Unstable systems stabilizing each other through adaptation - part II

It is now well realized [1] that two unstable dynamical systems, attempting to stabilize each other using the error between their outputs for adjusting their parameters, may not always succeed. The fact that adaptation may result in instability makes the mathematical problem a very interesting one. Since similar problems are arising in many other branches of science (e.g psychology, biology, medicine etc), obtaining precisely the conditions under which stability is achieved is also assuming greater importance. This paper may be considered as a first attempt to discuss the many aspects of this intriguing and difficult problem. Work carried out during the past year has shown that the interaction of two n
^th
order systems, results in a 4n
^th
order differential equation, whose stability has to be investigated. Even in the simple case when n = 2, determining necessary and sufficient conditions for stability is a formidable undertaking. However, through extensive simulation studies and the theoretical analysis of special cases suggested by them, numerous insights have been obtained. The objective of this paper is to convey these insights, and discuss wherever possible, their implications to higher order systems.