Chaos in Delay Differential Equations With
Chaos in Delay Differential Equations With
Chaos in Delay Differential Equations With
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Alfonso Ruiz-Herrera
Departamento de Matemática Aplicada, Facultad de Ciencias
Universidad de Granada, 18071 Granada, Spain
1. Introduction. For the last decades, delay differential equations have been con-
sidered as a natural framework to model many real world phenomena. These equa-
tions usually arise in models of population dynamics, neural networks or electrical
engineering since the use of time delay naturally appears to express the maturation
period of a concrete species, the transmission times along nerve axons, etc.
In this note we consider the system
0
x (t) = x(t)(−a(t) − b(t)x(t) + c(t)y(t))
(1)
y 0 (t) = y(t)(d(t) − e(t)x(t) − f (t)y(t − τ ))
where τ > 0 and all the coefficients are positive and T -periodic. This system was
originally proposed by R. May in [25], the autonomous version, in order to describe
the evolution of two species, a herbivore (y) and a carnivore (x), sharing the same
environment. In system (1) we introduce the dependence on time to model the
seasonal effect of the environment, see [2] or [8]. Apart from this consideration, the
reader can find the biological interpretation of the parameters of model (1) in [25]
(also see [3]).
For system (1) we give sufficient conditions for the presence of chaotic dynamics.
Roughly speaking, by chaos in (1) we understand that the Poincaré operator, that
is,
P : D ⊂ R × C([−τ, 0], R) −→ R × C([−τ, 0], R)
z = (ξ, η) 7→ (x(T ; z), yT (z))
with yT (z)(t) := y(T + t; z), ((x(t; z), y(t; z)) is the solution of (1) with initial
condition at z) admits an invariant set Λ semi conjugate to the Bernoulli shift with
infinitely many periodic points and sensitive dependence on the initial conditions.
From the point of view of applications, these facts have deep consequences. For
instance, it is impossible to describe the dynamics in (1) from experimental data
since small errors in the initial values can produce great changes in the future
1633
1634 ALFONSO RUIZ-HERRERA
(sensitive dependence). At this point it is important to observe that system (1) has
been extensively studied, see [2],[3], [16], [17], [22], [32]. However, to the best of
the author’s knowledge, it is the first time that the presence of complex dynamics
is detected analytically.
In relation to our method of proof, we combine the concept of topological horse-
shoes developed by Kennedy and Yorke in [12] (see also [37]) together with an
infinite dimensional variant of the notion of Stretching Along Paths developed by
Papini, Pireddu and Zanolin in [28], [29], [30]. This approach has a topological
character and involves the study of certain geometrical properties of the Poincaré
operator associated with (1). As an advantage, we point out that no hiperbolicity
conditions are required.
There is a broad literature concerning the presence of chaotic dynamics in delay
differential equations using different tools such as hyperbolicity and shadowing,
Conley index or Lefschetz fixed point theorem, see [5], [6] [18], [19], [20], [21],
[34],[35]. The first analytical results in this field were given by An der Heinden
and Walther in [6], [34]. In these results, the presence of chaotic dynamics was a
consequence of the classical Li-Yorke’s paper [23] since the dynamics in the infinite
dimensional space was essentially described by an interval map. In contrast, our
construction in this note is made directly in the infinite dimensional space and is
not based on any reduction of dimension.
Very recently, Wójcik and Zgliczyński in [36] also studied the presence of complex
dynamics in delay differential equations from a topological point of view. In this
interesting paper, the authors mainly proved that the notion of covering relation
introduced in [37] can be applied in delay differential equations with small delays (in
the sense of perturbation). In relation to [36], our results do not involve small/large
delays or any computer assisted proof.
each two-sided sequence (si )i∈Z ∈ {0, 1}Z , there exists a corresponding sequence
(ωi )i∈Z ∈ DZ such that
ωi ∈ Ksi and ωi+1 = ψ(ωi ) for all i ∈ Z, (2)
and, whenever (si )i∈Z is a k-periodic sequence (that is, si+k = si , ∀ i ∈ Z) for some
k ≥ 1, there exists a k-periodic sequence (ωi )i∈Z ∈ DZ satisfying (2).
As mentioned in the introduction, our definition of chaos guarantees natural
properties of chaotic dynamics such as sensitive dependence on the initial conditions
or the presence of an invariant set Λ topologically transitive and semi-conjugate
with the Bernoulli shift. Besides these properties, it is important to observe that
our definition ensures the presence of infinitely many periodic points. Notice that
this property is not generally assumed in many definitions of chaos, (see Theorem
2.2 in [26] for the proofs of these properties and [1], [13] for the main definitions of
chaos).
Once these remarks have been done, we properly introduce the notion of Stretch-
ing Along Paths.
Definition 2.2. Let P : D ⊂ R × C([−τ, 0], R) −→ R × C([−τ, 0], R) be a compact
operator and consider a rectangle A = [a, b] × [c, d] ⊂ R2 , a constant m > 0, and
a closed set H ⊂ XA,m . We say that (H, P ) stretches XA,m along the paths and
write
(H, P ) : XA,m −→X
m A,m ,
if for every continuous path γ : [0, 1] −→ XA,m such that γ(0) ∈ {(x, y) ∈ XA,m :
x = a} and γ(1) ∈ {(x, y) ∈ XA,m : x = b}, there exists a subinterval [t0 , t00 ] ⊂ [0, 1]
satisfying that
• γ(t) ∈ H, P (γ(t)) ∈ XA,m for all t ∈ [t0 , t00 ],
• P (γ(t0 )) ∈ {(x, y) ∈ XA,m : x = a} and P (γ(t00 )) ∈ {(x, y) ∈ XA,m : x = b} or
viceversa.
The following result links the previous notion with our definition of chaos.
Theorem 2.3. Assume that there exist two disjoint compact sets K0 , K1 ⊂ A, a
constant m, and a compact operator P such that
(XK0 ,m , P ) : XA,m −→X
m A,m ,
(XK1 ,m , P ) : XA,m −→X
m A,m .
Then P induces chaotic dynamics on two symbols relatively to P (XA,m ) ∩ XK1 ,m
and P (XA,m ) ∩ XK0 ,m .
Before starting the proof of this theorem, we need the following results taken
from [26] and [28].
Lemma 2.4. Under the conditions of the previous theorem, given j ≥ 1 and a
(j + 1)-tuple (s0 , ..., sj ) with si ∈ {0, 1} and sj = s0 we have that
(I, P j ) : XA,m −→X
m A,m ,
where
I := {ω ∈ XKs0 ,m : P i (ω) ∈ XKsi ,m for all i = 0, ..., j − 1}.
Proof. This lemma is essentially Lemma A. 1 in [26].
1636 ALFONSO RUIZ-HERRERA
To finish this section we notice that the hypotheses of Theorem 2.3 can be mod-
ified to derive a result which is stable with respect to small perturbations of the
operator P . With this respect, we have the next result.
Corollary 1. Let A, K0 , K1 and m be as in Theorem 2.3. Suppose that there exists
a constant r > 0 so that (XKi ,m , P ) : XA,m −→X
m A,m , for i = 0, 1 with
K0 , K1 ⊂ ]a+r, b−r[×[c, d] , P ev (XK0 ,m )∩A, P ev (XK1 ,m )∩A ⊂ [a, b]×]c+r, d−r[
and P (XK0 ,m ), P (XK1 ,m ) ⊂ {(x, y) ∈ R × C([−τ, 0], R) : |y(t) − y(0)| < m − r}.
Then every compact operator Pε : XA,m → Pε (XA,m ) is chaotic provided that
||Pε (x, y) − P (x, y)|| ≤ ε, for all (x, y) ∈ XA,m
r
for all 0 < ε < 4.
e −→X
(XK0 ,m , Pε ) : XA,m e ,
m A,m
e −→X
(XK1 ,m , Pε ) : XA,m m A,m
e
with
Ae := [a + r, b − r] × [c, d].
Finally we use Theorem 2.3 to finish the proof.
CHAOS IN DELAY DIFFERENTIAL EQUATIONS 1637
Observe that T -periodic systems as (3), i.e. “piecewise autonomous systems” are
employed in many biological models see [8], [9], [10], [14], [15], [24], [27], [31]. In
particular, in population dynamics such a type of equations are known as systems
with seasonal succession, see [8] or [10].
Proposition 1. Consider system (3) with all parameters fixed except d1 , T1 , T2 , τ
and suppose that
3d2 a2 5d2
< < . (8)
4f2 c2 4f2
Then there exist a constant T2∗ and three maps d∗1 (Te2 ), T1∗ (de1 , Te2 ), and τ ∗ (de1 , Te1 , Te2 )
with the following properties:
The Poincaré map associated to (3) with parameters (T2 , d1 , T1 , τ ) is chaotic pro-
vided 0 < T2 < T2∗ , d1 > d∗1 (T2 ), T1 > T1∗ (d1 , T2 ), and 0 < τ < τ ∗ (d1 , T1 , T2 ).
Remark 1. In the proof we give precise estimates of T2∗ , d∗1 (T2 ), T1∗ (d1 , T2 ) and
τ ∗ (d1 , T1 , T2 ) depending on the coefficients of the system.
Next we state the main result of this paper.
1638 ALFONSO RUIZ-HERRERA
Theorem 3.1. Fix the parameters in (3) such that condition (8) is satisfied and
such that 0 < T2 < T2∗ , d1 > d∗1 (T2 ), T1 > T1∗ (d1 , T2 ) and 0 < τ < τ ∗ (d1 , T1 , T2 ).
Then there exists > 0 such that if the distance in L1T between the previous pa-
rameters in (3) and the coefficients of (1) is smaller than , the Poincaré operator
associated to (1) is chaotic.
In our setting, the assumption of Theorem 3.1 means that the coefficient a(t)
satisfies Z T1 Z T
|a(t) − a1 |dt + |a(t) − a2 |dt < ,
0 T1
and so on (for the other coefficients).
There are some remarks to be made concerning Theorem 3.1. The coefficients in
(1) considered in the previous theorem are not necessarily “piecewise constant” and
can be taken as smooth as we like. The reader can see Theorem 3.1 as a result of
global continuation of topological horseshoes with respect to τ .
d2 3d2
A = [α − l, α + l] × , .
2f2 2f2
Next we estimate d∗1 , (see (4) as a system without delay). Take d1 such that
d1
e1 > (α+l). It is clear that if (x1 (t), y1 (t)) is a solution of (4) with (x1 (t), y1 (t)) ∈ A
then we can compute the derivative of the first component with respect to the second
one. In such a case
dx1 x1 (−a1 + c1 y1 ) (α + l) max{| −a1 + 3c2f1 d2 2 |, | −a1 + c2f1 d2
2
|}
= ≤ d2
= K.
dy1 y1 (d1 − e1 x1 ) 2f2 (d1 − e1 (α + l))
Thus, if
d2 l
K < , (14)
f2 2
CHAOS IN DELAY DIFFERENTIAL EQUATIONS 1639
we deduce that the solutions of (4) with initial conditions at (α − 2l , dl22 ) and (α +
l d2 d2
2 , l2 ) leave the rectangle A across the sides [α − l, α + l] × { 2f2 } and [α − l, α +
3d2
l] × { 2f2 }. We illustrate this behavior with the following figure.
A
(d1 /e1 , a1 /c1 )
d2
y(t1 ) − y(t1 − τ ) ≤ |y(t1 ) − y(0)| + |y(t1 − τ ) − y(0)| ≤ if t1 < τ
2f2
and so y(t1 ) < 3d
2f2 . (Observe that y(t) > 0)
2
The previous property together with τ < min{T2 , 4fd22R } enables us to conclude (18).
d2
Notice that easily one can also obtain that y(t) > 2f 2
for all t > 0.
Next we prove the following stretching property: given a continuous path
γ : [0, 1] −→ XA,m
d2
with γ(0) ⊂ {(x, y) ∈ XA,m : y(0) = 2f 2
} and γ(1) ⊂ {(x, y) ∈ XA,m : y(0) =
3d2
2f2 }, there exists a subinterval [ζ1 , ζ2 ] ⊂ [0, 1] so that Φ(γ([ζ1 , ζ2 ])) ⊂ XA,m with
Φ(γ(ζ1 )) ⊂ {(x, y) ∈ XA,m : x = α − l} and Φ(γ(ζ2 )) ⊂ {(x, y) ∈ XA,m : x = α + l}.
d2
First of all we observe that Φ(γ([0, 1])) ⊂ {(x, y) ∈ R × C([−τ, 0], R) : 2f 2
≤
3d2
y(0) ≤ f2 }. At this moment, it is enough to prove that Φ(γ(1)) ⊂ {(x, y) ∈
R × C([−τ, 0], R) : x > α + l} and Φ(γ(0)) ⊂ {(x, y) ∈ R × C([−τ, 0], R) : x < α − l}.
To see the first claim, we recall that by the definition of T2 , y(t, γ(1)) ⊂ [ 5d 2 3d2
4f2 , 2f2 ]
for all 0 < t < T2 . To finish the proof, we notice that by (10) and (11),
5d2 3d2
(α − l) + T2 min{x(−a2 − b2 x + c2 y) : (x, y) ∈ A ∩ { ≤y≤ }} > α + l.
4f2 2f2
The other claim is proved analogously.
To conclude the proof of Proposition 1, we apply Theorem 2.3 with XK0 ,m , XK1 ,m
and P .
Proof of Theorem 3.1. Notice that by the previous construction, the conditions
of Corollary 1 hold.
To finish this paper we illustrate how to apply our results in concrete examples.
Specifically we have to proceed in the following way:
• Fix the parameters in system (3) except d1 , T1 , T2 and τ and assume that (8)
holds.
• Take T2 , α, and l satisfying (4)-(13).
• Take d1 > d∗1 (see (15)).
• Compute P1 = P er(α − 2l , df22 ), P2 = P er(α + 2l , df22 ) and take T1 > P5P 1 P2
1 −P2
.
∗
• Compute η and take 0 < τ < τ .
With these parameters we can use Proposition 1 and Theorem 3.1 . The compu-
tation of P1 = P er(α − 2l , df22 ), P2 = P er(α + 2l , df22 ) is not an easy task because
CHAOS IN DELAY DIFFERENTIAL EQUATIONS 1643
it involves an improper integral. However the reader can find properties and nu-
merical estimates in [4], [33] and [7]. Apart from the fourth step, the rest of the
parameters are easy to obtain.
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