Infinite-period bifurcation

Fig. 26.1a). At first, multistage ignitions and extinctions occur followed by a relaxation (long period) mode [7]. Oscillations die a few degrees below the ignition temperature at a saddle-loop infinite-period homoclinic orbit bifurcation point. This is an example where both ignition and extinction are oscillatory. 

FIGURE I Steady-state bifurcation diagrams for variations in the reactant partial pressures, (a) Partial two-parameter bifurcation diagram representing the projection of turning points, Hopf bifurcation points, and apparent triple points. (b)-(Q One parameter sections of the steady-state ffe surface. The vertical axes are the steady-state (k and range from 0 to I. The horizontal axes correspond to the appropriate axis of the two parameter diagram (a). Steady-states are stable or unstable for solid or dashed curves respectively and periodic branches are denoted by pairs of chained curves which represent the minimum and maximum values of ffe on the limit cycle. The periodic branches all terminate in Hopf bifurcations or, when a saddle is present, homoclinic (infinite period) bifurcations. (b)-(e) a, = 0.017, 0.019, 0.021, 0.025 (f)-(i) oti = 0.031, 0.028, 0.024, 0.022. 

Bifurcation diagram for equations (7.198) and (7.199) with one Hopf bifurcation point, two periodic limit points and one homoclinical orbit (infinite period bifurcation point)... 

Figure 12 (A-2) shows one Hopf bifurcation point, one periodic limit point and the stable limit cycle terminates at a homoclinical orbit (infinite period bifurcation).

This limit cycle represents a trajectory that starts at the static saddle point and ends after one period at the same saddle point. This trajectory is called the homoclinical orbit and will occur at some critical value jiuc- It has an infinite period and therefore this bifurcation point is called infinite period bifurcation . For p < hc the limit cycle disappears. This is the second most important type of dynamic bifurcation after Hopf bifurcation. 

Limit cycles (periodic solutions) emerging from the Hopf bifurcation point and terminating at another Hopf bifurcation point or at a homoclinical orbit (infinite period bifurcation point) represent the highest degree of complexity in almost all two- dimensional autonomous systems. 

Recently Tambe et al. (284) extended this model and included two different types of adsorption sites for A and B, while permitting the conversion of sites from one type to the other. The authors used the same coverage dependency and the same parameters as Pikios and Luss (283). Introducing the possibility of adsorption on different sites generated a qualitatively new dynamic behavior for the system characterized by a finite amplitude/ infinite period bifurcation that yielded a homoclinic orbit. This new feature was observed when the equilibration between the two types of sites was slow compared to the other reactions. However, if equilibration is fast and the equilibrium constant is assumed to be one, this model is equivalent to the one discussed by Pikos and Luss (283). 

As p decreases, the limit cycle r = 1 develops a bottleneck at 8 = njl that becomes increasingly severe as > 1. The oscillation period lengthens and finally becomes infinite at p = 1, when a fixed point appears on the circle hence the term infinite-period bifurcation. For p < 1, the fixed point splits into a saddle and a node. 

This section deals with a physical problem in which both homoclinic and infinite-period bifurcations arise. The problem was introduced back in Sections 4.4 and 4.6. At that time we were studying the dynamics of a damped pendulum driven by a constant torque, or equivalently, its high-tech analog, a superconducting Josephson junction driven by a constant current. Because we weren t ready for two-dimensional systems, we reduced both problems to vector fields on the circle by looking at the heavily overdamped limit of negligible mass (for the pendulum) or negligible capacitance (for the Josephson junction). 

Putting it all together, we arrive at the stability diagram shown in Figure 8.5.10. Three types of bifurcations occur homoclinic and infinite-period bifurcations of periodic orbits, and a saddle-node bifurcation of fixed points. 

Consider the driven pendulum + atj) + sintl) = I. By numerical computation of the phase portrait, verify that if a is fixed and sufficiently small, the system s stable limit cycle is destroyed in a homoclinic bifurcation as / decreases. Show that if a is toolarge, the bifurcation is an infinite-period bifurcation instead. 

Note that from the figure alone, the saddle-loop bifurcation cannot be distinguished from another infinite period bifurcation, a saddle node with infinite period. ... 

Fig. 11.1 5 The experimentally determined bifnrcation structure of the chlorite-iodide reaction. The two constraints used here are the ratio of input concentrations, [CIO2 ]o/[I ]o> and the logarithm of the reciprocal residence time, log k(,. Notation filled triangles, supercritical Hopf bifurcations filled circles, saddle-node infinite period bifurcations in the region between these two kind of bifurcations, stable oscillations were observed. Open circles, excitable dynamics the smallest circles correspond to perturbations of 2 x 10 M in NaC102 the next smallest, 6 X 10 M in NaC102 the next smallest, 1.25 x 10 X x oMn - o io-2 1...

The oscillation appears or disappears with a finite amphtude but with an infinite period. If a limit cycle disappears via this bifurcation, then an excitable steady state can appear simultaneously... 

Noszticzius, Z. Stirling, P. Wittmann, M. 1985. Measurement of Bromine Removal Rate in the Oscillatory BZ Reaction of Oxalic Acid. Transition from Limit Cycle Oscillations to Excitability via Saddle-Node Infinite Period Bifurcation, J. Phys. Chem. 89, 4914-4921. 

Figure 7.9 Bifurcation diagram for Eqs. (7.15FT7.17). (A) A case with two Hopf bifurcation points (B) a case with two Hopf bifurcation points and one periodic limit point (C) a case with one Hopf bifurcation point, two periodic limit points, and one homoclinical orbit (infinite period bifurcation point). Solid curve stable branch of the bifurcation diagram dashed curve saddle points filled circles stable limit cycles open circles unstable limit cycles HB = Hopf bifurcation point PLP = periodic limit point.

The case of Figure 7.9C has one Hopf bifurcation point and one periodic limit point, and the stable Kmit cycle terminates at a homoclinical orbit (infinite period bifurcation). For /x = /X3, we get a case of an unstable steady state surrounded by a stable limit cycle similar to the case in Figure 7.10. However, in this case, as fj. decreases below /X3, the limit cycle grows until we reach a limit cycle that passes through the static saddle point, as... 

As a second application we study the effect of mixing on oscillations or, more precisely, on infinite period bifurcations. For this purpose the following reaction scheme will be investigated ... 

There a different kind of differential-equation model of excitability which might turn out to have significantly different properties [103-109]. Such media have three fixed points instead of just one and make the transition from excitability to spontaneous oscillation by a saddle-node bifurcation through infinite period ( SNIPER ), e.g., as revealed by the period and amplitude of the Belousov-Zhabotinsky reagent s bulk oscillations [24, Figure 1 110-114], In such media spontaneous oscillation and excitability are indeed mutually exclusive alternatives as commonly supposed, rather than independent, typically coexisting properties of the medium (as in the most profusely... 

Theorem 12.8. For all p> 0, the system on the Klein bottle has exactly two periodic orbits with negative multipliers. If fo (p) does not vanish identically, each of these two periodic orbits undergoes a period-doubling bifurcation infinitely many times as +0. 

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