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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 393875, 8 pages
http://dx.doi.org/10.1155/2011/393875
Research Article

On a Maximal Number of Period Annuli

1Information Systems and Management Institute, Ludzas Street 91, 1019 Riga, Latvia
2Institute of Mathematics and Computer Science, University of Latvia Rainis Boulevard 29, 1459 Riga, Latvia

Received 30 October 2010; Accepted 30 December 2010

Academic Editor: ElenaΒ Braverman

Copyright Β© 2011 Yelena Kozmina and Felix Sadyrbaev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider equation π‘₯ξ…žξ…ž+𝑔(π‘₯)=0, where 𝑔(π‘₯) 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 equationπ‘₯ξ…žξ…ž+𝑔(π‘₯)=0,(1.1) where 𝑔(π‘₯) is an odd degree polynomial with simple zeros.

The equivalent differential systemπ‘₯β€²=𝑦,𝑦′=βˆ’π‘”(π‘₯)(1.2) has critical points at (𝑝𝑖,0), where 𝑝𝑖 are zeros of 𝑔(π‘₯). 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 problemπ‘₯ξ…žξ…ž+𝑔(π‘₯)=0,π‘₯(0)=0,π‘₯(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 equationπ‘₯ξ…žξ…ž+𝑓(π‘₯)π‘₯β€²+𝑔(π‘₯)=0.(1.4)

The LiΓ©nard equation with a quadratical term π‘₯ξ…žξ…ž+𝑓(π‘₯)π‘₯β€²2+𝑔(π‘₯)=0(1.5) can be reduced to the form (1.1) by Sabatini's transformation [3]ξ€œπ‘’βˆΆ=Ξ¦(π‘₯)=π‘₯0𝑒𝐹(𝑠)𝑑𝑠,(1.6) where ∫𝐹(π‘₯)=π‘₯0𝑓(𝑠)𝑑𝑠. Since 𝑑𝑒/𝑑π‘₯>0, this is one-to-one correspondence and the inverse function π‘₯=π‘₯(𝑒) is well defined.

Lemma 1.5 (see [3, Lemma 1]). The function π‘₯(𝑑) is a solution of (1.5) if and only if 𝑒(𝑑)=Ξ¦(π‘₯(𝑑)) is a solution to π‘’ξ…žξ…ž+𝑔(π‘₯(𝑒))𝑒𝐹(π‘₯(𝑒))=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(π‘₯) is an odd degree polynomial with simple zeros and with a negative coefficient at the principal term (so 𝑔(βˆ’βˆž)=+∞ and 𝑔(+∞)=βˆ’βˆž). A zero 𝑧 is called simple if 𝑔(𝑧)=0 and 𝑔′(𝑧)β‰ 0.

The graph of a primitive function ∫𝐺(π‘₯)=π‘₯0𝑔(𝑠)𝑑𝑠 is an even degree polynomial with possible multiple local maxima.

The function 𝑔(π‘₯)=βˆ’π‘₯(π‘₯2βˆ’π‘2)(π‘₯2βˆ’π‘ž2) 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(π‘₯) 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(π‘₯) 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(π‘₯). Suppose that any other local maximum of 𝐺(π‘₯) 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(π‘₯) has π‘š pairs of non-neighboring points of maxima then π‘š nontrivial period annuli exist.

Consider, for example, (1.1), where 𝑔(π‘₯)=βˆ’π‘₯(π‘₯+3)(π‘₯+2.2)(π‘₯+1.9)(π‘₯+0.8)(π‘₯βˆ’0.3)(π‘₯βˆ’1.5)(π‘₯βˆ’2.3)(π‘₯βˆ’2.9).(2.1) The equivalent system has alternating β€œsaddles” and β€œcenters”, and the graph of 𝐺(π‘₯) is depicted in Figure 1.

393875.fig.001
Figure 1
393875.fig.002
Figure 2: The phase portrait for (1.1), where 𝐺(π‘₯) 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(π‘₯). Points of local maxima π‘₯𝑖 and π‘₯𝑗 of 𝐺(π‘₯) are non-neighboring if the interval (π‘₯𝑖,π‘₯𝑗) contains at least one point of local maximum of 𝐺(π‘₯).

Definition 3.1. Two non-neighboring points of maxima π‘₯𝑖<π‘₯𝑗 of 𝐺(π‘₯) will be called a regular pair if 𝐺(π‘₯)<min{𝐺(π‘₯𝑖),𝐺(π‘₯𝑗)} at any other point of maximum lying in the interval (π‘₯𝑖,π‘₯𝑗).

Theorem 3.2. Suppose 𝑔(π‘₯) is a polynomial which satisfies the condition A. Let 𝐺(π‘₯) be a primitive function for 𝑔(π‘₯) and 𝑛 a number of local maxima of 𝐺(π‘₯).
Then, the maximal possible number of regular pairs is π‘›βˆ’2.

Proof. By induction, let π‘₯1,π‘₯2,…,π‘₯𝑛 be successive points of maxima of 𝐺(π‘₯), π‘₯1<π‘₯2<β‹―<π‘₯𝑛.
(1)Let 𝑛=3. The following combinations are possible at three points of maxima:(a)𝐺(π‘₯1)β‰₯𝐺(π‘₯2)β‰₯𝐺(π‘₯3), (b)𝐺(π‘₯2)<𝐺(π‘₯1),𝐺(π‘₯2)<𝐺(π‘₯3), (c)𝐺(π‘₯1)≀𝐺(π‘₯2)≀𝐺(π‘₯3), (d)𝐺(π‘₯2)β‰₯𝐺(π‘₯1),𝐺(π‘₯2)β‰₯𝐺(π‘₯3).
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(π‘₯) the maximal number of regular pairs is π‘›βˆ’2. Without loss of generality, add to the right one more point of maximum of the function 𝐺(π‘₯). We get a sequence of 𝑛+1 consecutive points of maximum π‘₯1,π‘₯2,…,π‘₯𝑛, π‘₯𝑛+1, π‘₯1<π‘₯2<…<π‘₯𝑛<π‘₯𝑛+1. Let us prove that the maximal number of regular pairs is π‘›βˆ’1. For this, consider the following possible variants.
(a)The couple π‘₯1,π‘₯𝑛 is a regular pair. If 𝐺(π‘₯1)>𝐺(π‘₯𝑛) and 𝐺(π‘₯𝑛+1)>𝐺(π‘₯𝑛), then, beside the regular pairs in the interval [π‘₯1,π‘₯𝑛], only one new regular pair can appear, namely, π‘₯1,π‘₯𝑛+1. Then, the maximal number of regular pairs which can be composed of the points π‘₯1,π‘₯2,…,π‘₯𝑛,π‘₯𝑛+1, is not greater than (π‘›βˆ’2)+1=π‘›βˆ’1. If 𝐺(π‘₯1)≀𝐺(π‘₯𝑛) or 𝐺(π‘₯𝑛+1)≀𝐺(π‘₯𝑛), then the additional regular pair does not appear. In a particular case 𝐺(π‘₯2)<𝐺(π‘₯3)<β‹―<𝐺(π‘₯𝑛)<𝐺(π‘₯𝑛+1) and 𝐺(π‘₯1)>𝐺(π‘₯𝑛) the following regular pairs exist, namely, π‘₯1 and π‘₯3, π‘₯1 and π‘₯4,…, π‘₯1 and π‘₯𝑛, and the new pair π‘₯1 and π‘₯𝑛+1 appears, totally π‘›βˆ’1 pairs.(b)Suppose that π‘₯1,π‘₯𝑛 is not a regular pair. Let π‘₯𝑖 and π‘₯𝑗 be a regular pair, 1≀𝑖<𝑗≀𝑛, and there is no other regular pair π‘₯𝑝,π‘₯π‘ž such that 1≀𝑝≀𝑖<π‘—β‰€π‘žβ‰€π‘›. Let us mention that if such a pair π‘₯𝑖,π‘₯𝑗 does not exist, then the function 𝐺(π‘₯) does not have regular pairs at all and the sequence {𝐺(π‘₯π‘˜)}, π‘˜=1,β‹―,𝑛, is monotone. Then, if 𝐺(π‘₯𝑛+1) is greater than any other maximum, there are exactly (𝑛+1)βˆ’2=π‘›βˆ’1 regular pairs.
Otherwise, we have two possibilities:
either 𝐺(π‘₯𝑖)β‰₯𝐺(π‘₯𝑝), 𝑝=1,…,π‘–βˆ’1, or 𝐺(π‘₯𝑗)β‰₯𝐺(π‘₯π‘ž), π‘ž=𝑗+1,…,𝑛.
In the first case, the interval [π‘₯1,π‘₯𝑖] contains 𝑖 points of maximum of 𝐺(π‘₯), 𝑖<𝑛, and hence the number of regular pairs in this interval does not exceed π‘–βˆ’2. There are no regular pairs π‘₯𝑝,π‘₯π‘˜ for 1≀𝑝<𝑖,  𝑖<π‘˜β‰€π‘›+1. The interval [π‘₯𝑖,π‘₯𝑛+1] contains (𝑛+1)βˆ’(π‘–βˆ’1) points of maximum of 𝐺(π‘₯), and hence the number of regular pairs in this interval does not exceed (𝑛+1)βˆ’(π‘–βˆ’1)βˆ’2=π‘›βˆ’π‘–. Totally, there are no more regular pairs than (π‘–βˆ’2)+(π‘›βˆ’π‘–)=π‘›βˆ’2.
In the second case, the number of regular pairs in [π‘₯𝑖,π‘₯𝑗] does not exceed π‘—βˆ’(π‘–βˆ’1)βˆ’2=π‘—βˆ’π‘–βˆ’1. In [π‘₯𝑗,π‘₯𝑛+1],there are no more than (𝑛+1)βˆ’(π‘—βˆ’1)βˆ’2=π‘›βˆ’π‘— regular pairs. The points π‘₯𝑝, 𝑝=1,…,π‘–βˆ’1, π‘₯π‘ž, 𝑗<π‘žβ‰€π‘› do not form regular pairs, by the choice of π‘₯𝑝 and π‘₯π‘ž. The points π‘₯𝑝, 𝑝=1,…,𝑖, together with π‘₯𝑛+1 (it serves as the 𝑖+1th point in a collection of points) form not more than (𝑖+1)βˆ’2=π‘–βˆ’1 regular pairs. Totally, the number of regular pairs is not greater than (π‘—βˆ’π‘–βˆ’1)+(π‘›βˆ’π‘—)+(π‘–βˆ’1)=π‘›βˆ’2.

4. Existence of Polynomials with Optimal Distribution

Theorem 4.1. Given number 𝑛, a polynomial 𝑔(π‘₯) can be constructed such that (a)the condition (A) is satisfied,(b)the primitive function 𝐺(π‘₯) has exactly 𝑛 points of maximum and the number of regular pairs is exactly π‘›βˆ’2.

Proof. Consider the polynomial ξ‚€1𝐺(π‘₯)=βˆ’π‘₯+21π‘₯βˆ’23π‘₯+23π‘₯βˆ’25π‘₯+25π‘₯βˆ’27π‘₯+27π‘₯βˆ’2.(4.1) It is an even function with the graph depicted in Figure 3.
Consider now the polynomial πΊπœ€ξ‚€1(π‘₯)=βˆ’π‘₯+21+πœ€ξ‚ξ‚€π‘₯βˆ’23π‘₯+23π‘₯βˆ’25π‘₯+25π‘₯βˆ’27π‘₯+27π‘₯βˆ’2,(4.2) where πœ€>0 is small enough. The graph of πΊπœ€(π‘₯) with πœ€=0.2 is depicted in Figure 4.
Denote the maximal values of 𝐺(π‘₯) and πΊπœ€(π‘₯) to the right of π‘₯=0  π‘š+1,π‘š+2. Denote the maximal values of 𝐺(π‘₯) and πΊπœ€(π‘₯) to the left of π‘₯=0π‘šβˆ’1,π‘šβˆ’2. One has for 𝐺(π‘₯) that π‘š+1=π‘šβˆ’1<π‘šβˆ’2=π‘š+2. One has for πΊπœ€(π‘₯) that π‘š+1<π‘šβˆ’1<π‘š+2<π‘šβˆ’2. Then, there are two regular pairs (resp., π‘šβˆ’1 and π‘š+2, π‘š+2 and π‘šβˆ’2).
For arbitrary even 𝑛 the polynomial πΊπœ€ξ‚€1(π‘₯)=βˆ’π‘₯+21π‘₯βˆ’23π‘₯+23π‘₯βˆ’2⋯π‘₯+2π‘›βˆ’12π‘₯βˆ’2π‘›βˆ’12,(4.3) is to be considered where the maximal values π‘š+1,π‘š+2,…,π‘š+𝑛/2 to the right of π‘₯=0 form ascending sequence, and, respectively, the maximal values π‘šβˆ’1,π‘šβˆ’2,…,π‘šβˆ’π‘›/2 to the left of π‘₯=0 also form ascending sequence. One has that π‘š+𝑖=π‘šβˆ’π‘– for all 𝑖. For a slightly modified polynomial πΊπœ€ξ‚€1(π‘₯)=βˆ’π‘₯+21+πœ€ξ‚ξ‚€π‘₯βˆ’23π‘₯+23π‘₯βˆ’2…π‘₯+2π‘›βˆ’12π‘₯βˆ’2π‘›βˆ’12,(4.4) the maximal values are arranged as π‘š+1<π‘šβˆ’1<π‘š+2<π‘šβˆ’2<β‹―<π‘š+𝑛/2<π‘šβˆ’π‘›/2.(4.5) Therefore, there exist exactly π‘›βˆ’2 regular pairs and, consequently, π‘›βˆ’2 nontrivial period annuli in the differential equation (1.1).
If 𝑛 is odd, then the polynomial 𝐺(π‘₯)=βˆ’π‘₯2(π‘₯βˆ’1)(π‘₯+1)(π‘₯βˆ’2)(π‘₯+2)β‹―(π‘₯βˆ’(π‘›βˆ’1))(π‘₯+(π‘›βˆ’1))(4.6) with 𝑛 local maxima is to be considered. The maxima are descending for π‘₯<0 and ascending if π‘₯>0. The polynomial with three local maxima is depicted in Figure 5.
The slightly modified polynomial 𝐺(π‘₯)=βˆ’π‘₯2(π‘₯βˆ’1βˆ’πœ€)(π‘₯+1)(π‘₯βˆ’2)(π‘₯+2)β‹―(π‘₯βˆ’(π‘›βˆ’1))(π‘₯+(π‘›βˆ’1))(4.7) has maxima which are not equal and are arranged in an optimal way in order to produce the maximal (π‘›βˆ’2) regular pairs.
The graph of πΊπœ€(π‘₯) with πœ€=0.2 is depicted in Figure 6.

393875.fig.003
Figure 3: 𝐺(π‘₯) (solid) and 𝐺′(π‘₯)=𝑔(π‘₯) (dashed).
393875.fig.004
Figure 4: 𝐺(π‘₯) (solid line), πΊπœ€(π‘₯) (dashed line), and 𝐺(π‘₯)βˆ’πΊπœ€(π‘₯) (dotted line).
393875.fig.005
Figure 5: 𝐺(π‘₯) (solid) and 𝐺′(π‘₯)=𝑔(π‘₯) (dashed).
393875.fig.006
Figure 6: 𝐺(π‘₯) (solid), πΊπœ€(π‘₯) (dashed), and 𝐺(π‘₯)βˆ’πΊπœ€(π‘₯) (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.

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