Double limit cycles

Furthermore, Wegmann and Rossler experimenting with this reaction, observed oscillations corresponding to a) limit cycle, b) double limit cycle, and c) endogenous chaos and d) screw type chaos, see Fig. III.ll, a-d, respectively. 

Fig. III.ll. Experimental observations of a Limit cycle, b Double limit cycle, c Endogenous chaos, d Screw type chaos. Abscissa electrochemical potential, ordinate potential of bromide ion. (After Wegmann and Rossler (1978))...

D) if there is a semi-stable (double) limit cycle, the system may not have simultaneously an unstable separatrix of a saddle which tends to the cycle as t -> -hoo and a stable separatrix of a saddle which tends to the cycle as t —00, as shown in Fig. 8.1.4 and... 

To conclude this section, let us elaborate further on the restrictions (D) and (E). In case (D) the surface corresponding to the double cycle is of codimension-one, and therefore, it divides a neighborhood of the non-rough system Xq into two regions and D. Assume that in the double limit cycle is decomposed into two limit cycles, and that it disappears in D. The situation in -D is simple — all systems there are structurally stable and, moreover, of the same type. As for D the situation is less trivial if (D) is violated, then it is obvious that besides structurally stable systems in there are structurally unstable ones whose non-roughness is due to the existence of a heteroclinic trajectory between two saddles, as shown in Fig. 8.1.6(a). Moreover, this picture takes place in any neighborhood of Xq- In other words, in the region, there exists a countable number of the associated bifurcation surfaces of codimension-one which accumulate to In such cases the surface is said to be unattainable from one side. 

Since A < 0, the Poincare map is decreasing. The new feature in this case is that such maps may have orbits of period two, which correspond to the so-called double limit cycles. They may appear via a period-doubling bifurcation (a fixed point with a multiplier equal to —1) or via a bifurcation of a double homoclinic loop. The latter corresponds to the period-two point of the Poincare map at 2/ = 0 (see Fig. 13.3.2). 

In the region Z i, there are no limit cycles. On the curve L4, upon moving from D towards D2j a stable limit cycle is born from a simple separatrix loop. An imstable double-loop limit cycle bifurcates from a double separatrix loop with (To > 0 on L2. Thus, in the region D3, there are two limit cycles one stable and the other is unstable. The unstable double limit cycle merges with the stable limit cycle on the curve Li. After that only one single-circuit unstable limit cycle remains in region D4. It adheres into the homoclinic loop on the curve L3. 

Period-n Limit Cycles As a increases, the system undergoes an infinite sequence of successive period-doublings via pitchfork bifurcations. In general,... 

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. 

The new period-2 limit cycle is born with a CFM equal to + 1. This decreases rapidly as we reduce rN further. Eventually, the CFM approaches — 1 again a second period doubling occurs at tn = 0.137 307 1, where a period-2 with zp = 3.408 gives way to a period-4 solution with rp = 6.816. The initial splitting in the phase space trajectory again occurs in the vicinity of the maximum temperature rise and the double circuit gives way to four loops as shown in the sequence in Fig. 13.22. 

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-peri-odic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations. 

The basins of attraction of the coexisting CA (strange attractor) and SC are shown in the Fig. 14 for the Poincare crosssection oyf = O.67t(mod27t) in the absence of noise [169]. The value of the maximal Lyapunov exponent for the CA is 0.0449. The presence of the control function effectively doubles the dimension of the phase space (compare (35) and (37)) and changes its geometry. In the extended phase space the attractor is connected to the basin of attraction of the stable limit cycle via an unstable invariant manifold. It is precisely the complexity of the structure of the phase space of the auxiliary Hamiltonian system (37) near the nonhyperbolic attractor that makes it difficult to solve the energy-optimal control problem. 

Next we display 13 single, double, or multiple plots drawn by our MATLAB program neurocycle.m of (a) the acetylcholine concentration profile in compartment (II) above the phase plot of the acetylcholine concentration in compartment (I) versus that in compartment (II), or (b) the limit cycle plot, or (c) the plot of all 8 profiles. We include interpretative comments on the solution s behavior in each case. 

Our next-higher value of hf = 0.004552 exhibits a double-loop limit cycle, or a period-two periodic attractor in the phase plot of Figure 4.55. 

Larger-amplitude single period with a one double-loop limit cycle... 

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 expression (70) means that we have discovered, within the dynamical system OO, the dynamical subsystem O O ( = TM ) which is in a limit cycle. It means the thermodynamic equilibrium within the double cycle O O, thus for its temperatures it is valid that T w = T W,... 

C.G.Steinmetz, T.Geest and R.Larter, Universality in the Peroxidase-Oxidase Reaction Period Doubling, Chaos, Period Three, and Unstable Limit Cycles, The Journal of Physical Chemistry, 97, 5649-5653(1993). 

At c = 2.5 the attractor is a simple limit cycle. As c is increased to 3.5, the limit cycle goes around twice before closing, and its period is approximately twice that of the original cycle. This is what period-doubling looks like in a continuous-time system In fact, somewhere between c = 2.5 and 3.5, a period-doubling bifurcation of cycles must have occurred. (As Figure 10,6.6 suggests, such a bifurcation... 

Period-doubling in the Lorenz equations) Solve the Lorenz equations numerically for cr = 10, b-j, and r = 148.5. You should find a stable limit cycle. Then repeat the experiment for r = 147.5 to see a period-doubled version of this cycle. (Whenplotting your results, discard the initial transient, and use the xy projections of the attractors.)... 

Coexisting chaos and limit cycle) Consider the double-well oscillator (12.5,1) with parameters S = 0.15, F =0.3, and (O — 1. Show numerically that the system has at least two coexisting attractors a large limit cycle and a smaller strange attractor. Plot both in a Poincare section,... 

This textbook is aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. The presentation stresses analytical methods, concrete examples, and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors. 

Mathematical models of the reaction yield various solutions. Some of the solutions obtained are One singular point, 3 singular points, oscillating limit cycle, double periodic oscillations, chaotic oscillations. 

One of the significant qualitative characteristics of two dimensional solutions could be related to the concept of convexity of the area bordered by the closed solution, the limit cycle. One can easily visualize the x(r) and y(t ) representation in which both v and y exhibit smooth oscillations for convex, and oscillations with multiple periods for concave limit cycles, Fig. IV.3. Clearly one example of the latter oscillations may well be double oscillations discussed in Section III. 

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