Poincare-Bendixson theorem

The proof of this theorem is subtle, and requires some advanced ideas from topol- 

The Poincare-Bendixson theorem can be difficult to apply in practice. One convenient case occurs when the system has a simple representation in polar coordinates, as in the following example. 

When /z=0, there s a stable limit cycle at r=l, as discussed in Example 7.1.1. Show that a closed orbit still exists for p 0, as long as p is sufficiently small. 

Solution We seek two concentric circles with radii and such that r 0 on the outer circle and r 0 on the inner circle. Then the annulus 0 r  

When polar coordinates are inconvenient, we may still be able to find an appropriate trapping region by examining the system s nullclines, as in the next example. 

Another useful rule which can frequently guide us to situations where oscillatory solutions will be found is the Poincare-Bendixson theorem. This states that if we have a unique stationary state which is unstable, or multiple stationary states all of which are unstable, but we also know that the concentrations etc. cannot run away to infinity or become negative, then there must be some other non-stationary atractor to which the solutions will tend. Basically this theorem says that the concentrations cannot just wander around for an infinite time in the finite region to which they are restricted they must end up somewhere. For two-variable systems, the only other type of attractor is a stable limit cycle. In the present case, therefore, we can say that the system must approach a stable limit cycle and its corresponding stable oscillatory solution for any value of fi for which the stationary state is unstable. 

This quick test does not, however, tell us that there will be only one stable limit cycle, or give any information about how the oscillatory solutions are born and grow, nor whether there can be oscillations under conditions where the stationary state is stable. We must also be careful in applying this theorem. If we consider the simplified version of our model, with no uncatalysed step, then we know that there is a unique unstable stationary state for all reactant concentrations such that /i 1. However, if we integrate the mass-balance equations with /i = 0.9, say, we do not find limit cycle behaviour. Instead the concentration of B tends to zero and that for A become infinitely large (growing linearly with time). In fact for all values of fi less than 0.90032, the concentration of A becomes unbounded and so the Poincare-Bendixson theorem does not apply. 

Conserving mass explicit incorporation of reactant consumption 

Our approach of regarding a and / as functions of n has served us well. It has allowed us to identify stationary-state solutions and how they vary with the reactant concentration and the uncatalysed reaction rate. By examining the local stability of these solutions we have also been able to obtain simple 

We may now allow for reactant consumption explicitly by restoring the exponential decay, p = p0e- °. In dimensionless terms this means recognizing that p is a time-dependent parameter, p = /i0e . The governing rate equations are 

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Because we have only a single stationary state, we can use the Poincare-Bendixson theorem to recognize that sustained oscillatory responses will be found at least over the whole range of K-p parameter space corresponding to instability. (Although we must also check that the concentration a and... 

While the Poincare-Bendixson theorem yields the existence of limit cycles, it is often important to know when limit cycles do not exist. For two-dimensional systems, a result in this direction which complements the Poincare-Bendixson theorem is called the Dulac criterion. Its proof is a direct application of the classical Green s theorem in the plane (after an assumption that the theorem is false) and will not be given here a good reference is [ALGM]. 

Since 1 is a local attractor, to prove the theorem it remains only to show that it is a global attractor. This is taken care of by the Poincare-Bendixson theorem. As noted previously, stability conditions preclude a trajectory with positive initial conditions from having 0 or 2 in its omega limit set. The system is dissipative and the omega limit set is not empty. Thus, by the Poincare-Bendixson theorem, the omega limit set of any such trajectory must be an interior periodic orbit or a rest point. However, if there were a periodic orbit then it would have to have a rest point in its interior, and there are no such rest points. Hence every orbit with positive initial conditions must tend to j. (Actually, two-dimensional competitive systems cannot have periodic orbits.) Figure 5.1 shows the X1-X2 plane. 

In the previous chapter it was shown that the simple chemostat produces competitive exclusion. It could be argued that the result was due to the two-dimensional nature of the limiting problem (and the applicability of the Poincare-Bendixson theorem) or that this was a result of the particular type of dynamics produced by the Michaelis-Menten hypothesis on the functional response. This last point was the focus of some controversy at one time, inducing the proposal of alternative responses. In this chapter it will be shown that neither additional populations nor the replacement of the Michaelis-Menten hypothesis by a monotone (or even nonmonotone) uptake function is sufficient to produce coexistence of the competitors in a chemostat. This illustrates the robustness of the results of Chapter 1. It will also be shown that the introduction of differing death rates (replacing the parameter D by D, in the equations) does not change the competitive exclusion result. 

Everything in the statement has been established except the claim of global asymptotic stability of E2 in the first case. Since the system is two-dimensional, the Poincare-Bendixson theorem provides a proof of the global claim. 

If the inequality is reversed then the rest point E. is unstable - a repeller. The Poincar -Bendixson theorem then allows one to conclude that there exists a limit cycle. Unfortunately, there may (theoretically) be several limit cycles. If all limit cycles are hyperbolic then there is at least one asymptotically stable one, for if there are multiple limit cycles the innermost one must be asymptotically stable. Moreover, since all trajectories eventually lie in a compact set, there are only a finite number of limit cycles and the outermost one must be asymptotically stable. Since the system is (real) analytic, one could also appeal to results for such systems. For example, Erie, Mayer, and Plesser [EMP] and Zhu and Smith [ZSJ show that if E is unstable then there exists at least one limit cycle that is asymptotically stable. Stability of limit cycles will be discussed in the next section. We make a brief digression to outline the principal parts of this theory, and then return to the food-chain problem. 

Proof of Theorem 5.3. Condition (3.4) makes E locally asymptotically stable. By the Poincare-Bendixson theorem, it is necessary only to show that with condition (3.4) there are no limit cycles. Suppose there were a limit cycle. However, there is at most a finite number of limit cycles and each must contain < in its interior. Hence there is a periodic trajectory V that contains no other periodic trajectory in its interior. Intuitively speaking, r is the trajectory closest to the rest point. The constant term in the formula given in Lemma 5.1 is negative. The corollary shows that P is asymptotically stable. This is a contradiction, since the rest point is asymptotically stable - that is, between the two there must be an unstable periodic orbit. ... 

When the inequality in (3.4) is reversed, there will be a periodic orbit (by an application of the Poincare-Bendixson theorem). By our assumption of hyperbolicity, this orbit must be asymptotically orbitally stable since it is so from the inside. These comments establish the next result. 

Proof Solutions of (2.7) are bounded in L. Indeed, Q <0 for all large Q independent of x and x < 0 when Q is near P - that is, when x is large. The result is now a standard application of the Poincare-Bendixson theorem using the Dulac criterion (discussed in Chapter 1) to eliminate nontrivial periodic orbits and cycles of steady states in L. Indeed, since... 

Theorem C.7 bears a strong resemblance to the Poincare-Bendixson theorem stated in Chapter 1. It will be used in Chapter 4 for the case where (C.l) is a competitive system, that is, for a system (C.l) where -/ is cooperative. Note that the omega (alpha) limit set of a competitive system is the alpha (omega) limit set of the time-reversed cooperative system, so Theorems C.5, C.6, and C.7 apply to competitive systems. Unlike cooperative systems, competitive systems can have attracting periodic orbits. For more on the Poincare-Bendixson theory of competitive and cooperative systems in see [S3], [SWl], and [ZS]. 

For vector fields on the plane, the Poincare-Bendixson theorem states that if a trajectory is confined to a closed, bounded region and there are no fixed points in the region, then the trajectory must... 

The repeller drives all neighboring trajectories into the shaded region, and since this region is free of fixed points, the Poincare-Bendixson theorem applies. 

The Poincare-Bendixson theorem is one of the central results of nonlinear dy-... 

Cycle graphs) Suppose x = f(x) is a smooth vector field on. An improved version of the Poincare-Bendixson theorem states that if a trajectory is trapped in a compact region, then it must approach a fixed point, a limit cycle, or something exotic called a cycle graph (an invariant set containing a finite number of fixed points connected by a finite number of trajectories, all oriented either... 

We begin the analysis of (4), (5) by constructing a trapping region and applying the Poincare-Bendixson theorem. Then we ll show that the chemical oscillations arise from a supercritical Hopf bifurcation. 

We can t apply the Poincare-Bendixson theorem yet, because there s a fixed point... 

When (7) holds, the Poincare-Bendixson theorem implies the existence of a closed orbit somewhere in the punctured box. ... 

The complete set of equations with three variables, Eqs. (7.123)-(7.125), including the buffer step, may have a unique unstable solution. In view of the Poincare-Bendixson theorem, this is a necessary and sufficient condition for the occurrence of oscillations. For this set of equations, the solution is considered to be in the so-called reaction simplex S ... 

One very important mathematical result facilitates the analysis of two-dimensional (i.e., two concentration variables) systems. The Poincare Bendixson theorem (Andronov et al., 1966 Strogatz, 1994) states that if a two-dimensional system is confined to a finite region of concentration space (e.g., because of stoichiometry and mass conservation), then it must ultimately reach a steady state or oscillate periodically. The system cannot wander through the concentration space indefinitely the only possible asymptotic solution, other than a steady state, is oscillations. This result is extremely powerful, but it holds only for two-dimensional systems. Thus, if we can show that a two-dimensional system has no stable steady states and that all concentrations are bounded—that is, the system cannot explode—then we have proved that the system has a stable periodic solution, whether or not we can find that solution explicitly. 

By the Poincare-Bendixson theorem, the system will have a periodic limit cycle solution when eq. (14.34) holds. This inequality defines a surface in the [CIO2]-[I2HMA] plane that separates the regions of stable oscillatory and steady states. The experimental range of oscillation is well described by this equation, except when the initial concentration of CIO2 is so low that it is nearly consumed in one oscillatory period, violating the assumption that (CIO2) is a constant. 

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