A first cubic upper bound on the local reachability index for some positive 2-D systems

Abstract

The calculation of the smallest number of steps needed to deterministically reach all local states of an $$n\hbox {th}$$-order positive 2-D system, which is called local reachability index ($$I_{LR}$$) of that system, was recently tackled by means of the use of a suitable composition table. The greatest index $$I_{LR}$$ obtained in the previous literature was $$n+3\left( \left\lfloor n/ 2\right\rfloor \right) ^2$$ for some appropriated values of n. Taking as a basis both a combinatorial approach of such systems and the construction of suitable geometric sets in the plane, an upper bound on $$I_{LR}$$ depending on the dimension n for a new family of systems is characterized. The 2-D influence digraph of this family of order $$n\ge 6$$ consists of two subdigraphs corresponding to a unique source s. The first one is a cycle involving the first $$n_1$$ vertices and is connected to the another subdigraph through the 1-arc $$(2, n_1+ n_2)$$, being the natural numbers $$n_1$$ and $$n_2$$ such that $$n_1>n_2\ge 2$$ and $$n-n_1-n_2\ge 1$$. The second one has two main cycles, a cycle where only the remaining vertices $$n_1+1, \ldots , n$$ appear and a cycle containing only the vertices $$n_1+1, \ldots , n_1+n_2-1$$. Moreover, the last vertices are connected through the 2-arc $$(n_1+n_2-1, n)$$. Furthermore, if $$n\ge 12$$ and is a multiple of 3, for appropriate $$n_1$$ and $$n_2$$, the $$I_{LR}$$ of that family is at least cubic, exactly, it must be $$\frac{n^3+9n^2+45n+108}{27}$$, which shows that some local states can be deterministically reached much further than initially proposed in the literature.