Qualitative stability of certain nonlinear systems

Abstract

To certain nonlinear dynamical systems naturally correspond simplicial complexes. This correspondence is a generalization of the familiar relationship between the interaction matrix of a linear dynamical system and the signed digraph of that matrix. By defining stability in terms of attractor regions (as opposed to attractor trajectories), we can specify qualitative linear algebraic conditions involving the simplicial complex which insure stability of the nonlinear system. The analysis uses only signs of coefficients in the dynamical system. A model of E. Lorenz [3] is an example of a three-dimensional system which fulfills the stability conditions and which is known to have a strange attractor within the attractor region. In summary, the linear qualitative tests described (Theorems 2 and 3) can be applied to certain nonlinear dynamical systems to yield preliminary information about the global stability of such systems.