On existence of infinitely many homoclinic solutions.

Klaudiusz Wójcik^{1}, Piotr Zgliczyński^{2} ^{3} September 2, 1999

Jagiellonian University, Institute of Mathematics, Reymonta 4, 30-059 Kraków, Poland. E-mail: wojcik@im.uj.edu.pl, zgliczyn@im.uj.edu.pl

Abstract

In this note we prove some sufficient condition for the existence of homoclinic solutions in nonautonomous ODE’s. As an application we show that there exist infinitely many (geometrically distinc) homoclinic solutions to the trivial 0 solution in the planar system

ż = z̄(1 + e^{iκt}|z|^{2})

for 0 < κ sufficiently small.

Keywords: isolating segments, fixed point index, homoclinic solutions

1 Introduction

The concept of an periodic isolating segment was introduced by Srzednicki (see [S1]) to study the existence of periodic solutions for a time periodic nonautonomous ordinary differential equation. In concrete applications the isolating segments are manifolds with corners in the extended phase space, such that at any point at the boundary of the segment the vector field is directed either outward or inward with respect of the segment (compare the notion of isolating block in the Conley index theory [C]). In [SW] a new method of detection of chaos in dynamical systems generated by nonautonomous ODE’s was introduced. By chaos we mean the existence of a compact invariant set such that the Poincaré map is semiconjugated to the shift on k symbols and the counterimage by the semiconjugacy of any periodic point in the shift contains a periodic point of the Poincaré map. The method presented in [SW] is based on the Ważewski Retract Theorem (see [C]) and the Lefschetz Fixed Point Theorem (see [D]). The more complete description of the dynamics was obtained in [WZ] by adapting the result of [Z1] to the context of [SW]. The continuation theorem proved in [WZ] enables us to dig deeper in the structure of periodic solutions than the Lefschetz Fixed Point Theorem used originally in [SW]. This continuation result allows us to to prove that the topological model continues to that of the original equation. In this paper we show how by using isolating segments one can extract the existence of homoclinic solutions in nonautonomous ODE’s.

1partially supported by Polish KBN grant 2 P 03A 028 17

2supported by Polish grant 2 P03A 021 15, 2P03A 029 12

3current addres:CDSNS, Georgia Institute of Technology, Atlanta, GA 30322 USA

2 Proper pairs.

Let f : ℝ × ℝ^{n} → ℝ^{n} be C^{1} time dependent vector field. The system

(∗) ẋ = f(t, x)

generates a local process φ as follows. For x(t_{0}, x_{0}; ·) the solution of (∗) such that x(t_{0}, x_{0}; t_{0}) = x_{0} we put

φ_{(t_{0},τ)}(x_{0}) = x(t_{0}, x_{0}; t_{0} + τ).

If f is T -periodic with respect to t then φ is a T -periodic local process and in order to determine all T -periodic solutions of the equation (∗) it suffices to look for fixed points of φ_{(0,T)} (called the Poincaré map).

We will use the following notation: by π_{1} : ℝ × ℝ^{n} → ℝ and π_{2} : ℝ × ℝ^{n} → ℝ^{n} we denote the projections and for every Z ⊂ ℝ × ℝ^{n} and t ∈ ℝ we put

Z_{t} = {x ∈ ℝ^{n} : (t, x) ∈ Z}.

Let (U, U^{−}) be a pair of subsets of ℝ × ℝ^{n} (i.e. U^{−} ⊂ U ⊂ ℝ × ℝ^{n}) and (B, B^{−}) be a pair of subsets of ℝ^{n}. We call (U, U^{−}) a proper pair (for the process φ) with the base (B, B^{−}) if:

(i) B and B^{−} are compact ENR’s,

(ii) there exists a homeomorphism

h : ℝ × (B, B^{−}) ⟶ (U, U^{−})

such that π_{1} = π_{1} ∘ h,

(iii) for every σ ∈ ℝ and x ∈ ∂U_{σ} there exists a t ∈ ℝ such that φ_{(σ,t)}(x) ∉ U_{σ+t},

(iv) U^{−} = {(σ, x) ∈ U : ∃s_{n} > 0 s_{n} → 0 : φ_{(σ,s_{n})}(x) ∉ U_{σ+s_{n}}}.

Let (U, U^{−}) be a proper pair, U_{a} = U_{b}, U_{a}^{−} = U_{b}^{−} for some a, b ∈ ℝ, a < b. Define a homeomorphism

h_{a,b} : (U_{a}, U_{a}^{−}) ⟶ (U_{b}, U_{b}^{−}) = (U_{a}, U_{a}^{−})

by h_{a,b}(x) = π_{2}(h(b, π_{2}h^{−1}(a, x))) for x ∈ U_{a}.

Geometrically, h_{a,b} moves a point x ∈ U_{a} to U_{b} = U_{a} along the arc h([a, b] × {π_{2}h^{−1}(a, x)}). Consider the automorphism

μ_{U_{a,b}} : H(U_{a}, U_{a}^{−}) ⟶ H(U_{a}, U_{a}^{−})

induced by h_{a,b} in the singular homology with rational coefficients. Recall that its Lefschetz number is defined as

Lef(μ_{U_{a,b}}) = ∑_{n=0}^{∞}(−1)^{n}tr H_{n}(h_{a,b}).

In particular, if μ_{U_{a,b}} = id_{H(U_{a},U_{a}^{−})} then Lef(μ_{U_{a,b}}) is equal to the Euler characteristic χ(U_{a}, U_{a}^{−}).

3 Main result.

The following result, up to details was proved in [PW] (see [SW], [WZ] for the periodic case)

Lemma 1 Let (Z, Z^{−}), (U, U^{−}) be proper pairs with the same base (B, B^{−}) such that U ⊂ Z. Assume that there are a_{1} < ... < a_{l} ∈ ℝ such that for all i, j ∈ {1, ..., l}

(U_{a_{i}}, U_{a_{i}}^{−}) = (Z_{a_{i}}, Z_{a_{i}}^{−}) = (U_{a_{j}}, U_{a_{j}}^{−}).

If moreover

(I) there is t_{Z} ∈ ℝ_{+} such that |a_{i}| < t_{Z} (i = 1, ..., l) and for all |t| ≥ t_{Z}

(U_{t}, U_{t}^{−}) = (Z_{t}, Z_{t}^{−}) = (B, B^{−}),

(II) there exists the authomorphism F : H(Z_{a_{1}}, Z_{a_{1}}^{−}) → H(Z_{a_{1}}, Z_{a_{1}}^{−}) such that

F = μ_{Z_{a_{i},a_{i+1}}} ,
F^{2} = id_{H(Z_{a_{1}} ,Z_{a_{1}}^{−})} = μ_{U_{a_{i},a_{i+1}}} ,

for i = 1, ..., l − 1.

(III) Lef(F) ≠ χ(B, B^{−}),

(IV) μ_{U_{a_{l},t_{Z}}} ∘ μ_{U_{−t_{Z} ,a_{1}}} = id_{H(B,B^{−})},

then there exists x ∈ U_{a_{1}} such that for all t ∈ ℝ φ_{(a_{1},t−a_{1})}(x) ∈ Z_{t} and for any i ∈ {1, ..., l − 1} there is s_{i} ∈ (a_{i}, a_{i+1}) such that φ_{(a_{1},s_{i}−a_{1})}(x) ∉ U_{s_{i}}.

The main result of this section is

Proposition 2 Assume that there are the proper pairs (Z_{n}, Z_{n}^{−}), (U_{n}, U_{n}^{−}) (n ∈ ℕ) with the bases (B_{n}, B_{n}^{−}) which fulfill assumptions (I), (II), (III) and (IV) for n ∈ ℕ with the same a_{1} < ... < a_{l} ∈ ℝ. Suppose that the equation (∗) has trivial 0 solution which is contained in Z_{n} for n ∈ ℕ. If

(A) Z_{n+1} ⊂ Z_{n} for n ∈ ℕ,

(B) sequence t_{Z_{n}} increase to +∞,

(C) if |t| < t_{Z_{n}} then

((Z_{n+1})_{t}, (Z_{n+1}^{−})_{t}) = ((Z_{n})_{t}, (Z_{n}^{−})_{t}),
((U_{n+1})_{t}, (U_{n+1}^{−})_{t}) = ((U_{n})_{t}, (U_{n}^{−})_{t}),

(D) diam(B_{n}) decrease to 0,

then there exists x ∈ U_{a_{1}} such that solution through x is homoclinic to 0 solution, for all n ∈ ℕ is contained in Z_{n} and for i ∈ {1, ..., l − 1} there is s_{i} ∈ (a_{i}, a_{i+1}) such that φ_{(a_{1},s_{i}−a_{1})}(x) ∉ (U_{n})_{s_{i}}.

Proof: It follows by Lemma 1 that there is a sequence x_{n} ∈ (U_{n})_{a_{1}} such that for t ∈ ℝ φ_{(a_{1},t−a_{1})}(x_{n}) ∈ (Z_{n})_{t} and for i ∈ {1, ..., l − 1} there is s_{i} ∈ (a_{i}, a_{i+1}) such that φ_{(a_{1},s_{i}−a_{1})}(x_{n}) ∉ (U_{n})_{s_{i}}. Since for m, n ∈ ℕ (U_{n})_{a_{1}} = (U_{m})_{a_{1}} is compact, so we can assume that there is x ∈ (U_{n})_{a_{1}} such that x_{n} → x. By assumption (D) φ_{(a_{1},t)}(x) → 0 if |t| → +∞, so the result follows.

4 A Chaotic Equation

In the present section we consider the planar nonautonomous equation of the variable z ∈ ℂ

ż = (1 + e^{iϕt}|z|^{2})z̄,  (1)

for some ϕ ∈ ℝ.

Theorem 3 The equation (1) has infinitely many geometrically distinc solutions homoclinic to 0 solution for sufficiently small parameter values 0 < ϕ.

The reminder of this section will be devoted to presentation of a proof of the above theorem. Our proof of Th.3 consists in the construction of proper pairs (Z_{n}, Z_{n}^{−}), (U_{n}, U_{n}^{−}) satisfying the conditions (A) − (D) in Prop.2. In [SW] (see [WZ]) was proved that for sufficiently small 0 < ϕ there are a compact set I ⊂ ℝ^{2} invariant with respect to the Poincaré map and a continuous surjective map g : I → Σ_{2} = {U, W }ℤ such that the Poincaré map φ_{(0,T)} is semiconjugated to the shift map in the set I. The proof of this result is based on the constructions of two periodic isolating segments U and W satisfying assumptions of Th.2 in [SW]. The segment W is a twisted prism with the square base centered at origin, with the side of lenght 6. The segment U is a regular square-based prism with broadening ends. Its cross sections U_{t} corresponding to t near the center of the interval [0, \frac{2π}{ϕ}] have the side of lenght \frac{2}{3} and they are broadended to the lenght 6 when t approaches to 0 or \frac{2π}{ϕ}. Notice that a proper pair can be easily obtained by gluing translated copies of a periodic isolating segments.

Let c ∈ Σ_{2} be such that the symbol W appears finite times in c. It suffices to prove that g^{−1}(c) contains a solution homoclinic to 0 solution. Let a_{1} < ... < a_{l} ∈ ℤ be such that the symbol W appears in c on the positions a_{i}.

Put r_{1} = 1/3 and

B_{r_{1}} = {(x, y) ∈ ℝ^{2} : |x| ≤ r_{1}, |y| ≤ r_{1}},
B_{r_{1}}^{−} = {(x, y) ∈ B_{1} : |x| = r_{1}}.

It follows by result of [SW] (see also section 5 in [WZ]) that there are proper pairs (Z_{1}, Z_{1}^{−}), (U_{1}, U_{1}^{−}) with the base (B_{r_{1}}, B_{r_{1}}^{−}) for which all assumptions of lemma 1 hold. Suppose that Z_{1},...,Z_{n} and U_{1},...,U_{n} are defined. We have to construct the proper pairs (Z_{n+1}, Z_{n+1}^{−}) and (U_{n+1}, U_{n+1}^{−}).

The equation (1) coincides with the system of two planar equations:

\{ \begin{matrix}ẋ = x + (x^{2} + y^{2})(x cos(ϕt) + y sin(ϕt)), \\ ẏ = −y + (x^{2} + y^{2})(x sin(ϕt) − y cos(ϕt)). \end{matrix}  (2)

By F we denote the vector-field in the extended phase space ℝ^{3} generated by the right-hand side of the system (2), i.e.

F (t, x, y) = ( \begin{matrix}1 \\ x + (x^{2} + y^{2})(x cos(ϕt) + y sin(ϕt)) \\ −y + (x^{2} + y^{2})(x sin(ϕt) − y cos(ϕt)) \end{matrix} )

Let r > 0 and put

Ξ_{r}^{1}(t, x, y) = \frac{1}{r^{2}} x^{2} − 1,

Ξ_{r}^{2}(t, x, y) = \frac{1}{r^{2}} y^{2} − 1,

and

K_{r} = {(t, x, y) ∈ ℝ^{3} : Ξ_{r}^{i}(t, x, y) ≤ 0, i = 1, 2},
K_{r}^{−} = {(t, x, y) ∈ ℝ^{3} : Ξ_{r}^{1}(t, x, y) = 0, Ξ_{r}^{2}(t, x, y) ≤ 0},
K_{r}^{+} = {(t, x, y) ∈ ℝ^{3} : Ξ_{r}^{1}(t, x, y) ≤ 0, Ξ_{r}^{2}(t, x, y) = 0}.

Lemma 4 For an arbitrary ϕ > 0,

F (t, x, y) · ∇Ξ_{r}^{1}(t, x, y) > 0 (r ≤ \frac{1}{3}, (t, x, y) ∈ K_{r}^{−}),  (3)
F (t, x, y) · ∇Ξ_{r}^{2}(t, x, y) < 0 (r ≤ \frac{1}{3}, (t, x, y) ∈ K_{r}^{+}).  (4)

Proof of Lemma 4: This is Lemma 3 in [SW]. Let a > 0 be another parameter. For |t| ∈ [t_{n}, 2t_{n}] put

Π_{a}^{1}(t, x, y) = \frac{1}{(r_{n} + at_{n} − a|t|)^{2}} x^{2} − 1,

Π_{a}^{2}(t, x, y) = \frac{1}{(r_{n} + at_{n} − a|t|)^{2}} y^{2} − 1.

and

P_{a} = {(|t|, x, y) ∈ [t_{n}, 2t_{n}] × ℝ^{2} : Π_{a}^{i}(t, x, y) ≤ 0, i = 1, 2},
P_{a}^{−} = {(|t|, x, y) ∈ [t_{n}, 2t_{n}] × ℝ^{2} : Π_{a}^{1}(t, x, y) = 0, Π_{a}^{2}(t, x, y) ≤ 0},
P_{a}^{+} = {(|t|, x, y) ∈ [t_{n}, 2t_{n}] × ℝ^{2} : Π_{a}^{1}(t, x, y) ≤ 0, Π_{a}^{2}(t, x, y) = 0}.

Lemma 5 Assume that a = \frac{r_{n}}{2t_{n}}. Then

F (t, x, y) · ∇Π_{a}^{1}(t, x, y) > 0 ((t, x, y) ∈ P_{a}^{−}),  (5)
F (t, x, y) · ∇Π_{a}^{2}(t, x, y) < 0 ((t, x, y) ∈ P_{a}^{+}).  (6)

Proof of Lemma 5: Assume that t ∈ [t_{n}, 2t_{n}].

F (t, x, y) · ∇Π_{a}^{1}(t, x, y) = \frac{2ax^{2}}{(r_{n} + at_{n} − at)^{3}} +

\frac{2}{(r_{n} + at_{n} − at)^{2}} (x^{2} + (x^{2} + y^{2})(x^{2} cos(ϕt) + xy sin(ϕt))) ,

F (t, x, y) · ∇Π_{a}^{2}(t, x, y) = \frac{2ay^{2}}{(r_{n} + at_{n} − at)^{3}} +

\frac{2}{(r_{n} + at_{n} − at)^{2}} (−y^{2} + (x^{2} + y^{2})(xy sin(ϕt) − y^{2} cos(ϕt))) .

Let (t, x, y) ∈ P_{a}^{−}, hence |x| = r_{n} + at_{n} − at and |y| ≤ r_{n} + at_{n} − at. It follows that

F (t, x, y) · ∇Π_{a}^{1}(t, x, y) > 2 − 8(r_{n} + at_{n} − at)^{2} ≥ 2 − 8r_{n}^{2} ≥ 0,

since r_{n} < r_{1} = 1/3. By a similar argument we conclude that if (t, x, y) ∈ P_{a}^{+} then

F (t, x, y) · ∇Π_{a}^{2}(t, x, y) < \frac{4a}{r_{n}} − 2 + 8r_{n}^{2} ≤ −1 + 8(1/3)^{2} < 0.

Put t_{n+1} = 2t_{n}, a = \frac{r_{n}}{2t_{n}} and r_{n+1} = \frac{r_{n}}{2} . Define Z_{n+1} and U_{n+1} by ((Z_{n+1})_{t}, (Z_{n+1})_{t}^{−}) = ((Z_{n})_{t}, (Z_{n})_{t}^{−}),

((U_{n+1})_{t}, (U_{n+1})_{t}^{−}) = ((U_{n})_{t}, (U_{n})_{t}^{−}),

if |t| ≤ t_{n},

((Z_{n+1})_{t}, (Z_{n+1})_{t}^{−}) = ((U_{n+1})_{t}, (U_{n+1})_{t}^{−}) = (B_{r_{n+1}}, B_{r_{n+1}}^{−}),

if |t| ≥ t_{n+1},

((Z_{n+1})_{t}, (Z_{n+1})_{t}^{−}) = ((U_{n+1})_{t}, (U_{n+1})_{t}^{−}) = (B_{r_{n}+at_{n}−a|t|}, B_{r_{n}+at_{n}−a|t|}^{−}),

if |t| ∈ (t_{n}, t_{n+1}). It follows by the previous lemmas that all assumptions of Prop.1 hold.

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