A generalized index theory for non-Hamiltonian system (Q6165620)

Inspired by the construction of the ``conjugate index'' provided by \textit{M. Musso} et al. [Topol. Methods Nonlinear Anal. 25, No. 1, 69--99 (2005; Zbl 1101.58012)] for the study of the distribution of conjugate points along a semi-Riemannian geodesic and later by \textit{A. Portaluri} and \textit{L. Wu} [J. Differ. Equations 269, No. 9, 7253--7286 (2020; Zbl 1444.58007)] in the more general self-adjoint case, in this paper the authors introduce a new topological invariant named ``degree-index'' defined in terms of the Brouwer degree of an appropriate determinant map of a boundary matrix, which provides a possible substitute for the Maslov index in the non-Hamiltonian framework. Then they prove the equality between the Morse index and the degree index in this non-self-adjoint framework thanks to a new abstract trace formula. Since the theory of the Morse index in the classical framework ensures equality between the spectral properties of the second variation of the Lagrangian functional and the oscillation properties of the solution space of the associated limit value problem, the theory of the ``degree-index'' makes it possible to transpose a similar theory result in the non-Hamiltonian context, which can be useful for studying the dynamic properties of dissipative systems. Finally, they apply their theoretical results to certain 1D reaction-diffusion systems.