On Invariant Surfaces And Bifurcation of Periodic Solutions of Ordinary Differen

Robert John Sacker

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■ . We write this as w = a(M)w + e(n)|w|^w + Y«v(s, w, w, m) "♦■ Gif (3. 12) ' h7 - ■where v = a(s, u)v} + b(s, n)\j, a and b vectors. Con- sider the vector differential equation Y + [S'(^) •*■ {Ck-l)a(n) + -ta(^)}I]Y = hCsj^) where prime denotes transpose, I is the identity matrix and h is periodic in s. For n sufficiently small this equation has a unique periodic solution yCs, ^) and the transformation w = C •*■ Y • YCs, ^)C^C'^ (3. 13) carries (3. 12) Into C = aCu)C + pCn)|C|'C + Y • Cv(s, c..., C, m) - hCs, u)c^c^] + Gi, applying this for k = 1, t = 0> h = a(s, ia) and k = 0, -t, = 1, h = t)(s, ij) we obtain the form After carrying out the above transformations on the second equation of (3. 5') we write that equation as Y = S(ia)Y + v(s, C, C»^) + H3 (3. 1^) where v = riCs, n)c^ + ^'zQQ + ^'^V' ^o^ M sufficiently small the vector differential equation Y + CikaCn) + ^, a(^)3I - S(h)]y = h(s, |a) with h periodic in s, has a unique periodic solution yCsj^j. ) and the transformation ^-.

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