Unstable limit cycle

The limit cycle O is an isolated closed trajectory having the property that dll other trajectories O in its neighborhood are certain spirals winding themselves onto C either for t - oo in which case C is called a stable limit cycle) or for t- - — ao (an unstable limit cycle). 

Recall that a Hopf bifurcation is termed supercritical if its bifurcation diagram is as shown schematically in Fig. 6.2.2a. Correspondingly, in this case a stable limit cycle is born around the equilibrium, unstable hereon, only at a critical (bifurcation) value of the control parameter A = Ac. In contrast, in the subcritical case (Fig. 6.2.2b), the equilibrium is surrounded by limit cycles already for A < Ac, with an unstable limit cycle separating the stable one from the still stable equilibrium. At the bifurcation A = Ac the unstable limit cycle dies out with the equilibrium, unstable hereon, surrounded by a stable limit cycle. Thus the main feature of the subcritical case (as opposed to the supercritical one) is that a stable equilibrium and a stable limit cycle coexist in a certain parameter range, with a possibility to reach the limit cycle through a sufficiently strong perturbation of the equilibrium. 

There are no unstable limit cycles in this model, and the oscillatory solution born at one bifurcation point exists over the whole range of stationary-state instability, disappearing again at the other Hopf bifurcation. Both bifurcations have the same character (stable limit cycle emerging from zero amplitude), although they are mirror images, and are called supercritical Hopf bifurcations. 

Fig. 4.9. The development of oscillatory amplitude Ae and period T across the range of instability, 4.2 x 10 3 = n < n < jx = 0.0195, for the pool chemical model with k = 2x 10-3 and y = 0.21, typical of a system with a subcritical Hopf bifurcation at which an unstable limit cycle emerges at The broken curves give the limiting forms predicted by eqns (4.59)—(4.61).

As p increases further, the unstable limit cycle grows within the stable cycle. Eventually the two coalesce (at psu in Fig. 5.4(b)), whereupon limit cycle behaviour disappears. 

FlO. 5.4. The birth and growth of oscillatory solutions for the thermokinetic model with the full Arrhenius temperature dependence, (a) The Hopf bifurcations /x and ft are both supercritical, with [12 < 0, and the stable limit cycle born at one dies at the other, (b) The upper Hopf bifurcation is subcritical, with fl2 > 0. An unstable limit cycle emerges and grows as the dimensionless reactant concentration ft increases—at /rsu this merges with the stable limit cycle born at the lower supercritical Hopf bifurcation point ft. ... 

Fig. 5.5. A typical phase portrait for a system with ft < ft < ft, showing a stable stationary-state solution (singular point) surrounded first by an unstable limit cycle (broken curve) and then by a stable limit cycle (solid curve). The unstable limit cycle separates those initial conditions, corresponding to points in the parameter plane lying within the ulc, which are attracted to the stationary state from those outside the ulc, which are attracted on to the stable limit cycle and hence which lead to oscillations.

We now know that if a system on the upper branch of the isola, just below the Hopf bifurcation point, is given a small perturbation which remains within the unstable limit cycle, it will decay back to the upper solution. If, however, the perturbation is larger, so we move to a point outside the cycle, we will not be able to get back to the upper solution the system must move to the other stable state, with no reactant consumption. 

Fig. 8.6. Typical arrangement of local stabilities and development of unstable limit cycle, from a subcritical Hopf bifurcation, appropriate to cubic autocatalysis with decay and no autocatalyst inflow and with 9/256 < k2 < 1/16. The unstable limit cycle grows as t, decreases below t m, and terminates by means of the formation of a homoclinic orbit at rf . Stable stationary states, including the zero conversion branch 1 — a, = 0, are indicated by solid curves, unstable states and limit cycles by broken curves.

Fig. 8.7. Supercritical Hopf bifurcation for cubic autocatalysis with decay and /) = 0, appropriate for small dimensionless decay rate constant k2 < 9/256. A stable limit cycle emerges and grows as the residence time is increased above t s. At higher residence times, this disappears at rj , by merging with an unstable limit cycle born from a homoclinic orbit at t. (With non-zero autocatalyst inflow, (i0 > 0, the stable limit cycle itself may form a homoclinic orbit at long tres.)...

Even for the present simple case, for which the inflow does not contain the autocatalyst, we have seen a variety of combinations of stable and unstable stationary states with or without stable and unstable limit cycles. Stable limit cycles offer the possibility of sustained oscillatory behaviour (and because we are in an open system, these can be sustained indefinitely). A useful way of cataloguing the different possible combinations is to represent the different possible qualitative forms for the phase plane . The phase plane for this model is a two-dimensional surface of a plotted against j8. As these concentrations vary in time, they also vary with respect to each other. The projection of this motion onto the a-/ plane then draws out a trajectory . Stationary states are represented as points, to which or from which the trajectories tend. If the system has only one stationary state for a given combination of k2 and Tres, there is only one such stationary point. (For the present model the only unique state is the no conversion solution this would have the coordinates a,s = 1, Pss = 0.) If the values of k2 and tres are such that the system is lying at some point along an isola, there will be three stationary states on the phase... 

Fig. 8.8. Phase plane representations of the birth (or death) of limit cycles through homoclinic orbit formation. In the sequence (a)-fb)-(c) the system has two stable stationary states (solid circles) and a saddle point. As some parameter is varied, the separatrices of the saddle join together to form a closed loop or homoclinic orbit (b) this loop develops as the parameter is varied further to shed an unstable limit cycle surrounding one of the stationary states. The sequence (d)-(e)-(f) shows the corresponding formation of a stable limit cycle which surrounds an unstable stationary state. (In each sequence, the limit cycle may ultimately shrink on to the stationary state it surrounds—at a Hopf bifurcation point.)...

These requirements specify two loci one of them, labelled DH l in Fig. 8.12, emanates from the points / = 0, k2 = 9/256, as located in 8.3.6. This curve cuts through the parameter space for isola and mushroom patterns, but always lies below the curve A. (In fact it intersects A at the common point P0 = i(33/2 - 5), k2 = rg(3 - /3)4(1 -, /3)2 where the locus H also crosses.) In the vicinity of DH x, the stationary-state curve has only one Hopf point. This changes from a subcritical bifurcation (unstable limit cycle emerging) for conditions to the right of the curve to supercritical (stable limit cycle emerging) to the left. 

FlG. 8.14. The different phase plane portraits identified for cubic autocatalysis with decay (a) unique stable state (b) unique unstable stationary state with stable limit cycle (c) unique stable state with unstable and stable limit cycles (d) two stable stationary states and saddle point (e) stable and unstable states with saddle point (f) stable state, saddle point, and unstable state surrounded by stable limit cycle (g) two unstable states and a saddle point, all surrounded by stable limit cylcle (h) two stable states, one surrounded by an unstable limit cycle, and a saddle point (i) stable state surrounded by unstable limit cycle, unstable state, and saddle point, all surrounded by stable limit cycle (j) stable state, unstable state, and saddle point, all surrounded by stable limit cycle (k) stable state, saddle point, and unstable state, the latter surrounded by concentric stable and unstable limit cycles (1) two stable states, one surrounded by concentric unstable and stable limit cycles, and a saddle point. 

FIGU RE 10 Illustration of the disappearance of a limit cycle via a turning point on a periodic branch near a subcritical Hopf bifurcation, (a) A stable limit cycle surrounding an unstable focus (b) the unstable focus undergoes a subcritical Hopf bifurcation and leaves an inner unstable limit cycle surrounding a stable focus (c) the two limit cycles combine into a metastable configuration and disappear altogether as the parameter is further increased. 

This is called Hopf bifurcation. Figure 10 (A-2) shows two Hopf bifurcation points with a branch of stable limit cycles connecting them. Figure 13 (A-2) shows a schematic diagram of the phase plane for this case when g = g. In this case a stable limit cycle surrounds an unstable focus and the behavior of the typical trajectories are as shown. Figure 11 (A-2) shows two Hopf bifurcation points in addition to a periodic limit point (PLP) and a branch of unstable limit cycles in addition to the stable limit cycles branch. 

In Figure 15 there is an unstable limit cycle surrounding a stable focus and the unstable limit cycle is surrounded by a stable limit cycle. 

Example 2. There exist r 3 but no rj2 slow relaxations. In the above example let us replace the boundary loop by an unstable limit cycle... 

Fig. 5.10. The four possible types of Hopf bifurcation (a) a stable steady-state (sss) becomes unstable (uss) as a parameter fi is increased through the bifurcation point (/x ) and a stable limit cycle (sic) emerges - the growth of the limit cycle is indicated by plotting the maximum and minimum of the variable as it undergoes the oscillatory motion around the limit cycle (b) the scenario is reversed, with the steady-state losing stability and a stable limit cycle emerging as the parameter is reduced (a) and (b) are termed supercritical Hopf bifurcations. In (c) and (d) there is an unstable limit cycle emerging to surround the stable part of the steady-state branch this is characteristic of a subcritical Hopf bifurcation.

Stable, limit cycle. The latter occurs in the Salnikov case and the modified bifurcation diagram is shown in Fig. 5.11(b). The stable limit cycle born at the lower Hopf point overshoots the upper Hopf point but is extinguished by colliding with the unstable limit cycle born at the upper Hopf point which also grows in amplitude as )jl is increased. Over a, typically narrow range, then there are two limit cycles, one unstable and one stable around the (stable) steady-state point. If we start with the system at some large value of /r, so we settle onto the steady-state locus, and then decrease the parameter, we will first swap to oscillations at the Hopf point /r - At this point there is a stable limit cycle available as the system departs from the now unstable steady-state, but this stable limit cycle is not born at this point and so already has a relatively large amplitude. We would expect to... 

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