial integrals. Likewise we study the problem of determining a stationary complex analytic vector field  from a given finite subset of terms in the formal power series

V(z,w)=λ(z2+w2)+k=3Hk(z,w),Hk(az,aw)=akHk(z,w),V(z, w) = \lambda (z^2 + w^2) + \sum^\infty_{k=3} H_k (z, w), H-k (az, aw) = a^kH_k(z,w),

and from the subsidiary condition

Γ(V)=k=1G2k(z2+w2)k+1,\Gamma (V) = \sum^\infty_{k=1} G_{2k}(z^2 + w^2)^{k+1},

 where G2kG_{2k} is the Liapunov constant. The particular case when

V(z,w)=f0(z,w)f0(0,0)V (z,w) = f_0(z,w) – f_0(0,0)

and (f0,DC2)(f_0, D \subset \mathbb C^2) is a canonic element in the neigbourhood of the origin of the complex analytic first integral FF is analyzed. The results are applied to the quadratic planar vector fields. In particular we constructed the all quadratic vector field tangent to the curve

(yq(x))2p(x)=0.(y – q (x))^2 – p(x) = 0.

where qq and pp are polynomials of degree kk and m2km ≤ 2k respectively. We showed that the quadratic differential systems admits a limit cycle of this type only when the algebraic curve is of the fourth degree. For the case when k>5k > 5 it proved that there exist an unique quadratic vector field tangent to the given curve and it is Darboux's integrable.

Cite this article

Natalia Sadovskaia, Rafael O. Ramírez, Inverse problems in the theory of analytic planar vector . Rev. Mat. Iberoam. 14 (1998), no. 3, pp. 481–517

DOI 10.4171/RMI/243