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Extra resources for A course in mathematical analysis. - part.2 Differential equations

Example text

WS(X) Figure 9. Generic case : The b i f u r c a t i o n set is given by #(A) = O, u(~ o) v(~ o) - b(k) XA, near F 33 P-H-case We : suppose Then that X], t also Let -u(t,l), the has {H = ~} contains a saddle a loop F with at s (see the v(t,l) be the eigenvalues a hyperbolic introduction of this saddle. saddle s. of the chapter). We have u(t,i) = v(t,~) 1 - t~(i) + o(t). Let also I~(1) = I(~,~) = f WD(l ) (integral of the loop). F Then, the bifurcation set is given by : I~(I) = O, with ~(io) ~ O.

Consider U~ the linear coordinate : (x,y) ~ (~(~)x, ~(A)y). change, Applying depending on the parameter it to XA, we obtain : % : 26 a i 3 3 2 2 Y ~x + ~ (~i ~ x + >2 ~x + #i + By (v + ~bx + ~ cx + O(x3)) (u~). (xA) = a_ ay + ~2y2Q(~x'~Y'~)) Taking I ~ I ~ 2, that U% of Sign (~) = Sign (~) = Sign b(O) is a C a equivalence the same Observe form (7), which but transforms which enjoys i and I~(~)I = ~ the family X% , we see into a new one the desired properties on G(x,%). aG the sign of b = ~x (0,0), which that we have only possibly changed aG is coherent with the assumption ~x (0,0) # + 2 ~ .

So (In fact the other case we is easier ourselves to weak to study because there is no line of double cycles arriving at the bifurcation point TSC). Recallf2the r2 = r2(A°) saddle ratios = A22 (A°) points persist rl(~), r2(1) < 1 assumptions r2(A ) < 1 and in Sl(1), and r(1) made on X Io : rl = rl(A°) rlr 2 = r(lo) = r < I. s2(A), r(1) < I). Taking transversals the ~. same a i ~ we can figure define 19~ upwards). the shift functions orientations s(/), of ~ S For ~ near Io the with eigenvalues having i(I) 1 are II(A) .

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A course in mathematical analysis. - part.2 Differential equations by Goursat E.


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