Period doubling scenario

It is universal for a large class of period doubling scenarios. Physical examples of this route to chaos include the driven pendulum (Baker and Gollub (1990)) and ion traps (Blumel (1995b)). 

In order to gain some experience with the new concepts introduced above, we will now discuss the stability properties of the fixed points and periodic orbits of the logistic mapping. The following is also a more in-depth presentation of the period doubling scenario briefly discussed in Section 1.2. 

Furthermore, the map is unimodal, like the logistic map. This suggests that the chaotic state shown in Figure 12.4.1 may be reached by a period-doubling scenario. Indeed such period-doublings were found experimentally (Coffman et al. 1987), as shown in Figure 12.4.4. 

Let us imagine a scenario for which a supercritical Hopf bifurcation occurs as one of the parameters, fi say, is increased. For fi < fi, the stationary state is locally stable. At fi there is a Hopf bifurcation the stationary state loses stability and a stable limit cycle emerges. The limit cycle grows as ft increases above fi. It is quite possible for there to be further bifurcations in the system if we continue to vary fi. With three variables we might expect to have period-doubling sequences or transitions to quasi-periodicity such as those seen with the forced oscillator of the previous section. Such bifurcations, however, will not be signified by any change in the local stability of the stationary state. These are bifurcations from the oscillatory solution, and so we must test the local stability of the limit cycle. We now consider how to do this. 

We also now know that complex oscillations evolve as simple limit cycles become unstable, bifurcating to more complex limit cycles. Only a small number of bifurcation sequences account for all known scenarios. We have seen examples of mixed-mode sequences (H2 -I- O2) and period-doubling cascades (CO -I- O2). A third route involving quasi-periodic responses is known and arises in some chemical system [88], but has not yet been observed in combustion systems (except in some special studies in which the ambient temperature or some other parameter is forced to vary in some sinusoidal or other periodic manner [89]). The important lesson then... 

The appearance of aperiodic oscillations beyond a point of accumulation of a cascade of period-doubling bifurcations is one of the best-known scenarios for the emergence of chaos (Feigenbaum, 1978 Berge et al, 1984). It is also along this way that chaos arrives in the multiply regulated enzyme model. Another example of this type of irregular... 

The Willamowski-RSssler rate equations yield both periodic and chaotic attractors as the system parameters are tuned [10,21]. One common scenario for the appearance of a chaotic attractor is a cascade of ppriod-doubling bifurcations. As can be seen in the bifurcation diagram in Figure 2, the Willamowski-Rossler reaction possesses such a bifurcation sequence the chaotic attractor is followed by a reverse cascade of period doublings leading, ultimately, to a steady state. 

In the 1/1 entrainment region each side of the resonance horn terminates at points C and D respectively. These points are codimension-two bifurcations and correspond to double +1 multipliers. As the saddle-node curve at the right horn boundary rises from zero amplitude towards point D, one multiplier remains at unity (the criterion for a saddle-node bifurcation) as the other free-multiplier of the saddle-node increases until it is also equal to unity upon arrival at point D. The same thing occurs for the left boundary of the resonance horn. The arc CD is also a saddle-node bifurcation curve but is different from those on the sides of the resonance horn. As arc CD is crossed from below, the period 1 saddle combines not with its companion stable node, but with the unstable node that was in the centre of the phase locked torus. As the pair collides, the invariant circle is lost and only the stable node remains. Exactly the same scenario is observed for the 1/2 resonance horn as well. 

In relation to add input from 1980 to 1990, the buffering capacity of 1.8 Mio km or 15% of the acid-sensitive forest soils tends to become saturated in the next 25-100 years. Under the assumptions of the IS92a scenario, this share more than doubles and increases to 4.0 Mio km or 34% between 2040 and 2050. For 1980-1990, the mean buffering capacity of these sols based on our methodology is supposed to last for 65 years more. Under changed inputs this period tends to decrease for 2040-2050 to... 

With the final shutdown of Unit 2 of the Ignalina NPP at the end of 2009 and without constructing a new nuclear power plant, demand for primary energy resources would increase only by approximately 25% during the period until 2025 according to the basic scenario, however total demand for fossil fuel would increase almost 1.7 times within 20 years, that is from 6 million toe in 2005 to 10.5 million toe in 2025. Natural gas demand would double - from 2.4 million toe to... 

---