Read e-book online Computational Methods in Bifurcation Theory and Dissipative PDF

By M. Kubicek, M. Marek

ISBN-10: 3642859577

ISBN-13: 9783642859571

ISBN-10: 3642859593

ISBN-13: 9783642859595

"Dissipative constructions" is an idea which has lately been utilized in physics to debate the formation of constructions prepared in house and/or time on the cost of the power flowing into the approach from the surface. The space-time structural association of organic platforms ranging from the subcellular point as much as the extent of ecological structures, coherent buildings in laser and of elastic balance in mechanics, instability in hydro­ plasma physics, difficulties dynamics resulting in the improvement of turbulence, habit of electric networks and chemical reactors shape only a brief record of difficulties taken care of during this framework. Mathematical versions developed to explain those structures tend to be nonlinear, frequently shaped by way of advanced platforms of algebraic, traditional range­ ential, or partial differential equations and contain a couple of personality­ istic parameters. In difficulties of theoretical curiosity in addition to engineering perform, we're interested by the dependence of ideas on parameters and especially with the values of parameters the place qualitatively new kinds of recommendations, e.g., oscillatory suggestions, new desk bound states, and chaotic attractors, look (bifurcate). Numerical options to figure out either bifurcation issues and the depen­ dence of steady-state and oscillatory strategies on parameters are built and mentioned intimately during this textual content. The textual content is meant to function a operating guide not just for college students and learn staff who're drawn to dissipative constructions, but additionally for working towards engineers who take care of the issues of making types and fixing advanced nonlinear systems.

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Extra resources for Computational Methods in Bifurcation Theory and Dissipative Structures

Example text

3). By changing the value of a, the number of solutions will change at a limit point by two, but in most situations the number of solutions does not change when passing through a bifurcation point. A large number of numerical procedures can be used for finding the dependence x(a). The simplest is the sequential one. The interval of a values is divided by the nodes ao, a 1 , ••• , ap into sections. , by the Newton method). If the points {a j} are sufficiently close, one or two iterations usually suffice to obtain the solution for each aj • That is how we can obtain solutions on one branch; we can not cross a limit point, and when passing through a bifurcation point, the behavior is more or less random.

If all Floquet multipliers are inside the unit circle, the studied periodic solution is locally orbitally stable. As in the case of steady-state solutions, we can construct and study the dependence of periodic solutions on the parameter ex and plot branches of periodic solutions in the solution diagram. The amplitude or period of the solution can be chosen, for example, as the characteristic to be plotted. When a simple periodic solution (limit cycle) ceases to be attracting, one of the possibilities is that an attracting two-dimensional torus_ bifurcates.

Anaerobic digestion is a natural process of decomposition of organic waste. However, it is also being used increasingly for producing methane from agricultural, industrial, and community organic waste materials. Multiple steady states and oscillations which are predicted by the model of Example 5 have been observed in experiments. Now we shall present several examples concerned with distributed parameter systems. EXAMPLE 6: Reaction-diffusion system. 2. 95J). 32 1 Introduction A sequence of reactions taking place under open-system conditions considered in the following form: IS kl A~X k-l k2 B+XC;Y+D kJ 2X + Y k~3X k.

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Computational Methods in Bifurcation Theory and Dissipative Structures by M. Kubicek, M. Marek

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