On a Maximal Number of Period Annuli

Abstract

We consider equation 𝑥^+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.

1. Introduction

Consider equation𝑥^+𝑔(𝑥)=0,(1.1) where 𝑔(𝑥) is an odd degree polynomial with simple zeros.

The equivalent differential system𝑥′=𝑦,𝑦′=−𝑔(𝑥)(1.2) has critical points at (𝑝_𝑖,0), where 𝑝_𝑖 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]). A central region is the largest connected region covered with cycles surrounding 𝑂.

Definition 1.2 (see [1]). A period annulus is every connected region covered with nontrivial concentric cycles.

Definition 1.3. We will call a period annulus associated with a central region a 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 a nontrivial period annulus.

For example, there are four central regions and three nontrivial period annuli in the phase portrait depicted in Figure 2.

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 problem𝑥^+𝑔(𝑥)=0,𝑥(0)=0,𝑥(1)=0,(1.3) 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 equation𝑥^+𝑓(𝑥)𝑥′+𝑔(𝑥)=0.(1.4)

The Liénard equation with a quadratical term 𝑥^+𝑓(𝑥)𝑥′²+𝑔(𝑥)=0(1.5) can be reduced to the form (1.1) by Sabatini's transformation [3]𝑢∶=Φ(𝑥)=^𝑥0𝑒^𝐹(𝑠)𝑑𝑠,(1.6) where 𝐹(𝑥)=∫^𝑥0𝑓(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 function 𝑥(𝑡) is a solution of (1.5) if and only if 𝑢(𝑡)=Φ(𝑥(𝑡)) is a solution to 𝑢^+𝑔(𝑥(𝑢))𝑒^{𝐹(𝑥(𝑢))}=0.(1.7)

Our task in this article is to define the maximal number of nontrivial period annuli for (1.1). (A)We suppose that 𝑔(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 function 𝐺(𝑥)=∫^𝑥0𝑔(𝑠)𝑑𝑠 is an even degree polynomial with possible multiple local maxima.

The function 𝑔(𝑥)=−𝑥(𝑥²−𝑝²)(𝑥²−𝑞²) is a sample.

We discuss nontrivial period annuli in Section 2. 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 of regular 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 that 𝑔(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 if 𝐺(𝑥) has 𝑚 pairs of non-neighboring points of maxima then 𝑚 nontrivial period annuli exist.

Consider, for example, (1.1), where 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).(2.1) The equivalent system has alternating “saddles” and “centers”, and the graph of 𝐺(𝑥) is depicted in Figure 1.

Figure 1 

Figure 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 Figure 2.

3. Polynomials

Consider a polynomial 𝐺(𝑥). Points of local maxima 𝑥_𝑖 and 𝑥_𝑗 of 𝐺(𝑥) are non-neighboring if the interval (𝑥_𝑖,𝑥_𝑗) contains at least one point of local maximum of 𝐺(𝑥).

Definition 3.1. Two non-neighboring points of maxima 𝑥_𝑖<𝑥_𝑗 of 𝐺(𝑥) will be called a regular pair if 𝐺(𝑥)<min{𝐺(𝑥_𝑖),𝐺(𝑥_𝑗)} at any other point of maximum lying in the interval (𝑥_𝑖,𝑥_𝑗).

Theorem 3.2. Suppose 𝑔(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 is 𝑛−2.

Proof. By induction, let 𝑥₁,𝑥₂,…,𝑥_𝑛 be successive points of maxima of 𝐺(𝑥), 𝑥₁<𝑥₂<⋯<𝑥_𝑛.
(1)Let 𝑛=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) provides a regular pair. In this case, therefore, the maximal number of regular pairs is 1.
(2) Suppose that for any sequence of 𝑛>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,…,𝑥_𝑛, 𝑥_𝑛+1, 𝑥₁<𝑥₂<…<𝑥_𝑛<𝑥_𝑛+1. Let us prove that the maximal number of regular pairs is 𝑛−1. For this, consider the following possible variants.
(a)The couple 𝑥₁,𝑥_𝑛 is a regular pair. If 𝐺(𝑥₁)>𝐺(𝑥_𝑛) and 𝐺(𝑥_𝑛+1)>𝐺(𝑥_𝑛), then, beside the regular pairs in the interval [𝑥₁,𝑥_𝑛], only one new regular pair can appear, namely, 𝑥₁,𝑥_𝑛+1. Then, the maximal number of regular pairs which can be composed of the points 𝑥₁,𝑥₂,…,𝑥_𝑛,𝑥_𝑛+1, is not greater than (𝑛−2)+1=𝑛−1. If 𝐺(𝑥₁)≤𝐺(𝑥_𝑛) or 𝐺(𝑥_𝑛+1)≤𝐺(𝑥_𝑛), then the additional regular pair does not appear. In a particular case 𝐺(𝑥₂)<𝐺(𝑥₃)<⋯<𝐺(𝑥_𝑛)<𝐺(𝑥_𝑛+1) and 𝐺(𝑥₁)>𝐺(𝑥_𝑛) the following regular pairs exist, namely, 𝑥₁ and 𝑥₃, 𝑥₁ and 𝑥₄,…, 𝑥₁ and 𝑥_𝑛, and the new pair 𝑥₁ and 𝑥_𝑛+1 appears, totally 𝑛−1 pairs.(b)Suppose that 𝑥₁,𝑥_𝑛 is not a regular pair. Let 𝑥_𝑖 and 𝑥_𝑗 be a regular pair, 1≤𝑖<𝑗≤𝑛, and there is no other regular pair 𝑥_𝑝,𝑥_𝑞 such that 1≤𝑝≤𝑖<𝑗≤𝑞≤𝑛. Let us mention that if such a pair 𝑥_𝑖,𝑥_𝑗 does not exist, then the function 𝐺(𝑥) does not have regular pairs at all and the sequence {𝐺(𝑥_𝑘)}, 𝑘=1,⋯,𝑛, is monotone. Then, if 𝐺(𝑥_𝑛+1) is greater than any other maximum, there are exactly (𝑛+1)−2=𝑛−1 regular pairs.
Otherwise, we have two possibilities:
either 𝐺(𝑥_𝑖)≥𝐺(𝑥_𝑝), 𝑝=1,…,𝑖−1, or 𝐺(𝑥_𝑗)≥𝐺(𝑥_𝑞), 𝑞=𝑗+1,…,𝑛.
In the first case, the interval [𝑥₁,𝑥_𝑖] 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 pairs 𝑥_𝑝,𝑥_𝑘 for 1≤𝑝<𝑖,  𝑖<𝑘≤𝑛+1. The interval [𝑥_𝑖,𝑥_𝑛+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 of regular pairs in [𝑥_𝑖,𝑥_𝑗] does not exceed 𝑗−(𝑖−1)−2=𝑗−𝑖−1. In [𝑥_𝑗,𝑥_𝑛+1],there are no more than (𝑛+1)−(𝑗−1)−2=𝑛−𝑗 regular pairs. The points 𝑥_𝑝, 𝑝=1,…,𝑖−1, 𝑥_𝑞, 𝑗<𝑞≤𝑛 do not form regular pairs, by the choice of 𝑥_𝑝 and 𝑥_𝑞. The points 𝑥_𝑝, 𝑝=1,…,𝑖, together with 𝑥_𝑛+1 (it serves as the 𝑖+1th point in a collection of points) form not more than (𝑖+1)−2=𝑖−1 regular 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 number 𝑛, a polynomial 𝑔(𝑥) can be constructed such that (a)the condition (A) is satisfied,(b)the primitive function 𝐺(𝑥) has exactly 𝑛 points of maximum and the number of regular pairs is exactly 𝑛−2.

Proof. Consider the polynomial 𝐺(𝑥)=−𝑥+12𝑥−21𝑥+32𝑥−32𝑥+52𝑥−52𝑥+72𝑥−72.(4.1) It is an even function with the graph depicted in Figure 3.
Consider now the polynomial 𝐺_𝜀(𝑥)=−𝑥+12+𝜀𝑥−21𝑥+32𝑥−23𝑥+52𝑥−52𝑥+72𝑥−72,(4.2) where 𝜀>0 is small enough. The graph of 𝐺_𝜀(𝑥) with 𝜀=0.2 is depicted in Figure 4.
Denote the maximal values of 𝐺(𝑥) and 𝐺_𝜀(𝑥) to the right of 𝑥=0  𝑚⁺¹,𝑚⁺². Denote the maximal values of 𝐺(𝑥) and 𝐺_𝜀(𝑥) to the left of 𝑥=0𝑚⁻¹,𝑚⁻². One has for 𝐺(𝑥) that 𝑚⁺¹=𝑚⁻¹<𝑚⁻²=𝑚⁺². One has for 𝐺_𝜀(𝑥) that 𝑚⁺¹<𝑚⁻¹<𝑚⁺²<𝑚⁻². Then, there are two regular pairs (resp., 𝑚⁻¹ and 𝑚⁺², 𝑚⁺² and 𝑚⁻²).
For arbitrary even 𝑛 the polynomial 𝐺_𝜀(𝑥)=−𝑥+12𝑥−12𝑥+32𝑥−32⋯𝑥+2𝑛2−1𝑥−2𝑛2−1,(4.3) is to be considered where the maximal values 𝑚⁺¹,𝑚⁺²,…,𝑚^+𝑛/2 to the right of 𝑥=0 form ascending sequence, and, respectively, the maximal values 𝑚⁻¹,𝑚⁻²,…,𝑚^−𝑛/2 to the left of 𝑥=0 also form ascending sequence. One has that 𝑚^+𝑖=𝑚^−𝑖 for all 𝑖. For a slightly modified polynomial 𝐺_𝜀(𝑥)=−𝑥+12+𝜀𝑥−21𝑥+32𝑥−32…𝑥+2𝑛2−1𝑥−2𝑛2−1,(4.4) the maximal values are arranged as 𝑚⁺¹<𝑚⁻¹<𝑚⁺²<𝑚⁻²<⋯<𝑚^+𝑛/2<𝑚^−𝑛/2.(4.5) Therefore, there exist exactly 𝑛−2 regular pairs and, consequently, 𝑛−2 nontrivial period annuli in the differential equation (1.1).
If 𝑛 is odd, then the polynomial 𝐺(𝑥)=−𝑥²(𝑥−1)(𝑥+1)(𝑥−2)(𝑥+2)⋯(𝑥−(𝑛−1))(𝑥+(𝑛−1))(4.6) 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 𝐺(𝑥)=−𝑥²(𝑥−1−𝜀)(𝑥+1)(𝑥−2)(𝑥+2)⋯(𝑥−(𝑛−1))(𝑥+(𝑛−1))(4.7) 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 of 𝐺_𝜀(𝑥) with 𝜀=0.2 is depicted in Figure 6.

Figure 3 

𝐺(𝑥) (solid) and 𝐺′(𝑥)=𝑔(𝑥) (dashed).

Figure 4 

𝐺(𝑥) (solid line), 𝐺𝜀(𝑥) (dashed line), and 𝐺(𝑥)−𝐺𝜀(𝑥) (dotted line).

Figure 5 

𝐺(𝑥) (solid) and 𝐺′(𝑥)=𝑔(𝑥) (dashed).

Figure 6 

𝐺(𝑥) (solid), 𝐺𝜀(𝑥) (dashed), and 𝐺(𝑥)−𝐺𝜀(𝑥) (dotted).

Acknowledgments

This work has been supported by ERAF project no. 2010/0206/2DP/2.1.1.2.0/10/APIA/VIAA/011 and Latvian Council of Science Grant no. 09.1220.