TRANSVERSAL HOMOCLINICS IN NONLINEAR

SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Michal Fečkan

Department of Mathematical Analysis Comenius University, Mlynská dolina 842 48 Bratislava, Slovakia E-mail: Michal.Feckan@ fmph.uniba.sk

Abstract. Bifurcation of transversal homoclinics is studied for a pair of ordinary differential equations with periodic perturbations when the first unperturbed equation has a manifold of homoclinic solutions and the second unperturbed equation is vanishing. Such ordinary differential equations often arise in perturbed autonomous Hamiltonian systems.

1. Introduction

Let us consider the system of ordinary differential equations given by

(1.1)  \begin{matrix}ẋ = f(x, y) + ϵh(x, y, t, ϵ), \\ ẏ = ϵ(Ay + g(y) + p(x, y, t, ϵ) + ϵq(y, t, ϵ)) , \end{matrix}

where x ∈ ℝ^{n}, y ∈ ℝ^{m}, ϵ ≠ 0 is sufficiently small, A is an m × m matrix, and all mappings are smooth, 1-periodic in the time variable t ∈ ℝ and such that

(i) f(0, ·) = 0, g(0) = 0, g_{y}(0) = 0, p(0, ·, ·, ·) = 0. Here g_{y} means the derivative of g with respect to y. Similar notations are used below .

(ii) The eigenvalues of A and f_{x}(0, ·) lie off the imaginary axis .

(iii) There exists a smooth mapping γ(θ, y, t) ≠ 0, where θ ∈ ℝ^{d−1}, d ≥ 1 and y is small, such that

γ̇(θ, y, t) = f(γ(θ, y, t), y), γ(θ, y, t) = O(e^{−c_{1}|t|})
γ_{y}(θ, y, t) = O(e^{−c_{1}|t|}), γ_{yy}(θ, y, t) = O(e^{−c_{1}|t|})

for a constant c_{1} > 0, and uniformly for θ, y. Moreover, we suppose

d = dim W^{s}(y) ∩ W^{u}(y) = dim T_{γ(θ,y,t)}W^{s}(y) ∩ T_{γ(θ,y,t)}W^{u}(y) . EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 9, p. 1 Here W^{s(u)}(y) is the stable (unstable) manifold to x = 0 of ẋ = f(x, y), respectively, and T_{z}W^{s(u)}(y) is the tangent bundle of W^{s(u)}(y) at z ∈ W^{s(u)}(y), respectively.

This work supported by Grant GA-MS 1/6179/99. This paper is in final form and no version of it will be submitted for publication elsewhere

Consequently, assumption (iii) means that equation ẋ = f(x, y) has a nondegenerate homoclinic manifold [5,7,10]

W_{h}(y) = W^{s}(y) ∩ W^{u}(y) = {γ(θ, y, t) | θ ∈ ℝ^{d−1}, t ∈ ℝ} .

We suppose that \overline{W}_{h}(y) are compact. We are interested in homoclinic solutions of (1.1) near the family W_{h}(y). Moreover, we search for transversal such solutions to show chaos for (1.1) [2,5,10].

Systems like (1.1) are investigated in [3], where the existence of chaos is proved, but the situation of this note is not included in [3]. Usually such systems occur in perturbed Hamiltonian systems [7,10], but in this note, equation ẋ = f(x, y) has not to be necessary Hamiltonian in x uniformly for y small. For proving our results, we follow [3]. Related results are studied also in the papers [1,8,11,12].

2. Transversal Homoclinics

We take in (1.1) the following change of variables

x(t) = γ(θ, ϵy(t), t) + ϵz(t) ,
y ↔ ϵy, t ↔ t + α ,

then by (iii) we get

(2.1) \begin{aligned}ż & = f_{x}(γ(θ, 0, t), 0)z + h(γ(θ, 0, t), 0, t + α, 0) \\ & − γ_{y}(θ, 0, t)p(γ(θ, 0, t), 0, t + α, 0) + O(ϵ), \\ ẏ & = ϵ((A + p_{y}(γ(θ, 0, t), 0, t + α, 0))y + p_{ϵ}(γ(θ, 0, t), 0, t + α, 0) \\ & + q(0, t + α, 0) + p_{x}(γ(θ, 0, t), 0, t + α, 0)z + O(e^{−c_{1}|t|}) + O(ϵ)) \\ & + p(γ(θ, 0, t), 0, t + α, 0) . \end{aligned}

Now we consider the variational equation given by

(2.2)  u̇ = f_{x}(γ(θ, 0, t), 0)u .

According to (iii), we note that the system

\{\frac{∂}{∂θ_{i}} γ(θ, 0, t)\}_{i=1}^{d−1} ∪ γ̇(θ, 0, t)

is a family of bounded solutions of (2.2), where θ = (θ_{1}, θ_{2}, ⋯, θ_{d−1}). We can assume that these vectors are linearly independent. Then this family represents a basis of bounded solutions of (2.2). Let U_{θ}(t) denote a fundamental solution of (2.2) with u_{θj}(t) the jth column of U_{θ}(t) and define U_{θ}^{⊥}(t) = (U_{θ}(t)^{−1})^{∗}, where ∗ is a transposition with respect to a scalar product ⟨·, ·⟩ on ℝ^{n}. We can suppose that u_{θj}(t) and u_{θj+d}^{⊥}(t), j = 1, 2, ⋯, d form bases of the bounded solutions of (2.2) and of the adjoint equation

(2.3)  u̇ = −f_{x}(γ(θ, 0, t), 0)^{∗}u ,

respectively, where u_{θj}^{⊥}(t) is the jth column of U_{θ}^{⊥}(t). Moreover, we can assume the smoothness of U_{θ}(t) on both θ and t. We note that U_{θ}^{⊥}(t) is a fundamental solution of (2.3).

Now by following [3], we get the following result.

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 9, p. 2 Theorem 2.1. Let us define a mapping

M: ℝ^{d−1} × ℝ → ℝ^{d}, M = (M_{1}, M_{2}, ⋯, M_{d}) ,

by

M_{l}(θ, α) = ∫_{−∞}^{∞} ⟨u_{θl+d}^{⊥}(t), h(γ(θ, 0, t), 0, t + α, 0)⟩ dt (2.4)

− ∫_{−∞}^{∞} ⟨u_{θl+d}^{⊥}(t), γ_{y}(θ, 0, t)p(γ(θ, 0, t), 0, t + α, 0)⟩ dt .

If there is a simple root (θ_{0}, α_{0}) of M(θ, α) = 0, i.e. M(θ_{0}, α_{0}) = 0 and the matrix M_{(θ,α)}(θ_{0}, α_{0}) is nonsingular, then (1.1) has for any ϵ ≠ 0 sufficiently small a transversal homoclinic solution near γ(θ_{0}, 0, · + α_{0}) × 0.

Proof. Since the proof is very similar as of Theorem 2.10 of [3], so we only sketch it here. Let us define the following Banach spaces

Z = {z ∈ C(ℝ, ℝ^{n})| |z| = sup_{t} |z(t)| < ∞} ,

Y_{θ} = {h ∈ Z| ∫_{−∞}^{∞}⟨h(t), u_{θi+d}^{⊥}(t)⟩ dt = 0 for any i = 1, 2, …, d} ,

X = {v ∈ C(ℝ, ℝ^{m})| |v| = sup_{t} |v(t)| < ∞} .

We need the following two results.

Claim 1. ([3]) The nonhomogeneous equation

ż = f_{x}(γ(θ, 0, t), 0)z + h(t), h ∈ Z

has a solution z ∈ Z if and only if h ∈ Y_{θ}. The solution is unique if it satisfies

∫_{−∞}^{∞}⟨z(t), u_{θi}(t)⟩ dt = 0 for any i = 1, 2, …, d. This solution is smooth in θ and h.

Claim 2. ([3]) For ϵ ≠ 0 sufficiently small, the nonhomogeneous equation

ẏ = ϵ((A + p_{y}(γ(θ, 0, t), 0, t + α, 0))y + w), w ∈ X

has a unique solution in X which we denote t → y(t, α, θ, ϵ). This solution satisfies

|y| ≤ c_{2}|w| for a constant c_{2} > 0, and |\frac{∂y}{∂α} | = O(ϵ|w|). If in addition ∫_{−∞}^{∞} |w(s)| ds <

∞ then |y| ≤ c_{3}|ϵ| ∫_{−∞}^{∞} |w(s)| ds for a constant c_{3} > 0.

Now by using the standard way of Lyapunov-Schmidt like in [3], we can solve (2.1) to get the statement of the theorem. □

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 9, p. 3

We note that usually we start with a system of the form

ẋ = f_{1}(x, y) + ϵh_{1}(x, y, t, ϵ) , (2.5)

ẏ = ϵg_{1}(x, y, t, ϵ) .

Then we suppose that f_{1}(x, y) = 0 has a smooth solution x = ψ(y) and by changing the variables, we can suppose that f_{1}(0, y) = 0. Then we consider the equation

ẏ = ϵg_{1}(0, y, t, ϵ) and we take its averaged equation ẏ = ϵ ∫_{0}^{1} g_{1}(0, y, t, 0) dt (see [9]).

Let y = 0 be a hyperbolic root of ∫_{0}^{1} g_{1}(0, y, t, 0) dt = 0, i.e. ∫_{0}^{1} g_{1}(0, 0, t, 0) dt = 0 and

the matrix ∫_{0}^{1} g_{1y}(0, 0, t, 0) dt has no eigenvalues on the imaginary axis. By taking

in (2.5) the usual averaging change of variables of the form y ↔ y + ϵH(y, t), where H is smooth and 1-periodic in t, we arrive at the system like (1.1). So let us take y(t) = v(t) + ϵH(v(t), t) in (1.1). Then we get

ẋ = f(x, v) + ϵ(f_{y}(x, v)H(v, t) + h(x, v, t, 0)) + O(ϵ^{2})

= f_{1}(x, v) + ϵh_{1}(x, v, t, ϵ) , (2.6)

v̇ = ϵ(I + ϵH_{v}(v, t))^{−1}(Av + g(v + ϵH(v, t)) − H_{t}(v, t)
+ ϵAH(v, t) + p(x, v + ϵH(v, t), t, ϵ) + ϵq(v + ϵH(v, t), t, ϵ))
= ϵg_{1}(x, v, t, ϵ) .

The unperturbed equation of (2.6) has the same form as for (1.1). For the mapping M = (M_{1}, M_{2}, ⋯, M_{d}) of (2.4) in terms of (2.6), we have

M_{l}(θ, α) = − ∫_{−∞}^{∞} ⟨u_{θl+d}^{⊥}(t), f_{y}(γ(θ, 0, t), 0)H(0, t + α) + γ_{y}(θ, 0, t)H_{t}(0, t + α)⟩ dt

+ ∫_{−∞}^{∞} ⟨u_{θl+d}^{⊥}(t), h_{1}(γ(θ, 0, t), 0, t + α, 0) − γ_{y}(θ, 0, t)g_{1}(γ(θ, 0, t), 0, t + α, 0)⟩ dt .

Assumption (iii) for ω(t) = γ_{y}(θ, 0, t)H(0, t + α) gives

(2.7)  ω̇(t) = f_{x}(γ(θ, 0, t), 0)ω(t)
+ f_{y}(γ(θ, 0, t), 0)H(0, t + α) + γ_{y}(θ, 0, t)H_{t}(0, t + α) .

Since ω ∈ Z, equation (2.7) and Claim 1 imply

f_{y}(γ(θ, 0, t), 0)H(0, t + α) + γ_{y}(θ, 0, t)H_{t}(0, t + α) ∈ Y_{θ} .

Hence we get

M_{l}(θ, α) = ∫_{−∞}^{∞} ⟨u_{θl+d}^{⊥}(t), h_{1}(γ(θ, 0, t), 0, t + α, 0)⟩ dt (2.8)

− ∫_{−∞}^{∞} ⟨u_{θl+d}^{⊥}(t), γ_{y}(θ, 0, t)g_{1}(γ(θ, 0, t), 0, t + α, 0)⟩ dt .

When f_{1}(0, ·) = 0 in (2.5), then (2.8) expresses the mapping M in terms of (2.5) without using its averaged form (1.1).

Generally, when f_{1}(ψ(y), y) = 0 and γ(θ, y, t) are homoclinics to the hyperbolic fixed points x = ψ(y) of ẋ = f_{1}(x, y), and y = y_{0} is a hyperbolic root of the equation

∫_{0}^{1} g_{1}(ψ(y), y, t) dt = 0, then the mapping M = (M_{1}, M_{2}, ⋯, M_{d}) has the form

M_{l}(θ, α) = ∫_{−∞}^{∞} ⟨u_{θl+d}^{⊥}(t), h_{1}(γ(θ, y_{0}, t), y_{0}, t + α, 0)⟩ dt (2.9)

− ∫_{−∞}^{∞} ⟨u_{θl+d}^{⊥}(t), γ_{y}(θ, y_{0}, t)g_{1}(γ(θ, y_{0}, t), y_{0}, t + α, 0)⟩ dt ,

where (2.2) has to be replaced by

u̇ = f_{x}(γ(θ, y_{0}, t), y_{0})u .

3. An Example

Let us consider the system

z̈ = z − (v^{2} + v̇^{2})z(z^{2} + w^{2} + u) + ϵδv̇ ,

ẅ = w − (v^{2} + v̇^{2})w(z^{2} + w^{2} + u) , (3.1)

u̇ = (1 + v^{2} + v̇^{2})u + ϵw^{2} ,
v̈ + v = ϵ((1 − v^{2})v̇ + w) ,

where δ is a constant and ϵ is a small parameter. By taking the polar coordinates

v = y sin ϕ, v̇ = y cos ϕ ,

(3.1) possesses the form

x′_{1} = x_{2}/g_{2}(y, ϕ, x, ϵ) ,
x′_{2} = (x_{1} − y^{2}x_{1}(x_{1}^{2} + x_{3}^{2} + x_{5}) + ϵδy cos ϕ)/g_{2}(y, ϕ, x, ϵ) ,

x′_{3} = x_{4}/g_{2}(y, ϕ, x, ϵ) , (3.2)

x′_{4} = (x_{3} − y^{2}x_{3}(x_{1}^{2} + x_{3}^{2} + x_{5}))/g_{2}(y, ϕ, x, ϵ) ,
x′_{5} = ((1 + y^{2})x_{5} + ϵx_{3}^{2})/g_{2}(y, ϕ, x, ϵ) ,
y′ = ϵ((1 − y^{2} sin^{2} ϕ)y cos^{2} ϕ + x_{3} cos ϕ)/g_{2}(y, ϕ, x, ϵ) ,

where ′ = \frac{d}{dϕ}, x = (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) and

g_{2}(y, ϕ, x, ϵ) = 1 − ϵ((1 − y^{2} sin^{2} ϕ) cos ϕ sin ϕ + \frac{x_{3}}{y} sin ϕ) .

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 9, p. 5

6 MICHAL FEČKAN

Of course, we suppose that y ≠ 0. The unperturbed equation of (3.2) has the form x′_{1} = x_{2} ,

x′_{2} = x_{1} − y^{2}x_{1}(x_{1}^{2} + x_{3}^{2} + x_{5}) ,

x′_{3} = x_{4} , (3.3)

x′_{4} = x_{3} − y^{2}x_{3}(x_{1}^{2} + x_{3}^{2} + x_{5}) ,
x′_{5} = (1 + y^{2})x_{5} .

By putting r(t) = sech t, for (3.3) we have [5,6]

γ(θ, y, t) = \frac{\sqrt{2}}{y}( sin θr(t), sin θṙ(t), cos θr(t), cos θṙ(t), 0) ,

(3.4) u_{θ3}^{⊥}(y, t) = ( − sin θr̈(t), sin θṙ(t), − cos θr̈(t), cos θṙ(t), 0) ,

u_{θ4}^{⊥}(y, t) = ( − cos θṙ(t), cos θr(t), sin θṙ(t), − sin θr(t), 0) .

Now we consider the equation

y′ = ϵ\frac{(1 − y^{2} sin^{2} ϕ)y cos^{2} ϕ}{1 − ϵ(1 − y^{2} sin^{2} ϕ) cos ϕ sin ϕ}

= ϵ((1 − y^{2} sin^{2} ϕ)y cos^{2} ϕ + O(ϵ))

and its first-order averaging is given by

y′ = ϵy(\frac{1}{2} − \frac{y^{2}}{8} ) .

y_{0} = 2 is a simple root of \frac{1}{2} − \frac{y^{2}}{8} = 0. Hence we take y = 2 in the formulas (3.4).

In the notation of (2.5), we have h_{1}(x, 2, ϕ, 0) = (x_{2}, x_{1} − 4x_{1}(x_{1}^{2} + x_{3}^{2} + x_{5}), x_{4}, x_{3} − 4x_{1}(x_{1}^{2} + x_{3}^{2} + x_{5}), 5x_{5})g_{3}(x, ϕ) + 2δ(0, cos ϕ, 0, 0, 0) + (0, 0, 0, 0, x_{3}^{2}) , g_{3}(x, ϕ) = (1 − 4 sin^{2} ϕ) sin ϕ cos ϕ + \frac{x_{3}}{2} sin ϕ , g_{1}(x, 2, ϕ, 0) = 2(1 − 4 sin^{2} ϕ) cos^{2} ϕ + x_{3} cos ϕ .

We see that

γ_{y}(θ, 2, t) = −γ(θ, 2, t)/2 .

Since h_{1}(γ(θ, 2, t), 2, t + α, 0) = γ̇(θ, 2, t)g_{3}(γ(θ, 2, t), t + α)

+ 2δ(0, cos(t + α), 0, 0, 0) + \frac{1}{2}(0, 0, 0, 0, cos^{2} θr(t)^{2}) ,

u_{θ2}(t) = γ̇(θ, 2, t), ⟨u_{θ2}(t), u_{θi+2}^{⊥}(t)⟩ = 0, i = 1, 2 ⟨γ_{y}(θ, 2, t), u_{θ4}^{⊥}(t)⟩ = 0 ,

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 9, p. 6 the formula (2.9) has after several calculations [4] now the form

M_{1}(θ, α) = 2δπ sech \frac{π}{2} sin θ sin α + \frac{2π\sqrt{2}}{3} cosech π cos 2α
+ \frac{10π\sqrt{2}}{3} cosech 2π cos 4α + \frac{5π}{24} sech \frac{π}{2} cos θ cos α ,
M_{2}(θ, α) = 2δπ sech \frac{π}{2} cos θ cos α .

For finding a simple root of M(θ, α) = 0, we suppose δ ≠ 0 and take θ = −π/2 while α ≠ ±π/2 must be a simple zero of the equation

(3.5)  δ = \sqrt{2} \frac{cosech π cos 2α + 5 cosech 2π cos 4α}{3 sech (π/2) sin α} = Ω(α) .

Function Ω(α) is odd and it is satisfying

Ω(α) = Ω(π − α), Ω(α) = −Ω(π + α), lim_{α→0_{+}} Ω(α) = +∞ .

Furthermore, Ω has on (0, π) only three critical points α_{1}, α_{2} = π−α_{1}, α_{3} = π/2 for some α_{1} ≃ 1.378. Moreover, Ω attains on (0, π) its global minimum at α_{1}, α_{2} and a local maximum at α_{3}. We note that Ω(α_{1}) = Ω(α_{2}). Consequently as Ω(π/2) < 0, (3.5) has a simple zero for any δ.

Summarizing, by applying Theorem 2.1 and results of the papers [2,5], we arrive at the following result.

Theorem 3.1. Let δ ≠ 0 be fixed. Equation (3.1) has chaos for any ϵ ≠ 0 sufficiently small.

We note that for any compact interval [a_{1}, a_{2}] ⊂ ℝ, 0 ∉ [a_{1}, a_{2}], there is an ϵ_{0} > 0 such that (3.1) has chaos for any δ ∈ [a_{1}, a_{2}] and 0 < |ϵ| < ϵ_{0}. On the other hand, the function M_{2}(θ, α) is vanishing for δ = 0, and we should derive higher-degenerate Melnikov mapping to get a reasonable bifurcation result as δ is crossing 0. We do not follow this line in this paper.

When w = u = 0 in (3.1), we get the simpler system

z̈ = z − (v^{2} + v̇^{2})z^{3} + ϵδv̇ , (3.6)

v̈ + v = ϵ(1 − v^{2})v̇ .

Then (3.3) has the form

(3.7)  x′_{1} = x_{2}, x′_{2} = x_{1} − y^{2}x_{1}^{3} .

(3.7) has a homoclinic γ(y, t) = \frac{\sqrt{2}}{y}(r(t), ṙ(t)). So now we have d = 1 and u_{2}^{⊥}(y, t) = ( − r̈(t), ṙ(t)). The Melnikov function has now the form

M(α) = 2δ ∫_{−∞}^{∞} cos(t + α)ṙ(t) dt = 2δπ sech \frac{π}{2} sin α .

We see that α_{0} = 0 is a simple root of M(α) = 0 for δ ≠ 0. Consequently, (3.6) is chaotic for δ ≠ 0 fixed and ϵ ≠ 0 sufficiently small. Hence (3.1) has, in addition to Theorem 3.1, also “trivial” chaos of (3.6) with w = u = 0.

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 9, p. 7

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