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Publications about 'chaos'
Articles in journal or book chapters
and E.D. Sontag.
Chaos in simple periodically-forced biological models.
Note: Submitted. Preprint: biorxiv 10.1101/145201.[PDF]
What complicated dynamics can arise in the simplest biochemical systems, in response to a periodic input? This paper discusses two models that commonly appear as components of larger sensing and signal transduction pathways in systems biology: a simple two-species negative feedback loop, and a prototype nonlinear integral feedback. These systems have globally attracting steady states when unforced, yet, when subject to a periodic excitation, subharmonic responses and strange attractors can arise via period-doubling cascades. These behaviors are similar to those exhibited by classical forced nonlinear oscillators such as those described by van der Pol or Duffing equations. The lack of entrainment to external oscillations, in even the simplest biochemical networks, represents a level of additional complexity in molecular biology.
F. Albertini and E.D. Sontag.
Some connections between chaotic dynamical systems and control systems.
In Proc. European Control Conf. , Vol 1, Grenoble, July 1991,
Keyword(s): chaotic systems,
This paper shows how to extend recent results of Colonius and Kliemann, regarding connections between chaos and controllability, from continuous to discrete time. The extension is nontrivial because the results all rely on basic properties of the accessibility Lie algebra which fail to hold in discrete time. Thus, this paper first develops further results in nonlinear accessibility, and then shows how a theorem can be proved, which while analogous to the one given in the work by Colonius and Klieman, also exhibits some important differences. A counterexample is used to show that the theorem given in continuous time cannot be generalized in a straightforward manner.
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