Bifurcation results

The quasiperiodic route to chaos is historically important. It arises from a succession of Hopf birfurcations. As already noted, a single Hopf bifurcation results in a limit cycle. The next Hopf bifurcation produces a phase flow tliat can be represented on tire surface of a toms (douglmut). This flow is associated witli two frequencies if tire ratio of tliese frequencies is irrational tlien tire toms surface is densely covered by tire phase trajectory, whereas if... 

Stability, Bifurcations, Limit Cycles Some aspects of this subject involve the solution of nonlinear equations other aspects involve the integration of ordinaiy differential equations apphcations include chaos and fractals as well as unusual operation of some chemical engineering eqmpment. Ref. 176 gives an excellent introduction to the subject and the details needed to apply the methods. Ref. 66 gives more details of the algorithms. A concise survey with some chemical engineering examples is given in Ref. 91. Bifurcation results are closely connected with stabihty of the steady states, which is essentially a transient phenomenon. 

Buckling of Cooling-Tower Shells Bifurcation Results Cole, Peter P. Abel, John F. Billington, David P. 

Simulation and Bifurcation Results Discussion for two Industrial FCC Units... 

Fig. 21. Theoretical dynamic phase diagram for forced oscillations on Pt(l 10), evaluated from Eqs. (4), (5a), and (6). The calculations have been performed up to larger forcing amplitudes than were experimentally accessible (compare with Fig. 17). For small A, the bands characterizing entrainment and quasiperiodicity are discernible, whereas for larger A more complex bifurcations result. Types of bifurcations ns, Neimark-Sacker pd, period doubling snp, saddle-node of periodic orbits. (From Ref. 69.)...

In the next section, we apply sequential bifurcation to the other two supply chain configurations (the Current and Next generations) and we interpret these sequential bifurcation results through our knowledge of the simulated real system. 

We label the factors such that all factors have the same meaning in the three simulation models. To achieve this, we introduce dummy factors for the Current and the Next Generation models that represent those factors that are removed as the supply chain is changed. Such dummy factors have zero effects but simplify the calculations and interpretations of the sequential bifurcation results. 

With the surface of section technique, it can be observed that a limit cycle can undergo, for example, a Hopf bifurcation. When the stability analysis of the Poincare section is carried out, the real part of a complex conjugate pair of eigenvalues is seen to pass from negative to positive a small perturbation added to the limit cycle will evolve away from the cycle in an oscillatory fashion. This type of bifurcation results, then, in the appearance of a second... 

Figure 26 Generation of a torus attractor via two Hopf bifurcations. The first Hopf bifurcation converts a stable fixed point (a focus) into an unstable focus. A stable limit cycle generally originates at this bifurcation point. A second Hopf bifurcation occurs, rendering the limit cycle unstable, and giving rise to a stable torus. Each Hopf bifurcation results in one additional frequency of oscillation in the system.

Throughout this chapter I have taken the point of view that the meandering of spiral waves in excitable media can and should be examined from the perspective of bifurcation theory. With this approach, it has been possible to show that the organizing center for spiral dynamics is a particular codimension-two bifurcation resulting from the interaction of a Hopf bifurcation from rotating waves with symmetries of the plane. From this observation has followed a simple ordinary-differential-equation model for spiral meandering. 

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