Invariance for ordinary differential equations: Correction

James A. Yorke
1968 Mathematical Systems Theory  
of the University of Illinois has pointed out that Method II can be blocked in a way not covered in lines 9 and 10 of page 227 (Math. Systems Theory 1 (1967), 225-228): if sgrp A consists entirely of permutations and resets, then T will be the ideal of resets and V the group of units; Method II will then produce a first component that is permutation-reset, and hence no simpler than the original automaton. To salvage the proof we eliminate the resets from this first component by modifying the
more » ... hod as follows: Let st B1 = V instead of T, then whenever u is in T, let p' = p (instead of u), and r' = p-1 (the state to which u resets), instead of p(r). This method (call it IIA) then suffices to decompose a permutationreset automaton into a permutation automaton followed by a reset automaton; since the methods as originally stated bring an arbitrary automaton to a cascade of permutation-reset automata, Method IIA finishes the job.
doi:10.1007/bf01703268 fatcat:2dou3bbcfvbdxeho7x3vqpmwsu