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Extra resources for Bifurcations of Planar Vector Fields: Proceedings of a Meeting held in Luminy, France, Sept. 18–22, 1989
Example text
1 One parameter family of Poincard maps A simple form for the return maps Let X . represent a generic saddle loop bifurcation at # = 0. Like in the previous section we suppose that the saddle point has a fixed position s and that 3'(0) = 7 < 1 where 7(#) is the ratio AI(#___~) with - ~ i ( # ) and t2(#) the negative and positive eigenvalues of X . at s. Also, as in introduction, we call a(t, ) the shift function of the invariant manifolds, calculated on some transversal ~ with respect to the stable manifold and with a positive orientation toward the inside of the saddle loop F.
To prove this Lipschitz property take Ul,U2 E B¢(K) and let v~ = Tul,v2 = Tu2. We need to calculate: v~(x) X 5* v~(x) X 5* Therefore, consider the expression (12) in function of the independent variables (x, U, V') with V = u(x a/'Y) and U' =: u'(xV'~): v'(x) xS, = F(x, u(xl/'~), u'(xl/'Y)) with F(x, U, U') -- (1 + U)'Y-'(1 + ¢(xl/'Y(1 + U ) ) U t x a X~ 4 - - ( 1 + U)'Y(1 + U + xl/~u#)~#(x'/q'(1 + U)) 7 Now : v'~(x) q(x) X 5' (15) X5 t = F ( x , U 2, U2) ' - F(x, U1, U~) may be estimated by the mean value theorem.
The chosen values of correspond to k = 1 / x / ~ and k = 1/2 respectively. 4561286509. Thus, the system g=~'~ defined by ic = - y , ~) = x + x 3 + )`12e2y + ex 2 + exy + ,~42~2x2y has a first order bifurcation function which vanishes identically and a second order bifurcation function with two zeros. The results of the numerical experiment are shown 42 in Figure 1. In the figure the horizontal coordinate is the distance ~ along the positive x-axis while the vertical coordinate is the value of the p a r a m e t e r e.
