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Topological horseshoes for perturbations of singular difference equations
View the table of contents for this issue, or go to the journal homepage for more 2006 Nonlinearity 19 795
(http://iopscience.iop.org/0951-7715/19/4/002)
Nonlinearity 19 (2006) 795–811 doi:10.1088/0951-7715/19/4/002
Topological horseshoes for perturbations of singular
difference equations
Ming-Chia Li1and Mikhail Malkin2
1Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan 2Department of Mathematics, Nizhny Novgorod State University, Nizhny Novgorod, Russia
E-mail:[email protected]@uic.nnov.ru
Received 26 August 2005, in final form 23 December 2005 Published 14 February 2006
Online atstacks.iop.org/Non/19/795
Recommended by L Bunimovich
Abstract
In this paper, we study solutions of difference equations λ(yn, yn+1, . . . ,
yn+m) = 0, n ∈ Z, of order m with parameter λ, and consider the case
when λ has a singular limit depending on a single variable as λ → λ0, i.e.
λ0(y0, . . . , ym)= ϕ(yN), where N is an integer with 0 N m and ϕ is
a function. We prove that if ϕ has k simple zeros then for λ close enough to
λ0, the difference equation has a k-horseshoe among its solutions, that is, the
dynamics is conjugate to the full shift with k symbols. Moreover, we show that these horseshoes change continuously in the uniform topology as λ varies. As applications of these results, we establish the horseshoe structure in families of generalized H´enon-like maps and of Arneodo–Coullet–Tresser maps near their anti-integrable limits as well as in steady states for certain lattice models.
Mathematics Subject Classification: 39A05, 37B45, 37B10, 54F15, 54H20
1. Introduction
In the theory of dynamical systems, there are several approaches which aim to establish the presence of chaotic behaviour for orbits under appropriate hypotheses. Among others, a typical method is to find homoclinic points and then discover a horseshoe structure (for hyperbolic horseshoes see e.g. [3,25,26]; for merely topological horseshoes see [14,15]). Recently, there appeared the so-called anti-integrable limit approach for systems with generating functions inspired by the concept of Aubry and Abramovici in [6]; see also [9,20,23]. For example, in [23], Qin obtained a two-horseshoe orbit structure in a family of generalized H´enon-like maps
Hλ(x1, x2, . . . , xm−1, xm)= (x2, x3, . . . , xm, g(x1, . . . , xm)+ λϕ(xm)) (1)
with sufficiently large λ (i.e. ‘near’ the anti-integrable limit for λ0 = ∞), where ϕ(x) =
x(x − 1) and g(x) is a C1 function on Rm with bounded partial derivatives on Rm and
(∂/∂x1)g(x1, . . . , xm)not vanishing, in which case Hλ is one-to-one. (Let us note in this
connection that results on the stability of hyperbolic sets are known mainly for diffeomorphisms on manifolds, and only few are concerned with non-invertible smooth maps, e.g. [16,24] for the latter).
The above family (1) can be regarded as a singular system at the anti-integrable limit
λ0 = ∞. Although there is no corresponding map Hλ0 at this limit value of parameter,
one may use the usual elimination process for Hλ with finite λ to get a difference equation
λ(yn, yn+1, . . . , yn+m)= 0, n ∈ Z (see section4) and so, one obtains a well-defined function
λ0at this limit λ0 = ∞. It is of the form ϕ(yn)= yn(yn− 1) = 0, and thus its solutions are
all bi-sequences of zeros and ones.
In this paper, we extend Qin’s example and Aubry and Abramovich’s anti-integrable limit approach to m-th order difference equations λ(yn, yn+1, . . . , yn+m) = 0, n ∈ Z, that have
a singular limit of the form λ0(y0, . . . , ym)= ϕ(yN)as λ → λ0, for some integer N with
0 N m and some function ϕ. We prove that if ϕ has k simple zeros, then for λ close enough to λ0, the difference equation has a topological k-horseshoe, i.e. a closed (in the product
topology) shift-invariant set λfor which the restriction of the shift map σ is conjugate to σ|k,
the two-sided full shift on k symbols; see the precise statement of theorem3.3. Moreover, we prove that the conjugacy map ψλ: k→ λdepends continuously on λ not only in the product
topology, but also in the uniform topology; more precisely, the family of maps λ→ ψλ(x)
from the parameter space (close to λ0) to ∞forms an equicontinuous family in x∈ k. The
latter implies that the full orbit structure varies continuously starting from the ‘rigid horseshoe’ (i.e. from k) as λ varies.
Let us remark that our assumptions require neither invertibility of maps associated with the difference equation nor even continuity everywhere. For example, our arguments being applied to the generalized H´enon-like maps, allow us to extend the result of [23] so as to consider ϕ and g to be C1functions on Q= [s
1, s2]\V and Qm, respectively, for some real
numbers s1< s2and open set V ⊂ [s1, s2]. This setting could be of use, for example, when ϕ
is not differentiable or even not continuous at some points (such as for tent maps or Lorenz-like maps, in which case we may include these points in V ). In such a situation we are interested only in those orbits under the map Hλwhich always stay in ([s1, s2]\V )m, and we show below
that if ϕ has k simple zeros in int([s1, s2]\V ) then for |λ| sufficiently large, there still remain
enough orbits to create a full shift on k symbols. Also note that for wide classes of polynomial maps, the non-wandering set is bounded (see [12,17]), and thus, for appropriate s1, s2, the set
[s1, s2]mcontains all non-wandering orbits.
The paper is organized as follows. In section2, we prove a special version of the implicit function theorem in Banach spaces (theorem2.1) in order to control simultaneously infinitely many branches of the implicit function while constructing later a conjugacy map from the full shift at the exceptional value of the parameter to solutions of difference equations at close values. Then we get some consequences from theorem2.1for the maps on l∞.
In section3, we apply the results of section2to families of difference equations in the case when at the exceptional value of parameter they are reduced to functions of a single variable with k 2 simple zeros. Here we especially pay attention to the usual situation when the difference equations for non-exceptional values of parameter correspond to well-defined maps
fλ (possibly non-invertible), in which case we prove that for fλwith parameter λ close to
the exceptional one, there is a closed invariant set λsuch that the inverse limit of fλ|λis
conjugate to the full shift on k symbols; see more properties in theorem3.3. In section4, with the help of this result we obtain a topological horseshoe structure near anti-integrable limits in
several families of generalized H´enon-like maps (see subsection4.1) and in the families of the Arneodo–Coullet–Tresser maps (see subsection4.2) and of quadratic volume preserving maps (see subsection4.3). As another application, in subsection4.4we obtain chaotic behaviour for steady-state solutions in lattice models of discrete versions for some PDE of evolution type (see also [2] for Afraimovich and Chow’s approach to chaotic behaviour of steady states in lattice systems using homoclinic structure).
2. A version of the implicit function theorem and its consequences for shift commuting functions on ∞
The results of the paper will be based on simultaneous control of infinitely many branches of appropriate implicit functions. To this end we provide the following ‘uniform version’ of the implicit function theorem. In the statement and proof of this version, we will use notations
U (x, r)and U [x, r] for the open and closed ball, respectively, of radius r centred at the point
x∈ X, where X is a metric space.
Theorem 2.1. Let E be a metric space, and G and H be Banach spaces. Let a be a point in
E and B be a subset of G. Denote V0= U(a, δ0) and W0 =
b∈BU (b, η0) for some δ0>0
and η0>0, and assume that F : V0× W0→ H is a function such that
(i) F (a, b)= 0 for all b ∈ B;
(ii) F is continuous on V0× W0and moreover, the family of functions F (·, y), y ∈ W0, is
equicontinuous on the domain V0, i.e. for any > 0 there exists δ > 0 such that for any
y ∈ W0 one hasF (x1, y)− F (x2, y) < provided x1, x2 ∈ V0 with ρ(x1, x2) < δ,
where ρ is the metric on E;
(iii) the partial derivative operator D2F (x, y) exists for any point (x, y) ∈ V0 × W0 and
moreover, it is continuous at{a} × B uniformly in b ∈ B in the following sense: for any >0 there exists δ > 0 and η > 0 such that for any b∈ B, D2F (x, y)−D2F (a, b) <
provided x∈ U(a, δ) and y ∈ U(b, η); and
(iv) the operator D2F at any point (a, b) ∈ {a} × B is invertible, and the inverse,
(D2F )−1, is uniformly bounded, i.e. there is a constant M > 0 such that for any
b∈ B, (D2F (a, b))−1 M.
Then there exist 0 < ¯δ < δ0, 0 < ¯η < η0 and a family of continuous functions
ψb : U (a, ¯δ)→ U(b, ¯δ), b ∈ B, such that for any b ∈ B,
ψb(a)= b and F (x, ψb(x))= 0 for all x ∈ U(a, ¯δ).
Furthermore, the functions ψb are uniquely determined (that is, if F (x, y) = 0 and
(x, y) ∈ U(a, ¯δ) × U(b, ¯η) for some b ∈ B then necessarily y = ψb(x)) and form an
equicontinuous family. Moreover, for any x∈ U(a, ¯δ), the map b → ψb(x) is injective on B.
We postpone the proof of theorem2.1to the appendix.
Remark 2.2. It is easy to see that assumption (ii) of the theorem is surely satisfied provided
E is a normed space, D1F (x, y)exists, and D1F (x, y) is bounded above on V0 × W0.
Also, both assumptions (ii) and (iii) are satisfied if F (x, y) is C2 on V0× W0with bounded
derivatives D2D2F (x, y)and D1D2F (x, y)and moreover, in this case, the resultant functions
ψb(x)are C1.
Let ∞be the space of bounded real sequences endowed with the normy = sup{|yn| :
n ∈ Z} for y = (yn), yn ∈ R, i.e. with the topology of uniform convergence and let
Corollary 2.3. Let the assumptions of theorem2.1be satisfied for the case when G= H = ∞ and B is σ -invariant: σ (B)= B. If for any x ∈ U(a, ¯δ), the function F (x, ·) commutes with σ , i.e. F (x, σ y)= σ ◦ F (x, y), then so is the map b → ψb(x), i.e. ψσ (b)(x)= σ ◦ ψb(x). Proof. Denote the origin in ∞ by 0 = (· · · 0.00 · · ·). Along with the conclusions of theorem2.1, the commutative assumption implies that
F (x, σ◦ ψb(x))= σ ◦ F (x, ψb(x))= σ (0) = 0,
while σ -invariance implies that
F (x, ψσ (b)(x))= 0.
Moreover, we haveσ ◦ ψb(x)− σ (b) = ψb(x)− b < ¯η. Hence, by the uniqueness of ψb
in theorem2.1, we get σ◦ ψb(x)= ψσ (b)(x).
In the following theorem (as well as in the next section) we will consider mainly subsets of ∞ endowed with the product (or Tichonov) topology onRZ, i.e. with the topology of pointwise convergence. In such a case we will supply the notation of the appropriate sets with subscript prod, for example: ∞,prod, Bprod, etc. Since in the following theorem, the coordinate
x∈ E will be fixed, it is convenient to re-denote ψb(x)by ¯ψx(b).
Theorem 2.4. Let the assumptions of theorem2.1be satisfied for the case when G= H = ∞ and let B be a subset of [s1, s2]Zfor some real numbers s1 < s2. If for any x ∈ U(a, δ0),
the function F (x,·) is continuous with respect to the product topologies on the domain V0
and codomain ∞, then there exists 0 < ˆδ < ¯δ ( ¯δ being from the statement of theorem2.1) such that for any (fixed) x∈ U(a, ˆδ), the map b → ¯ψx(b):= ψb(x) from Bprod to ∞,prod is
continuous.
Proof. The equicontinuity property from theorem2.1implies that there is 0 < ˆδ < ¯δ such that for any x∈ U(a, ˆδ) and any b ∈ B, one has
ψb(x)− b <
¯η 3,
where¯η is from the statement of theorem2.1.
Fix x∈ U(a, ˆδ). Since ∞,prod is separable, it is sufficient to show that given a sequence
b(m)= (bn(m))∞n=−∞∈ B, m 0, which tends to b∗ = (b∗n)∞n=−∞∈ B in the product topology,
one has ¯ψx(b(m))
prod
−→ ¯ψx(b∗)as m→ ∞.
By theorem2.1, for any b∈ B, one has ¯ψx(b)−b < ¯η and thus ¯ψx(b)∈ [s1− ¯η, s2+¯η]Z.
Since [s1− ¯η, s2+¯η]Zis compact in the product topology, it follows that the sequence ¯ψx(b(m))
has an accumulation point, say y∗, in [s1− ¯η, s2+ ¯η]prodZ . We must show that y∗ = ¯ψx(b∗).
Without loss of generality, we may suppose (by switching to an appropriate subsequence) that ¯ψx(b(m))
prod
−→ y∗ as m → ∞. Since b(m) −→ bprod ∗, it follows that for any N ∈ N there is
M∈ N such that for m > M one has
b(m) n − bn∗ < ¯η 3 for all − N n N, and ¯ψx(b(m))n− b∗n ¯ψx(b(m))n− bn(m) + b (m) n − bn∗ < 2¯η 3 , where the last inequality follows from theorem2.1. Letting N → ∞, we get
y∗− b∗ 2¯η
On the other hand, because of the continuity of F (x,·) with respect to the product topology, one has
0= F (x, ¯ψx(b(m)))
prod
−→ F (x, y∗).
Therefore, F (x, y∗)= 0 and by using (2) one gets from theorem2.1that y∗ = ¯ψx(b∗).
We summarize the above results as follows.
Theorem 2.5. Let the assumptions of theorem2.1be satisfied for the case when G= H = ∞ and let B be a σ -invariant and compact subset of [s1, s2]Zprod for some real numbers s1 < s2.
If for any x∈ U(a, δ0), the function F (x,·) commutes with the shift map σ and is continuous
with respect to the product topologies on the domain V0and codomain ∞, then there exists
0 < ˆδ < ¯δ ( ¯δ being from the statement of theorem2.1) such that for any (fixed) x∈ U(a, ˆδ), the map b→ ¯ψx(b):= ψb(x) is a homeomorphism which conjugates σ|Bprodto σ| ¯ψx(B)prod.
Moreover, the conjugacy map depends continuously on x, not only in the product topology but also in the ∞topology; more precisely, the family of maps x → ψb(x) from U (a, ˆδ) to ∞
forms an equicontinuous family in b∈ B.
Proof. Since the domain is compact and the codomain is Hausdorff, the result follows
immediately from the conclusions of theorems2.1and2.4and corollary2.3.
Remark 2.6. Note that in theorem2.4, the assumption on continuity of F (x,·) is with respect to the product topology and hence it is surely satisfied when each component of F is a continuous function depending on only finitely many coordinates of the domain (i.e. a finitary code function by terminology of the information theory):
(F (x, y))n= fn(x, yn+k(n), yn+k(n)+1, . . . , yn+K(n))
for each x ∈ V0, n ∈ Z, where k(n) and K(n) are some functions from Z to itself, and fn
is a continuous function fromRK(n)+k(n)+2toR. If, in addition, k(n) and K(n) are constant
(i.e. independent of n) then F (x,·) commutes with the shift map (in which case F is called a stationary code function).
Remark 2.7. One can check that theorems2.4and 2.5still hold for a more general case when the spaces G and H are d
∞:= ( ∞)d, i.e. the direct product of d copies of ∞, the set B is a
subset of ([s1, s2]Z)d and σd : d∞→ d∞is the direct product of d copies of the shift map σ . 3. Perturbation of zero-dimensional chaotic systems for difference equations
Let us consider a difference equation of the form
λ(yn, yn+1, . . . , yn+m)= 0, (3)
where λ is a parameter from a metric space E and the function λis defined on a closed subset
Qm+1⊂ Rm+1, where Q= [s
1, s2]\V for some real numbers s1< s2and some open (possibly
empty) set V ⊂ R. We assume that for each λ ∈ E the function λ: Qm+1→ R is C1and is
continuous in λ on E and also that the partial derivatives ∂iλ(x1, . . . , xm+1), i = 1, . . . , m+1,
(x1, . . . , xm+1)∈ Qm+1 are continuous in λ on E, where ∂iλis the partial derivative of λ
Given a λ∈ E, let Yλbe the set of solutions of the difference equation (3), i.e. the set of
bi-sequences y= (yn)= (. . . , y−1, y0, y1, . . .)such that for any n∈ Z
1. yn∈ Q; and
2. (m + 1) consecutive components yn, yn+1, . . . , yn+mof y satisfy (3).
It is easy to see that Yλis a closed subset of the space ∞. Note that Yλ,prod is also a closed
subset of [s1, s2]Zprod, and since the latter space is compact (by the Tichonov theorem), Yλ,prod
is compact.
Note that the shift map σ , being considered as a map from ∞to ∞, is an isometric linear operator, while σ : ∞,prod → ∞,prod is a homeomorphism. It is evident that for any λ∈ E, the set Yλ,prodis σ -invariant and the restriction σ|Yλ,prod is a homeomorphism on a compact
space. Thus, one can define the topological entropy for solutions of the difference equation (3) as htop(σ|Yλ,prod).
In our applications, the difference equations for non-exceptional values of λ actually will be associated with well-defined maps. So we will need the following definition. We denote by σmthe shift map acting on m∞.
Definition 3.1. We say that for a given λ, the difference equation (3) corresponds to a map fλ : Pλ → Rm, where Pλ is a compact subset ofRm and fλ is continuous, if σ|Yλ,prod is
conjugate to σm| ˜Pλ, where ˜Pλ = {p = (pn)∞n=−∞ ∈ PλZ : pn+1 = fλ(pn)}, i.e. the space of
full orbits for fλin Pλendowed with the product topology.
We are concerned now with the case when the difference equation (3) corresponds to a map
fλ: Pλ→ Rm. Let Tλ: Yλ→ ˜Pλdenote the corresponding conjugacy map (we have omitted
here the subscript prod, and we will do so again later if no doubts appear). So ˜Pλis a compact
and σm-invariant set. Let us denote Kλ+ =
∞
n=0fλ−n(Pλ). Then Kλ+ is compact and fλ
-invariant, and σm| ˜Pλcan be regarded as the inverse limit lim
←−(K + λ, fλ)by identifying a sequence (. . . , p−2, p−1, p0)∈ lim ←−(K + λ, fλ)with (. . . , p−2, p−1, p0, fλ(p0), fλ2(p0), . . .)∈ ˜Pλ. Note that Kλ:= π0( ˜Pλ)= ∞ n=0f n
λ(Kλ+)(because of the compactness of Kλ+), where by
π0 we denote the projection to the 0th coordinate. Obviously, π0 is a semi-conjugacy map
from σ| ˜Pλto fλ|Kλ, i.e. fλ◦ π0 = π0◦ σ. It is easy to see that if, in addition, fλ: Pλ→ Rm
is one-to-one, then π0is in fact a conjugacy. In any case, for a (possibly non-invertible) map
fλ, we may claim that π0is a semi-conjugacy which is an ‘almost conjugacy’; more precisely,
for π0: ˜Pλ→ Kλthe following holds.
(i) π0is surjective and so Kλis compact and fλ-invariant;
(ii) π0yields a one-to-one correspondence between the sets of periodic points for σm| ˜Pλand
fλ|Kλ; and
(iii) π0 preserves the topological entropy, i.e. for any compact fλ-invariant set L ⊂ Kλ,
htop(fλ|L) = htop(σm|π0−1(L)).
Indeed, items (i) and (ii) are obvious while (iii) follows from the Bowen inequality in [7]
htop(fλ|L) htop(σm|π0−1(L)) htop(fλ|L) + sup p∈L
htop(σm, π0−1(p)),
since for any p ∈ L we have that htop(σm, π0−1(p)) = 0 because the diameter of the sets
(σm)n(π0−1(p))tends to zero as n→ +∞ (here we have chosen a metric compatible with the
product topology).
We will call a map satisfying property (iii) above an entropy-preserving map. Notice that there are other properties of π0 which recall the properties of a real conjugacy, because
˜
lim
←−(K +
λ, fλ) = lim
←−(Kλ, fλ)). Roughly speaking, one may say that both systems have the
same chaotic properties (with respect to several definitions of chaos); see the precise statements in [18]. So we have the following commutative diagram:
Yλ Tλ −→ ˜Pλ π0 −→ Kλ σ ↓ σm↓ ↓ fλ Yλ−→ Tλ ˜ Pλ−→ π0 Kλ.
In this situation, the topological entropy htop(σ|Yλ,prod)is equal to htop(fλ), the topological
entropy of the map fλ, which is meant as htop(fλ|Kλ+), the topological entropy of restriction
of fλ to the maximal fλ-(forward) invariant set Kλ+. Indeed, we have htop(σ|Yλ,prod) =
htop(σ| ˜Pλ) = htop(fλ|Kλ) because Tλ is a conjugacy and π0 is entropy-preserving by
property (iii); further, the topological entropy htop(fλ|Kλ)is the same as htop(fλ|Kλ+), because
Kλ= π0( ˜Pλ)=
∞
n=0f n
λ(Kλ+); see corollary 8.6.1 in [27].
Remark 3.2. Usually (see examples below and also [15,17,23]), the conjugacy Tλ−1
from P˜λ to Yλ is of the form (pn)∞n=−∞ → (yn)∞n=−∞ = (π(pn))n=−∞∞, where
π:Rm→ R is the projection to a chosen coordinate, while T
λ can be expressed by pn =
G(yn−N, yn+1−N, . . . , yn+m−1−N)for some integer N and diffeomorphism G :Rm→ Rm.
Now according to notations from the previous section, we define F : E× QZ→ ∞by
F (λ, y)= (λ(yn, yn+1, . . . , yn+m))∞n=−∞. (4)
Then F (λ,·) is σ-invariant; indeed,
F (λ, σ (y))= F (λ, (yn+1)∞n=−∞)= (λ(yn+1, . . . , yn+m+1))∞n=−∞
= σ(λ(yn, . . . , yn+m)∞n=−∞)= σ(F (λ, y)).
Note that Yλ is precisely the kernel of F (λ,·), i.e. the set of bi-sequences y satisfying
F (λ, y)= 0.
Let λ0 ∈ E be the initial value of parameter λ, i.e. λ0 corresponds to the unperturbed
difference equation (3) (in the previous section, such a value is denoted by a). Let B be a subset of Yλ0. We now check the assumptions of theorem2.1for our settings. By the definition
of Yλthe assumption (i) is readily satisfied. Given y = (yn)∈ QZ, λ∈ E, and integers n ∈ Z,
1 i m + 1, we denote for brevity ∂iλ(yn)= ∂iλ(yn, yn+1, . . . , yn+m). Then the partial
derivative operator D2F (λ, y)exists at any point (λ, y) ∈ E × QZ and corresponds to the
following bi-infinite band matrix
D2F (λ, y) = · · · ... ... ... ... ... ... · · · · · · ∂1λ(yn) ∂2λ(yn) · · · · ∂m+1λ(yn) 0 · · · · · · 0 ∂1λ(yn+1) ∂2λ(yn+1) · · · · ∂m+1λ(yn+1) · · · · · · ... ... ... ... ... ... · · · ← the nth row. ↑ the nth column
Since λ(x1, . . . , xm+1)as well as the the partial derivatives ∂iλ(x1, . . . , xm+1), i =
is compact, it follows that both F (λ, y) and D2F (λ, y)are uniformly continuous on E× QZ.
Therefore the assumptions (ii) and (iii) of theorem2.1are also satisfied.
As for (a more delicate) assumption (iv), we need to consider some special properties of the set B in order to guarantee (iv), and we are now going to do this. We consider the case when the unperturbed difference equation involves only one variable. More precisely, let
λ0(y0, . . . , ym)= ϕ(yN),
where 0 N m and ϕ : Q → R is a C1 function. In addition, we suppose that ϕ has simple zeros at d1, d2, . . . , dk in int(Q), i.e. ϕ(di) = 0 for all 1 i k. Then for any
d = (din)∞n=−∞∈ {d1, . . . , dk}Z, the matrix D2F (λ0, d)reduces to the diagonal matrix of the
form D2F (λ0, d)= σN· diag(· · · , ϕ(din), ϕ (d in+1), ϕ (d in+2),· · ·),
whose entries are
(D2F (λ0, d))n,j= ϕ(din), if j = n + N, 0, otherwise. Therefore, (D2F (λ0, d))−1= σ−N· diag · · · , 1 ϕ(din) , 1 ϕ(din+1) , 1 ϕ(din+2) ,· · · and consequently (D2F (λ0, d))−1 min 1ik|ϕ (d i)| −1 .
Hence the assumption (iv) of theorem2.1is satisfied for B = {d1, . . . , dk}Z, and we are able
to use the results of section2(note that the additional assumptions contained in theorems2.4
and2.5are surely satisfied). So we may use the conjugacy ¯ψλ|B from theorem2.5in order to
get a closed σ -invariant subset λ := ¯ψλ(B)of Yλ,prod such that σ|λis conjugate to σ|k,
the full shift on k symbols. As an immediate consequence we have the following theorem. For simplicity, we consider here one-parameter families with parameter λ∈ E := [λ0, λ1], where
λ0might be either a real number or infinity, in which case E is the metric space of one point compactification with the metric ρ(λ, λ)= |1λ−λ1|; ∞1 = 0. We summarize our arguments
as follows.
Theorem 3.3. Let
λ(yn, yn+1, . . . , yn+m)= 0 (5)
be a difference equation with parameter λ ∈ [λ0, λ1] and let the function λ : Qm+1 → R,
where Q = [s1, s2]\ V for some numbers s1 < s2 and some open set V ⊂ [s1, s2], be
such that it is C1 for each λ and is continuous in λ and so are the partial derivatives ∂i
λ,
i= 1, . . . , m + 1. Suppose that for λ = λ0, the function λ0 depends on only one variable: λ0(x1, x2, . . . , xm+1)= ϕ(xN), where N is an integer with1 N m + 1 and ϕ : Q → R is a C1function with k simple zeros in the interior of Q.
Then there exists ¯δ > 0 such that for any λ∈ [λ0, λ0+ ¯δ) there is a closed σ -invariant
subset λof Yλ, the set of solutions for (5) in the product topology, such that
(i) σ|λ is topologically conjugate to σ|k, the full shift on k symbols; in particular,
htop(σ|Yλ) log k;
(ii) the conjugacy map ¯ψλ: λ→ λis the identity map for λ= λ0and is continuous in λ;
moreover, the map λ→ ¯ψλ(x) from[λ0, λ0+ ¯δ) to ∞(in the uniform topology) forms an
(iii) if, in addition, the difference equation (5) for given λ corresponds to a map fλ: Pλ→ Rm,
then htop(fλ) log k and one has the following commutative diagram:
k ¯ψλ −→ Yλ Tλ −→ ˜Pλ π0 −→ Kλ σ ↓ σ ↓ σm↓ ↓ fλ k −→ ¯ψλ Yλ −→ Tλ ˜ Pλ −→ π0 Kλ,
where ψ¯λ is injective, Tλ is bijective and π0 is a (surjective) projection which is
entropy preserving and is bijective when restricted to the set of periodic points; here
˜ Pλ = {p = (pn)∞n=−∞ ∈ PλZ : pn+1 = fλ(pn)} and Kλ = ∞ i=0f i λ( ∞ n=0fλ−n(Pλ)). In
particular, the map θλ:= π0◦ Tλ◦ ¯ψλsemiconjugates the full shift on k symbols to the
restriction of fλto the closed fλ-invariant subset λ:= θλ(k) and thus the map fλ|λ
is transitive and its periodic points are dense in λ. Moreover, the inverse limit of fλ|λ
is conjugate to σ|k. If fλ : Pλ → Rm is injective, then the the above semiconjugacy
θλ|kis in fact a conjugacy. 4. Examples
4.1. The generalized H´enon-like maps
We now apply the above results to generalized H´enon-like maps (for relevant properties of H´enon-like maps see [10,12,15,21,23,26]). Let Hλbe the following m-dimensional (m 2)
one-parameter family of H´enon-like maps.
Hλ(x1, x2, . . . , xm−1, xm)= (x2, x3, . . . , xm, g(x1, . . . , xm)+ λϕ(xm)), (6)
where λ∈ R is a parameter, g : Rm→ R is a C1function, and ϕ :R → R is a C1function
with k simple zeros. We do not require that ∂1g(x1, . . . , xm) = 0 and so Hλneed not to be a
diffeomorphism onRm(even locally). Furthermore, we allow the domains of C1functions ϕ
and g to be Q= [s1, s2]\V and Qm, respectively, for some real numbers s1< s2and open set
V ⊂ [s1, s2]. In such a situation we are interested only in those orbits under Hλwhich stay
always in ([s1, s2]\V )m, and we show below that if ϕ has k simple zeros in int([s1, s2]\V ) then
for|λ| sufficiently large, there still remain enough orbits to create a full shift on k symbols. Note that if V = ∅ and ∂1g(x1, . . . , xm) = 0 on Qmthen Hλis injective for any λ; this can be
verified directly by the contrary.
For an initial point p= (x0,1, x0,2, . . . , x0,m), the nth iteration Hλn(p)of p under Hλwill
be also denoted by (xn,1, xn,2, . . . , xn,m). Then for any n∈ Z, we have the following system
consisting of m(m− 1) + 1 equations (the index i for labelling the equations below takes the values 1, 2, . . . , m − 1): xn+i,1= xn+i−1,2, (1, i) xn+i,2= xn+i−1,3, (2, i) .. . ... xn+i,m−1 = xn+i−1,m, (m− 1, i)
xn+i,m= g(xn+i−1,1, . . . , xn+i−1,m)+ λϕ(xn+i−1,m), (m, i)
xn+m,1= xn+m−1,2. (1, m + 1)
By the elimination process with respect to the first coordinate inRm, we get
xn+m,1= xn+m−1,2= · · · = xn+1,m
= g(xn,1, xn,2, . . . , xn,m)+ λϕ(xn,m)
Let π1 : Rm → R denote the projection to the first coordinate. Then for any p =
(x0,1, x0,2, . . . , x0,m) ∈ Rm, the projection π1(Hλn(p)) = xn,1 := yn of the full orbit of p
under Hλsatisfies the following difference equation.
λ(yn, yn+1, . . . , yn+m):=
1
λyn+m−
1
λg(yn, yn+1, . . . , yn+m−1)− ϕ(yn+m−1)= 0. (7)
Conversely, solutions (yn)∞n=−∞of (7) uniquely determine the full orbits of Hλ; indeed, it is
easily seen from the above system of equations that the correspondence y = (yn)∞n=−∞ →
p= (pn)∞n=−∞∈ (R
m)Zwith p
n= (yn, yn+1, . . . , yn+m−1)provides the full orbits of Hλ. So
we have that the correspondence map Tλprovides a conjugacy between the shift map acting
on Yλ, the set of solutions of (7) with yn∈ Q for all n ∈ Z, and ˜Pλ, the set of full orbits of Hλ
in Pλ:= Qm, both sets being considered with respect to the product topologies.
Yλ σ −→ Yλ (Tλ(y))n= (yn, . . . , yn+m−1) Tλ↓ ↓ Tλ ˜ Pλ−→ σm ˜ Pλ (Tλ−1(p))n= π1(pn).
Note that if Hλis injective then σm: ˜Pλ→ ˜Pλis conjugate to the H´enon map itself.
Now let λ→ ∞. Then for λ0:= ∞, we have the limit difference equation
λ0(yn, yn+1, . . . , yn+m)= −ϕ(yn+m−1)= 0,
which does not correspond to any map but satisfies our assumptions above. Thus theorem3.3
implies the following.
Proposition 4.1. Let Hλbe a one-parameter family of H´enon-like maps of the form (6), where
ϕ and g are C1 functions on Q = [s
1, s2]\V and Qm, respectively, for some real numbers
s1< s2and open set V ⊂ [s1, s2]. If ϕ has k simple zeros in int(Q), then for|λ| sufficiently
large the statements of theorem3.3hold (with fλ= Hλ); in particular, by item (iii), there is a
closed Hλ-invariant subset λof Qmsuch that the inverse limit of Hλ|λis conjugate to the
full shift on k symbols. If V = ∅ and ∂1g(x1, . . . , xm) = 0 on Qmthen in fact, Hλ|λitself is
conjugate to the full shift on k symbols.
Remark 4.2. It is allowed that the function (6) might have discontinuities. For example, one may include Lorenz-type maps instead of the usual quadratic maps there, more precisely, if one considers H´enon-like maps of the form
Hλ(x1, x2, . . . , xm−1, xm)= (x2, x3, . . . , xm, g(x1, . . . , xm)+ λx
−λ− 1
2 (1 + sgn(x− 1/2))). (8)
Then by repeating the above arguments one gets for λ0 = ∞ the equations
λ0(yn, yn+1, . . . , yn+m)= yn+m−1−
1
2(1 + sgn(yn+m−1− 1 2))= 0,
which have two simple zeros{0, 1}.
There are families of H´enon-like maps for which the above propositions do not apply directly; nevertheless, they might be of use when adapted by appropriate scaling. For example, consider the (original) H´enon maps in the form Hλ(x, y)= (y, bx + λ − ay2)with a > 0 and
parameter λ → +∞. Then after rescaling ˆx = x/√λand ˆy = y/√λ, we have for the new coordinates,
ˆ
Hλ(ˆx, ˆy) = ( ˆy, b ˆx +
√
and the corresponding difference equation reads ˆλ(yn, yn+1, yn+2)= 1 √ λyn+2− b √ λyn− (1 − ay 2 n+1)= 0.
So we are able to use theorem3.3(with λ0 = +∞, k = 2, and s1, s2any two numbers such
that s1 <−
√
aand s2 >
√
a) to provide a conjugacy between orbits of the full shift on two symbols and orbits of Hλwith λ sufficiently large.
The same arguments work for a family of m-dimensional H´enon-like maps of the form
Hλ(x1, x2, . . . , xm)= (x2, x3, . . . , xm, g(x1, . . . , xm)+ λ− ax2jm), (9)
where a > 0, j ∈ N, and g(x1, . . . , xm)is a real polynomial with deg(g) < 2j . After rescaling
ˆxi = xi/ 2j √ λfor i= 1, . . . , m, we have ˆ Hλ(ˆx1,ˆx2, . . . ,ˆxm)= ˆx2,ˆx3, . . . ,ˆxm, 1 2j √ λg( 2j √ λˆx1, . . . , 2j √ λˆxm)+ λ 2j √ λ(1− a ˆx 2j m) and the corresponding difference equation reads
ˆλ(yn, yn+1, . . . , yn+m):= 2j √ λ λ yn+m− 1 λg( 2j √ λyn, . . . , 2j √ λyn+m−1)+ (1− ayn2j+m−1)= 0.
Since deg(g) < 2j , we have that (1/λ)g(2j√
λyn, . . . ,
2j
√
λyn+m−1) → 0 as λ → +∞ and
since the function ϕ(x) := 1 − ax2j has two simple zeros{−√2j
a−1, √2ja−1}, we may apply
theorem3.3with constants s1 <−
2j
√
a−1and s2>
2j
√
a−1and get the following result.
Proposition 4.3. Let Hλbe a one-parameter family of H´enon-like maps of the form (9) where
a > 0, j ∈ N and g(x1, . . . , xm) is a real polynomial withdeg(g) < 2j . Then for λ > 0
sufficiently large the statements of theorem3.3hold (with fλ= Hλ, k= 2); in particular, by
item (iii) there is a closed Hλ-invariant subset λ⊂ Rmfor which the inverse limit of Hλ|λ
is conjugate to the full shift on two symbols. 4.2. The Arneodo–Coullet–Tresser maps
Now we consider the family of the so-called ACT maps f : R3 → R3 (due to Arneodo,
Coullet and Tresser; refer to [11]); they are of the form
f (x, y, z)= (ax − b(y − z), bx + a(y − z), cx − dxk+ ez), (10)
where a, b, c, d, e∈ R are parameters and k 2. If (a2+b2)e = 0, then f is a diffeomorphism
with the inverse
f−1(x, y, z)= ˆx,−bx + ay a2+ b2 +ˆz, ˆz , where ˆx = ax+ by a2+ b2 andˆz = z− c ˆx + d ˆxk e .
If bd = 0, then there are interesting dynamical properties and bifurcations in several regions of the parameter space; see [11]. For an initial point p= (x0, y0, z0), denote the nth iteration
of p under f by (xn, yn, zn). Then for any n∈ Z, we have the following system consisting of
7 equations (the index i for labelling the equations below takes the values 1 and 2):
xn+i= axn+i−1− b(yn+i−1− zn+i−1), (1, i)
yn+i= bxn+i−1+ a(yn+i−1− zn+i−1), (2, i)
zn+i= cxn+i−1− dxkn+i−1+ ezn+i−1, (3, i)
Assuming b = 0, From equations (1, 1) and (2, 1), we can express yn+1in terms of xnand
xn+1. Then plugging yn+1into equation (1, 2), we can express zn+1 in terms of xn, xn+1and
xn+2. The expressions are
yn+1= a2+ b2 b xn− a bxn+1, zn+1 = a2+ b2 b xn− 2a b xn+1+ 1 bxn+2.
Similarly, from equations (1, 2), (2, 2) and (1, 3), we can express zn+2in terms of xn+1, xn+2
and xn+3. Then, plugging the expressions of zn+1 and zn+2 into equation (3, 2), we get the
difference equation dxnk+1−a 2e+ b2e b xn+ a2+ b2− bc + 2ae b xn+1− 2a + e b xn+2+ 1 bxn+3= 0. (11)
Suppose the parameters a, b and e to be fixed and let c→ ∞, d → ∞ in such a way that
c/d = constant := A > 0. Denote λ = 1/d, then c = A/λ and for the x-coordinate of orbit
under the ACT map f = fλ, we have the difference equation
xn+1(xnk−1+1 − A) + λ −a2e+ b2e b xn+ a2+ b2+ 2ae b xn+1− 2a + e b xn+2+ 1 bxn+3 = 0. (12) The difference equation (12) corresponds to the map fλfor λ = 0 because between solutions
x= (xn)∞n=−∞of (12) and full orbits p= (pn)∞n=−∞of fλwe have a conjugacy x→ p, given
by pn= xn, a2+ b2 b xn−1− a bxn, a2+ b2 b xn−1− 2a b xn+ 1 bxn+2 ,
while the inverse p→ x is given by xn = π1(pn). For the limit value λ0 = 0 of parameter,
we have
ϕ(xn+1):= xn+1(xnk−1+1 − A) = 0.
Thus, the function ϕ has at least two simple zeros; more precisely ϕ has two simple zeros {0, k−1√
A} when k is even, and three simple zeros {0, ±k−1√A} when k is odd. Thus we are able
to use theorem3.3for the above family of the ACT maps (with s1<−
k−1√A
and s2>
k−1√A
) and have the following result.
Proposition 4.4. Suppose that for the family of the ACT maps (10), the parameters a, b = 0
and e are fixed while c → ∞ and d → ∞ in such a way that c/d = constant > 0. Let
λ= 1/d. Then for all |λ| sufficiently small, the ACT map f = fλhas a closed invariant set
λsuch that fλ|λis conjugate to the full shift on either two or three symbols depending on
whether k is even or odd, respectively.
There are also other parameter routes in families of ACT maps, for which the above arguments apply. Namely, if we consider a and c to be fixed while e→ 0, and b → ∞ and
d → ∞ in such a way that b/d = constant > 0, then similarly along the lines of the previous
proposition, we will have the following.
Proposition 4.5. Suppose that for the family of the ACT maps (10), the parameters a and c are
fixed while e→ 0, b → ∞ and d → ∞ in such a way that b/d = constant > 0. Let λ = 1/d.
Then for all sufficiently small|e| and |λ|, the ACT map f = fe,λhas a closed invariant set
e,λsuch that fe,λ|e,λis conjugate to the full shift on either two or three symbols depending
One can also replace the variable xiby k−1 √ |c| ˆxi so that (11) becomes d k−1|c|kˆxk n+1− k−1|c| a2e+ b2e b λˆxn+ a2+ b2− bc + 2ae b ˆxn+1− 2a + e b ˆxn+2+ 1 bˆxn+3 = 0. By using the above method, we have the following result.
Proposition 4.6. Suppose that for the family of the ACT maps (10), the parameters a, b =
0, d = 0, e are fixed. Then for all sufficiently small |c| with c = 0, the ACT map f = fchas
a closed invariant set csuch that fc|cis conjugate to the full shift on either two or three
symbols provided that k is even or k is odd and d < 0, respectively. 4.3. Quadratic volume preserving maps
In this subsection, we consider the family of volume preserving maps f :R3→ R3defined by
f (x, y, z)= (η + αx + βy + z + Q(x, y), x, y), (13)
where η, α, β are real parameters and Q(x, y) = ax2+ bxy + cy2 is a quadratic form. As shown by Lomeli and Meiss in [19], generically every quadratic volume preserving map is conjugate to a map of the above form, while for volume preserving quadratic automorphisms (i.e. for diffeomorphisms with quadratic inverse) in generic form, one may have a + b + c= 1 and β= 0.
It is easy to see that the corresponding difference equation for (13) is of the form
η+ αxn+ βxn−1+ xn−1+ axn2+ bxnxn−1+ cxn2−1− xn+1= 0.
By using the above method, we have the following result.
Proposition 4.7. Suppose that for the family of quadratic volume preserving maps (13),
the parameters η, β, b, c are fixed while α → ∞ and a → ∞ in such a way that
a/α = constant < 0. Let λ = 1/a. Then for all sufficiently small |λ|, the map f = fλ
has a closed invariant set λsuch that fλ|λis conjugate to the full shift on two symbols.
Notice that for quadratic automorphisms, the presented method does not apply because the equality a + b + c= 1 implies that in the anti-integrable limit both variables xnand xn−1should
be involved, and so the limit function contains two variables. In a forthcoming paper [13], we consider such a case when at the singular value of parameter, the limit function depends on two (not necessarily consecutive) variables.
4.4. Steady states in lattice models
Finally, we consider steady states in lattice models which are discrete versions of partial differential equations of evolution type (for lattice models and their chaotic and stability properties see section 4.3 of [3] and [1,2,4,8,22]). The discrete versions are obtained by replacing derivatives with respect to time and space by appropriate differences (see [22]). In general, the time–space discrete version of these PDEs are of the form
utn+1= f (utn)+ g(ut n−s, u t n−s+1, . . . , u t n+s), (14)
where t∈ Z is the time variable and n ∈ Z is the space one. The function f is called the local map and g is called the interaction of finite size s.
If we look for the steady-state (or stationary) solutions ut
n of (14), then utn must be
independent of the time coordinate t, i.e., ut
n := xnfor all t ∈ Z. In this case, equation (14)
can be reduced to the difference equation
Thus, by using theorem3.3, we have the following result on the chaotic structure of steady-state solutions.
Proposition 4.8. Let the function x → x − f (x) have k simple zeros and be of class C1in a neighbourhood U of these zeros, and let g be of class C1 in U2s+1. Then for sufficiently small , there exists a closed (in the product topology) σ -invariant subset of the set of
steady-state solutions for (14) such that σ| is topologically conjugate to σ|k, the full shift
on k symbols, and the conjugacy map can be chosen to be continuous in with respect not only to the product topology but also to the uniform topology.
Let us remark that unlike in [2,4,22] we do not require that the functions f and g are defined and smooth on the wholeR and R2s+1, respectively, with bounded partial derivatives.
There are several partial differential equations of evolution type whose discrete versions can be considered as lattice dynamical systems of the form (14). For instance, the one-dimensional nonlinear reaction diffusion equation
ut = h(u) + uxx, (16)
where is the diffusion coefficient, has its discrete version in the following form:
utn+1= utn+ δh(utn)+ δη−2 (u t n+1− 2u t n+ u t n−1),
where δ and η are the discretization steps of time and space, respectively, h is a nonlinear function and t and n are the time and space variables, respectively. Hence the steady-states are solutions of the difference equation
h(un)+ η−2 (un+1− 2un+ un−1)= 0. (17)
The well-known particular cases of reaction diffusion equation are:
1. Kolmogorov–Petrovsky–Piskunov equation with h(u)= αu(1 − u), where α > 0; 2. Huxley equation with h(u)= αu(1 − u)(u − a), where 0 < a < 1 and α > 0; 3. Frenkel–Kontorowa model with h(u)= sin(u), refer to [5,6].
As another example, we can consider the Ginzburg–Landau equation,
ut = h(u) + ( + iδ)uxx,
whose real version has the form (16) with h(u)= u(α−βu2)and real parameters , α, β. Since the function h for the Kolmogorov–Petrovsky–Piskunov equation has two simple zeros while for the Huxley equation and Ginzburg–Landau equation it has three simple zeros, it follows that proposition4.8applies for these systems whenever the parameter is sufficiently small. Therefore, the steady-state solutions of these systems contain chaotic dynamical behaviour conjugate to full shift on two or three symbols, respectively.
As one can notice, the results of section3can be presented in the setting of high dimensional difference equations. To this end we can consider the difference equation (5) with coordinates
yn ∈ Rd, the operator λwith domain (Qd)m+1⊂ (Rd)m+1and co-domainRd, and the shift
map σd acting on d∞. So, the high dimensional reaction diffusion equation
∂u
∂t = h(u) + Au,
where A is the coupling matrix, can be also considered in our context. For example, for the two dimensional FitzHugh–Nagumo equation, the function h is of the form
where 0 < θ < 1 and a, b, c, d > 0. Thus, proposition4.8is applicable here because in certain regions of parameters, the function h has three zeros, namely,
(0, 0) and u±,b cu± , where u±=1 2 1 + θ± (1− θ)2−4ab cd ,
with the non-zero Jacobian determinant at these zeros.
Acknowledgments
We are grateful to the referees for helpful suggestions and to the Institute of Mathematics at Academia Sinica of Taiwan for hospitality during our visit. We thank Professors V Afraimovich, Yi-Chiuan Chen, Bau-Sen Du and L P Shilnikov for their helpful discussions. MCL was partially supported by NSC Grant 93-2115-M-018-001 and MM was partially supported by RFBR Grants 05-01-00501 and 05-01-00558.
Appendix
Here we give the proof of our uniform version of the implicit function theorem.
Proof of Theorem2.1. For any b∈ B, we define a function gb : V0× U[b, η0]→ G by
gb(x, y)= y − (D2F (a, b))−1F (x, y).
Then for any b∈ B,
gb(a, b)= b and D2gb(a, b)= 0.
From now on, we denote Tb= D2F (a, b). By assumptions (iii) and (iv), there are 0 < δ1 < δ0
and 0 < η1 < η0such that for any b∈ B and any (x, y) ∈ U[a, δ1]× U[b, η1] one has
D2F (x, y)− Tb 12M−1 12Tb−1−1,
and therefore
D2gb(x, y) = I − Tb−1· D2F (x, y) = Tb−1· Tb− Tb−1· D2F (x, y)
T−1
b · Tb− D2F (x, y) 12.
This implies with that the help of the mean value theorem applied to gb(x,·), that for any
b∈ B, x ∈ U[a, δ1], and any two points y1, y2∈ U[b, η1],
gb(x, y1)− gb(x, y2) 12y1− y2. (18)
We now fix η1and by using assumptions (ii) and (iv), we choose 0 < δ2 < δ1such that
for any b∈ B and any x ∈ U[a, δ2],
F (x, b) = F (x, b) − F (a, b) < 1 2T −1 b −1η1, and therefore gb(x, b)− b = Tb−1· F (x, b) Tb−1F (x, b) < 1 2η1. (19)
Thus for any b∈ B, x ∈ U[a, δ1] and y ∈ U[b, η1], one has by using (18) and (19), that
gb(x, y)− b gb(x, y)− gb(x, b) + gb(x, y)− b < 12y − b +21η1 η1. (20)
The inequality (20) combining with (18) implies that for any b ∈ B and any (fixed)
x ∈ U[a, δ2], the map y→ gb(x, y)is a contraction of the complete metric space U [b, η1]
say ψb(x), and so gb(x, ψb(x))= ψb(x)or, equivalently, F (x, ψb(x)) = 0. Also, we have
obtained that for any x∈ U[a, δ2], the element y= ψb(x)is the only solution of the equation
F (x, y)= 0 in the region y ∈ U[b, η1]. Note that in fact ψb(x)∈ U(b, η1)due to the strict
inequality in (20).
Next, we prove the equicontinuity property. Let b∈ B and x, x∈ U[a, δ2]. Then
ψb(x)− ψb(x) = gb(x, ψb(x))− gb(x, ψb(x)) gb(x, ψb(x))− gb(x, ψb(x)) + gb(x, ψb(x))− gb(x, ψb(x)) 1 2ψb(x)− ψb(x ) + g b(x, ψb(x))− gb(x, ψb(x)). Thus ψb(x)− ψb(x) 2gb(x, ψb(x))− gb(x, ψb(x)) = 2T−1 b · (F (x, ψb(x))− F (x, ψb(x))) 2MF (x, ψb(x))− F (x, ψb(x))
and the result follows from the continuity assumption (ii).
Finally, take ¯η = η1/2. By using the equicontinuity property for the family of functions
ψb(x), we can choose 0 < ¯δ < δ2such that for any b∈ B one has
b − ψb(x) < ¯η whenever x ∈ U(a, δ). (21)
Then the sets U (b,¯η), b ∈ B, are disjoint; indeed, otherwise one would have y∗ ∈
U (b,¯η) ∩ U(b,¯η) for some y∗ ∈ G and b, b ∈ B with b = b, and hence b − b b − y + y − b < 2 ¯η = η
1 which implies that b and b would be different solutions
of the equation F (a, y) = 0 in the region y ∈ U(b, η1), a contradiction to the uniqueness
of solutions. Since, by (21), ψb ∈ U(b, ¯η) for any x ∈ U(a, ¯δ), it follows that the map
b→ ψb(x)is injective.
The proof of theorem2.1is completed.
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