In the present note, the chaotic behaviour of a class of infinite system of linear ODEs (with variable coefficients) describing the population of neoplastic cells divided into subpopulations characterized by different levels of resistance to drugs is discussed. The result of [1] is extended to a wider class of sequences defining the parameters of the system. (~) 2003 Elsevier Science Ltd. All rights reserved.
Keywords--Chaos, Infinite system of linear ODEs, Gene amplification.
TOPOLOGICAL CHAOS
In the past two decades, it was observed that some processes in physiology and medicine display chaotic behaviour . However, as Hill [4] noticed, "By now these words ("chaos" and "chaos theory") have appeared in more than 7000 mathematical and scientific books, dictionaries, and papers. Unfortunately the technical meanings are themselves confused, and there is no agreement on definition." The most popular and widely utilized definition of chaos (for discrete in time dynamical systems) is due to Devaney [5]: a map f : X -~ X, where (X, d) is a metric space, is chaotic on X if it has three ingredients--it is topologically transitive, its set of periodic points is dense in X, and it satisfies the condition of sensitive dependence on initial conditions (SDIC).
These properties describe irreducibility (indecomposability), some regularity, and unpredictability, respectively.
The paper demonstrates that under some weak assumptions the last condition (SDIC) is implied by the remaining two. Therefore, despite its sense of intuitiveness, the condition SDIC should be left out of Devaney's definition of chaos. Moreover, the result of Banks et al. shows