Dynamics Reported: A Series in Dynamical Systems and Their by Peter W. Bates, Christopher K. R. T. Jones (auth.), Urs

By Peter W. Bates, Christopher K. R. T. Jones (auth.), Urs Kirchgraber, Hans Otto Walther (eds.)

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Due to the circle symmetry there is a neighbourhood of the origin filled with ({Jrinvariant circles. It is well known that in general the orbit structure of ({J is much more complicated. For a tentative picture see Arnol'd and Avez [2, p. 91]. We may assume that arbitrarily near the fixed point there exist ({J2-invariant circles filled with periodic points (otherwise we perturb again). So ({J2 has nonisolated periodic points and is not generic in the sense of part (b) above. With the next small perturbation we can destroy such a circle of periodic points so as to obtain, say, exactly 37 hyperbolic periodic orbits and 37 elliptic periodic orbits (with the invariant manifolds having transverse intersection).

The flow of the modified system is used to get a contraction. Therefore the graph of this Lipschitz function is pushed onto W+ by this flow. If t(Uo) stays in U then it actually satisfies the unmodified problem and t(u o) is pushed onto w eu = W+ n U. If uo¢K and uo¢ W", apply a moving cone argument to get t(uo)eK for some t > 0 and reapply the above. If UoE W S then the property is trivial. 24 Semilinear Partial Differential Equations-P. W. Bates and C. K. R. T. Jones (P3) Let V c U and K the cone associated with the splitting xes Et> xu.

We assume moreover that these eigenvalues are not roots of unity. By a linear change of coordinates we can put dqJ(O) on Jordan normal form, so we may 46 Formally Symmetric Normal Forms-H. W. Broer and F. Takens assume that dqJ(O) = ( c~s oc -smoc sin oc ) cosoc with oc/2n irrational. In this case dqJ(O) is semi-simple. The group G(dqJ(O)), as introduced in section 1, consists of all rotations around the origin. Hence, after suitably modifying the coordinates, there is a locally integrable approximation qJ(i) which commutes near the origin with all rotations (with respect to these new coordinates).

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