Bifurcations and chaos

Non-linear (fynamics turned out to be extremely sensitive to coupling with some external parameters (representing the neighbourhood ). 

If K were small (0 1, extremely poor grass), the rabbit population would 

This mathematical phenomenon was carefully investigated and the results were really amazing. Further increase in K introduces further qualitative changes. First, for 3 3.44948 the oscillations have period two (b iircation), then at 

44948 3.5441 the oscillations have period four (next bifurcation, the four- 

the next surprise exceeding K = 3.56994 we obtain populations that do not exhibit any regularity (no limit cycle, just chaos). A further surprise is that this is not the end of the surprises. Some sections of K began to rahibit odd-peiiod behaviour, separated by some sections of chaotic behaviour. 

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WIGGINS, S. Global Bifurcations and Chaos. Springer-Verlag, New York, Berlin, Heidelberg, 1988... 

MSN. 151.1. Prigogine, Why irreversibility The formulation of classical and quantum mechanics for non-integrable systems, Int. J. Bifurcation and Chaos 5, 3-16 (1995). 

The prototype potential surface invoked in chemical kinetics is a two-dimensional surface with a saddle equilibrium point and two exit channels at lower energies. The classical and quantal dynamics of such surfaces has been the object of many studies since the pioneering works by Wigner and Polanyi. Recent advances in nonlinear dynamical systems theory have provided powerful tools, such as the concepts of bifurcations and chaos, to investigate the classical dynamics from a new point of view and to perform the semiclassical... 

This appendix gives an introduction to multiplicity of steady states and to bifurcation and chaos in Chemical and Biological Engineering. 

Another important point is what may be called feedback of information , such as countercurrent operation, recycle or continuous circulation (e.g., the novel autothermic circulating fluidized presented by Elnashaie and co-workers). This feedback of information can give rise to bifurcation and chaos as well [46-50]. 

These simple and mostly intuitive arguments apply mainly to chemical and biological reaction engineering systems. For other systems such as fluid flow systems, the sources and causes of bifurcation and chaos can be quite different. It is well established that the transition from laminar flow to turbulent flow is a transition from nonchaotic to chaotic behavior. The synergetic interaction between hydrodynamically induced bifurcation and chaos and that resulting from chemical and biological reaction/diffusion is not well studied and calls for an extensive multidisciplinary research effort. 

Research Areas Modeling, Simulation and Optimization of Chemical and Biological Processes, Clean Fuels (Hydrogen, Biodiesel and Ethanol), Fixed and Fluidized Bed Catalytic Reactors, Nonlinear Dynamics, Bifurcation and Chaos,... 

Said had always felt more mathematically inclined in his work on modeling problems and bifurcation and chaos in chemical/biological engineering ... 

V. Belykh, I. Belykh, and E. Mosekilde Hyperbolic Plykin attractor can exist in neuron models. Int. J. Bifurcation and Chaos 2005,15 3567-3578. 

R. Seydel (Ed.) Proc. of Bifurcation and Chaos Analysis, Algorithms, Applications . International Series of Numerical Mathematics, Vol. 97, pp. 231-236. Basel Birkhauser Verlag. 

Bifurcation and chaos can be expressed infinite difference equations of the form... 

S. Wiggins, Global Bifurcation and Chaos—Analytical Methods, Springer, New York, 1988. 

Lindenberg K, Romero AH, Sancho JM. (1998) Nonclassical kinetics in constrained geometries Initial distribution effects. International Journal of Bifurcation and Chaos 8 853-868. 

Onken, H. U., Schiith, F., and Wicke, E., Bifurcation and Chaos Analysis, Algorithms and Applications. Wiirzbiug, 1990. 

A. J. Szeri and L. G. Leal, The onset of chaotic oscillations and rapid growth of a gas bubble at subcritical conditions in an incompressible liquid, Phys. Fluids A3, 551-5 (1991) Z. C. Feng and L. G. Leal, Bifurcation and chaos in shape and volume oscillations of a periodically driven bubble with two-to-one internal resonance, J. Fluid Mech. 266, 209-42 (1994) L. G. Leal and Z. C. Feng, Nonlinear bubble dynamics, Annu. Rev. Fluid Mech. 29, 201-42 (1997). 

E.M. Izhikevich, Neural excitability, spiking and bursting. International Journal of Bifurcation and Chaos 10 (2000) 1171-1267. 

Geest, T., C.G. Steinmetz, R. Carter L.F. Olsen. 1992. Period-doubling bifurcations and chaos in an enzyme reaction. J. Phys. Chem. 96 5678-80. 

Glass, L., M.R. Guevara, A. Shrier R. Perez. 1983. Bifurcation and chaos in a periodically stimulated cardiac oscillator. Physica 7D 89-101. 

Yokoi, H., Adamatzky, A., De Lacy Costello, B., MeUiuish, C. Excitable Chemical Medium Controller for a Robotic Hand Closed-Loop Experiments. International Journal of Bifurcation and Chaos 14, 3347-3354 (2004)... 

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