# Bifurcations of Planar Vector Fields by J.P. Francoise, R. Roussarie

By J.P. Francoise, R. Roussarie

A set of papers on differential equations and dynamical platforms which stresses themes on the subject of Hilbert's sixteenth challenge. between those are an answer to Dulac's challenge, effects at the multiplicity of singular cycles and effects at the zeroes of abelian integrals.

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**Example text**

Consider the \I ave equation It is easily seen that the functions u,,(x, y) = (x - y)", 11 = I, 2, ... satisfy the equation for each 11. ::; N. , for k = I, 2, ... i (2k - k=1 •)2k- l I)! \-;·-plane containing the origin. Hence. by the extended principle of superposition, the preceding series is also a solution of the wave equation. As a matter of fact, we notice that the series converges to the function u(x, y) = sin(x - y), which is easily seen to satisfy the wave equation. There is a variation of the principle of superposition concerning solution of a homogeneous differential equation that depends on a parameter.

When x 2 + 2y 2 = 4. + 2xy11 = 0, 11 = c" sin(x + I), when y 2 = 2x + 2 20. r11r 21. ) 11, xu,. 22. rn,. 23. Show that no solution exists for the differential equation of Problelll 22, which assullles the prescribed value ef>(_y) on the circle x 2 + y 2 = a 2 , unless cf> is of the 2 form qJ(x) = kc-x • where k is a constant. If cf> has the indicated form. show that the problem has infinitely many solutions. I.

1. In Problems 7 through 13, find the general solution of the given equation. 7. 9. + + Xllx Jllx Xllx - xy11,. - 2x11 x + 2 2 )'lly - xu = 2e" + x u, - xyu,. + 2y11 2 10. y. xu,. = 2 8. l. Jllr = = 2 x y. 0. 11. y ux - 12. (x 13. xyux - 14. Let Lu = Au, + Bur + Cu= + Du = 0 be a linear equation in three variables x, J', ::, where A, B, C, Dare constants. Introduce the new variables + y)(u, - 11r) + = 11 = y . «YJ. x 2 uy - yu = xy. ~ = a 1x + h1Y + I/ = a1 x + h2Y + ( = G3X + h3y + CI;: C2Z C3Z where the deterlllinant of the coefficients a;.