By Antonio Ambrosetti (auth.), J. P. Aubin C.E.R.E.M.A.D.E., A. Bensoussan C.E.R.E.M.A.D.E., I. Ekeland C.E.R.E.M.A.D.E. (eds.)
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We wish finally to mention that, by means of continuation arguments, Alexander and ~allet-Paret Yor~[ 13 ] (see also recent work of Yorke and [14 ]) have proved the existence of global branches of periodic solutions of (H) bifurcating from stationary points. A much weaker results, for second order hamiltonian systems, was given earlier by Berger (see [15 ]). In the last section of this exposition, we will consider a situatmn similar to the one in [15 ]. Assuming strong positivity on the Hessian matrix of the potential energy, and suitable growth conditions, we prove the existence of a continuous branch of periodic solutions, which are "increasing in half period".
Clarke and I. Ekeland, one shows a similar dual principle in the finite dimensionnal case, by discretisation. From this result, we infer an algorithm (which has been computed), to find periodic solutions to non-linear differential equations. Finally, we illustrate this method by some numerical results concerning the equation - x 3 + a (cos t + sin t) o x(·) E:R KEY-WORDS Nm1ERICAL ANALYSIS, NON LINEAR DIFFERENTIAL EQUATION, NON CONVEX OPTI:USATION, FORCED OSCILLATION. 45 PERIO~IC SOLUTIONS OF ~fILTONIAN SYSTE~S HAVING PRESCRIBED lHNIMAL PERIOD Giovanni l1ancini Universita di Trieste O.
9» is bounded from below. 3», implies m m o um has minimal period 27T : period T = 27TA • VG(u(~)) is a solution of (H) with minimal We now consider, following [17 I, the superquadratic case, namely S >2 • In this case J A is unbounded from below (as well as from above) but J: 7T G(u) is dominating near u =0 and so J A has a strict local minimum at zero. 4. IO) where d >0 is some pure constant. 10) by comparing J A with suitable simpler functionals. 8». Thus n E:R2N GI (n) <;G(n) <;G 2 (n) TV Denote 21T A f21T = - 2' 0
Advances in Hamiltonian Systems by Antonio Ambrosetti (auth.), J. P. Aubin C.E.R.E.M.A.D.E., A. Bensoussan C.E.R.E.M.A.D.E., I. Ekeland C.E.R.E.M.A.D.E. (eds.)