Tipping cycles

Instability in Jacobians is determined by the presence of an eigenvalue lying in the right half plane. The coefficients of the characteristic polynomial contain information related to the specific matrix elements that play a greater destabilising role. Yet the destabilising circuits, or cycles, constructed by multiplying these elements together, form only a subset of all the cycles comprising a given system. This paper looks at the destabilising cycles in three sign-restricted forms in terms of sets of the matrix elements to explore how sign structure affects how the elements contribute to instability. This leads to quite rich combinatorial structure among the destabilising cycle sets as set size grows within the coefficients of the characteristic polynomial.