Connections to fixed points and Sil’nikov saddle-focus homoclinic orbits in singularly perturbed systems
We consider a singularly perturbed system depending on two parameters with two (possibly the same) normally hyperbolic center manifolds. We assume that the unperturbed system has an orbit that connects a hyperbolic fixed point on one center manifold to a hyperbolic fixed point on the other. Then we...
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| author | Battelli, F. Palmer, K. J. Баттеллі, Ф. Палмер, К. Дж. |
| author_facet | Battelli, F. Palmer, K. J. Баттеллі, Ф. Палмер, К. Дж. |
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| datestamp_date | 2020-03-18T19:46:36Z |
| description | We consider a singularly perturbed system depending on two parameters with two (possibly the same) normally hyperbolic center manifolds. We assume that the unperturbed system has an orbit that connects a hyperbolic fixed point on one center manifold to a hyperbolic fixed point on the other. Then we prove some old and new results concerning the persistence of these connecting orbits and apply the results to find examples of systems in dimensions greater than three that possess Sil’nikov saddle-focus homoclinic orbits. |
| first_indexed | 2026-03-24T02:36:53Z |
| format | Article |
| fulltext |
UDC 517.9
F. Battelli* (Marche Polytech. Univ. Ancona),
K. J. Palmer** (Nat. Taiwan Univ., Taiwan)
CONNECTIONS TO FIXED POINTS AND SIL’NIKOV
SADDLE-FOCUS HOMOCLINIC ORBITS IN SINGULARLY
PERTURBED SYSTEMS
ПОЄДНАННЯ НЕРУХОМИХ ТОЧОК ТА СIДЛОВI
ФОКУСНI ГОМОКЛIНIЧНI ОРБIТИ СIЛЬНIКОВА
В СИНГУЛЯРНО ЗБУРЕНИХ СИСТЕМАХ
We consider a singularly perturbed system depending on two parameters with two (possibly the same)
normally hyperbolic centre manifolds. We assume that the unperturbed system has an orbit connecting a
hyperbolic fixed point on one centre manifold to a hyperbolic fixed point on the other. Then we prove
some old and new results concerning the persistence of these connecting orbits and apply the results to find
examples of systems in dimensions greater than three which possess Sil’nikov saddle-focus homoclinic
orbits.
Розглянуто сингулярно збурену систему, що залежить вiд двох параметрiв та має два (можливо,
однаковi) нормально гiперболiчнi центрованi многовиди. При цьому припускається, що незбурена
система має орбiту, яка поєднує гiперболiчну нерухому точку на одному центрованому многовидi
з гiперболiчною нерухомою точкою на iншому. Доведено деякi вiдомi та новi результати щодо
збереження цих орбiт та наведено приклади систем розмiрностi бiльше, нiж три, що мають сiдловi
фокуснi гомоклiнiчнi орбiти Сiльнiкова.
1. Introduction. In this paper, which continues [1], we consider a singularly perturbed
system like:
ẋ = εf(x, y, λ, ε),
ẏ = g(x, y, λ, ε)
(1)
where x ∈ Rm, y ∈ Rn, λ and ε are small real parameters and f(x, y, λ, ε), g(x, y, λ, ε)
are Cr-functions in their arguments bounded with their derivatives, r ≥ 1. We suppose
that the following conditions hold:
(i) for any x ∈ Rm, the equation
g(x, y, 0, 0) = 0
has Cr-solutions y = v±(x) (that may coincide) such that v±(x) and its derivatives are
bounded on R,
(ii) the infimums over x ∈ Rm of the moduli of the real parts of the eigenvalues
of the Jacobian matrix gy(x, v±(x), 0) are greater than a positive number δ0. Moreover
gy(x, v+(x), 0) and gy(x, v−(x), 0) have the same number of eigenvalues with positive
(and hence also negative) real parts.
As explained in more detail in Section 2, conditions (i) and (ii) imply the existence of
centre manifolds y = v+(x, λ, ε) and y = v−(x, λ, ε) for the perturbed system together
*Supported by GNAMPA-INdAM and MIUR (Italy).
**Supported by MIUR (Italy) and NSC (Taiwan).
c© F. BATTELLI, K. J. PALMER, 2008
28 ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
CONNECTIONS TO FIXED POINTS AND SIL’NIKOV SADDLE-FOCUS HOMOCLINIC ORBITS ... 29
with their associated centre stable and centre unstable manifolds. We also assume the
following conditions:
(iii) there exists ξ0 ∈ Rm such that the equation
ẏ = g(ξ0, y, 0, 0)
has a solution y0(t) satisfying
y0(t) → v−(ξ0) as t→ −∞, y0(t) → v+(ξ0) as t→∞,
(iv) ẏ0(t) is the unique bounded solution of the linear variational system:
ẏ = gy(ξ0, y0(t), 0, 0)y (2)
up to a scalar multiple.
Given these conditions, we would expect the perturbed system to have orbits
connecting the two centre manifolds. However, in this paper, we are particularly
interested in orbits which connect fixed points lying on the centre manifolds. So we
need an additional condition:
(v) both equations on the centre manifolds
ẋ = F±(x) = f(x, v±(x), 0, 0)
have the same hyperbolic fixed point ξ0 and the matrices F±x (ξ0) have the same number
of eigenvalues, counted with mutiplicities, with positive (resp. negative) real parts and
no eigenvalues with zero real part such that if Q+ is the projection onto the stable
subspace of F+
x (ξ0) and Q− is the projection onto the stable subspace of F−x (ξ0), then
RQ+ ∩NQ− = {0}. Under this condition, the perturbed system
ẋ = F±(x, λ, ε) = f(x, v±(x, λ, ε), λ, ε) (3)
has a hyperbolic fixed point ξ±0 (λ, ε) and
q±(λ, ε) =
(
ξ±0 (λ, ε), v±(ξ±0 (λ, ε), λ, ε
)
is a hyperbolic fixed point of system (1). We make the additional assumption that for
sufficiently small λ
ξ+0 (λ, 0) = ξ−0 (λ, 0) = ξ0(λ).
Our objects in this paper are the following:
(a) to extend the result of [1] concerning the existence of solutions of (1) that connect
a fixed point on one centre manifold to a fixed point on the other centre manifold to a
more degenerate case;
(b) to give a general class of singularly perturbed systems in dimensions greater than
three which possess Sil’nikov saddle-focus homoclinic orbits.
In [1] in the homoclinic case, following on from the work of Szmolyan [2] and
Beyn – Stiefenhofer [3], we already gave a nondegeneracy condition (which corresponds
to condition (vi) in Theorem 2 below) under which there is a curve λ = λ(ε) in the
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
30 F. BATTELLI, K. J. PALMER
parameter space along which system (1) has a connecting orbit. This condition says that
the connecting orbit y0(t) in the system
ẏ = g(ξ0(λ), y, λ, 0) (4)
breaks as λ passes through 0. In this paper, in a slightly more general situation, we give
a different and shorter proof of this theorem (see Theorem 2). We proceed in two steps,
first we use the results of [4] to find the connecting orbits between the centre manifolds
and then employ a further argument to pick out from these orbits the ones that connect
the fixed points.
More importantly, we can use these techniques to treat degenerate cases. Such a
degeneracy would arise if y0(t) does not break so that there is a one-parameter family
y(t, λ) of homoclinic orbits for (4). Then in Theorem 3 we need to add two additional
conditions: one condition says that the centre stable and centre unstable manifolds
intersect transversally along (ξ0, y0(t)) when λ = ε = 0 and the other is that a certain
Melnikov function have a simple zero. Under these conditions we can again prove there
is a curve λ = λ(ε) in the parameter space along which system (1) has a connecting
orbit. Note that there are other kinds of degeneracies that could arise such as what we
called the Cherry and Duffing cases in [5] and [4] in which the centre stable and centre
unstable manifolds do not intersect transversally along (ξ0, y0(t)) when λ = ε = 0.
However, in this paper, we confine ourselves to what appears to be the simplest kind of
degeneracy.
The theory of Sil’nikov saddle-focus homoclinic orbits is developed in [6] and [7].
Such orbits have been found in special systems (for example, see [8 – 10] and see [7] for
others) but not many general classes of systems with such orbits have been found, apart
from that of Rodriguez [11]. However, Rodriguez looked only at three-dimensional
systems. In four dimensions two extra conditions must be verified. In [1] we exhibited
a class of systems in 4 dimensions with saddle-focus homoclinic orbits. In this paper,
with the help of [12], we show the conditions we gave in [1] can be weakened (in [1]
we used too strong a condition to ensure Deng’s condition (D4) was satisfied) and we
also give a result in n dimensions.
Now we summarize the contents of the paper. In Section 2, we recall the main
result from [4] where we construct bifurcation equations, the zeros of which are initial
values of solution of (1) which lie in the intersection of the global centre stable manifold
corresponding to y = v+(x, λ, ε) and the global centre unstable manifold corresponding
to y = v−(x, λ, ε). Then in Section 3 we prove Theorems 2 and 3 as described above
and we give examples of the application of both. Then in Section 4, we prove the
theorem concerning Sil’nikov saddle-focus homoclinic orbits and give an example of it
as well.
2. Heteroclinic connections between the centre manifolds. In this section we
recall the main result of [4], where under the conditions (i) – (iv) of the Introduction, we
construct bifurcation equations, the zeros of which are initial values of solution of (1)
which lie in the intersection of the global centre stable manifold corresponding to y =
= v+(x, λ, ε) and the global centre unstable manifold corresponding to y = v−(x, λ, ε).
First we observe that conditions (i) and (ii) of the Introduction imply the existence
of ε0 > 0, λ0 > 0 and functions v±(x, λ, ε) which are defined for x ∈ Rm, |λ| ≤ λ0
and |ε| ≤ ε0 such that v±(x, 0, 0) = v±(x) and the manifolds y = v±(x, λ, ε) are
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CONNECTIONS TO FIXED POINTS AND SIL’NIKOV SADDLE-FOCUS HOMOCLINIC ORBITS ... 31
invariant for the flow of (1) (see for example [13 – 15]). Moreover v±(x, λ, ε) are Cr
and bounded with their derivatives. We will refer to y = v±(x, λ, ε) as global centre
manifolds for system (1). We use the notation x±c (t, ξ, λ, ε) for the solution of the initial
value problem
ẋ = F±(x, λ, ε) := f(x, v±(x, λ, ε), λ, ε), x(0) = ξ.
In this situation there also exist global centre stable and unstable manifolds. The global
centre stable manifold Mcs consists of those solutions (x(t), y(t)) such that |y(t) −
− v+(x(t), λ, ε)| → 0 as t→∞ and the global centre unstable manifold Mcu consists
of those solutions (x(t), y(t)) such that |y(t)− v−(x(t), λ, ε)| → 0 as t→ −∞.
Next, according to condition (ii), for all x ∈ Rm, the linear systems
ẏ = gy(x, v±(x), 0, 0)y
have exponential dichotomies on R with constant K, exponent δ0 and projections, say,
P 0
±(x). Moreover rankP 0
+(x) = rankP 0
−(x) = p, p being the number of eigenvalues
of gy(x, v±(x), 0, 0) with negative real parts.
Then from (ii), (iii) and the roughness of exponential dichotomies, it follows that for
any δ with 0 < δ < δ0 linear system (2) has an exponential dichotomy on R+ and R−
respectively with constants K and δ and respective projections P+(t) = Y (t)P+Y
−1(t)
and P−(t) = Y (t)P−Y −1(t), Y (t) being the fundamental matrix with Y (0) = I. Note
that since from (iv) ẏ0(t) is, up to a scalar multiple, the unique bounded solution, it
follows that RP+ ∩ NP− is the one-dimensional subspace spanned by ẏ0(0). Also
RP+ + NP− has codimension 1 and if we let ψ0 be a unit vector orthogonal to
RP+ +NP−, then the solution ψ(t) of the adjoint equation
ẏ = −g∗y(ξ0, y0(t), 0, 0)y (5)
with ψ(0) = ψ0 is, up to a scalar multiple, the unique bounded solution.
In Theorem 4 in [4] the following result, concerning the existence of heteroclinic
connections between the two centre manifolds, has been proved. Such solutions lie in
the intersection of the global centre stable and unstable manifolds.
Theorem 1. Let f and g be bounded Cr functions, r ≥ 2, with bounded deri-
vatives, satisfying conditions (i) – (iv) of the Introduction. Then there exist ε0, λ0, α0,
Cr−1 functions ∆(ξ, λ, ε) Z(ξ, λ, ε) defined for |ξ − ξ0| ≤ α0, |λ| ≤ λ0, |ε| ≤ ε0 and
a neighborhood O of (ξ0, y0(0)) such that if (ξ, η) satisfies
∆(ξ, λ, ε) = 0, η = Z(ξ, λ, ε) (6)
then (ξ, η) lies in the intersection of the global centre stable manifold corresponding
to y = v+(x, λ, ε) and the global centre unstable manifold corresponding to y =
= v−(x, λ, ε) and, conversely, if (ξ, η) is in O and lies in this intersection and satisfies
〈η − y0(0), ẏ0(0)〉 = 0, then (6) holds. Moreover,
∆(ξ0, 0, 0) = 0, Z(ξ0, 0, 0) = y0(0),
∆ξ(ξ0, 0, 0) = −
∞∫
−∞
ψ∗(t)gx(ξ0, y0(t), 0, 0)dt,
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
32 F. BATTELLI, K. J. PALMER
∆λ(ξ0, 0, 0) = −
∞∫
−∞
ψ∗(t)gλ(ξ0, y0(t), 0, 0)dt,
∆ε(ξ0, 0, 0) =
= −
∞∫
−∞
ψ∗(t)
{
gx(ξ0, y0(t), 0, 0)
t∫
0
f(ξ0, y0(τ), 0, 0)dτ + gε(ξ0, y0(t), 0, 0)
}
dt,
and there exist β > 0 (β < δ0), T > 0, µ1 > 0, µ2 > 0 and N1 > 0 such that the
solution (x(t), y(t)) = (x̂(t, ξ, λ, ε), ŷ(t, ξ, λ, ε)) of (1) such that (x(0), y(0)) = (ξ, η)
satisfies
eβ|t∓T |
∣∣x(t)− x±c (ε(t∓ T ), ξ̂±(ξ, λ, ε), λ, ε)
∣∣ ≤ N1|ε| ≤ µ1,
eβ|t∓T |
∣∣y(t)− v±(x(t), λ, ε)
∣∣ ≤ µ2,
(7)
for t ≥ T and t ≤ −T respectively, where the ξ̂±(ξ, λ, ε) are Cr−1 functions satisfying
ξ̂±(ξ, λ, 0) = ξ,
ξ̂+,ε(ξ, λ, 0) = Tf(ξ, v+(ξ, λ, 0), λ, 0)+
+
∞∫
0
f(ξ, ŷ(t, ξ, λ, 0), λ, 0)− f(ξ, v+(ξ, λ, 0), λ, 0) dt,
(8)
ξ̂−,ε(ξ, λ, 0) = −Tf(ξ, v−(ξ, λ, 0), λ, 0)−
−
0∫
−∞
f(ξ, ŷ(t, ξ, λ, 0), λ, 0)− f(ξ, v−(ξ, λ, 0), λ, 0) dt.
Moreover,
x̂(t, ξ0, 0, 0) = ξ0, ŷ(t, ξ0, 0, 0) = y0(t), (9)
x̂(t, ξ, λ, ε), ŷ(t, ξ, λ, ε) are Cr−1 in (ξ, λ, ε) and if σ satisfies 0 < rσ < β, the
k-th order derivatives of x̂(t, ξ, λ, ε)− x±c (ε(t∓ T ), ξ̂±(ξ, λ, ε), λ, ε) and ŷ(t, ξ, λ, ε)−
− v±(x̂(t, ξ, λ, ε), λ, ε), satisfy estimates similar to (7) with β − kσ instead of β and
possibly different constants Ck instead of µ1, µ2.
Remarks. (i) Estimate (7) with N1|ε| and those concerning the derivatives of
x̂(t, ξ, λ, ε) − x±c (ε(t ∓ T ), ξ̂±(ξ, λ, ε), λ, ε) and ŷ(t, ξ, λ, ε) − v±(x̂(t, ξ, λ, ε), λ, ε)
are not explicitly stated in [4], Theorem 4. However they follow from Eq. (63) and
Theorems 1 and 2 in [4].
(ii) Eqns. (6) are what we call the bifurcation equations. Zeros (ξ, η) of these
equations are initial values of solutions (x̂(t, ξ, λ, ε), ŷ(t, ξ, λ, ε)) of (1) which lie in the
intersection of the global centre stable manifold corresponding to y = v+(x, λ, ε) and
the global centre unstable manifold corresponding to y = v−(x, λ, ε). The functions
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CONNECTIONS TO FIXED POINTS AND SIL’NIKOV SADDLE-FOCUS HOMOCLINIC ORBITS ... 33
ξ̂±(ξ, λ, ε) tell us in which leaves of the stable (resp. unstable) foliation these solutions
lie (see [4] for details).
3. Heteroclinic orbits connecting fixed points. In this section we assume that in
addition to conditions (i) – (iv), condition (v) also holds. Our aim is to find a curve in the
parameter space along which there exist heteroclinic orbits connecting the fixed point
q−(λ, ε) to q+(λ, ε). First we construct the slow stable manifold of q+(λ, ε) and the
slow unstable manifold of q−(λ, ε). By slow, we mean the stable and unstable manifolds
lying inside the respective centre manifolds. Then we prove our two main theorems on
the existence of heteroclinic orbits. Finally we give examples of the application of both
theorems.
3.1. The slow stable and unstable manifolds. First we describe the local stable
manifold of the fixed point ξ+0 (λ, ε) of the equation ẋ = F+(x, λ, ε) and the unstable
manifold of the fixed points ξ−0 (λ, ε) of the equation ẋ = F−(x, λ, ε). As in [1],
we can prove that the following holds: let ρ be a sufficiently small positive number.
Then there exists ρ1, 0 < ρ1 < ρ such that if ε and λ are sufficiently small and if
ξ+ ∈ RQ+, ξ
− ∈ NQ−, with |ξ+| ≤ ρ1, |ξ−| ≤ ρ1, there exist unique solutions
x(t) = u+(t, ξ+, λ, ε) of equation ẋ = F+(x, λ, ε) and x(t) = u−(t, ξ−, λ, ε) of
equation ẋ = F−(x, λ, ε), that are defined for t ≥ 0 and t ≤ 0 respectively, such that∣∣u+(t, ξ+, λ, ε)− ξ+0 (λ, ε)
∣∣ ≤ ρ for t ≥ 0,∣∣u−(t, ξ−, λ, ε)− ξ−0 (λ, ε)
∣∣ ≤ ρ for t ≤ 0
and
Q+
[
u+(0, ξ+, λ, ε)− ξ+0 (λ, ε)
]
= ξ+,
(I−Q−)
[
u−(0, ξ−, λ, ε)− ξ−0 (λ, ε)
]
= ξ−.
(10)
Moreover, denoting with α > 0 a positive number which strictly bounds from below the
absolute values of the real parts of the eigenvalues of both matrices F±x (ξ0, 0, 0), then
u±(t, ξ±, λ, ε) − ξ±0 (λ, ε) and their first derivatives with respect to (ξ±, λ, ε) satisfy
exponential estimates like ∣∣u(t)∣∣ ≤ Le−α(t−s)|u(s)| (11)
where t ≥ s when u(t) = u+(t, ξ+, λ, ε) − ξ+0 (λ, ε) and t ≤ s when u(t) = u−(t, ξ−,
λ, ε)− ξ−0 (λ, ε). Also
u+
ξ+(0, 0, 0, 0) = Q+ and u−ξ−(0, 0, 0, 0) = I−Q−. (12)
Next we note that, by uniqueness,
u+(t, 0, λ, ε) = ξ+0 (λ, ε), u−(t, 0, λ, ε) = ξ−0 (λ, ε) (13)
and that
x±c (t, u±(τ, ξ±, λ, ε), λ, ε) = u±(t+ τ, ξ±, λ, ε) (14)
for any ξ+ ∈ RQ+ and ξ− ∈ NQ− with |ξ±| < ρ1. Moreover, differentiating (13) we
see that
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
34 F. BATTELLI, K. J. PALMER
u±λ (0, 0, λ, 0) =
∂ξ±0
∂λ
(λ, 0) (15)
for any λ ∈ R.
3.2. Two conditions ensuring existence of homoclinic orbits to fixed points. In
this subsection we prove two theorems on the existence of heteroclinic orbits. The first
is essentially the same as the theorem proved in [1]. Our proof, though, here is different
in that we proceed in two stages. Note that we do prove additional properties of the
homoclinic solution in that paper. The second considers a more degenerate case which
involves a Melnikov function. We hope this second theorem gives a guide as to how
one might handle such degenerate cases.
We recall that α > 0 is a positive number which bounds from below the absolute
value of the eigenvalues of both matrices F±x (ξ0, 0, 0).
We first prove the following theorem.
Theorem 2. Let f and g be Cr functions (r ≥ 2), bounded together with their
derivatives and satisfying conditions (i) – (v). Suppose also that the condition
(vi)
∫ ∞
−∞
ψ∗(t)
[
gx(ξ0, y0(t), 0, 0)ξ′0(0) + gλ(ξ0, y0(t), 0, 0)
]
dt 6= 0
holds. Then there exists a Cr−1-function λ(ε) with λ(0) = 0 such that for ε sufficiently
small and nonnegative, system (1) with λ = λ(ε) has a heteroclinic solution p(t, ε) =
= (x(t, ε), y(t, ε)), that is,
p(t, ε) 6= q±(λ(ε), ε)
but
p(t, ε) → q±(λ(ε), ε) as t→ ±∞. (16)
Moreover,
p(t, 0) = (ξ0, y0(t)), (17)
and
supt≥0 |x(t, ε)− ξ+0 (λ(ε), ε)|eεαt = O(ε),
supt≤0 |x(t, ε)− ξ−0 (λ(ε), ε)|e−εαt = O(ε),
supt≥0 |y(t, ε)− v+(ξ+0 (λ(ε), λ(ε), ε)|eεαt = O(1),
supt≤0 |y(t, ε)− v−(ξ−0 (λ(ε), λ(ε), ε)|e−εαt = O(1),
supt∈R |y(t, ε)− y0(t)| = o(1)
(18)
as ε → 0. Furthermore for 1 ≤ k ≤ r − 1, positive constants Ĉk exist such that the
following hold for t ≥ 0 ( for v+), t ≤ 0 ( for v−):∣∣∣D(k)
2 [x(t, ε)− ξ±0 (λ(ε), ε)]
∣∣∣ eεα|t| ≤ Ĉk,∣∣∣D(k)
2 [y(t, ε)− v±(ξ±0 (λ(ε), ε), λ(ε), ε)]
∣∣∣ eεα|t| ≤ Ĉk,
(19)
where D(k)
2 denotes the k-th derivative with respect to ε.
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CONNECTIONS TO FIXED POINTS AND SIL’NIKOV SADDLE-FOCUS HOMOCLINIC ORBITS ... 35
Proof. Our aim is to find a solution (x(t), y(t)) of (1) such that
lim
t→±∞
∣∣x(t)− ξ±0 (λ, ε)
∣∣ = 0 and lim
t→±∞
∣∣y(t)− v±(ξ±0 (λ, ε), λ, ε)
∣∣ = 0. (20)
According to Theorem 1, if ∆(ξ, λ, ε) = 0 with |ξ − ξ0|, |λ|, and |ε| sufficiently small,
we can find a solution (x(t), y(t)) =
(
x̂(t, ξ, λ, ε), ŷ(t, ξ, λ, ε)
)
of (1) that satisfies
supt≥0 e
βt
∣∣x(t+ T )− xc(εt, ξ̂+(ξ, λ, ε), λ, ε)
∣∣ ≤ µ1,
supt≥0 e
βt
∣∣y(t+ T )− v+(x(t+ T ), λ, ε)
∣∣ ≤ µ2
(21)
and
supt≤0 e
−βt
∣∣x(t− T )− xc(εt, ξ̂−(ξ, λ, ε), λ, ε)
∣∣ ≤ µ1,
supt≤0 e
−βt
∣∣y(t− T )− v−(x(t− T ), λ, ε)
∣∣ ≤ µ2
(22)
with µ1 = N1|ε| (see (7)) and the other statements of Theorem 1 hold.
Now, if we can find ξ+ ∈ RQ+ and ξ− ∈ NQ− such that
ξ̂+(ξ, λ, ε) = u+(0, ξ+, λ, ε) and ξ̂−(ξ, λ, ε) = u−(0, ξ−, λ, ε), (23)
then
xc(εt, ξ̂+(ξ, λ, ε), λ, ε) = u+(εt, ξ+, λ, ε),
and
x(t+ T )− ξ+0 (λ, ε) =
[
x(t+ T )− xc(εt, ξ̂+(ξ, λ, ε), λ, ε)
]
+
+
[
u+(εt, ξ+, λ, ε)− ξ+0 (λ, ε)
]
,
(24)
y(t+ T )− v+(ξ+0 (λ, ε), λ, ε) =
[
y(t+ T )− v+(x(t+ T ), λ, ε)
]
+
+
[
v+(x(t+ T ), λ, ε)− v+(ξ±0 (λ, ε), λ, ε)
]
.
Thus, for t → ∞, (20) follows from (7) and (11). A similar argument applies when
t→ −∞.
So assume that
∆(ξ, λ, ε) = 0,
u+(0, ξ+, λ, ε)− ξ̂+(ξ, λ, ε) = 0,
u−(0, ξ−, λ, ε)− ξ̂−(ξ, λ, ε) = 0
(25)
has aCr−1 solution (ξ, ξ+, ξ−, λ) =
(
ξ(ε), ξ+(ε), ξ−(ε), λ(ε)
)
∈ Rm×RQ+×NQ−×
×R, such that (
ξ(0), ξ+(0), ξ−(0), λ(0)
)
= (ξ0, 0, 0, 0). (26)
Then the Cr−1-function
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36 F. BATTELLI, K. J. PALMER
x(t, ε) = x̂(t, ξ(ε), λ(ε), ε), y(t, ε) = ŷ(t, ξ(ε), λ(ε), ε)
is, for 0 < ε < ε0, a solution of (1) with λ = λ(ε), such that
lim
t→±∞
x(t, ε) = ξ±0 (λ(ε), ε), lim
t→±∞
y(t, ε) = v±(ξ±0 (λ(ε), ε), λ(ε), ε)).
Next from (24) and (11), for t ≥ T∣∣x(t, ε)− ξ0(λ(ε), ε)
∣∣eεαt ≤ N1|ε|+ L
∣∣ξ̂+(ξ(ε), λ(ε), ε)− ξ0(λ(ε), ε)
∣∣ = O(ε).
Moreover, from (9) we obtain
x(t, 0) = x̂(t, ξ0, 0, 0) = ξ0, y(t, 0) = ŷ(t, ξ0, 0, 0) = y0(t)
and hence the estimate
sup
0≤t≤T
∣∣x(t, ε)− ξ+0 (λ(ε), ε)
∣∣eεαt = O(ε)
follows from the continuity and ξ+0 (0, 0) = ξ0. A similar argument applies when t ≤ 0.
This proves the first two estimates in (18). The third and the fourth estimates in (18)
follow from a similar argument, using again (24) when |t| ≥ T and the continuous
dependence on the data when |t| ≤ T. Note that in this case Theorem 1 implies only
that the difference
∣∣y(t, ε)− v±(x(t, ε), λ(ε), ε)
∣∣eεαt is bounded.
Next, using (24), (11) and Theorem 1, we see that (19) holds.
Now we prove the last estimate in (18). We have∣∣y(t, ε)− y0(t)
∣∣ ≤ ∣∣y(t, ε)− v±(x(t, ε), λ(ε), ε)
∣∣+
+
∣∣v±(x(t, ε), λ(ε), ε)− v±(ξ±0 (ε), λ(ε), ε)
∣∣+
+
∣∣v±(ξ±0 (ε), λ(ε), ε)− v±(ξ0, 0, 0)
∣∣ +
∣∣y0(t)− v±(ξ0, 0, 0)
∣∣,
where now we write ξ±0 (ε) = ξ±0 (λ(ε), ε). Then, from (21) and the first inequality in
(18), we see that there exists a constant M1 such that for t ≥ T we have∣∣y(t, ε)− v+(x(t, ε), λ(ε), ε)
∣∣ ≤ µ2e
−β(t−T ),∣∣v+(x(t, ε), λ(ε), ε)− v+(ξ+0 (ε), λ(ε), ε)
∣∣ ≤M1εe
−εα(t−T ).
Also, from the smoothness of the three functions v+(x, λ, ε), ξ+0 (ε), λ(ε) and from (26),
there exists a constant M2∣∣v+(ξ+0 (ε), λ(ε), ε)− v+(ξ0, 0, 0)
∣∣ ≤M2ε.
So, given ρ > 0, there exists T̄ρ = T + β−1 log(µ2ρ
−1) > 0 such that, for t ≥ T̄ρ we
have ∣∣y(t, ε)− y0(t)
∣∣ ≤ ρ+ (M1 +M2)ε+
∣∣y0(t)− v+(ξ0, 0, 0)
∣∣.
Since lim
|t|→∞
y0(t) = v+(ξ0, 0, 0) there exists Tρ > T̄ρ such that for t ≥ Tρ,
∣∣y0(t) −
− v+(ξ0, 0, 0)
∣∣ < ρ. So we have proved that, given ρ > 0, there exists Tρ > 0 such that
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CONNECTIONS TO FIXED POINTS AND SIL’NIKOV SADDLE-FOCUS HOMOCLINIC ORBITS ... 37
for t ≥ Tρ, we have ∣∣y(t, ε)− y0(t)
∣∣ ≤ ρ+ (M1 +M2)ε.
By similar arguments, we can show this inequality holds for t ≤ −Tρ also. Next since
y(t, 0) = y0(t), from continuity we see that
lim
ε→0
sup
|t|≤Tρ
∣∣y(t, ε)− y0(t)
∣∣ = 0.
Then for any ρ > 0 there exists ε̄ > 0 such that for 0 ≤ ε ≤ ε̄ we have sup
t∈R
∣∣y(t, ε) −
− y0(t)
∣∣ < ρ. Hence the last equality in (18) follows.
So all we need to prove is that system (25) has a Cr−1 solution (ξ, ξ+, ξ−, λ) =
= (ξ(ε), ξ+(ε), ξ−(ε), λ(ε)) that satisfies (26).
Now, from ξ̂±(ξ, λ, 0) = ξ, we see that, when ε = 0, system (25) reads:
∆(ξ, λ, 0) = 0,
u+(0, ξ+, λ, 0)− ξ = 0,
u−(0, ξ−, λ, 0)− ξ = 0
(27)
and has the solution (ξ, ξ+, ξ−, λ) = (ξ0, 0, 0, 0) (see (13) and Theorem 1). Then the
linear part of the left-hand side of (27) at the point (ξ0, 0, 0, 0) is given by, according to
(13), (12):
L : (ξ, ξ+, ξ−, λ) 7→
∆ξ(ξ0, 0, 0)ξ + ∆λ(ξ0, 0, 0)λ
ξ+ + ξ′0(0)λ− ξ
ξ− + ξ′0(0)λ− ξ
hence if (ξ, ξ+, ξ−, λ) belongs to its kernel we must have
∆ξ(ξ0, 0, 0)ξ + ∆λ(ξ0, 0, 0)λ = 0,
ξ+ + ξ′0(0)λ− ξ = 0,
ξ− + ξ′0(0)λ− ξ = 0.
(28)
Subtracting the second equation from the third we see that ξ+ = ξ− and hence
ξ+ = ξ− = 0 since ξ+ ∈ RQ+, ξ
− ∈ NQ− and RQ+ ∩ NQ− = {0}. Thus
(ξ, ξ+, ξ−, λ) ∈ NL if and only if
∆ξ(ξ0, 0, 0)ξ + ∆λ(ξ0, 0, 0)λ = 0,
ξ = ξ′0(0)λ.
Plugging the second equation into the first we see that λ has to satisfy
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38 F. BATTELLI, K. J. PALMER
[
∆ξ(ξ0, 0, 0)ξ′0(0) + ∆λ(ξ0, 0, 0)
]
λ = 0.
From Theorem 1 and assumption (vi) we see that the above equation has only the solution
λ = 0 and hence (28) has the unique solution (ξ, ξ+, ξ−, λ) = (0, 0, 0, 0). Thus L is
invertible and then, for 0 < ε < ε0, sufficiently small, (25) has a unique Cr-solution
(ξ, ξ+, ξ+, λ) =
(
ξ(ε), ξ+(ε), ξ−(ε), λ(ε)
)
such that (
ξ(0), ξ+(0), ξ−(0), λ(0)
)
= (ξ0, 0, 0, 0).
This concludes the proof of the theorem.
Now we prove the second theorem in this subsection.
Theorem 3. Let f and g be Cr+2 functions (r ≥ 2), bounded together with their
derivatives and satisfying conditions (i) – (v). Suppose also that the three conditions,
(vii)
∫ ∞
−∞
ψ∗(t)gx(ξ0, y0(t), 0, 0)dt 6= 0;
(viii) the stable and unstable manifolds respectively of the hyperbolic equilibria
v+(ξ0(λ), λ, 0) and v−(ξ0(λ), λ, 0) of
ẏ = g(ξ0(λ), y, λ, 0)
intersect near y0(0) so that there is a solution y0(t, λ) → v±(ξ0(λ), λ, 0) as t → ±∞
with y(0, λ) depending continuously on λ and y0(0, 0) = y0(0);
(ix) if we let Q±(λ) be the projection on to the stable subspace of the linear system
ẋ = F±x (ξ0(λ), λ, 0)x (29)
along the unstable subspace, Q(λ) the projection on to RQ+(λ) along NQ−(λ) and
ψ(t, λ) the unique (up to a multiplicative constant), bounded solution of the adjoint
linear system
ẏ + g∗y(ξ0(λ), y0(t, λ), λ, 0)y = 0,
then the Melnikov function
M(λ) =
∞∫
−∞
ψ(t, λ)∗
gε(ξ0(λ), y0(t, λ), λ, 0) + gx(ξ0(λ), y0(t, λ), λ, 0) ×
×
Q(λ)
t∫
−∞
f(ξ0(λ), y0(τ, λ), λ, 0)dτ +
∂ξ−0
∂ε
(λ, 0)
−
− (I−Q(λ))
∞∫
t
f(ξ0(λ), y0(τ, λ), λ, 0)dτ − ∂ξ+0
∂ε
(λ, 0)
dt
has a simple zero at λ = 0
hold. Then we get the same conclusions as in Theorem 2.
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CONNECTIONS TO FIXED POINTS AND SIL’NIKOV SADDLE-FOCUS HOMOCLINIC ORBITS ... 39
Proof. As in the proof of Theorem 2, all we need to prove is that system (25) has a
Cr−1 solution (ξ, ξ+, ξ−, λ) =
(
ξ(ε), ξ+(ε), ξ−(ε), λ(ε)
)
that satisfies (26).
First note by solving the equation
〈y0(t, λ)− y0(0), ẏ0(0)〉 = 0
for a continuous function t = t(λ) with t(0) = 0 and then replacing y0(t, λ) by y0(t+
+ t(λ), λ), we can assume without loss of generality that〈
y0(0, λ)− y0(0), ẏ0(0)
〉
= 0.
Now, since for λ sufficiently small the point (ξ0(λ), y0(0, λ)) belongs to the neigh-
borhood whose existence is stated in Theorem 1 and (ξ0(λ), y0(0, λ)) also belongs
to the intersection of the centre stable and the centre unstable manifolds and satisfies〈
y0(0, λ)− y0(0), ẏ0(0)
〉
= 0, it follows from Theorem 1 that
∆(ξ0(λ), λ, 0) = 0, Z(ξ0(λ), λ, 0) = y0(0, λ). (30)
Hence
y0(t, λ) = ŷ(t, ξ0(λ), λ, 0) (31)
because both functions satisfy the same equation and have the same value at t = 0.
Next, from (7), (8) we see that x̂(t, ξ0(λ), λ, 0) = ξ̂±(ξ0(λ), λ, 0) = ξ0(λ) and hence,
using (31) and the properties of ŷ(t, ξ, λ, ε), we get∣∣∣∣ ∂k
∂λk
[y0(t, λ)− v±(ξ0(λ), λ, 0)]
∣∣∣∣ ≤ Cke
−(β−kσ)|t|
for k = 0, . . . , r − 1 and t ≥ 0 in the case of v+ and t ≤ 0 in the case of v−. Thus
y0(0, λ) is Cr+1 and its derivatives with respect to λ are bounded. Moreover, using the
inequality with k = 0 we see that y0(t, λ) → v±(ξ0(λ), λ, 0) as t→ ±∞ exponentially
and uniformly with respect to λ.
Now from assumption (ii), it follows that
ẏ = gy(ξ0(λ), y0(t, λ), λ, 0)y
has an exponential dichotomy on both R+ and R− with projections P+(λ) and P−(λ)
respectively that can be assumed to be Cr+1 (see, for example, [16 – 18]). Also we can
take P+(0) = P+ and P−(0) = P− (see Section 2). Moreover
RP+(λ) ∩NP−(λ) = span
{
ẏ0(0, λ)
}
,
since this intersection certainly contains the subspace on the right and has dimension at
most 1 by continuity. It follows that NP ∗+(λ)∩RP ∗−(λ) is the subspace of initial values
of bounded solutions of the adjoint system
ẏ = −g∗y
(
ξ0(λ), y0(t, λ), λ, 0
)
y. (32)
Then if we define ψ(t, λ) to be the solution of the adjoint system with
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40 F. BATTELLI, K. J. PALMER
ψ(0, λ) = ψ0(λ) ∈ NP ∗+(λ) ∩RP ∗−(λ),
where ψ0(λ) is Cr+1 with
ψ(0, 0) = ψ0,
then, since y0(t, λ) is Cr+1 and is bounded together with its derivatives, it follows from
[19] that ψ(t, λ) can be assumed to be Cr+1 and it and its derivatives tend to zero
exponentially and uniformly with respect to λ as |t| → ∞. Thus, in particular, M(λ)
is Cr−1. (We pause here to observe that if ψ(t, λ) is a bounded solution of the adjoint
system (32), then ψ(t− t(λ), λ) is a bounded solution of
ẏ = −g∗y(ξ0(λ), y0(t− t(λ), λ), λ, 0)y
and hence M(λ) does not change if we replace y0(t, λ) by y0(t − t(λ), λ), which was
the original y0(t, λ), and ψ(t, λ) by ψ(t− t(λ), λ).)
This being said we go back to the problem of solving equation (25). Well, in this
case, from (8), (13) and (30) we see that equation (27) has the solution
ξ = ξ0(λ), ξ+ = 0, ξ− = 0, λ = 0,
and the kernel of the linear map obtained by differentiating the left-hand side of (27)
with respect to (ξ+, ξ−, ξ, λ) at the point (ξ+, ξ−, ξ, λ) = (0, 0, ξ0(λ), λ) consists of
those (ξ+, ξ−, ξ, λ) such that
∆ξ(ξ0(λ), λ, 0)ξ + ∆λ(ξ0(λ), λ, 0)λ = 0,
u+
ξ+(0, 0, λ, 0)ξ+ + u+
λ (0, 0, λ, 0)λ− ξ = 0,
u−ξ−(0, 0, λ, 0)ξ− + u−λ (0, 0, λ, 0)λ− ξ = 0
that is, using (15),
∆ξ(ξ0(λ), λ, 0)ξ + ∆λ(ξ0(λ), λ, 0)λ = 0,
u+
ξ+(0, 0, λ, 0)ξ+ + ξ′0(λ)λ− ξ = 0,
u−ξ−(0, 0, λ, 0)ξ− + ξ′0(λ)λ− ξ = 0.
(33)
Subtracting the third equation from the second we obtain
u+
ξ+(0, 0, λ, 0)ξ+ = u−ξ−(0, 0, λ, 0)ξ−.
When λ = 0 the above equation is equivalent to (see (12))
ξ+ = ξ− = 0
since ξ+ ∈ RQ+ and ξ− ∈ NQ−. Thus we obtain the same conclusion for small λ 6= 0,
because of continuity. Then (ξ+, ξ−, ξ, λ) satisfies (33) if and only if ξ+ = ξ− = 0 and
the kernel of the linear map consists of those (ξ+, ξ−, ξ, λ) such that ξ+ = ξ− = 0 and
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CONNECTIONS TO FIXED POINTS AND SIL’NIKOV SADDLE-FOCUS HOMOCLINIC ORBITS ... 41
∆ξ(ξ0(λ), λ, 0)ξ + ∆λ(ξ0(λ), λ, 0)λ = 0,
ξ = ξ′0(λ)λ.
Plugging the second equation in the first and using ∆(ξ0(λ), λ, 0) = 0 we conclude that
the solution of system (33) is the one dimensional space spanned by (ξ+, ξ−, ξ, λ) =
= (0, 0, ξ′0(λ), 1).
We use the Crandall – Rabinowitz theorem as given in [20] (Theorem 4.1), a suitable
version of which can be stated as follows:
Proposition 1. Let F : E×R 7→ G, (z, ε) 7→ F(z, ε), be a Cr mapping (r ≥ 2),
where E and G are Banach spaces. Suppose that there exists a Cr function φ(λ) defined
on an interval I such that φ′(λ) 6= 0 and such that
F(φ(λ), 0) = 0
and L(λ) = Fz(φ(λ), 0) is Fredholm of index zero with nullspace spanned by φ′(λ).
Define
d(λ) = ψ(λ)
(
Fλ(φ(λ), 0)
)
,
where ψ(λ) ∈ G∗ is a Cr function such that N (ψ(λ)) = R(L(λ)). Then if d(λ) has a
simple zero at λ0, the equation
F(z, ε) = 0
has a solution z(ε) for ε sufficiently small. Moreover, z(0) = φ(λ0), z(ε) is a Cr−1
function and Fz(z(ε), ε) is invertible if ε 6= 0.
Let F(ξ+, ξ−, ξ, λ, ε) : RQ+ × NQ− × Rm × R × R → R × R2m be the map
defined by the left-hand side of (25). We have seen that
F(0, 0, ξ0(λ), λ, 0) = 0
and that its linearization at (ξ+, ξ−, ξ, λ) = (0, 0, ξ0(λ), λ) with ε = 0: L(λ) =
= F(ξ+,ξ−,ξ,λ)(0, 0, ξ0(λ), λ, 0) has the one dimensional kernel
NL(λ) = span
{
(0, 0, ξ′0(λ), 1)
}
.
To apply the Crandall – Rabinowitz theorem, we need to determine a vector Ψ(λ) such
that RL(λ) = {Ψ(λ)}⊥. Now, RL(λ) consists of those (σ, u1, u2) ∈ R ×Rm ×Rm
for which (ξ+, ξ−, ξ, µ) ∈ RQ+ ×NQ− ×Rm ×R exists such that
∆ξ(ξ0(λ), λ, 0)ξ + ∆λ(ξ0(λ), λ, 0)λ = σ,
u+
ξ+(0, 0, λ, 0)ξ+ + ξ′0(λ)λ− ξ = u1,
u−ξ−(0, 0, λ, 0)ξ− + ξ′0(λ)λ− ξ = u2.
(34)
Subtracting the third equation from the second we see that (ξ+, ξ−) has to be a solution
of
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42 F. BATTELLI, K. J. PALMER
u+
ξ+(0, 0, λ, 0)ξ+ − u−ξ−(0, 0, λ, 0)ξ− = u1 − u2. (35)
Now Q+(λ) : Rm → Rm is the projection on to the stable space of the linear system
u̇ = F+
x (ξ0(λ), λ, 0)u (36)
along the unstable space. Moreover since u+(t, ξ+, λ, 0) satisfies the equation ẋ =
= F+(x, λ, 0) together with the first equality in (10), we see that u+
ξ+(t, 0, λ, 0) is a
solution of (36) which is bounded on R+ (because of (11)) and satisfies
Q+u
+
ξ+(0, 0, λ, 0) = Q+.
Hence u+
ξ+(0, 0, λ, 0) in an isomorphism from RQ+ onto RQ+(λ) with Q+ as its
inverse. Similarly u−ξ−(0, 0, λ, 0) in an isomorphism from NQ− onto NQ−(λ) with
I − Q− as its inverse. So, using (35) and the fact that RQ(λ) = RQ+(λ), NQ(λ) =
= NQ−(λ), we get
u+
ξ+(0, 0, λ, 0)ξ+ = Q(λ)u+
ξ+(0, 0, λ, 0)ξ+ = Q(λ)(u1 − u2)
and
u−ξ−(0, 0, λ, 0)ξ− = (I−Q(λ))u−ξ−(0, 0, λ, 0)ξ− = (I−Q(λ))(u2 − u1).
Then, from the last two equations in (34) we obtain
ξ = ξ′0(λ)λ−
[
(I−Q(λ))u1 +Q(λ)u2
]
.
Plugging this equality into the first equation in (34) and using again the first equality in
(30) we obtain
σ = −∆ξ(ξ0(λ), λ, 0)
[
(I−Q(λ))u1 +Q(λ)u2
]
.
Thus the range of L(λ) is the nullspace of the linear functional Ψ(λ), defined by
Ψ(λ)
σu1
u2
= σ + ∆ξ(ξ0(λ), λ, 0)
[
(I−Q(λ))u1 +Q(λ)u2
]
.
Finally, we evaluate Fε(0, 0, ξ0(λ), λ, 0). Using (13) we see that
Fε(0, 0, ξ0(λ), λ, 0) =
∆ε(ξ0(λ), λ, 0)
∂ξ+0
∂ε
(λ, 0)− ∂ξ̂+
∂ε
(ξ0(λ), λ, 0)
∂ξ−0
∂ε
(λ, 0)− ∂ξ̂−
∂ε
(ξ0(λ), λ, 0)
.
Thus our Melnikov function is
d(λ) = Ψ(λ)Fε(0, 0, ξ0(λ), λ, 0) =
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CONNECTIONS TO FIXED POINTS AND SIL’NIKOV SADDLE-FOCUS HOMOCLINIC ORBITS ... 43
= ∆ε(ξ0(λ), λ, 0)−∆ξ(ξ0(λ), λ, 0)
{
(I−Q(λ))
[
∂ξ̂+
∂ε
(ξ0(λ), λ, 0)− ∂ξ+0
∂ε
(λ, 0)
]
+
+Q(λ)
[
∂ξ̂−
∂ε
(ξ0(λ), λ, 0)− ∂ξ−0
∂ε
(λ, 0)
]}
.
This turns out to be difficult to calculate. We replace it by
d̃(λ) =
[
ψ∗0(λ)ψ0
]
d(λ).
Since ψ∗0(λ)ψ0 = 1 when λ = 0, we see that d̃(λ) has a simple zero at λ = 0 if and
only if d(λ) has. Then, setting
∆̃(ξ, λ, ε) =
[
ψ∗0(λ)ψ0]∆(ξ, λ, ε
)
,
we see that
∆̃ξ(ξ, λ, ε) =
[
ψ∗0(λ)ψ0
]
∆ξ(ξ, λ, ε), ∆̃ε(ξ, λ, ε) =
[
ψ∗0(λ)ψ0
]
∆ε(ξ, λ, ε)
and
d̃(λ) = ∆̃ε(ξ0(λ), λ, 0)−
−∆̃ξ(ξ0(λ), λ, 0)
{(
I−Q(λ)
)[∂ξ̂+
∂ε
(ξ0(λ), λ, 0)− ∂ξ+0
∂ε
(λ, 0)
]
+
+Q(λ)
[
∂ξ̂−
∂ε
(ξ0(λ), λ, 0)− ∂ξ−0
∂ε
(λ, 0)
]}
.
To calculate further we observe that from the equality
f(ξ0(λ), v±(ξ0(λ), λ, 0), λ, 0) = 0
(see assumption (v)), and (8), (31) it follows that
∂ξ̂+
∂ε
(ξ0(λ), λ, 0) =
∞∫
0
f(ξ0(λ), y0(t, λ), λ, 0)dt,
∂ξ̂−
∂ε
(ξ0(λ), λ, 0) = −
0∫
−∞
f(ξ0(λ), y0(t, λ), λ, 0)dt.
Also, as in the proof of Theorem 1 in [4], we can derive the formulae
∆̃ξ(ξ0(λ), λ, 0) = −
∞∫
−∞
ψ∗(t, λ)gx(ξ0(λ), y0(t, λ), λ, 0)dt
and
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44 F. BATTELLI, K. J. PALMER
∆̃ε(ξ0(λ), λ, 0) = −
∞∫
−∞
ψ∗(t, λ)
{
gε(ξ0(λ), y0(t, λ), λ, 0)+
+gx(ξ0(λ), y0(t, λ), λ, 0)
t∫
0
f(ξ0(λ), y0(τ, λ), λ, 0)dτ
}
dt.
Then it follows that
d̃(λ) = −M(λ).
Thus, if M(λ) has a simple zero at λ = 0, from Proposition 1 we obtain the
existence of a Cr−1−solution (ξ+, ξ−, ξ, λ) = (ξ+(ε), ξ−(ε), ξ(ε), λ(ε)) of (25) that
satisfies (26). The proof is complete.
Remark. We can also show the solution p(t, ε) found in Theorems 2 and 3 has the
following properties:
(a) ṗ(t, ε) is not in the tangent space to the stable fibre through p(t, ε), provided
that
Q
−∂ξ+0
∂ε
(0, 0) +
∂ξ−0
∂ε
(0, 0) +
∞∫
−∞
f(ξ0, y0(t), 0, 0)dt
6= 0, (37)
where Q is the projection with the same range as Q+ and the same nullspace as Q−,
and where, according to Theorem 3 in [4], vectors in the tangent space to the stable fibre
at p(t, ε) are the initial values of the solutions of the variational system along p(t, ε)
which approach zero as t→∞ at the exponential rate β − σ.
To prove the statement we first note from (7) and the equation after (23) that for
t ≥ T ∣∣x(t, ε)− u+(ε(t− T ), ξ+(ε), λ(ε), ε)
∣∣ ≤ µ1e
−β(t−T ),∣∣y(t, ε)− v+(u+(ε(t− T ), ξ+(ε), λ(ε), ε), λ(ε), ε)
∣∣ ≤ µ2e
−β(t−T ).
We next prove that u+(0, ξ+(ε), λ(ε), ε) 6= ξ+0 (λ(ε), ε) = u+(0, 0, λ(ε), ε) (see (13)).
Since u+(0, ξ+(0), λ(0), 0) = u+(0, 0, 0, 0) we compute
d
dε
[
u+(0, ξ+(ε), λ(ε), ε)− u+(0, 0, λ(ε), ε)
]∣∣
ε=0
= Q+
dξ+
dε
(0)
(see (12)). To calculate
dξ+
dε
(0), we differentiate the last two equations in (25) with
ξ± = ξ±(ε), λ = λ(ε), ξ = ξ(ε) with respect to ε at ε = 0 to get
dξ+
dε
(0) + ξ′0(0)λ′(0) +
∂ξ+0
∂ε
(0, 0) = ξ′(0) + ξ̂+,ε(ξ0, 0, 0),
dξ−
dε
(0) + ξ′0(0)λ′(0) +
∂ξ−0
∂ε
(0, 0) = ξ′(0) + ξ̂−,ε(ξ0, 0, 0).
Subtracting these equations, we get
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dξ+
dε
(0)− dξ−
dε
(0) = −∂ξ
+
0
∂ε
(0, 0) +
∂ξ−0
∂ε
(0, 0) +
∞∫
−∞
f(ξ0, y0(t), 0, 0)dt
from which it follows that
dξ+
dε
(0) = Q
−∂ξ+0
∂ε
(0, 0) +
∂ξ−0
∂ε
(0, 0) +
∞∫
−∞
f(ξ0, y0(t), 0, 0)dt
and hence that
Q+
dξ+
dε
(0) = Q
−∂ξ+0
∂ε
(0, 0) +
∂ξ−0
∂ε
(0, 0) +
∞∫
−∞
f(ξ0, y0(t), 0, 0)dt
6= 0
sinceQ andQ+ have the same range. Thus u+(0, ξ+(ε), λ(ε), ε) 6= ξ+0 (λ(ε), ε) if ε > 0
is sufficiently small.
Now write xc(t) for u+(t + εT, ξ+(ε), λ(ε), ε). Note that ẋc(0) 6= 0 since xc(t) is
not an equilibrium. Then for t ≥ 0∣∣x(t+ T, ε)− xc(εt)
∣∣ ≤ µ1e
−βt,∣∣y(t+ T, ε)− v+(x(t+ T, ε), λ(ε), ε)
∣∣ ≤ µ2e
−βt.
Note that from these inequalities we also obtain for t ≥ 0∣∣y(t+ T, ε)− v+(xc(εt), λ(ε), ε)
∣∣ ≤ (N̂µ1 + µ2)e−βt
where N̂ is an upper bound for the norm of vx(x, λ, ε). Then since
ẋ(t, ε) = εf(x(t, ε), y(t, ε), λ(ε), ε),
εẋc(εt) = εf(xc
(
εt), v+(xc(εt), λ(ε), ε), λ(ε), ε
)
,
we see that for t ≥ 0∣∣ẋ(t+ T, ε)− εẋc(εt)
∣∣ ≤ N |ε|
[
(N̂ + 1)µ1 + µ2
]
e−βt.
Now ẋc(t) is a solution of
ẋ = F+
x (xc(t), λ(ε), ε)x
and so for all t ∣∣ẋc(t)
∣∣ ≥ |ẋc(0)|e−N |t|,
where N is a bound on |F+
x (x, λ, ε)|. It follows that for t ≥ 0∣∣ṗ(t+ T, ε)
∣∣ ≥ ∣∣ẋ(t+ T, ε)
∣∣ ≥ ∣∣εẋc(εt, ε)
∣∣−N |ε|
[(
N̂ + 1
)
µ1 + µ2
]
e−βt ≥
≥
∣∣εẋc(0)
∣∣e−Nεt −N |ε|
[
(N̂ + 1)µ1 + µ2
]
e−βt.
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46 F. BATTELLI, K. J. PALMER
If Nε < β, it follows that eNεt|ṗ(t, ε)| does not tend to 0 as t→∞. So if Nε < β−σ,
it follows that ṗ(t, ε) cannot be in the tangent space to the stable fibre at p(t, ε).
(b) Similarly we can prove that ṗ(t, ε) is not in the tangent space to the unstable
fibre through p(t, ε), provided that
(I−Q)
−∂ξ+0
∂ε
(0, 0) +
∂ξ−0
∂ε
(0, 0) +
∞∫
−∞
f(ξ0, y0(t), 0, 0)dt
6= 0,
where, according to Theorem 3 in [4], vectors in the tangent space to the unstable fibre
at p(t, ε) are the initial values of the solutions of the variational system along p(t, ε)
which approach zero as t→ −∞ at the exponential rate β − σ (see Theorem 1).
(c) It is in general position, that is, the tangent spaces to the stable manifold Ws
of the hyperbolic equilibrium q+(λ(ε), ε) and to the unstable manifold Wu of the
hyperbolic equilibrium q−(λ(ε), ε) of the system
ẋ = εf(x, y, ε, λ(ε)),
ẏ = g(x, y, ε, λ(ε))
at p(t, ε) intersect in the one-dimensional subspace spanned by ṗ(t, ε).
The proof of this fact goes as in [1, p. 46], so we will not repeat it here.
(d) From Section 3.1 in [5] it follows that, if
∞∫
−∞
ψ∗(t)gx(ξ0, y0(t), 0, 0)dt 6= 0,
then Tp(0,ε)Mcs and Tp(0,ε)Mcu intersect transversely.
3.3. Examples. Here we give examples of the application of Theorems 2 and 3.
First, for an example of Theorem 2, we consider the following system, with x, y ∈ R:
ẋ = εf(x, y, ẏ, λ, ε),
ÿ = −g(y) + λh1(x, λ, ε)ẏ + εh(x, y, ẏ, λ, ε)
where the functions involved are sufficiently smooth. Setting y1 = y, y2 = ẏ we obtain
the system
ẋ = εf(x, y1, y2, λ, ε),
ẏ1 = y2,
ẏ2 = −g(y1) + λh1(x, λ, ε)y2 + εh(x, y1, y2, λ, ε).
(38)
We assume the following conditions hold: there exists a point ξ0 ∈ R such that
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f(ξ0, 0, 0, 0, 0) = 0, g(0) = 0,
fx(ξ0, 0, 0, 0, 0) < 0, g′(0) < 0,
h1(ξ0, 0, 0) 6= 0
(39)
and the second order equation ÿ+g(y) = 0 has the solution p(t) 6= 0 which is homoclinic
to y = 0, that is:
lim
|t|→∞
p(t) = lim
|t|→∞
ṗ(t) = 0.
Note that, taking G(y) =
∫ y
0
g(u)du then any solution of ÿ + g(y) = 0 satisfies
1
2
ẏ2 +G(y) = const. In particular
1
2
ṗ2(t) +G(p(t)) = 0. Hence we see that p(t) 6= 0
for all t ∈ R, since otherwise, if p(t∗) = 0 for some t∗, then we also have ṗ(t∗) = 0
and hence p(t) = 0 since both y = p(t) and y = 0 are solutions of the Cauchy problem:
ÿ + g(y) = 0,
y(t∗) = ẏ(t∗) = 0.
As a consequence, either p(t) < 0 or p(t) > 0 for all t ∈ R.
When λ = ε = 0, the equations are decoupled and we have the single normally
hyperbolic centre manifold (y1, y2) = v(x) = 0, that persists for λ 6= 0 also, that
is, v(x, λ, 0) = 0. The equation ẋ = f(x, 0, 0, 0, 0) on the centre manifold has the
exponentially stable fixed point x = ξ0, and the y-equation has for all x the homoclinic
orbit
y0(t) =
p(t)
ṗ(t)
.
Finally, the bounded solution of the adjoint system is
ψ(t) =
p̈(t)
−ṗ(t)
.
Hence we see that conditions (i) – (v) are satisfied and condition (vi) reads:
−
∞∫
−∞
ṗ(t)h1(ξ0, 0, 0)ṗ(t)dt = −h1(ξ0, 0, 0)
∞∫
−∞
ṗ(t)2dt 6= 0.
Thus we conclude that, if (39) holds, then for some λ = λ(ε), equation (38) has a
solution p(t, ε) =
(
x(t, ε), y(t, ε)
)
homoclinic to a fixed point of (38) with λ = λ(ε),
close to (ξ0, 0).Moreover, by the Remarks after Theorem 3, p(t, ε) is in general position,
and ṗ(t, ε) is not in the tangent space to the stable fibre at p(t, ε) if
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48 F. BATTELLI, K. J. PALMER
∞∫
−∞
f(ξ0, p(t), ṗ(t), 0, 0)dt 6= 0
(note that (39) implies that ξ0 is a stable fixed point for the equation ẋ = f(x, 0, 0, 0, 0)
and hence Q = I). For example if f(x, y, ẏ, λ, ε) = y − x we have ξ0 = 0 and
∞∫
−∞
f(ξ0, p(t), ṗ(t), 0, 0)dt =
∞∫
−∞
p(t)dt 6= 0
since either p(t) < 0 or p(t) > 0 for all t ∈ R.
As an example for Theorem 3, we consider the following system:
ẋ = ε(f(x) + λh0(y1) + εh(x, y)),
ẏ1 = y2,
ẏ2 = −(1 + λ)2g̃(y1) + k(x)y2
(40)
where x, y1, y2 are in R, the functions are sufficiently smooth and we assume that
f(0) = g̃(0) = h(0, 0) = k(0) = h0(0) = 0,
f ′(0) < 0, g̃′(0) < 0, k′(0) 6= 0,
and also that the equation ÿ + g̃(y) = 0 has a homoclinic orbit y0(t) = (p(t), ṗ(t))
associated with the saddle point (0, 0). Without loss of generality we can assume that
ṗ(0) = 0 and hence p(t) = p(−t) since both solve the Cauchy problem
ÿ + g̃(y) = 0,
y(0) = p(0), ẏ(0) = 0.
Then we assume that
∞∫
0
h0(p(τ))dτ 6= 0.
Now system (40) has the single normally hyperbolic centre manifold y = v(x, λ, ε) = 0
and, when λ = ε = 0, the system ẋ = f(x) on the centre manifold has the exponentially
stable fixed point ξ0 = 0. For small λ and ε, the equation on the centre manifold is
ẋ = f(x) + εh(x, 0) with no λ and so the equilibrium near 0 is ξ0(λ, ε) = ξ0(ε). Then
differentiating the equation f(ξ0(ε)) + εh(ξ0(ε), 0) = 0, we see that
ξ′0(0) = −h(0, 0)
f ′(0)
= 0.
Next observe that the y-equations in (40) with x = ξ0(λ, 0) = 0 and ε = 0 have the
homoclinic orbit
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y0(t, λ) =
p((1 + λ)t)
(1 + λ)ṗ((1 + λ)t)
and we have
ψ(t, λ) =
(1 + λ)2p̈((1 + λ)t)
−(1 + λ)ṗ((1 + λ)t)
.
As an expected consequence
∞∫
−∞
ψ∗(t, λ)gλ(x, y0(t, λ), 0, 0)dt = 2(1 + λ)2
∞∫
−∞
ṗ((1 + λ)t)p̈((1 + λ)t)dt =
= 2(1 + λ)
∞∫
−∞
ṗ(t)p̈(t)dt = 0.
Next we have
∞∫
−∞
ψ∗(t, 0)gx(ξ0, y0(t, 0), 0, 0)dt = −k′(0)
∞∫
−∞
ṗ(t)2dt 6= 0.
Then, the linear system (29) in this case reads: ẋ = f ′(0)x and has the exponentially
stable equilibrium x = 0. Hence Q(λ) = Q±(λ) = I and M(λ) reads:
M(λ) = −k′(0)
∞∫
−∞
(1 + λ)2ṗ2((1 + λ)t)
t∫
−∞
λh0(p((1 + λ)τ)dτdt =
= −λk′(0)
∞∫
−∞
ṗ2(t)
t∫
−∞
h0(p(τ))dτdt =
= −λk′(0)
∞∫
−∞
ṗ2(t)dt
0∫
−∞
h0(p(t))dt =
= −λk′(0)
∞∫
−∞
ṗ2(t)dt
∞∫
0
h0(p(t))dt
where we have used the fact that ṗ(t) and
∫ t
0
h(p(τ))dτ are odd functions. Thus M(λ)
has a simple zero at λ = 0 and hence we conclude that, for some λ = λ(ε), system (40)
has a homoclinic solution p(t, ε) = (x(t, ε), y(t, ε)) such that
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50 F. BATTELLI, K. J. PALMER
lim
|t|→∞
x(t, ε) = ξ0(ε), lim
|t|→∞
y(t, ε) = 0.
Moreover, by the Remarks after Theorem 3, p(t, ε) is in general position and the centre
stable and centre unstable manifolds intersect transversely along p(t, ε).
Note that Kokubu et al. [21] consider systems such as
ẋ = ε(1− x2),
ẏ1 = y2,
ẏ2 = xy2 − y1(y1 − a− λ).
Such a system satisfies our conditions (i) to (viii) if a 6= 0. However the Melnikov
function is identically zero and so our theorem cannot be applied.
4. Sil’nikov saddle-focus homoclinic orbits. Saddle-focus homoclinic orbits were
first studied by Sil’nikov [6]. In this section we review the definition of such orbits and
describe the equivalent definition developed in [12]. Then we state a theorem which
gives a general class of singularly perturbed systems in dimension n ≥ 4 which have
saddle-focus homoclinic orbits.
We consider an autonomous system
ẋ = F (x) (41)
in Rn, where n ≥ 3 and F is C1, with a hyperbolic equilibrium. We denote the stable
manifold by Ws and the unstable manifold by Wu. Also we denote by φ(t, ξ) the
solution x(t) of (1) with x(0) = ξ.
Here we give the definition of saddle-focus homoclinic orbit as given in Deng [7].
The first two conditions are:
(D1) the eigenvalues of F ′(q) having the smallest positive real part are µ± iω with
ω > 0 and
0 < µ < −Re(λ)
for all eigenvalues λ with negative real parts;
(D2) there is a homoclinic orbit p(t) to q, that is, p(t) 6= q and p(t) ∈ Ws ∩Wu,
such that
dim Tp(t)Ws ∩ Tp(t)Wu = 1.
These are the only conditions needed in 3 dimensions, although note that in 3
dimensions the second part of (D2) is automatically satisfied. In higher dimensions,
two additional conditions are needed. We denote by Wuu the strong unstable manifold
of the equilibrium q. This is a locally invariant manifold containing q whose tangent
space at q consists of the sum of the generalized eigenspaces of F ′(q) corresponding to
the eigenvalues with real part greater than µ. Solutions of (1) starting in this manifold
approach q as t→ −∞ at an exponential rate faster than µ. The two conditions are:
(D3) as t→ −∞, p(t) is asymptotically tangent to the linear span of the eigenvectors
of µ± iω;
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(D4) there is a submanifold M0 of Wu containing p(0) with dimM0 = dimWuu
such that
lim
t→∞
Tp(t)Mt = TqWuu,
where Mt = φ(t,M0).
If there is such a homoclinic orbit, Silnikov and Deng show the presence of chaotic
dynamics near it.
Now we discuss conditions (D3) and (D4). Note that Deng’s condition (D4), which
he refers to as the “strong inclination” condition, corresponds to Sil’nikov’s condition (D)
in [6]. In [12], we show that these two conditions can be formulated in terms of certain
subspaces related to the variational system
ẋ = F ′(p(t))x (42)
of which φx(t, p(0)) is the fundamental matrix with φx(0, p(0)) = I. Consider a homocli-
nic orbit p(t) = φ(t, p(0)) satisfying Deng’s condition (D1). Then choose ν so that
µ < ν < Re(λ)
for all eigenvalues λ of F ′(q) with real part greater than µ. We define the centre stable
subspace as
W cs =
{
ξ : sup
t≥0
e−νt
∣∣φx(t, p(0))ξ
∣∣ <∞}
.
We show in [12] that this definition is independent of the choice of ν and that
dimW cs = dimWs + 2 = n+ 2− dimWu. (43)
Next we define the strong unstable subspace as
Wuu =
{
ξ : sup
t≤0
e−νt
∣∣φx(t, p(0))ξ
∣∣ <∞}
.
We show in [12] that this definition is independent of the choice of ν and that
dimWuu = dimWu − 2.
In [12], we show the following.
Proposition 2. Suppose (41) has a hyperbolic equilibrium q satisfying (D1). Let
p(t) be an associated homoclinic orbit. Then
(i) (D3) holds if and only if p′(0) is not in the strong unstable subspace Wuu;
(ii) Deng’s condition (D4) and Sil’nikov’s condition (D) are equivalent to the condi-
tion
dim(Tp(0)Wu ∩W cs) = 2 or Tp(0)Wu +W cs = Rn.
We now prove the following theorem, which gives a general class of systems with
Sil’nikov saddle-focus homoclinic orbits.
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52 F. BATTELLI, K. J. PALMER
Theorem 4. Consider system (1) with m = 2 such that conditions (i) – (iv) hold
with v+(x) = v−(x) = v(x) and condition (v) holds with ξ±0 (λ, ε) = ξ0(λ, ε) such that
the derivative with respect to x of f(x, v(x), 0, 0) at ξ0 has eigenvalues µ ± iω with
µ > 0 and ω > 0. Further suppose either (vi) in Theorem 2 holds or (vii), (viii) and (ix)
in Theorem 3 hold. Then we deduce the existence of a function λ(ε) and a homoclinic
orbit p(t, ε) = (x(t, ε), y(t, ε)) for system (1) with λ = λ(ε) as in Theorem 2 and if we
assume in addition that
∞∫
−∞
f(ξ0, y0(t), 0, 0)dt 6= 0,
∞∫
−∞
ψ∗(t)gx(ξ0, y0(t), 0, 0)dt 6= 0,
then p(t, ε) is a Sil’nikov saddle-focus homoclinic orbit.
Proof. Now under condition (vi) or conditions (vii), (viii) and (ix) we get the
existence of λ(ε) and the solution p(t, ε) as described in Theorem 2.
Now we verify Deng’s conditions for system (1) with λ = λ(ε). First note that
q(λ(ε), ε) =
(
ξ0(λ(ε), ε), v(ξ0(λ(ε), ε), λ(ε), ε)
)
is an equilibrium for system (1) with λ = λ(ε). The corresponding variational matrix is
A =
εfx εfy
gx gy
,
where fx = fx(ξ0(λ(ε), ε), v(ξ0(λ(ε), ε), λ(ε), ε), λ, ε), etc. If we take
T =
Im 0
vx In
,
where vx = vx(ξ0(λ(ε), ε), λ(ε), ε), then we see that
T−1AT =
ε(fx + fyvx) εfy
0 gy − εfyvx
.
Now when ε = 0, the eigenvalues of fx + fyvx are µ ± iω and the eigenvalues of gy
have real parts with absolute value greater than δ0. So if ε is small enough, A has a pair
of eigenvalues ε(µ± iω+O(ε)) and the other eigenvalues have real parts with absolute
value greater than or equal to δ0. Hence, if ε is sufficiently small, (D1) is satisfied by
the equilibrium q(λ(ε), ε).
Next we note that (D2) follows from Theorem 2 and (a) in the Remark after
Theorem 3.
In regard to (D3), it follows from (b) in the Remark after Theorem 3, where we note
that here Q = 0, that ṗ(0, ε) is not in the tangent space to the unstable fibre at p(0, ε).
However according to Theorem 3 in [4], vectors in the tangent space to the unstable
fibre at p(0, ε) are the initial values of the solutions of the variational system along
p(t, ε) which approach zero as t → −∞ at the exponential rate β − σ. Now here we
choose ν so that εµ < ν < δ0 and we have β < δ0 (see Theorem 1). So we can assume
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ν > β − σ. It follows that Wuu is a subspace of the tangent space to the unstable fibre
at p(0, ε). Hence ṗ(0, ε) is not in Wuu.
Now we show that the last condition in Theorem 4 implies Deng’s condition (D4).
[Note: in [1], we used a stronger condition.] Now we know from Proposition 2 above
that Deng’s condition (D4) is equivalent to the transversality of the subspaces Tp(0,ε)Wu
and W cs, where Wu is the unstable manifold of the equilibrium q(λ(ε), ε) and W cs is
the centre stable subspace. We need to show that W cs = Tp(0,ε)Mcs and Tp(0,ε)Wu =
= Tp(0,ε)Mcu. For the first one, it suffices to show that Tp(0,ε)Mcs ⊂ W cs since both
subspaces have the same dimension. So we just need to show that the norm of a solution
starting in Tp(0,ε)Mcs is bounded by a constant times eδ0t/2 for t ≥ 0 since by taking
ε small enough we can choose ν so that ε(µ + O(ε)) < δ0/2 < ν < δ0. Using the
notation of Theorem 1 in [4], we see that, to prove this, it suffices to show that if
(x+(t, ζ+, ξ, λ, ε), y+(t, ζ+, ξ, λ, ε)) is a solution in Mcs(ξ0), then its derivatives with
respect to ξ and ζ+ are bounded by a constant times eδ0t/2 for t ≥ 0. However, this
follows from [4] (Theorem 1) provided ε is sufficiently small. In fact the result for
the ζ+-derivative directly follows from [4] (Theorem 1), since xc(εt, ξ, λ, ε) does not
depend on ζ+. As for the ξ-derivative, from [4] (Theorem 1) it follows that this does not
grow faster than
∂x+
c
∂ξ
(εt, ξ, λ, ε), which is the solution X(t) of
Ẋ = εFx(x(t))X, X(0) = I,
where F (x) = f(x, v(x, λ, ε), λ, ε), x(t) = x+
c (εt, ξ, λ, ε). For the second one, it
suffices to show that Tp(0,ε)Wu ⊂ Tp(0,ε)Mcu since both subspaces have the same
dimension. However it follows from [4] that Tp(0,ε)Mcu consists of the initial values of
solutions of the variational system which do not grow at too high an exponential rate as
t→ −∞. However, all the solutions beginning in Tp(0,ε)Wu tend to zero as t→ −∞.
So the inclusion follows. Thus by (d) in the Remark after Theorem 3 holds and the
proof of the theorem is complete.
Example. Consider the system
ẋ = ε[f(x) + y],
ẏ1 = y2 + [λ+
√
−G′(0)sin(x2)]y1,
ẏ2 = −G(y1) + λy2,
where x and y = (y1, y2) are in R2, and f, G are suitably smooth. f is chosen so that
0 is an unstable focus for
ẋ = f(x).
G satisfies G(0) = 0, G′(0) < 0 so that (0, 0) is a saddle for
ẏ1 = y2, ẏ2 = −G(y1).
Also we assume there is a solution y0(t) = (p(t), ṗ(t)) of this last equation such that
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
54 F. BATTELLI, K. J. PALMER
y0(t) → (0, 0) as t→ ±∞
and such that p(t) > 0 for all t. For example, if
G(y1) = −y1 + 2y3
1 ,
then p(t) = sech(t) > 0.
Conditions (i) – (v) of the Introduction are satisfied with v±(x, λ, ε) = 0. Now
∞∫
−∞
ψ∗(t)gλ(ξ0, y0(t), 0, 0)dt = −2
∞∫
−∞
(ṗ(t))2dt < 0.
So (vi) in Theorem 2 holds and Theorem 4 implies that there is a function λ(ε) with
λ(0) = 0 such that for ε sufficiently small, system
ẋ = ε
[
f(x) + y
]
,
ẏ1 = y2 +
[
λ(ε) +
√
−G′(0)sin(x2)
]
y1,
ẏ2 = −G(y1) + λ(ε)y2,
has a solution (x(t, ε), y(t, ε)) 6= (0, 0) satisfying (x(t, ε), y(t, ε)) → (0, 0) as t→ ±∞.
Next we find that the last two conditions in Theorem 4 read:
∞∫
−∞
y0(t)dt =
∞∫
−∞
p(t)
ṗ(t)
dt =
∞∫
−∞
p(t)dt
0
6= 0
and
∞∫
−∞
ψ∗(t)gx(0, y0(t), 0, 0)dt =
∞∫
−∞
[
p̈(t) −ṗ(t)
] 0
√
−G′(0)p(t)
0 0
dt =
=
∞∫
−∞
[
0
√
−G′(0)p(t)p̈(t)
]
dt = −
√
−G′(0)
0
∞∫
−∞
ṗ(t)2dt
6= 0.
So Theorem 4 applies and we deduce that our homoclinic orbit is in fact a Sil’nikov
saddle-focus homoclinic orbit.
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CONNECTIONS TO FIXED POINTS AND SIL’NIKOV SADDLE-FOCUS HOMOCLINIC ORBITS ... 55
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Received 08.10.07
ISSN 1027-3190. Укр. мат. журн., 2008, т. 60, № 1
|
| id | umjimathkievua-article-3135 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:36:53Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c5/f7a57752af2269605667bf06d0e91fc5.pdf |
| spelling | umjimathkievua-article-31352020-03-18T19:46:36Z Connections to fixed points and Sil’nikov saddle-focus homoclinic orbits in singularly perturbed systems Поєднання нерухомих точок та сiдловi фокуснi гомоклiнiчнi орбiти Сiльнiкова в сингулярно збурених системах Battelli, F. Palmer, K. J. Баттеллі, Ф. Палмер, К. Дж. We consider a singularly perturbed system depending on two parameters with two (possibly the same) normally hyperbolic center manifolds. We assume that the unperturbed system has an orbit that connects a hyperbolic fixed point on one center manifold to a hyperbolic fixed point on the other. Then we prove some old and new results concerning the persistence of these connecting orbits and apply the results to find examples of systems in dimensions greater than three that possess Sil’nikov saddle-focus homoclinic orbits. Розглянуто сингулярно збурену систему, що залежить вiд двох параметрiв та має два (можливо, однаковi) нормально гiперболiчнi центрованi многовиди. При цьому припускається, що незбурена система має орбiту, яка поєднує гiперболiчну нерухому точку на одному центрованому многовидi з гiперболiчною нерухомою точкою на iншому. Доведено деякi вiдомi та новi результати щодо збереження цих орбiт та наведено приклади систем розмiрностi бiльше, нiж три, що мають сiдловi фокуснi гомоклiнiчнi орбiти Сiльнiкова. Institute of Mathematics, NAS of Ukraine 2008-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3135 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 1 (2008); 28–55 Український математичний журнал; Том 60 № 1 (2008); 28–55 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3135/3018 https://umj.imath.kiev.ua/index.php/umj/article/view/3135/3019 Copyright (c) 2008 Battelli F.; Palmer K. J. |
| spellingShingle | Battelli, F. Palmer, K. J. Баттеллі, Ф. Палмер, К. Дж. Connections to fixed points and Sil’nikov saddle-focus homoclinic orbits in singularly perturbed systems |
| title | Connections to fixed points and Sil’nikov saddle-focus homoclinic orbits in singularly perturbed systems |
| title_alt | Поєднання нерухомих точок та сiдловi
фокуснi гомоклiнiчнi орбiти Сiльнiкова
в сингулярно збурених системах |
| title_full | Connections to fixed points and Sil’nikov saddle-focus homoclinic orbits in singularly perturbed systems |
| title_fullStr | Connections to fixed points and Sil’nikov saddle-focus homoclinic orbits in singularly perturbed systems |
| title_full_unstemmed | Connections to fixed points and Sil’nikov saddle-focus homoclinic orbits in singularly perturbed systems |
| title_short | Connections to fixed points and Sil’nikov saddle-focus homoclinic orbits in singularly perturbed systems |
| title_sort | connections to fixed points and sil’nikov saddle-focus homoclinic orbits in singularly perturbed systems |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3135 |
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