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Mathematical Medicine and Biology 1993 10(4):227-247; doi:10.1093/imammb/10.4.227
© 1993 by Institute of Mathematics and its Applications
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Transient periodicity and episodic predictability in biological dynamics

W. M. SCHAFFER, B. E. KENDALL, C. W. TIDD and L. F. OLSEN

Department of Ecology & Evolutionary Biology, The University of Arizona Tucson, Arizona 85721, USA
Department of Biochemistry, Odense University Odense, Denmark

Biological time series often contain passing episodes of nearly periodic dynamics. In chaotic systems, such transient periodicity can reflect the existence of semiperiodic saddles—nonstable invariant sets—contained in the attractor. Motion in the vicinity of such objects has a prominent periodic component. In addition, trajectories can become temporarily trapped in these neighbourhoods before exiting. The immediate dynamical antecedents (low-order preimages) of transient periodicity are well defined and, along with the saddles to which they map, correspond to regions of enhanced predictability under nonlinear forecasting. This suggests that it may be possible to forecast the onset of transient periodicity in systems for which overall predictability is low. The present paper reviews these concepts and applies them to biological phenomena at different levels of organization.

Keywords: chaos; epidemiology; epilepsy; nonlinear forecasting; SEIR model; semiattractors; transient periodicity


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