  Morse index theory provides an elegant and useful tool for describing several
aspects of a Lagrangian system in terms of its variational properties. In the
classical framework it provides an equality between the spectral properties of
a second order selfadjoint differential operator arising by the second
variation of the Lagrangian action functional and the oscillation properties of
the space of solutions of the associated boundary value problem. These
oscillation properties are described by means of the Maslov index, which is a
symplectic invariant intimately related to the Hamiltonian nature of the
problem.
  Driven by the desire to get an analogous result in the non-Hamiltonian
context for the investigation of the dynamical properties of dissipative
systems, in this paper we start to introduce a new topological invariant, the
so-called degree-index, defined in terms of the Brouwer degree of a suitable
determinant map of a boundary matrix, which provides one possible substitute of
the Maslov index in this non-Hamiltonian framework and finally we prove the
equality between the Morse index and the degree-index in this non-selfadjoint
setting through a new abstract trace formula.
  In the last part of the paper we apply our theoretical results to some 1D
reaction-diffusion systems.
