Conservative limit of two-dimensional spectral submanifolds

The paper considers two-dimensional spectral submanifolds (SSM) of equilibria of finite dimensional vector fields. SSMs are the smoothest invariant manifolds tangent to an invariant linear subspace of an equilibrium. The paper assumes that the vector field becomes conservative at the zero limit of a parameter. It is known that in the conservative limit there exists a unique sub-centre manifold. It is also known that the non-conservative system has a unique SSM under some conditions. However, it is not clear whether the sub-centre manifold is the limit of the SSM and if this limit is smoothly approached. In this paper, we show that the unique SSM continuously approaches the sub-centre manifold as the system tends to the conservative limit.