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An index theory for periodic solutions of convex Hamiltonian systems. (English) Zbl 0596.34023

Nonlinear functional analysis and its applications, Proc. Summer Res. Inst., Berkeley/Calif. 1983, Proc. Symp. Pure Math. 45/1, 395-423 (1986).
[For the entire collection see Zbl 0583.00018.]
This paper deals with the problem of finding, counting and describing periodic solutions of the 2n-dimensional system of ODEs: \(\dot x=JH'(x)\) where \(J=-J^*=J^{-1}\) defines a symplectic structure on \({\mathbb{R}}^{2n}\), and the Hamiltonian \(H: {\mathbb{R}}^{2n}\to {\mathbb{R}}\) is convex and \(C^ 3\). We study the fixed-energy problem, that is, we prescribe the energy level h, and we seek solutions x such that \(H(x(t))=h\) and \(x(t+T)=x(t)\) for all t. The hypersurface defined by the equation \(H(x)=h\) will henceforth be denoted by \(\Sigma\), and it will be assumed that it surrounds the origin. We associate with each periodic solution on \(\Sigma\) an index, which is a non-negative integer, and we relate this index with qualitative properties of the solution namely its length and stability. Here is a sample of the results we obtain: (a) set \(r=Inf\{| x| | \quad x\in \Sigma \}\quad and\quad R=Sup\{| x| | \quad x\in \Sigma \}.\) Then all periodic solutions x satisfy \(\int (Jx,\dot x)dt\geq \pi r^ 2\) (this is due to Croke and Weinstein) and there is at least one periodic solution with \(\int (Jx,\dot x)dt\leq \pi R^ 2;\) (b) assume now that \(\Sigma\) is (r,R)-pinched (this is a condition which is somewhat stronger than the preceding one; it means that \(\Sigma\) lies between two concentric balls with radii r and R, plus an analogous condition on the radii of curvature). If \(R/r<\sqrt{2}\), there is at least one periodic solution which is linearly stable. In (a) as in (b), the solution considered has index zero.

MSC:

34C25 Periodic solutions to ordinary differential equations
70H05 Hamilton’s equations

Citations:

Zbl 0583.00018