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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 393875, 8 pages
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: Elena Braverman
Copyright © 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.
We consider equation , where 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.
Consider equation where is an odd degree polynomial with simple zeros.
The equivalent differential system has critical points at , where are zeros of . Recall that a critical point 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 ). A central region is the largest connected region covered with cycles surrounding .
Definition 1.2 (see ). 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 problem or a given number of positive solutions  to the same problem.
Under certain conditions, period annuli of (1.1) give rise to limit cycles in a dissipative equation
The Liénard equation with a quadratical term can be reduced to the form (1.1) by Sabatini's transformation  where . Since , this is one-to-one correspondence and the inverse function is well defined.
Our task in this article is to define the maximal number of nontrivial period annuli for (1.1). (A)We suppose that is an odd degree polynomial with simple zeros and with a negative coefficient at the principal term (so and ). A zero is called simple if and .
The graph of a primitive function is an even degree polynomial with possible multiple local maxima.
The function 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 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 ). Suppose that in (1.1) is a polynomial with simple zeros. Assume that and () are nonneighboring points of maximum of the primitive function . Suppose that any other local maximum of in the interval is (strictly) less than .
Then, there exists a nontrivial period annulus associated with a pair .
It is evident that if has pairs of non-neighboring points of maxima then nontrivial period annuli exist.
There are three pairs of non-neighboring points of maxima and three nontrivial period annuli exist, which are depicted in Figure 2.
Consider a polynomial . Points of local maxima and of are non-neighboring if the interval contains at least one point of local maximum of .
Definition 3.1. Two non-neighboring points of maxima of will be called a regular pair if at any other point of maximum lying in the interval .
Theorem 3.2. Suppose is a polynomial which satisfies the condition A. Let be a primitive function for and a number of local maxima of .
Then, the maximal possible number of regular pairs is .
Proof. By induction, let be successive points of maxima of , .
(1)Let . The following combinations are possible at three points of maxima:(a), (b), (c), (d).
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 ordered points of maxima of the maximal number of regular pairs is . Without loss of generality, add to the right one more point of maximum of the function . We get a sequence of consecutive points of maximum , , . Let us prove that the maximal number of regular pairs is . For this, consider the following possible variants.
(a)The couple is a regular pair. If and , then, beside the regular pairs in the interval , only one new regular pair can appear, namely, . Then, the maximal number of regular pairs which can be composed of the points , is not greater than . If or , then the additional regular pair does not appear. In a particular case and the following regular pairs exist, namely, and , and ,, and , and the new pair and appears, totally pairs.(b)Suppose that is not a regular pair. Let and be a regular pair, , and there is no other regular pair such that . Let us mention that if such a pair does not exist, then the function does not have regular pairs at all and the sequence , , is monotone. Then, if is greater than any other maximum, there are exactly regular pairs.
Otherwise, we have two possibilities:
either , , or , .
In the first case, the interval contains points of maximum of , , and hence the number of regular pairs in this interval does not exceed . There are no regular pairs for , . The interval contains points of maximum of , and hence the number of regular pairs in this interval does not exceed . Totally, there are no more regular pairs than .
In the second case, the number of regular pairs in does not exceed . In ,there are no more than regular pairs. The points , , , do not form regular pairs, by the choice of and . The points , , together with (it serves as the th point in a collection of points) form not more than regular pairs. Totally, the number of regular pairs is not greater than .
4. Existence of Polynomials with Optimal Distribution
Theorem 4.1. Given number , a polynomial can be constructed such that (a)the condition (A) is satisfied,(b)the primitive function has exactly points of maximum and the number of regular pairs is exactly .
Proof. Consider the polynomial
It is an even function with the graph depicted in Figure 3.
Consider now the polynomial where is small enough. The graph of with is depicted in Figure 4.
Denote the maximal values of and to the right of . Denote the maximal values of and to the left of . One has for that . One has for that . Then, there are two regular pairs (resp., and , and ).
For arbitrary even the polynomial is to be considered where the maximal values to the right of form ascending sequence, and, respectively, the maximal values to the left of also form ascending sequence. One has that for all . For a slightly modified polynomial the maximal values are arranged as Therefore, there exist exactly regular pairs and, consequently, nontrivial period annuli in the differential equation (1.1).
If is odd, then the polynomial with local maxima is to be considered. The maxima are descending for and ascending if . The polynomial with three local maxima is depicted in Figure 5.
The slightly modified polynomial has maxima which are not equal and are arranged in an optimal way in order to produce the maximal () regular pairs.
The graph of with is depicted in Figure 6.
This work has been supported by ERAF project no. 2010/0206/2DP/188.8.131.52.0/10/APIA/VIAA/011 and Latvian Council of Science Grant no. 09.1220.
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