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What is Chaotic Dynamics

Chapter 3
Handbook of Research on Systems Biology Applications in Medicine
Chaotic motions are based on homoclinic (heteroclinic) structures which instability accompanied by local divergence and global contraction. Meanwhile, the transition from stability to instability requires the vanishing of stable equilibrium states and of stable periodic motions or sufficiently large increase in the periodic ones.
Published in Chapter:
Mathematical Description of Time Delays in Pathways Cross Talk
S. Nikolov (Institute of Mechanics and Biomechanics, Bulgaria)
Copyright: © 2009 |Pages: 47
DOI: 10.4018/978-1-60566-076-9.ch003
Abstract
In this chapter we investigate how the inclusion of time delay alters the dynamic properties of (a) delayed protein cross talk model, (b) time delay model of RNA silencing (also known as RNA interference), and (c) time delay in ERK and STAT interaction. The consequences of a time delay on the dynamics of those systems are analysed using Hopf’s theorem and Lyapunov-Andronov theory. Our analytical calculations predict that time delay acts as a key bifurcation parameter. This is confirmed by numerical simulations.
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Chapter 40
Chaos in Economics
Roughly speaking, refers to the dynamic behavior of certain equations F which possess: (a) a non-degenerate n-period point for each n =1, and (b) an uncountable set S ? J (=[0,1]) containing no periodic points and no asymptotically periodic points.
Published in Chapter: Chaos in Economics; From: Encyclopedia of Business Analytics and Optimization
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Chapter 43
Chaotic Neural Networks
Dynamics with specific features indicating complex behaviour, only produced in highly non-linear systems. These indicative features, formalized by the discipline of “Theory of Chaos”, are high sensibility to initial conditions, non-periodic behaviour, and production of a large number of different trajectories in the state space, according to the change of some meaningful parameter of the dynamical system (see bifurcation and diverse dynamics ahead). For some of the tools related to chaotic dynamics, see the related topic in the main text: Non-linear dynamics tools.
Published in Chapter: Chaotic Neural Networks; From: Encyclopedia of Artificial Intelligence
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