On the Maximum Number of Period Annuli for Second Order Conservative Equations

. We consider a second order scalar conservative diﬀerential equation who-se potential function is a Morse function with a ﬁnite number of critical points and is unbounded at inﬁnity. We give an upper bound for the number of nonglobal nontrivial period annuli of the equation and prove that the upper bound obtained is sharp. We use tree theory in our considerations.


Introduction
In this paper, we consider a second order conservative differential equation where U is a twice continuously differentiable function on the real line. The Equation (1.1) is equivalent to the planar system x = y, y = −U (x).
(1.2) Definition 1. Suppose U ∈ C 2 (R, R). A point x 0 ∈ R is a critical point of U if U (x 0 ) = 0. A critical point x 0 of U is nondegenerate, see [14], if U (x 0 ) = 0. The function U is said to be a Morse function, see [14], if all its critical points are nondegenerate.
In the present article, we consider functions U satisfying the following conditions. In what follows, we refer function U satisfying (A) and (B) as a Morse potential.
If U is a Morse potential, then ξ i (1 ≤ i ≤ n) are strict local minimum or maximum points for U . In what follows, we will assume that ξ 1 < · · · < ξ n . In view of (B), a Morse potential U has at least two strict local minimum points and at least one strict local maximum point; we denote by n min and n max the number of local minimum and maximum points of U , respectively. Then, n min + n max = n. The points (ξ i , 0), i ∈ {1, . . . , n}, are singular points of the planar system (1.2): a point (ξ i , 0) is a centre of (1.2) if U (ξ i ) > 0 and a saddle of (1.2) if U (ξ i ) < 0. The singular points of (1.2) form a sequence (ξ 1 , 0), . . . , (ξ n , 0) of alternating centres and saddles on the horizontal axis of the phase plane of (1.2).
We are looking for regions of the phase plane of (1.2) filled with nontrivial periodic orbits of (1.2). By region we mean a nonempty open connected subset of R 2 .
Definition 2. A maximal region covered with nontrivial periodic orbits of (1.2) is called a period annulus for (1.1). Maximality means that a period annulus is not contained in any other region covered with nontrivial periodic orbits of (1.2).
More generally, period annuli are similarly defined for autonomous planar systems as well. Some authors classify period annuli by the number of singular points enclosed by period annuli.
If U is a Morse potential, then U has at least three critical points and thus a trivial period annulus for (1.1) is a nonglobal period annulus for (1.1). For the Equation (1.1) with a Morse potential U , by the phase plane analysis of (1.2), a global period annulus exists if and only if (B1) is fulfilled; moreover, it is unique.
The number of trivial, nontrivial, global, nonglobal, and nonglobal nontrivial period annuli for (1.1) are denoted by N T , N N T , N G , N N G , and N N G,N T , respectively. By N we denote the number of all period annuli for (1.1). In view of Definitions 3 and 4, In this paper, we are interested in finding a sharp upper bound for N N G,N T .
We mention some references concerning period annuli for autonomous planar systems, in particular, for the systems (1.2). The period functions associated with central regions have been extensively studied, for instance, in [10,11,16,21]. Planar systems in which period annuli enclose limit cycles, or limit cycles enclose period annuli are studied, for instance, in [2,5,6,17,22]. In [8,9], the authors study the maximum number of nontrivial period annuli for a conservative equation with a polynomial potential; the results obtained in [8,9] are discussed in more detail at the end of this section.
Let us recall some concepts from the tree theory, see, for instance, [18,19,20], that will be used in our article. A tree is a connected acyclic graph. A rooted tree is a tree that has one node designated as a root. Let T be a rooted tree with at least two nodes and let u 0 be the root of T . Let u be a node of T that is different from u 0 . There exists exactly one path in T between u 0 and u, and thus there exists a unique node v of T such that the path contains the edge {v, u}. The node v is called a parent node of u and u is called a child node of v. In a rooted tree, a leaf node is a node with no child nodes. A binary tree is a rooted tree in which all nodes have at most two child nodes.
The present article is organized as follows. In Section 2, we define critical segments for a Morse potential U and explore their properties. In Section 3, we introduce the rooted tree G(U ) associated with a Morse potential U in such a way that non-root nodes of G(U ) are critical segments for U . We show that the leaf nodes and the non-root non-leaf nodes of G(U ) represent the trivial and the nonglobal nontrivial period annuli for (1.1), respectively. Let α be a real number. Then, α := max{k ∈ Z : k ≤ α} is the floor of α and α := min{k ∈ Z : k ≥ α} is the ceiling of α. In Section 4, we prove the first main theorem of our article.
Theorem 1. Let U be a Morse potential with n critical points. The number of nonglobal nontrivial period annuli for (1.1) satisfies the following inequality: In Section 5, at first, we prove that, for a Morse potential U , providing the inequality (1.4) becomes equality, the associated rooted tree G(U ) is a binary tree with additional structure. Then, we prove the second main theorem of our article.
Theorem 2. The following statements are valid.
(1) For every odd integer n (n ≥ 3), there is a Morse potential U with n critical points such that U satisfies (B1) and ensures equality in (1.4).
(2) For every even integer n (n ≥ 4), there is a Morse potential U with n critical points such that U satisfies (B2) and ensures equality in (1.4).
(3) For every even integer n (n ≥ 4), there is a Morse potential U with n critical points such that U satisfies (B3) and ensures equality in (1.4).
(4) For every odd integer n (n ≥ 5), there is a Morse potential U with n critical points such that U satisfies (B4) and ensures equality in (1.4).
The main theorems in our article provide a wide generalization of the results obtained in [8,9]. In the last two references, the authors explore the Equation (1.1) for a real polynomial U of degree n+1 (n is odd and n ≥ 5) such that the function U has n nondegenerate critical points, and the leading coefficient of U is negative. Hence, U satisfies (A) and (B4) and thus n max = n min + 1. It follows from Theorem 1 that The condition (B4) implies N N G,N T = N N T . Hence, we obtain the inequality N N T ≤ n max − 2, which was proved in [9]. In [8,9], the authors proved the sharpness of the last inequality.
It is worth mentioning that graph theory is often used to study Morse functions, for example, Reeb graphs are important for exploring Morse functions over manifolds, see, for instance, [4,7,12,13] and references therein. Reeb graphs are used also in computer graphics, see [4]. In [12], conditions are indicated under which Reeb graph is a tree. In [7,13], the authors provide the upper bound for the number of cycles in Reeb graphs and prove that the upper bound is sharp. (Compare with the main theorems in this article.) In our opinion, this article contributes to the application of graph theory to the study of Morse functions and differential equations.  Definition 6. Let E 1 = (a 1 , M 1 ), (b 1 , M 1 ) and E 2 = (a 2 , M 2 ), (b 2 , M 2 ) be two CS for U . We say E 1 is a parent CS of E 2 and E 2 is a child CS of E 1 if [a 2 , b 2 ] ⊂ (a 1 , b 1 ), the inequality M 2 < M 1 holds, and U (η) ≤ M 2 for every local maximum point η of U in the interval (a 1 , b 1 ).
Since E 1 is a CS for U , we see that a 1 or b 1 (may be both) is a local maximum point of U and U (a 1 ) = U (b 1 ) = M 1 . Assume that a 1 is a local maximum point of U . Hence, there exists a local maximum point a 1 of U in the interval (a 0 , b 0 ) such that U (a 1 ) > M 2 and thus the grandparent CS E 0 of E 2 is not a parent CS of E 2 . Example 1. In Figure 1, the points a 0 , b 1 , b 2 , and c 2 are local maximum points of U . The point P 1 = a 0 , U (a 0 ) does not generate any CS for U , the point generates exactly two CS E 2 and E 3 for U , and the point P 4 = c 2 , U (c 2 ) generates exactly two CS E 3 and E 4 for U . In the same figure, E 1 is a parent CS of E 2 , E 3 , and E 4 and the last three CS are child CS of E 1 .
In the next proposition, we collect properties of CS.
Proposition 1. Let U be a Morse potential.
(a) Let ξ be a local maximum point of U . The point ξ, U (ξ) generates at most two CS for U . Any CS for U is generated by either one or both of its endpoints.
(b) The function U has at least two CS and the number of CS for U is finite.
(c) A CS E 1 = (a 1 , M 1 ), (b 1 , M 1 ) for U has a child CS if and only if the interval (a 1 , b 1 ) contains at least one local maximum point of U . If a CS E 1 has a child CS, then E 1 has at least two child CS. A union of all child CS for E 1 is a closed line segment (a 2 , for U has a parent CS if and only if there exists a local maximum M 1 of U such that M 1 > M 2 and both the sets (e) Every CS for U has at most one parent CS. Proof.
is nonempty. Let b 2 be the element of U −1 (M ) closest to ξ on the right. Then, Hence, U has at least two CS. It follows from (a) that every local local maximum point of U generates at most two CS for U . Since U has a finite number of local maximum points, we conclude that the number of CS for U is finite. ( Let E 1 = (a 1 , M 1 ), (b 1 , M 1 ) be a CS for U . Suppose that U has a local maximum point in the interval (a 1 , b 1 ). Consider all local maximum points η 1 , . . . , η r (r≥1) in the interval (a 1 , b 1 ) and find M 2 = max U (η 1 ), . . . , U (η r ) .
If a CS E 1 has a child CS, then U has a local maximum point in the interval (a 1 , b 1 ). It follows from the above considerations that E 1 has at least two child CS and a union of The proof is similar to the one of (c).
Then, the following inequalities hold: contradicts to the hypothesis. If a 1 > a 1 , then the interval (a 1 , b 1 ) contains a point a 1 such that U (a 1 ) = M 1 > M 1 , which contradicts to E 1 is a CS. The case when b 1 is a local maximum point of U is considered similarly. Therefore, M 1 = M 1 =: M 1 . Let us prove that a 1 = a 1 and b 1 = b 1 . Suppose to the contrary that a 1 = a 1 or b 1 = b 1 . Suppose a 1 = a 1 . If a 1 < a 1 , then a 1 ∈ (a 1 , b 1 ) and U (a 1 ) = M 1 , which contradicts to E 1 is a CS. If a 1 > a 1 , then a 1 ∈ (a 1 , b 1 ) and U (a 1 ) = M 1 , which contradicts to E 1 is a CS. The case b 1 = b 1 also leads to a contradiction. Thereby, a 1 = a 1 and b 1 = b 1 and thus E 1 = E 1 .
(f) The proof follows from Definition 5 and the proof of (a).

Rooted trees associated with Morse potentials
Let U be a Morse potential. It follows from Proposition 1(b) that U has at least two CS and the number of CS for U is finite. Let θ be a real number and let M 0 be a real number greater than the maximum energy level of all CS for U . Let Denote by N CS the number of critical segments for U . We conclude that the graph G(U ) has N CS + 1 nodes and N CS edges. We designate the node O as a root of G(U ). Hence, G(U ) is a rooted graph with the root O.
Proposition 2. For a Morse potential U , the graph G(U ) is a rooted tree.
Suppose O ∈ {u 1 , . . . , u k }; then, the nodes u 1 , . . . , u k are CS for U . For every j ∈ {1, . . . , k}, let M j be an energy level for u j . Since {u 1 , u 2 } ∈ E(U ), then either u 1 is a parent CS of u 2 or u 2 is a parent CS of u 1 . Without loss of generality, we can assume that u 1 is a parent CS of u 2 . Hence, M 1 > M 2 . For the edge {u 2 , u 3 }, it follows from Proposition 1(e) that u 2 is a parent CS of u 3 and thus M 2 > M 3 . Continuing, for the edge {u k , u k+1 }, it follows from Proposition 1(e) that u k is a parent CS of u k+1 . Since u k+1 = u 1 , we see that M k > M 1 . Therefore, M 1 > M 2 > · · · > M k > M 1 which leads to a contradiction.
Suppose O ∈ {u 1 , u 2 , . . . , u k }. Without loss of generality, we can assume that O = u 1 . Then, the cycle C contains the edge {O, u 2 } and thus u 2 has no parent CS. For the edge {u 2 , u 3 }, it follows from Proposition 1(e) that u 2 is a parent CS of u 3 . Continuing, for the edge {u k−1 , u k }, it follows from Proposition 1(e) that u k−1 is a parent CS of u k . Finally, for the edge {u k , u k+1 }={u k , O}, u k has no parent CS. The contradiction obtained proves that G(U ) does not contain cycles. Since G(U ) has N CS + 1 nodes and N CS edges, it follows from [20, Theorem 2.4.1.] that G(U ) is a tree.
In Figure 1, a Morse potential U and the associated rooted tree G(U ) are depicted. It follows from Proposition 1, that the number N CS of CS for (1.1) is finite and N CS ≥ 2. To every CS E for U , we associate a period annulus A(E) as follows.
In view of (B), a trivial period annulus of (1.1) is a nonglobal period annulus of (1.1). Thereby, the correspondence E → A(E) defines a mapping A from the set C of all CS for U to the set P N G of all nonglobal period annuli for (1.1). Using the phase plane analysis of the planar system (1.2) and taking into account Proposition 1, we deduce that (I ) the mapping A : C → P N G is one-to-one and thus N CS = N N G ;

Proof of the first main theorem
Let X and Y be two nonempty sets. Denote by P(Y ) the power set of Y . A multivalued mapping ϕ from the set X to the set Y is a mapping from the set X to the set P(Y ) such that ϕ(x) is a nonempty subset of Y for every x ∈ X, see [3]. The multivalued mapping ϕ from X to Y is denoted by ϕ : . The cardinality of a finite set S is denoted by |S|. Assume that X and Y are nonempty finite sets and Y = {P 1 , . . . , P q }, where q ≥ 1. For a multivalued mapping ϕ : X ⇒ Y , Suppose that a Morse potential U has local maximum points η i (1 ≤ i ≤ n max ). Consider the points P i := η i , U (η i ) (1 ≤ i ≤ n max ) and the set Y := {P 1 , . . . , P nmax }. In accordance with Proposition 1(b), the function U has at least two CS and the number of CS for U is finite. Consider the set C = {E 1 , . . . , E N CS } of all CS for U . Let us define a multivalued mapping ϕ : C ⇒ Y as follows: for every j ∈ {1, . . . , N CS }, the value ϕ(E j ) is a subset of Y consisting of the points that generate E j . On account of Proposition 1(a), the cardinality ϕ(E j ) ∈ {1, 2} for every j ∈ {1, . . . , N CS }. For every i ∈ {1, . . . , n max }, the set ϕ − (P i ) consists of the CS for U generated by the point P i . Then, k i := ϕ − (P i ) (1 ≤ i ≤ n max ) is the number of the CS for U generated by the point P i . It follows from Proposition 1(a) that k i ∈ {0, 1, 2} for every i ∈ {1, . . . , n max }. Let J 1 , . . . , J t (t ≥ 1) be a partition of the set {1, . . . , n max }, where among the sets J 1 , . . . , J t (t ≥ 1) there may be empty sets; see [15, p. 130]. Since |C| = N CS , it follows from (4.1) that (4.2) Let us interpret the summands in the right-hand side of (4.2). For example, let us consider the summand S 1 := s1∈J1 ϕ − (P s1 ) . If J 1 = ∅, then the points P s1 (s 1 ∈ J 1 ) together generate exactly S 1 CS for U and The other summands in the right-hand side of (4.2) are interpreted similarly.
Example 2. Let us consider the Morse potential U depicted in Figure 1. Then, , and P 4 = c 2 , U (c 2 ) . For the multivalued mapping ϕ : C ⇒ Y defined above, we have ϕ( we see that the points P 1 and P 2 together generate exactly one CS for U and the points P 3 and P 4 together generate exactly three CS for U .
Let us prove Theorem 1.
In view of (3.2), we obtain The proof is complete.

Proof of the second main theorem
In a rooted tree, the depth of a node is the length of the unique path connecting the node and the root, see [18, p. 17]. The depth of a rooted tree T is the maximum depth of a node in T , see [18, p. 17]; the depth of T is denoted by depth(T ). For a node u of a rooted tree, the number of child nodes for u is denoted by children(u), see [18, p. 18]. If a rooted tree T has k nodes u 1 , . . . , u k , then, see [18,Lemma 1.44], children(u 1 ) + · · · + children(u k ) = k − 1.
In the next proposition, we will indicate the necessary conditions for the Morse potential to ensure equality in (1.4). (2) the number of non-root non-leaf nodes in G(U ) is n/2 − 1; (3) every non-root non-leaf node in G(U ) has two child nodes; Proof. Let U be a Morse potential with n critical points. It follows from Proposition 2 that G(U ) is a rooted tree. Assume N N G,N T = n 2 − 1. Let s stand for n 2 − 1. In view of Proposition 3, the number of non-root non-leaf nodes in G(U ) is s and the number of leaf nodes in G(U ) is n min . Suppose n = 3; then, s = 0. Since n = 3, it follows from (B) that (B1) holds and thus lim x→±∞ U (x) = +∞. Hence, the rooted tree G(U ) has three nodes: the root and two leaf nodes. Therefore, the statements (1)-(5) are fulfilled.
Proof. The assertion (b) is a consequence of (a) in view of Proposition 4. The assertion (c) is a trivial consequence of (b). Let us prove that (c) implies (a). Assume that the rooted tree G(U ) has k nodes, where k is defined by (5.2). On account of Proposition 3, we have k = N N G,N T + n min + 1.
Remark 2. The condition that the rooted tree G(U ) associated with a Morse potential U is a binary tree is necessary but not sufficient for the inequality (1.4) to become the equality. For example, the Morse potential U depicted in Figure 2 has n = 6 critical points and satisfies (B2); the associated rooted tree G(U ) is a binary tree with k = 5 nodes. Since (5.2) is not fulfilled, it follows from Corollary 2 that the inequality (1.4) is strict.
Let k be an integer greater than one and let s stand for k 2 − 1. Based on Proposition 4, to every k, we associate a binary tree T k as follows. Consider a binary tree T s+2 with s + 2 nodes and the depth s + 1. Then, T s+2 has s non-root non-leaf nodes and one leaf node. Add a leaf node to each non-root non-leaf node of T s+2 and add a leaf node to the root of T s+2 if k is odd. The binary tree T k obtained has k nodes, k 2 − 1 non-root non-leaf nodes, and k 2 leaf nodes, see Figure 3. We see that depth(T k ) = depth(T s+2 ) = k 2 and every non-root non-leaf node in T k has two child nodes. Among binary trees with k nodes, T k provides an example of a binary tree that has the minimum number k 2 of non-leaf nodes, see [19, p. 216], and the maximum number k Figure 3. Binary trees T k (2 ≤ k ≤ 11).
Remark 3. There exist Morse potentials U with n critical points such that the inequality (1.4) becomes equality, and the rooted tree G(U ) is a binary tree different from T k , where k is defined by (5.2).

Conclusions
In our paper, we consider a second order scalar differential equation with a Morse potential. We present the upper bound for the number of nonglobal nontrivial period annuli for the equation and, as a consequence, the upper bound for the number of all period annuli for the equation. We prove that these bounds are sharp, indicating examples of Morse potentials. In our reasoning, we assign a rooted tree to each Morse potential. The associated with Morse potentials rooted trees have at least three nodes, and every non-root non-leaf node, if any exists, has at least two child nodes. In further research, it would be interesting (a) to describe the Morse potentials with a given rooted tree, (b) to investigate the maximum number of period annuli for potentials with a finite limit at minus or plus infinity.