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then Br(z) is a repelling neighborhood of z, under the Euclidean norm.
The conditions (19) and (22) are computable numerically and the value r can be found from
numerical computation. Furthermore, if there exists a snapback point x0 in Br(z), i.e., x0 ∈
Br(z), and F (x0) = z for some integer > 1, then z is a snapback repeller. Hence, the map F
is chaotic in the sense of Marotto.
Sequential graphic-iteration scheme:
We recall an approach which is developed to exploit the existence of snapback
repeller, without estimating the repelling neighborhood.
In particular, it is a scheme
to construct homoclinic orbits for repelling fixed point x of F: {x −j : j ∈ N } with
F(x − 1) = x, F(x −j) = x −j+1, for j ≥ 2, and lim j→∞ F(x −j) = x.
Theorem (Liao & Shih, 2011). Assume that there exists a compact, connected, convex region
Ω = Π n Ω
i=1
i ⊂ R n, so that the C 1 map F = ( F 1, F 2, · · · , Fn) : R n → R n satisfies n
| ∂Fi (
| ∂Fi (
∂
x) | > 1 + ∑
x) |, for all i = 1, · · · , n, x ∈ Ω,
xi
∂x
j=1, j= i
j
and has a repelling fixed point x in Ω ⊂ R n. For i = 1, · · · , n, set
ˆ fi,(1)( ξ) := sup {Fi( x
∈ Ω
1, · · · , xi− 1, ξ, xi+1, · · · , xn) : xj
j, j ∈ { 1, · · · , n}/ {i}},
ˇ f
∈
i,(1)( ξ) := inf {Fi( x
Ω
1, · · · , xi− 1, ξ, xi+1, · · · , xn) : xj
j, j ∈ { 1, · · · , n}/ {i}},
for ξ ∈ R1. Also assume that ˆ fi,(1) and ˇ fi,(1) both have fixed points in Ω i, for all i = 1, · · · , n, and
x − +1 ∈ R n \ Ω, x − ∈ int(Ω)
hold, for some
≥ 2. Then there exist a sequence of nested regions {Ω( k) }∞ with Ω
k=1
( k+1) ⊆
Ω( k) ⊂ Ω, and preimages x −k− 1 ∈ Ω( k) of x under F, k ∈ N. If furthermore, Ω i,( k) → 0, as k → ∞, for all i = 1, · · · , n, then {x −k}∞ is a homoclinic orbit for x. Moreover, if k=1
det ( DF(x −k)) = 0, for 1 ≤ k ≤ − 1
holds, then x is a snapback repeller and F is chaotic in the sense of Marotto’s theorem.
Remark. (i) The existence of this homoclinic orbit guarantees the existence of the snapback
point without finding the repelling neighborhood. (ii) The conditions in the above theorem
are formulated for DF and the one-dimensional maps ˆ fi,(1)( ξ) and ˇ fi,(1)( ξ) (the upper and
lower maps), hence they are easy to examine in applications. For example, for TCNN map,
we can find explicit and computable conditions such that all conditions in the theorem are
satisfied.
4.2 Applications and extensions
We review some applications of snapback repeller and chaotic dynamics in (Marotto, 1979a,
1979b). Consider a two-dimensional mapping F : R2 → R2 of the form F( x, y) = ( f ( x), x), with f : R → R being differentiable.
Multidimensional Dynamics: From Simple to Complicated
519
Lemma (Marotto, 1979a). (i) If f has a stable periodic point z of period p, then F( x, y) =
( f ( x), x) has a stable periodic point ( z, y 0) of period p where y 0 = f p− 1( z). (ii) If f has a snapback repeller, then F( x, y) = ( f ( x), x) has a transversal homoclinic orbit.
Using these results, one can investigate the dynamics of the following difference equation:
xk+1 = f ( xk, bxk− 1),
(23)
where b, xk ∈ R and f : R2 → R is differentiable. We rewrite (23) into the following
two-dimensional system:
xk+1 = f ( xk, byk)
(24)
yk+1 = xk.
Moreover, when b = 0, (23) is the following scalar problem
xk+1 = f ( xk, 0).
(25)
It was shown that the dynamics of (23) or (24) are determined by those of (25), if b is close to
0:
Theorem (Marotto, 1979a). (i) If (25) has a stable periodic point x 0 of period p, then there
exists
> 0 such that (24) has a stable periodic point ( x( b), y( b)) of period p for all |b| < . In this case ( x( b), y( b)) is a uniquely defined, continuous function of b with x(0) = x 0. (ii) If (25) has a snapback repeller, then (24) has a transversal homoclinic orbit for all |b| < , for some
> 0.
Next, let us consider another class of two-dimensional map G : R2 → R2 which is determined
by two scalar equations f ( x) and g( y) where f , g : R → R are differentiable and G( x, y) is defined by G( x, y) = ( f ( x), g( y)).
Lemma (Marotto, 1979a). (i) If one of the mappings f and g has a snapback repeller and the
other has an unstable fixed point, then G( x, y) = ( f ( x), g( y)) has a snapback repeller. (ii) If one of the mappings f and g has a snapback repeller and the other has a stable fixed point,
then G( x, y) = ( f ( x), g( y)) has a transversal homoclinic orbit.
Now, we consider the dynamics for systems of the form:
xk+1 = f ( xk, byk)
(26)
yk+1 = g( cxk, yk),
where f , g : R2 → R are differentiable, and b, c ∈ R are close to 0. If b = c = 0, then (26) can be simplified to the uncoupled system:
xk+1 = f ( xk, 0)
(27)
yk+1 = g(0, yk).
(28)
Theorem (Marotto,1979a). (i) If one of the (27) and (28) has a snapback repeller and the other
has an unstable fixed point, then (26) has a snapback repeller for all |b|, |c| < , for some > 0.
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Discrete Time Systems
(ii) If one of the (27) and (28) has a snapback repeller and the other has a stable fixed point,
then (26) has a transversal homoclinic orbit for all |b|, |c| < , for some > 0.
Remark. By examining the simplified systems, the above results exhibit the dynamics of
system (24) or (26) under some small perturbations of certain parameters. However, these
theorems do not provide any indication about the estimate of .
Next, let us recall the Hénon map
xk+1 = yk + 1 − ax 2 k
yk+1 = bxk,
which can be equivalently written as
uk+1 = bvk + 1 − au 2 =: f ( u
k
k, bvk )
(29)
vk+1 = uk,
where f ( u, v) = v + 1 − au 2. It was shown in (Marotto, 1979b) that uk+1 = f ( uk, 0) has a snapback repeller, when a > 1.55. Hence (29) has a transversal homolinic orbit for all a > 1.55
and |b| < , for some > 0.
In (Li, et al., 2008), they considered a one-parameter family of maps Hλ on R n × R m with
H 0(x, y) = ( F(x), G(x)) and continuous F : R n → R n and G : R n → R m or H 0(x, y) =
( F(x), G(x, y)) with continuous maps F : R n → R n and G : R n × R m → R m. They used the covering relations method proposed by Zgliczy ński in (Zgliczy ński, 1996, 1997) to prove
that if n = 1 and F has a positive topological entropy, or if n > 1 and F has a snapback
repeller, then any small perturbation Hλ of H 0 has a positive topological entropy. Without
using hyperbolicity, the covering relations method still provides a way to verify the existence
of periodic points, the symbolic dynamics and the positive topological entropy. Moreover,
they also applied this method to obtain a new proof for García’s result (García, 1986) that if a
map has a snapback repeller then it has a positive topological entropy. One can obtain similar
results by using this method with other structure, such as a hyperbolic horseshoe.
Since the definition of snapback repeller proposed by Marotto relies on the norm, the
following definition independent of norm was proposed.
Definition (Li, et al., 2008). Let F : R n → R n be a C 1 function. A fixed point z for F is called a snapback repeller if (i) all eigenvalues of the derivative DF(z) are greater than one in absolute
value and (ii) there exists a sequence {x −i}i∈N such that x − 1 = z, lim i→∞ x −i = z, and for all i ∈ N, F(x −i) = x −i+1, F(x − 1) = z and det( DF(x −i)) = 0.
Remark. Although the above definition is independent of norm on the phase space, it requires
the existence of the pre-images for the repeller. The sequential graphic-iteration scheme
outlined above provides a methodology for such a construction.
Note that item (i) implies that there exist a norm
· ∗ on R n, r > 0 and s > 1, such that
F(x) − F(y) ∗ > s x − y ∗ for all x, y ∈ B∗r(z). Hence F is one-to-one on B∗r(z) and F( B∗r(z)) ⊃ B∗r(z). Therefore, if there exists a point x0 ∈ B∗r(z) such that F (x0) = z and det( DF (x0)) = 0 for some positive integer , then item (ii) of the above definition is satisfied.
In addition, in (Li & Chen, 2003), they showed that this norm can be chosen to be the Euclidean
Multidimensional Dynamics: From Simple to Complicated
521
norm on R n, under the condition that all eigenvalues of ( DF(z)) TDF(z) are greater than one.
However, this condition is more restrictive, due to that a repelling fixed point has the potential
to be a snapback repeller, without satisfying this condition.
Theorem (Li, et al., 2008). Let Hλ be a one-parameter family of continuous maps on R n × R m
such that Hλ(x, y) is continuous as a function of λ ∈ R l and (x, y) ∈ R n × R m. Assume that H 0(x, y) = ( F(x), G(x)), where F : R n → R n is C 1 and has a snapback repeller and G : R n → R m. Then Hλ has a positive topological entropy for all λ sufficiently close to 0.
Theorem (Li, et al.,2008). Let Hλ be a one-parameter family of continuous maps on R n × R m
such that Hλ(z) is continuous as a function of λ ∈ R l and (x, y) ∈ R n × R m. Assume that H 0(x, y) = ( F(x), G(x, y)), where F : R n → R n is C 1 and has a snapback repeller, G : R n ×
R m → R m, and G(R n × S) ⊂ int( S) for some compact set S ⊂ R m homeomorphic to the closed unit ball in R m. Then Hλ has a positive topological entropy for all λ sufficiently close
to 0.
Moreover, it was shown in (Li & Lyu,2009) that if F has a snapback repeller and G is a small C 1
perturbation of F, then G has a snapback repeller, positive topological entropy, as the implicit
function theorem is applied. Moreover, G is chaotic in the sense of Li-Yorke. More precisely,
Theorem (Li & Lyu, 2009). Let F be a C 1 map on R n with a snapback repeller. If G is a C 1
map on R n such that F − G + DF − DG
is small enough, where
·
is the operator
norm on the space of linear maps on R n induced by the Euclidean norm
· , then G has a
snapback repeller, exhibits Li-Yorke chaos, and has positive topological entropy.
Corollary (Li & Lyu, 2009). Let Fμ(x) be a one-parameter family of C 1 maps with x ∈ R n and μ ∈ R l. Assume that Fμ(x) is C 1 as a function jointly of x and μ and that Fμ has a 0
snapback repeller. Then map Fμ has a snapback repeller, exhibits Li-Yorke chaos, and has
positive topological entropy, for all μ sufficiently close to μ 0.
In (Shi & Chen, 2004, 2008), they generalized the definitions of expanding fixed point,
snapback repeller, homoclinic orbit, and heteroclinic orbit for a continuously differentiable
map from R n to general metric spaces as follows. Herein, Bdr(x) denotes the closed balls of
radius r centered at x ∈ X under metric d, i.e.
Bdr(z) := {x ∈ R n : d(x, z) ≤ r}.
In the following, we introduce the coupled-expanding map.
Definition (Shi & Chen, 2008). Let F : D ⊂ X → X be a map where ( X, d) is a metric space. If
there exists
≥ 2 subsets Vi, 1 ≤ i ≤ , of D with Vi ∩ Vj = ∂DVi ∩ ∂DVj for each pair of ( i, j), 1 ≤ i = j ≤ , such that
F( Vi) ⊃ ∪j=1 Vj, 1 ≤ i ≤ ,
where ∂DVi is the relative boundary of Vi with respect to D, then F is said to be
coupled-expanding in Vi, 1 ≤ i ≤ . Moreover, the map F is said to be strictly coupled-expanding
in Vi, 1 ≤ i ≤ , if d( Vi, Vj) > 0, for all 1 ≤ i = j ≤ .
Definition (Shi