Non-Existence of Limit Cycles We turn our attention now to the negative side of the problem of show ing limit cycles exist. Here are two theorems which can sometimes be used to show that a limit cycle does not exist. 1. Bendixson’s Criterion If f x and gy are continuous in a region R which is simply-connected (i.e., without holes), and ∂f ∂g + �= 0 at any point of R, ∂x ∂y then the system x � = f ( x, y) y� = g( x, y) (1) has no closed trajectories inside R. Proof. Assume there is a closed trajectory C inside R. We shall derive a contradiction, by applying Green’s theorem, in its normal (or ﬂux) form. This theorem says � C ( f i + gj) · n ds ≡ � C f dy − g dx = �� ( D ∂f ∂g + ) dx dy . (2) ∂x ∂y where D is the region inside the simple closed curve C. This however is a contradiction. Namely, by hypothesis, the integrand on the right-hand side is continuous and never 0 in R; thus it is either al ways positive or always negative, and the right-hand side of (2) is therefore either positive or negative. On the other hand, the left-hand side must be zero. For since C is a closed trajectory, C is always tangent to the velocity ﬁeld f i + gj deﬁned by the system. This means the normal vector n to C is always perpendicular to the velocity ﬁeld f i + gj, so that the integrand ( f i + gj) · n on the left is identically zero. This contradiction means that our assumption that R contained a closed trajectory of (1) was false, and Bendixson’s Criterion is proved. � Critical-point Criterion terior. A closed trajectory has a critical point in its in 18.03SC If we turn this statement around, we see that it is really a criterion for non-existence: it says that if a region R is simply-connected (i.e., without holes) and has no critical points, then it cannot contain any limit cycles. For if it did, the Critical-point Criterion says there would be a critical point inside the limit cycle, and this point would also lie in R since R has no holes. (Note carefully the distinction between this theorem, which says that limit cycles enclose regions which do contain critical points, and the PoincareBendixson theorem, which seems to imply that limit cycles tend to lie in re gions which don’t contain critical points. The difference is that these latter regions always contain a hole; the critical points are in the hole. Example 1 illustrated this. � x � = ax + by Example 2. For what a and d does have closed trajec y� = cx + dy tories? Solution. By Bendixson’s criterion, ries. a + d �= 0 ⇒ no closed trajecto What if a + d = 0? Bendixson’s criterion says nothing. We go back to our analysis of the linear system. The characteristic equation of the system is λ2 − ( a + d)λ + ( ad − bc) = 0 . Assume a + d = 0. Then the characteristic roots have opposite sign if ad − bc < 0 and the system is a saddle; the roots are pure imaginary if ad − bc > 0 and the system is a center, which has closed trajectories. Thus the system has closed trajectories ⇔ a + d = 0, ad − bc > 0.

Non-Existence of Limit Cycles

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