Two families of infinitely many homoclinics for singular strong force Hamiltonian systems

We are concerned with a planar autonomous Hamiltonian system q¨+∇V(q)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\ddot{q} +\nabla V (q)=0}$$\end{document}, where a potential V:R2\{ξ}→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V : \mathbb{R}^2 \backslash \{\xi\}\rightarrow \mathbb{R}}$$\end{document} has a single well of infinite depth at a point ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\xi}$$\end{document} and a unique strict global maximum 0 at a point a. Under a strong force condition around the singularity ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\xi}$$\end{document}, via minimization of an action integral and using a shadowing chain lemma together with simple geometrical arguments, we prove the existence of infinitely many homotopy classes of π1(R2\{ξ})\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\pi_1(\mathbb{R}^2 \backslash \{\xi\})}$$\end{document} containing at least two geometrically distinct homoclinic (to a) solutions.


Introduction
In this paper we are concerned with the second order Hamiltonian system where ·· = d 2 dt 2 , q ∈ R 2 and ∇V denotes the gradient of a potential V . We denote by | · | the norm in R 2 induced by the standard inner product (·, ·). Throughout the paper we assume that the potential V satisfies the following conditions: (V 1 ) there is ξ ∈ R 2 such that V ∈ C 1,1 (R 2 \{ξ}, R) and lim x→ξ V (x) = −∞, (V 2 ) there are a neighbourhood N ⊂ R 2 of the point ξ and a function U ∈ C 1 (N \ {ξ}, R) such that |U (x)| → ∞ as x → ξ and |∇U (x)| 2 ≤ −V (x) for all x ∈ N \ {ξ}, (V 3 ) V (x) ≤ 0 and V has a unique maximum at a point a ∈ R 2 \ {ξ}, In [5], under assumptions (V 1 )-(V ′ 4 ) and some geometric condition (⋆) on V due to Bolotin (see [2]), Caldiroli and Jeanjean proved the existence of infinitely many homoclinics of (HS), each one being characterized by a distinct winding number around the singularity ξ.
The aim of this work is to show by the use of minimization arguments that under hypotheses (V 1 )-(V 4 ) and somewhat stronger geometric condition than Bolotin's one, there are infinitely many homotopy classes of π 1 (R 2 \{ξ}) containing at least two geometrically distinct homoclinic solutions of (HS).
The existence of homoclinic orbits is an important problem in the study of the behaviour of dynamical systems. Their existence may give the horseshoe chaos (see, for example, [18] and the references therein). The presence of infinitely many geometrically distinct homoclinic or heteroclinic orbits is an indication of nonintegrability and chaotic behaviour for the system (HS) (see [2,3]).

Multiplicity results
At the beginning we set up notation and terminology. It is well known that the Sobolev space 16 (2014) Two families of homoclinics 303 Two families of homoclinics 3 equipped with the norm given by where 0 < ε ≤ 1 3 |a − ξ| and B ε (a) denotes the ball of radius ε around a. By (V 1 ), (V 3 ) and (V 4 ) it follows that α ε > 0. For q ∈ E, set An easy proof of this lemma can be found in [8].
To shorten notation, q(±∞) = lim t→±∞ q(t). Applying Lemma 2.1 one can prove that if I(q) < ∞, then q ∈ L ∞ (R, R 2 ) and q(±∞) = a (cf. [ which implies that q(t) ̸ = ξ for t ∈ R (cf. [ The rotation number (the winding number) rot ξ (q) of q around ξ is constant on every connected component of A and induces an isomorphism Equivalently, A is a sum of its path-connected components labeled by the integers.
We define the family F as follows. A set Z ⊂ A is a member of F if and only if • for each q ∈ Z and for each ψ ∈ C ∞ 0 (R, R 2 ) there exists δ > 0 such that if s ∈ (−δ, δ), then q + sψ ∈ Z. Let us remark that if q is a minimizer of I on a set Z ∈ F, then and consequently, q is a weak solution of (HS). Analysis similar to that in the proof of Proposition 3.18 in [13] shows that q is a classical solution of (HS). Finally, using (HS), (V 1 ) and (V 3 ) as in [12] givesq(±∞) = 0. Let l(s) be the line through a and ξ parameterized by s ∈ R in such a way that a and ξ correspond to s = 0 and s = 1, respectively. The line l divides R 2 into two half-planes Π ± . To be more precise, if ( ⃗ aξ, ⃗ e) is a positively oriented orthogonal basis in R 2 , then ⃗ e ∈ Π + . Let , the set of curves in A that do not achieve a in finite time.
In other words, q 0 and q 1 are homotopic in G if and only if they belong to the same path-connected component of G. The homotopy class of q ∈ G is denoted by [[q]], and Γ is the set of homotopy classes. The inclusion ι : G → A induces a surjective map In fact, for every [q] ∈ π 1 (R 2 \ {ξ}) the inverse image ι −1 * ([q]) contains infinitely many elements. We are going to describe the set Γ.
Lemma 2.2. Every homotopy class γ ∈ Γ can be represented by q ∈ G that has at most finitely many intersection points with the line l.
By standard transversality (or simplicial approximation) arguments there is a perturbation q 0 of q in G that has at most finitely many intersection points with the line l on the interval [−T, T ] and q 0 (−T ), q 0 (T ) / ∈ l. Let us introduce the polar coordinate system in R 2 with the pole a and the polar axis l whose orientation agrees with the orientation of the plane. In this coordinate system one has q 0 (t) = (r(t) cos φ(t), r(t) sin φ(t)). Clearly, there is no uniqueness of a function φ(t). Since q 0 (t) is continuous we can assume that φ(t) is continuous.
Given a homotopy class [[q]], assume that q has a minimal number, k > 0, of crossing points with the line l. Thus there are t 1 < t 2 < · · · < t k such that q(t i ) = l(s i ) for certain s i ∈ R, i = 1, . . . , k. We associate with q a word ω of length k as follows. If q crosses the line l at time t i , leaving Π − and entering Π + , then at the ith place in ω we will write If q crosses l living Π + and entering Π − , then we use lettersū,v,w, respectively. If [[q]] ∈ Γ, then the corresponding word ω has the following properties: • ω begins and ends at the letter u (with or without a bar, i.e.,ū, u), • two consecutive letters in ω are never the same, • every second letter in ω appears with a bar. The set of words satisfying the above conditions is denoted by Ω. Additionally, a contractible loop is represented by the empty word. For every ω ∈ Ω we defineω ∈ Ω as follows. We remove all bars from the word ω. Next we put bars over letters that appear in ω without bars and finally we write letters in the opposite order. For instance, if ω = uwuvu, thenω =ūvūwū. It is clear that ω is represented by a loop q(t) if and only ifω is represented by q(−t). The proof is left to the reader. Given a word ω ∈ Ω of length k. Assume that the letter u (with or without a bar) appears at ith and jth places in ω and there is no u at places with indices between i and j. We define a derived from ω sequence of words ω 1 ∪ω 2 as follows. The word ω 1 is a sequence of the first i elements of ω and ω 2 is a sequence of the last k − j + 1 elements of ω. Clearly, ω 1 , ω 2 ∈ Ω and the decomposition depends on the choice of i and j. This procedure can be iterated, and any sequence ω 1 ∪ · · · ∪ ω d obtained in this way is called a derived from ω sequence of words. Let u appear u(ω) times andū appearū(ω) times in a word ω.
Analogously to the proof of Lemma 3.2 in [9] one proves the following version of the shadowing chain lemma.
One can easily prove the following proposition. We can now formulate our main result.
• If (B1) holds, then for every k > m 0 there exists P k ∈ Γ µ k such that I(P k ) = λ µ k . Moreover, P k is a homoclinic solution of (HS). • If (B2) holds, then for every k > n 0 there exists Q k ∈ Γ ν k such that I(Q k ) = λ ν k . Moreover, Q k is a homoclinic solution of (HS).
In fact, every loop q satisfying rot ξ (q) ≥ 2 contains a subloop with the rotation number equal to 1. The proof of this topological property will be given in the appendix.
Sketch of the proof of Theorem 2.9. Let us observe that in case (B1) a closed orbitP existing by Proposition A.3 (see the appendix) can be chosen in such a way that rot a (P ) = 0, whereas in case (B2), there is a closed orbitQ such that rot a (Q) = 1. Lemma 4.2 of [5] implies the following inequalities: and (2.2) M. Izydorek and J. Janczewska For a fixed n > m 0 consider a derived from µ n sequence of words Then, for any i = 1, . . . , d, we have ω i = µ ji for some j i < n and ∑ d i=1 j i = n. Thus, by (2.1), Now the existence of a homoclinic solution P n ∈ Γ µn of (HS) follows from Theorem 2.7. Similar argumentation applied to ν n , n > n 0 , together with (2.2) gives the existence of a homoclinic solution Q n ∈ Γ νn of (HS).
Observe that one family of solutions is represented by words containing u's and v's, whereas another family is represented by words containing u's and w's. It would be interesting to know if, except the above quite specific families of solutions, there exist solutions represented by more "complicated" words, in particular, words containing three letters. To this purpose let us modify condition (B). Assume that (B ′ ) for i = 1, 2 there exist T i ∈ (0, ∞) and such that p 1 (0) = p 1 (T 1 ) = p 2 (0) = p 2 (T 2 ) and (B ′ 1) rot a (p 1 ) = 0, rot ξ (p 1 ) = 1 and Define two sequences of words in Ω as follows. For n ∈ N, • τ n = uwuv . . . u consists of 2n − 1 letters, u appears at every second place,w andv appear at every fourth place, • σ n = uvuw . . . u is obtained from τ n by interchangingw withv. Clearly, ρ τn = ρ σn = n, hence λ τn ≤ nλ u and λ σn ≤ nλ u for n ∈ N. Set k τ = sup{n : λ τn = nλ u } and k σ = sup{n : λ σn = nλ u }.
Proposition 2.12. If (B ′ ) is satisfied, then both numbers k τ and k σ are finite.
The proof is straightforward.
In addition to the families of solutions given by Theorem 2.9, there exists Q ∈ Λ τ k for which I(Q) = λ τ k , where k = k τ + 1. Moreover, Q is a homoclinic solution of (HS).
Vol. 16 (2014) Two families of homoclinics 309 Two families of homoclinics 9 Proof. Set k = k τ + 1. The inequality λ τ k < ρ τ k λ u follows from the definition of k τ . It is enough to show that for any derived from τ k sequence of words ω 1 ∪ · · · ∪ ω d with d > 1 one has Then the conclusion follows from Theorem 2.7. If ω 1 ̸ = u, then (2.3) follows from Corollary 2.6 applied to the word τ k . If ω 1 = u, then (2.3) is a consequence of Corollary 2.6 applied to σ kσ .