On bifurcations and chaos
in time delay feedback systems
A. Evangelisti, R. Genesio, A. Tesi
Dipartimento di Sistemi e Informatica, Università di Firenze
Via di Santa Marta 3 - 50139 Firenze - Italy
A number of models concerning biological processes is given in terms of nonlinear delay-differential equations. Many examples exist in ecology for describing certain population dynamics [1] and other models of the same class appear in the representation of different biological oscillators [2]. A well-known example of these is the Mackey-Glass equation for the control of white blood-cell production [2].
Since the study of such delay systems, having an infinite-dimensional state, is a difficult task by time domain approaches, and also their numerical simulation is very delicate, this paper presents a frequency method to investigate the related dynamics. The method is based on the harmomic balance technique and it follows from a general input-output approach to feedback nonlinear systems for predicting complex dynamics in terms of structural (not numerical) conditions [3]. These are based on an approximate analytical detection of local and global bifurcations, and many successful applications to finite dimensional systems have been given [4].
In order to present and illustrate clearly the method, the paper studies in detail a modified Mackey-Glass equation, where the usual nonlinearity is slightly changed preserving the qualitative system behaviours. The analysis results are easily derived and compared with numerical simulations showing the effectiveness of the method. In particular, since the Mackey-Glass dynamics has been extensively examined by researchers as a significant benchmark to compute measures of complexity and chaos, the above results can be useful also in this context.