# On a Maximal Number of Period Annuli

Authors: Yelena Kozmina; Felix Sadyrbaev
Journal: Abstract and Applied Analysis
 (2011)

---

## Abstract

We consider equationx′′+g(x)=0, where g(x) is a polynomial,
allowing the equation to have multiple period annuli. We detect the
maximal number of possible period annuli for polynomials of odd degree and
show how the respective optimal polynomials can be constructed.


---

## Body

## 1. Introduction

Consider equation(1.1)x′′+g(x)=0,
where g(x) is an odd degree polynomial with simple zeros.The equivalent differential system(1.2)x′=y,y′=-g(x)
has critical points at (pi,0), where pi are zeros of g(x). Recall that a critical point O of (1.2) is a center if it has a punctured neighborhood covered with nontrivial cycles.We will use the following definitions.Definition 1.1 (see [1]).
Acentral region is the largest connected region covered with cycles surrounding O.Definition 1.2 (see [1]).
Aperiod annulus is every connected region covered with nontrivial concentric cycles.Definition 1.3.
We will call a period annulus associated with a central regiona trivial period annulus. Periodic trajectories of a trivial period annulus encircle exactly one critical point of the type center.Definition 1.4.
Respectively, a period annulus enclosing several (more than one) critical points will be called anontrivial period annulus.For example, there are four central regions and three nontrivial period annuli in the phase portrait depicted in Figure2.Period annuli are the continua of periodic solutions. They can be used for constructing examples of nonlinear equations which have a prescribed number of solutions to the Dirichlet problem(1.3)x′′+g(x)=0,x(0)=0,x(1)=0,
or a given number of positive solutions [2] to the same problem.Under certain conditions, period annuli of (1.1) give rise to limit cycles in a dissipative equation(1.4)x′′+f(x)x′+g(x)=0.The Liénard equation with a quadratical term(1.5)x′′+f(x)x′2+g(x)=0
can be reduced to the form (1.1) by Sabatini's transformation [3](1.6)u:=Φ(x)=∫0xeF(s)ds,
where F(x)=∫0xf(s)ds. Since du/dx>0, this is one-to-one correspondence and the inverse function x=x(u) is well defined.Lemma 1.5 (see [3, Lemma 1]).
The functionx(t) is a solution of (1.5) if and only if  u(t)=Φ(x(t)) is a solution to(1.7)u′′+g(x(u))eF(x(u))=0.Our task in this article is to define the maximal number of nontrivial period annuli for (1.1).  (A)
We suppose thatg(x) is an odd degree polynomial with simple zeros and with a negative coefficient at the principal term (so g(-∞)=+∞ and g(+∞)=-∞). A zero z is called simple if g(z)=0 and g′(z)≠0.The graph of a primitive functionG(x)=∫0xg(s)ds is an even degree polynomial with possible multiple local maxima.The functiong(x)=-x(x2-p2)(x2-q2) is a sample.We discuss nontrivial period annuli in Section2. In Section 3, a maximal number of    regular pairs is detected. Section 4 is devoted to construction of polynomials g(x) which provide the maximal number ofregular pairs or, equivalently, nontrivial period annuli in (1.1).


## 2. Nontrivial Period Annuli

The result below provides the criterium for the existence of nontrivial period annuli.Theorem 2.1 (see [4]).
Suppose thatg(x) in (1.1) is a polynomial with simple zeros. Assume that M1 and M2 (M1<M2) are nonneighboring points of maximum of the primitive function G(x). Suppose that any other local maximum of G(x) in the interval (M1,M2) is (strictly) less than min{G(M1);G(M2)}.
Then, there exists a nontrivial period annulus associated with a pair(M1,M2).It is evident that ifG(x) has m pairs of non-neighboring points of maxima then m nontrivial period annuli exist.Consider, for example, (1.1), where
(2.1)g(x)=-x(x+3)(x+2.2)(x+1.9)(x+0.8)(x-0.3)(x-1.5)(x-2.3)(x-2.9).
The equivalent system has alternating “saddles” and “centers”, and the graph of G(x) is depicted in Figure 1.Figure 1Figure 2
The phase portrait for (1.1), where G(x) is as in Figure 1.There are three pairs of non-neighboring points of maxima and three nontrivial period annuli exist, which are depicted in Figure2.


## 3. Polynomials

Consider a polynomialG(x). Points of local maxima xi and xj of G(x) are non-neighboring if the interval (xi,xj) contains at least one point of local maximum of G(x).Definition 3.1.
Two non-neighboring points of maximaxi<xj of G(x) will be called a  regular pair  if G(x)<min{G(xi),G(xj)} at any other point of maximum lying in the interval (xi,xj).Theorem 3.2.
Supposeg(x) is a polynomial which satisfies the condition  A. Let G(x) be a primitive function for g(x) and n a number of local maxima of G(x).
Then, the maximal possible number of  regular pairs isn-2.Proof.
By induction, letx1,x2,…,xn be successive points of maxima of G(x), x1<x2<⋯<xn.
(1)
Letn=3. The following combinations are possible at three points of maxima:(a)
G(x1)≥G(x2)≥G(x3),(b)
G(x2)<G(x1),G(x2)<G(x3),(c)
G(x1)≤G(x2)≤G(x3),(d)
G(x2)≥G(x1),G(x2)≥G(x3).
Only the case (b) providesa regular pair. In this case, therefore, the maximal number of regular pairs is 1.
(2) Suppose that for any sequence ofn>3 ordered points of maxima of G(x) the maximal number of regular pairs is n-2. Without loss of generality, add to the right one more point of maximum of the function G(x). We get a sequence of n+1 consecutive points of maximum x1,x2,…,xn, xn+1, x1<x2<…<xn<xn+1. Let us prove that the maximal number of regular pairs is n-1. For this, consider the following possible variants.
(a)
The couplex1,xn is a regular pair. If G(x1)>G(xn) and G(xn+1)>G(xn), then, beside the regular pairs in the interval [x1,xn], only one new regular pair can appear, namely, x1,xn+1. Then, the maximal number of regular pairs which can be composed of the points x1,x2,…,xn,xn+1, is not greater than (n-2)+1=n-1. If G(x1)≤G(xn) or G(xn+1)≤G(xn), then the additional regular pair does not appear. In a particular case G(x2)<G(x3)<⋯<G(xn)<G(xn+1) and G(x1)>G(xn) the following regular pairs exist, namely, x1 and x3, x1 and x4,…, x1 and xn, and the new pair x1 and xn+1 appears, totally n-1 pairs.(b)
Suppose thatx1,xn is not a regular pair. Let xi and xj be a regular pair, 1≤i<j≤n, and there is no other regular pairxp,xq such that 1≤p≤i<j≤q≤n. Let us mention that if such a pair xi,xj does not exist, then the function G(x) does not have regular pairs at all and the sequence {G(xk)}, k=1,⋯,n, is monotone. Then, if G(xn+1) is greater than any other maximum, there are exactly (n+1)-2=n-1regular pairs.
Otherwise, we have two possibilities:
eitherG(xi)≥G(xp), p=1,…,i-1,orG(xj)≥G(xq), q=j+1,…,n.
In the first case, the interval[x1,xi] contains i points of maximum of G(x), i<n, and hence the number of regular pairs in this interval does not exceed i-2. There are no regular pairsxp,xk for 1≤p<i,  i<k≤n+1. The interval [xi,xn+1] contains (n+1)-(i-1) points of maximum of G(x), and hence the number of regular pairs in this interval does not exceed (n+1)-(i-1)-2=n-i. Totally, there are no more regular pairs than (i-2)+(n-i)=n-2.
In the second case, the number ofregular pairs in [xi,xj] does not exceed j-(i-1)-2=j-i-1. In [xj,xn+1],there are no more than (n+1)-(j-1)-2=n-jregular pairs. The points xp,  p=1,…,i-1, xq, j<q≤n do not form regular pairs, by the choice of xp and xq. The points xp, p=1,…,i, together with xn+1 (it serves as the i+1th point in a collection of points) form not more than (i+1)-2=i-1regular pairs. Totally, the number of regular pairs is not greater than (j-i-1)+(n-j)+(i-1)=n-2.


## 4. Existence of Polynomials with Optimal Distribution

Theorem 4.1.
Given numbern, a polynomial g(x) can be constructed such that(a)
the condition (A) is satisfied,(b)
the primitive functionG(x) has exactly n points of maximum and the number of  regular pairs is exactly n-2.Proof.
Consider the polynomial(4.1)G(x)=-(x+12)(x-12)(x+32)(x-32)(x+52)(x-52)(x+72)(x-72).
It is an even function with the graph depicted in Figure 3.
Consider now the polynomial(4.2)Gε(x)=-(x+12+ε)(x-12)(x+32)(x-32)(x+52)(x-52)(x+72)(x-72),
where ε>0 is small enough. The graph of Gε(x) with ε=0.2 is depicted in Figure 4.
Denote the maximal values ofG(x) and Gɛ(x) to the right of x=0  m1+,m2+. Denote the maximal values of G(x) and Gɛ(x) to the left of x=0m1-,m2-. One has for G(x) that m1+=m1-<m2-=m2+. One has for Gɛ(x) that m1+<m1-<m2+<m2-. Then, there are two regular pairs (resp., m1- and m2+, m2+ and m2-).
For arbitrary evenn the polynomial
(4.3)Gɛ(x)=-(x+12)(x-12)(x+32)(x-32)⋯(x+2n-12)(x-2n-12),
is to be considered where the maximal values m1+,m2+,…,mn/2+ to the right of x=0 form ascending sequence, and, respectively, the maximal values m1-,m2-,…,mn/2- to the left of x=0 also form ascending sequence. One has that mi+=mi- for all i. For a slightly modified polynomial
(4.4)Gɛ(x)=-(x+12+ɛ)(x-12)(x+32)(x-32)…(x+2n-12)(x-2n-12),
the maximal values are arranged as
(4.5)m1+<m1-<m2+<m2-<⋯<mn/2+<mn/2-.
Therefore, there exist exactly n-2regular pairs and, consequently, n-2 nontrivial period annuli in the differential equation (1.1).
Ifn is odd, then the polynomial
(4.6)G(x)=-x2(x-1)(x+1)(x-2)(x+2)⋯(x-(n-1))(x+(n-1))
with n local maxima is to be considered. The maxima are descending for x<0 and ascending if x>0. The polynomial with three local maxima is depicted in Figure 5.
The slightly modified polynomial(4.7)G(x)=-x2(x-1-ɛ)(x+1)(x-2)(x+2)⋯(x-(n-1))(x+(n-1))
has maxima which are not equal and are arranged in an optimal way in order to produce the maximal (n-2) regular pairs.
The graph ofGɛ(x) with ɛ=0.2 is depicted in Figure 6.Figure 3
G(x) (solid) and G′(x)=g(x) (dashed).Figure 4
G(x) (solid line), Gɛ(x) (dashed line), and G(x)-Gɛ(x) (dotted line).Figure 5
G(x) (solid) and G′(x)=g(x) (dashed).Figure 6
G(x) (solid), Gɛ(x) (dashed), and G(x)-Gɛ(x) (dotted).



---

*Source: 393875-2011-01-27.xml*