Melnikov function and Transversality of Manifolds

To prove the main results in the last subsection , we first detect the transversality of invariant manifolds for connecting orbits by investigating the Melnikov function.

First, we recall the results for the formula of Melnikov function, [5, 20, 27, 34].

Lemma 3.8 Consider the plane system:

y⁰= R0(y) + ¯εR1(y, ¯ε), (3.23)

where ¯ε≥ 0 and R⁰, R₁ ∈ C^r with r≥ 2. Suppose that y0¹ and y²₀ are two different hyperbolic saddle points of (3.23)|^ε=0¯ , and there exists a heteroclinic orbit y₀(t) of (3.23)|^¯ε=0connecting from y¹₀ to y₀². Then the Melnikov function of (3.23) is

M(y⁰) = Z ∞

−∞

e^{−R0tσ(s)ds}D(t)dt (3.24)

where σ(t) = tr^∂R_∂y⁰(y₀(t)) and D(t) = R₀(y₀(t))∧ R¹(y₀(t), 0).

According to formula (3.24), we can compute the Melnikov function of system (3.17) in the fol-lowing.

Lemma 3.9 Suppose, for some c₀ and w₀, system (3.17) has a heteroclinic orbit Γ := r(t; w₀) = (u₀(t; w₀), v₀(t; w₀)).

(1) For fixed w = w0 and varying c, the Melnikov function with respect to the heteroclinic orbit Γ is given by

M(c⁰) = Z ∞

−∞

e^−c0tv₀²(t; w0)dt.

In particular,M(c⁰)6= 0.

(2) For fixed c = c0 and varying w, the Melnikov function is given by M(w⁰) = a⁰(w0)

Z ∞

−∞

e^−c0tv0(t; w0)u0(t; w0)(1− u⁰(t; w0))dt. (3.25)

Proof. (1) For fixed w = w0, let F (u, v; c) be the vector field of system (3.17), i.e., F (u, v; c) =

Base on the results of Lemma 3.9, we now compute the first order Melnikov function of system (3.17) when w varies in different parameter regions.

Lemma 3.10 Assume a⁰(w0) 6= 0, then M(w⁰) 6= 0 for any c⁰ ∈ (0, 1/√

2c0∈ (1, 2). According to Remark 2.1, the heteroclinic orbit (u0(t; w0), v0(t; w0)) can be represented explicitly in the following:

M(w⁰)

where B(x, y) and Γ(x) are the Beta function and the Gamma function respectively. Note that B(x, y) = Γ(x)Γ(y)/Γ(x + y).

(3) If w0∈ H³(c0) then a(w0) = 2 +√

2c0∈ (2, 3). Similar to the proof of part (2), the heteroclinic orbit (u0(t; w0), v0(t; w0)) satisfies

2c0 ∈ (−2, −1) and the heteroclinic orbit (u0(t; w0), v0(t; w0)) satisfies

where−1 < γ < 0. Then we have

M(w⁰) = a⁰(w₀)a(w₀)(1− a(w⁰))²Γ(1 + γ)Γ(2− γ)/Γ(4) < 0.

The proof is complete.

However, if a⁰(w₀) = 0 in Lemma 3.10 then M(w⁰) = 0, and there give no information about the transversality of the invariant manifolds. Therefore we need to compute the higher order term of Melnikov function to detect the transversality of the invariant manifolds. The following we only investigate the second-order term of Melnikov functionM²(w0).

Lemma 3.11 Suppose, for some small c0 and w0, a⁰(w0) = 0 and system (3.17) has a heteroclinic

Proof. Without lost of generality, we may assume w₀= 0. For such fixed w near w₀, let’s write the system (3.17) in the following vector form

d

As w = 0, system (3.17) has a heteroclinic orbit r(t; w) connecting two equilibria, E₀¹ and E₀². Let L be a line segment transversal to r(t; 0) at r(0; 0). For sufficiently small w, there exists a unique bounded solution r^u(t; w) for t≤ 0 such that r^u(t; w) in the unstable manifold of one equilibrium E_w¹ and r^u(0; w)∈ L. For t ≤ 0, let’s define

z^u(t) := ∂

∂wr^u(t; w)|^w=0, 4^u(t) := z^u(t)∧ R⁰(r^u(t; 0)),

∂²

Differentiating equation (3.26) with respect to w, we have

Similarly, there exists a unique bounded solution r^s(t; w) for t≥ 0 such that r^s(t; w) in the unstable manifold of the other equilibrium E_w² and r^s(0; w)∈ L. For t ≥ 0, let’s define

By the proof of Lemma 3.10 and Lemma 3.11, we have the following corollary.

Corollary 3.12 Let’s assume the same assumptions as stated in Lemma 3.11. If a⁰⁰(w0)6= 0 then M²(w0)6= 0 for any c⁰∈ (0, 1/√

2) and w0∈ Hⁱ(c0), i = 1.· · · 6.

For more higher order terms of Melnikov function, the computation is similar but more complicated.

The following we only illustrate the general result, and skip the proof.

Lemma 3.13 Suppose, for some small c₀ and w₀, a⁽ⁱ⁾(w₀) = 0 for all 1≤ i < k, where k is a given positive integer, and system (3.17) has a heteroclinic orbit Γ₀:= (u₀(t; w₀), v₀(t; w₀)). For fixed c = c₀ and varying parameter w, the kth order Melnikov function is given by

M^k(w0) = a^(k)(w0) Z ∞

−∞

e^−c0tv0(t; w0)u0(t; w0)(1− u⁰(t; w0))dt.

As a consequence of previous lemmas and corollary, we have the following conclusions for the transver-sality of invariant manifolds.

Lemma 3.14 Let M t N be in the sense that manifolds M and N intersect transversally, and Cδ(c) := (c− δ, c + δ).

(1) Consider system (3.17) with c = c₀∈ (0, 1/√

2). The transversality of various invariant mani-folds of (3.17) along Γ(w) are illustrated in Table 2.

Region of w transversality of manifolds along Γ(w) w∈ H¹(c0) W₀^u(M₀^δ(w)) t W0^c(M₁^δ(w)) w∈ H²(c0) W₀^u(M₁^δ(w)) t W0^c(M₀^δ(w)) w∈ H³(c0) W₀^u(M_a^δ(w)) t W0^c(M₀^δ(w)) w∈ H⁴(c0) W₀^u(M₁^δ(w)) t W0^c(M_a^δ(w)) w∈ H⁵(c0) W₀^u(M_a^δ(w)) t W0^c(M₀^δ(w)) w∈ H⁶(c0) W₀^u(M_a^δ(w)) t W0^c(M₁^δ(w)) w∈ G¹(c0) W₀^cu(0, 0, w) t W0^c(M₁^δ(w)) w∈ G²(c0) W₀^cu(1, 0, w) t W0^c(M₀^δ(w)) w∈ G³(c0) W₀^cu(0, 0, w) t W0^c(M_a^δ(w)) w∈ G⁴(c0) W₀^cu(1, 0, w) t W0^c(M_a^δ(w))

Table 2: Transversalities of manifolds W₀^c(M_0,1,a^δ (w)), W₀^u(M_0,1,a^δ (w)), W₀^cu(0, 0, w) and W₀^cu(1, 0, w).

(2) Consider (2.10) with c = c0 ∈ (0, 1/√

2). The transversality of various invariant manifolds of

Region of w transversality of manifolds along Γ(w)× {c⁰} w∈ H¹(c0) W₀^u(M₀^δ(w))× {c⁰} t W0^c(M₁^δ(w)× C^δ(c0)) w∈ H²(c₀) W₀^u(M₁^δ(w))× {c⁰} t W0^c(M₀^δ(w)× C^δ(c₀)) w∈ H³(c0) W₀^u(M₀^δ(w)× {c⁰}) t W0^c(M_a^δ(w)× C^δ(c0)) w∈ H⁴(c₀) W₀^u(M₁^δ(w))× {c⁰} t W0^c(M_a^δ(w)× C^δ(c₀)) w∈ H⁵(c0) W₀^u(M_a^δ(w)× {c⁰}) t W0^c(M₀^δ(w)× C^δ(c0)) w∈ H⁶(c₀) W₀^u(M_a^δ(w))× {c⁰} t W0^c(M₁^δ(w)× C^δ(c₀)) w∈ G¹(c0) W₀^cu(0, 0, w, c0) t W0^c(M₁^δ(w)× C^δ(c0)) w∈ G²(c₀) W₀^cu(1, 0, w, c₀) t W0^c(M₀^δ(w)× C^δ(c₀)) w∈ G³(c0) W₀^cu(0, 0, w, c0) t W0^c(M_a^δ(w)× C^δ(c0)) w∈ G⁴(c₀) W₀^cu(1, 0, w, c₀) t W0^c(M_a^δ(w)× C^δ(c₀))

Table 3: Transversalities of manifolds W₀^c(M_0,1,a^δ (w)× C^δ(c₀)), W₀^u(M_0,1,a^δ (w))× {c⁰}, W₀^cu(0, 0, w, c0) and W₀^cu(1, 0, w, c0).

Now we begin the proof of the main theorems.

Proof of Theorem 3.5.

We only prove the first part of the theorem. The proof for the second part of the theorem is the same as the proof of Theorem 2.3 of [33] and omitted.

Assume w = (w₁, ..., w_n) is s-admissible with respect to c^∗, where s = (s₁, ..., s_n+1). We first claim that the singular local orbit 0−−−→ 1^w=0 −−−−→ s^{w=w1 2} is weakly shadowed by a true local orbit.

According to Table 1 and the assumption a(0) = a₁(c^∗), we know that 0∈ H¹(c^∗) and there exists a singular local orbit 0−→ 1 at w = 0. For the following local path 1 → s², the admissible conditions lead to s2= 0 or a. Therefore, there exists a singular local orbit 1−→ s²at w = w1if w1∈ H²(c^∗)∪ G²(c^∗) or H4(c^∗)∪ G⁴(c^∗) (in fact, w1∈ H²(c^∗) or H4(c^∗)). Take M⁰= W₀^u((0, 0, 0)× C^δ(c^∗)). According to Lemma 3.14, we have

M⁰t W0^c(M₁^δ(0)× C^δ(c^∗)).

Let N0 be their intersection. Since the phase space of system (2.10) is R⁴and dimensions of M⁰and W₀^c(M₁^δ(0)×C^δ(c^∗)) are 2 and 3 respectively, then dimN₀= 2 + 3−4 = 1. We now apply the exchange lemma 3.2 to the vicinity of the slow manifold M₁× C^δ(c^∗) along the slow orbit from (1, 0, 0, c^∗) to (1, 0, w₁, c^∗). Taking M^ε = W_ε^u((0, 0, 0)× C^δ(c^∗)) be such that M^ε → M⁰ as ε → 0. By condition (A1) and part (1) of Lemma 3.2, a portion M_p^ε

0 of M^εwill proceed near the singular orbit and leave

the vicinity of slow orbit close to W_ε^u(M₁^δ(w1)× C^δ(c^∗)). Note that dimW₀^u(M₁^δ(w1)× C^δ(c^∗)) = 3.

Thus, the singular orbit 0−−−→ 1^w=0 −−−−→ s^{w=w1 2} is weakly shadowed by a true orbit.

Next, we claim that the singular local orbit 0−−−→ 1^w=0 −−−−→ s^{w=w1 2}−−−−→ s^{w=w2 3} is weakly shadowed by a true local orbit. Two cases for s₂ = 0 or a are considered. For the case s₂ = 0, a portion M_p^ε

1 of W_ε^u((a(w₁), 0, w₁)× C^δ(c^∗)) will proceed near the singular orbit and leave the vicinity of slow orbit close to W_ε^u((s₃, 0)× (w²− δ, w²+ δ)× C^δ(c^∗)) or W_ε^cu((s₃, 0)× {w²} × {c^∗}). The details of proof can be found in [33] by using the Exchange Lemma with turning point (Lemma 3.2) and omitted. For the case s2= a, the admissible conditions imply s3= 1 or 0. Thus there exists a singular local orbit s2−→ s³ at w = w2 if w2∈ H⁵(c^∗) or H6(c^∗). From the admissible condition (A4), there is no turning point lying between w1 and w2. It’s also easy to see that W₀^u((1, 0)× (w¹− δ, w¹+ δ)× (C^δ(c^∗)) and W₀^c((a(w1), 0)× (w¹− δ, w¹+ δ)× C^δ(c^∗)) intersect transversally. Then, by Lemma 3.3, a portion M_p^ε₁ of W_ε^u((a(w1), 0)× (w¹− δ, w¹+ δ)× C^δ(c^∗)) will proceed near the singular orbit and leave the vicinity of slow orbit close to W_ε^u((s3, 0)× (w²− δ, w²+ δ)× C^δ(c^∗)). Note that dimW₀^u(M₁^δ(w1)× C_δ(c^∗)) =dimW₀^cu((s₃, 0)× {w²} × C^δ(c^∗)) = 3.

According to the above discussions, we can conclude inductively that the singular local orbit 0−−−→ 1^w=0 −−−−→ · · ·^w=w1 −−−−−−→ s^{w=wn−1 n}is weakly shadowed by a true local orbit. Generally, for 2 < i < (n+1), we consider the following two cases.

(1) Assume there exists a turning point lying between wi−1 and wi. By Lemma 3.2, a portion M_p^ε_i−1 of W_ε^u((si, 0, wi−1)× C^δ(c^∗)) will proceed near the singular orbit and leave the vicinity of slow orbit close to W_ε^u((si+1, 0)× (wⁱ− δ, wⁱ+ δ)× C^δ(c^∗)) or W_ε^cu((si+1, 0, wi)× C^δ(c^∗)).

(2) Assume there is no turning point lying between wi−1and wi. By Lemma 3.3, a portion M_p^ε_i−1 of W_ε^u((si, 0, wi−1)× C^δ(c^∗)) will leave the vicinity of slow orbit close to W_ε^u((si+1, 0)× (wⁱ− δ, wⁱ+ δ)× C^δ(c^∗)).

Furthermore, we have dimW₀^u((s_i+1, 0)× (wⁱ− δ, wⁱ+ δ)× C^δ(c^∗)) = 3 and dimW₀^cu((s_i+1, 0, w_i)× C_δ(c^∗)) = 3.

Finally, we prove that the true orbits obtained by above arguments are C¹ O(ε)-close to the unstable manifold W₀^u((sn+1, 0, wn)× C^δ(c^∗)). Since sn+1 = 1, the admissible conditions lead to wn ∈ H¹(c^∗)∪ G¹(c^∗) or H6(c^∗). Thus, there exists a singular local orbit sn −→ 1 at w = wⁿ. By condition (A1), we have a(w) < 1 for all w∈ [wⁿ, 1] and the singular orbit will approach to (1, 0, 1) as time goes infinity. Moreover, W₀^u((sn, 0, wn)× {c^∗}) intersects W0^c(M₁^δ(wn)× C^δ(c^∗)) transversally.

As a result, the true orbit will approach a neighborhood of (1, 0, 1, c^∗), near the singular orbit and C¹

O(ε)-close to the unstable manifold W₀^u((sn+1, 0, wn)×C^δ(c^∗)). The proof of the theorem is complete.

Proof of Theorem 3.6.

The proof of Theorem 3.6 is similar to that of Theorem 3.5. Let w = (w1, ..., wn) is s-admissible for c = c^∗ and s = (s1, ..., sn+1). We first claim that the singular local orbit 0 −−−→ 1^w=0 −−−−→ s^{w=w1 2} is strongly shadowed by a true orbit.

According to Table 1 and the assumption a(0) ≤ a⁶(c^∗), we know that 0 ∈ G¹(c^∗) and there exists a singular local orbit 0 −→ 1 at w = 0. For the following local path 1 → s², the admissible conditions lead to s2 = 0 or a. Therefore, there exists a singular local orbit 1 −→ s² at w = w1 if w₁ ∈ H²(c^∗)∪ G²(c^∗) or H₄(c^∗)∪ G⁴(c^∗) (in fact, w₁∈ H²(c^∗) or H₄(c^∗)). Take M⁰= W₀^u(0, 0, 0).

According to Lemma 3.14, we have

M⁰t W0^c{M1^δ(0)}.

Let N0 be their intersection. Since the phase space of system (2.3) is R³ and both dimensions of M⁰ and W₀^c(M₁^δ(0)) are 2, then dimN0 = 2 + 2− 3 = 1. We now apply the exchange lemma 3.2 to the vicinity of the slow manifold M1 along the slow orbit from (1, 0, 0) to (1, 0, w1). Taking M^ε= W_ε^u((0, 0, 0)) be such that M^ε→ M⁰ as ε→ 0. By condition (A1) and part (1) of Lemma 3.2, a portion M_p^ε₀ of M^ε will proceed near the singular orbit and leave the vicinity of slow orbit close to W_ε^u(M₁^δ(w1)). Note that dimW₀^u(M₁^δ(w1)) = 2. Thus, the singular orbit 0 −−−→ 1^w=0 −−−−→ s^{w=w1 2} is strongly shadowed by a true orbit.

Next, we claim that the singular local orbit 0−−−→ 1^w=0 −−−−→ s^{w=w1 2}−−−−→ s^{w=w2 3} is strongly shadowed by a true local orbit. Two cases for s2 = 0 or a are considered. For the case s2 = 0, a portion M_p^ε₁ of W_ε^u((a(w₁), 0, w₁)) will proceed near the singular orbit and leave the vicinity of slow orbit close to W_ε^u((s₃, 0)×(w²−δ, w²+ δ)) or W_ε^cu((s₃, 0)×{w²}). The details of proof can be found in [33] by using the exchange lemma with turning point (Lemma 3.2) and omitted. For the case s₂= a, the admissible conditions imply s3= 1 or 0. Thus there exists a singular local orbit s2−→ s³at w = w2if w2∈ H⁵(c^∗) or H6(c^∗). From the admissible condition (A4), there is no turning point lying between w1 and w2. It’s also easy to see that W₀^u((1, 0)× (w¹− δ, w¹+ δ)) and W₀^c((a(w1), 0)× (w¹− δ, w¹+ δ)) intersect transversally. Then, by Lemma 3.3, a portion M_p^ε₁ of W_ε^u((a(w1), 0)× (w¹− δ, w¹+ δ)) will proceed near the singular orbit and leave the vicinity of slow orbit close to W_ε^u((s3, 0)× (w²− δ, w²+ δ)). Note that dimW₀^u(M₁^δ(w1)) =dimW₀^cu((s3, 0)× {w²}) = 2.

According to the above discussions, we can conclude inductively that the singular local orbit 0−−−→^w=0 1−−−−→ · · ·^w=w1 −−−−−−→ s^{w=wn−1 n} is strongly shadowed by a true local orbit. Generally, for 2 < i < (n + 1), we consider the following two cases.

(1) Assume there exists a turning point lying between wi−1 and wi. By Lemma 3.2, a portion M_p^ε_i−1 of W_ε^u((si, 0, wi−1)) will proceed near the singular orbit and leave the vicinity of slow orbit close to W_ε^u((si+1, 0)× (wⁱ− δ, wⁱ+ δ)) or W_ε^cu((si+1, 0, wi)).

(2) Assume there is no turning point lying between w_i−1and w_i. By Lemma 3.3, a portion M_p^ε

i−1

of W_ε^u((s_i, 0, w_i−1)) will leave the vicinity of slow orbit close to W_ε^u((s_i+1, 0)× (wⁱ− δ, wⁱ+ δ)).

Furthermore, we have dimW₀^u((s_i+1, 0)× (wⁱ− δ, wⁱ+ δ)) = dimW₀^cu((s_i+1, 0, w_i)) = 2.

Finally, we prove that the true orbits obtained by above arguments are C¹O(ε)-close to the unsta-ble manifold W₀^u((sn+1, 0, wn)). Since sn+1= 1, the admissible conditions lead to wn ∈ H¹(c^∗)∪G¹(c^∗) or H6(c^∗). Thus, there exists a singular local orbit sn −→ 1 at w = wⁿ. By condition (A1), we have a(w) < 1 for all w ∈ [wⁿ, 1] and the singular orbit will approach to (1, 0, 1) as time goes infinity.

Moreover, W₀^u((sn, 0, wn)) intersects W₀^c(M₁^δ(wn)) transversally. As a result, the true orbit will ap-proach a neighborhood of (1, 0, 1), near the singular orbit and C¹O(ε)-close to the unstable manifold W₀^u((sn+1, 0, wn)). The proof of the theorem is complete.

The results of Theorem 3.7 can also be proved in the same way and omitted.

4 Wazewski Theorem and Lasalle Invariance Principle

4.1 Wazewski theorem

We now recall a variant of the Wazewski Theorem which is a formalization and extension of the shooting method in higher dimension (see [9], p. 563). Let usconsider the differential equation:

y⁰(s) = f (y(s)) (4.27)

where f : Rⁿ → Rⁿ is a Lipschitz continuous function. Suppose y(s; y₀) is the unique solution of (4.27) with initial value y(0) = y0. For convenience, set y(s; y0) = y0· s and let Y · S be the set of points y· s, where y ∈ Y ⊂ Rⁿ and s∈ S ⊂ R. Given W ⊆ Rⁿ, define the immediate exit set W⁻ of W by

W⁻={y⁰∈ W : ∀s > 0, y⁰· [0, s) * W }.

Given Σ⊆ W , let

Σ⁰={y⁰∈ Σ : ∃s⁰> 0 such that y0· s⁰∈ W }./ For y₀∈ Σ, define the exit time T (y⁰) of y₀by

T (y₀) = sup{s : y⁰· [0, s] ⊂ W }.

Note that y0· T (y⁰)∈ W⁻ and T (y0) = 0 if and only if y0∈ W⁻. The Wazewkski Theorem is then stated as the following.

Theorem 4.1 Consider (4.27) and suppose that

(i) if y0∈ Σ and y⁰· [0, s] ⊆ c`(W ), then y⁰· [0, s] ⊆ W ;

(ii) if y0∈ Σ, y⁰· s ∈ W , y⁰· s /∈ W⁻, then there is an open set Vsabout y0· s disjoint from W⁻; (iii) Σ = Σ⁰, Σ is a compact set and intersects a trajectory of y⁰ = f (y) only once.

Then the mapping F (y₀) = y₀·T (y⁰) is a homeomorphism from Σ to its image on W⁻. A set W ⊆ Rⁿ satisfying the conditions (i) and (ii) is called a Wazewski set