Instability of a classical dynamical system with the negative curvature region

The classical dynamical system can have the negative curvature region where orbital instability takes place. But it is not yet clear whether this system has instability in the large. To see this a dynamical system with the hamiltonian (equation) is studied by the method of the surface of section. The system does not have negative curvature for ε ^z.ast;≤0 but can have negative curvature locally for ε ^z.ast;≥0. For the case of ε ^z.ast;≤0 isolating integrals exist globally, but for the case of ε ^z.ast;≥0 the instability zone extends to larger region beyond that of negative curvature.