Brake Orbits of a Reversible Even Hamiltonian System Near an Equilibrium

In this paper, we consider the brake orbits of a reversible even Hamiltonian system near an equilibrium. Let the Hamiltonian system (HS) (x) over dot =JH'(x) satisfies H(0)=0, H'(0)=0, reversible and even conditions H(N-x)=H(x) and H(-x)=H(x) for all x is an element of R-2n. Suppose the quadratic form Q(x)=1/2 < H ''(0)x, x > is non-degenerate. Fix tau(0)>0 and assume that R-2n=E circle plus F decomposes into linear subspaces E and F which are invariant under the flow associated to the linear system (x) over dot =JH ''(0)x and such that each solution of the above linear system in E is tau 0-periodic whereas no solution in F \ {0} is tau(0)-periodic. Write sigma(tau(0))=sigma(Q)(tau(0)) for the signature of Q vertical bar(E). If sigma(tau(0))not equal 0, we prove that either there exists a sequence of brake orbits x(k)-> 0 with tau(k)-periodic on the hypersurface H-1(0) where tau(k)-> tau(0); or for each lambda close to 0 with lambda sigma(tau(0))>0 the hypersurface H-1(lambda) contains at least 1/2 vertical bar sigma(tau(0))vertical bar distinct brake orbits of the Hamiltonian system (HS) near 0 with periods near tau(0). Such result for periodic solutions was proved by Bartsch in 1997.