c
World Scientific Publishing Company
ANALYZING DISPLACEMENT TERM’S MEMORY
EFFECT IN A VAN DER POL TYPE BOUNDARY
CONDITION TO PROVE CHAOTIC VIBRATION
OF THE WAVE EQUATION
GOONG CHEN∗
Department of Mathematics, Texas A&M University, College Station, TX 77843, USA gchen@math.tamu.edu
SZE-BI HSU†
Department of Mathematics, National Tsing Hua University, Hsinchu 30043, Taiwan, R.O.C.
sbhsu@math.nthu.edu.tw TINGWEN HUANG
Department of Mathematics, Texas A&M University, College Station, TX 77843, USA twhuang@math.tamu.edu
Received December 30, 2000; Revised October 1, 2001
Consider the one-dimensional wave equation on a unit interval, where the left-end boundary condition is linear, pumping energy into the system, while the right-end boundary condition is self-regulating of the van der Pol type with a cubic nonlinearity. Then for a certain parameter range it is now known that chaotic vibration occurs. However, if the right-end van der Pol boundary condition contains an extra linear displacement feedback term, then it induces a memory effect and considerable technical difficulty arises as to how to define and determine chaotic vibration of the system. In this paper, we take advantage of the extra margin property of the reflection map and utilize properties of homoclinic orbits coupled with a perturbation approach to show that for a small parameter range, chaotic vibrations occur in the sense of unbounded growth of snapshots of the gradient. The work also has significant implications to the occurrence of chaotic vibration for the wave equation on a 3D annular domain.
Keywords: Chaotic wave vibration; memory effects; unbounded growth of total variations.
1. Introduction
First, we take this opportunity to express our great admiration toward Professor Leon O. Chua. Throughout his career so far, Professor Chua has made major contributions to many areas of electri-cal engineering, particularly, that of nonlinear cir-cuits and systems, and complexity theory. He has
an unusual, high appreciation of the role played by mathematics in the research and development of applied sciences and technology. The journal founded by him, the International Journal of Bi-furcation and Chaos in Applied Sciences and Engi-neering (IJBC), has fully bloomed into a premier journal in nonlinear science under his editorship. It is also the most favorite forum for us to publish our
∗Supported in part by DARPA grant F49620-01-1-0566, and a Texas A&M University Telecommunication Grant. †Supported in part by a grant from National Science Council of R.O.C.
mathematical papers on chaos. We just wish to say our appreciation, in a small way, “Thank you, Professor Chua”, by dedicating this paper to him on the occasion of his 65th birthday.
The problem we wish to address is the chaotic vibration of the wave equation. (Our first research article on this subject, [Chen et al., 1996], was pub-lished in IJBC in 1996.) Consider the following. The wave equation
∂2w(x, t)
∂x2 −
∂2w(x, t)
∂t2 = 0 , 0 < x < 1, t > 0 ,
(1) on a bounded interval (0, 1), where the speed of wave propagation is assumed to be one without loss of generality as far as the mathematical analysis herein is concerned. The boundary condition at the left-end satisfies
wt(0, t) =−ηwx(0, t), η > 0, η 6= 1, t > 0 , (2)
while that at the right-end satisfies
wx(1, t) = αwt(1, t)− βwt3(1, t)− γw(1, t) ,
t > 0, 0 < α < 1, β > 0, γ > 0 . (3) The initial conditions are given by
w(x, 0) = w0(x), wt(x, 0) = w1(x), 0 < x < 1 .
(4) Note that the boundary condition (2) signifies the pumping of energy into the system in a feedback way. The boundary condition (3) is similar to the van der Pol nonlinearity we studied in our earlier work [Chen et al., 1998a–1998d]. The major dis-tinction here is the presence of the term γw on the RHS (right-hand side) of (3). In elastic vibrations, this term γw usually corresponds to some elastic support of a vibrating string at x = 1; see [Chen & Zhou, 1993], for example. The rate of change of energy of the vibrating system is
d dtE(t) ≡ d dt 1 2 Z 1 0 [w2x(x, t) + wt2(x, t)]dx +γ 2w 2(1, t)
= (· · · integration by parts and simplification, using (1)–(3)) = ηwx2(0, t) + [αw2t(1, t)− βwt4(1, t)] . (5) Since ηwx2(0, t)≥ 0, for all t > 0 , (6) αwt2(1, t)− βw4t(1, t) ≥ 0, if |wt(1, t)| ≤ rα β, < 0, if|wt(1, t)| > rα β, (7)
we see that (6) injects energy into the system while (7) is self-regulating just like the usual van der Pol nonlinearity treated in [Chen et al., 1998a– 1998d].
Remark 1.1. How important is it to include the γw term in (3)? There are two major reasons that motivate us to consider it in this paper:
(1) In the standard PID (proportional, integral and differential) methodology of feedback con-trol, the feedback of position or displacement is of utmost importance in problems such as tracking. Here the γw term corresponds pre-cisely to the position or displacement term. (2) The γw term may arise due to reduction of
dimensionality for certain symmetry. Consider the following: The wave equation in a 3D an-nular domain Ω:
∆W (x, t)− 1
c2Wtt(x, t) = 0, x∈ Ω, t > 0 ,
Ω ={x|a < |x| < b} , (8) where a > 0, b > 0, x = (x1, x2, x3) and ∆ =
(∂2/∂x21) + (∂2/∂x22) + (∂2/∂x23). Let n denote the outward unit normal vector on ∂Ω, the boundary of Ω. The boundary condition on the inner shell |x| = a is assumed to be ∂W (x, t) ∂n = αWt(x, t)− βW 3 t(x, t)− k1W (x, t) , k1≥ 0, |x| = a, t > 0 , (9)
and that on the outer shell|x| = b is ∂W (x, t)
∂n =
1
ηWt(x, t)− k2W (x, t) ,
η > 0, k2 ≥ 0, |x| = b, t > 0 . (10)
The initial conditions satisfy
W (x, 0) = W0(|x|) ,
Wt(x, 0) = W1(|x|) , x ∈ Ω ,
(11)
again examine the rate of change of energy: d dtE(t)≡ d dt 1 2 Z Ω |∇W (x, t)|2+ 1 c2W 2 t(x, t) dx +k1 2 Z |x|=aW 2(x, t)dσ + k2 2 Z |x|=bW 2(x, t)dσ (12) = Z Ω ∇W (x, t) · ∇Wt(x, t) + 1 c2Wt(x, t)Wtt(x, t) dx + k1 Z |x|=aW (x, t)Wt(x, t)dσ + k2 Z |x|=bW (x, t)Wt(x, t)dσ
=· · · (integration by parts and simplification, utilizing (8)–(10)) = Z |x|=a[α− βW 2 t(x, t)]W2t(x, t)dσ + 1 η Z |x|=bW 2 t(x, t)dσ . (13)
In the above, dx = dx1dx2dx2 and dσ are,
respec-tively, the infinitesimal volume element on Ω and the infinitesimal surface element on ∂Ω. Again, from (13), we see that the boundary condition on the inner shell |x| = a is self-regulating of the van der Pol type, while that on the outer shell |x| = b injects energy into the system. Because (9)–(11) are independent of the angular variables (in the spherical coordinate system), the initial-boundary value problem (8)–(11) has rotational symmetry. So let us attempt the reduction of dimensionality by writing
W (x, t) = w(r, t)
r , r =|x| . (14)
Substitution of (14) into (8)–(11) leads to the fol-lowing initial-boundary value problem:
∂2w(r, t) ∂r2 − 1 c2 ∂2w(r, t) ∂t2 = 0, a < r < b, t > 0 , −wr(a, t) = αwt(a, t)− β a2w 3 t(a, t) − k1+ 1 a w(a, t) , wt(b, t) = ηwr(b, t) + η k2− 1 b w(b, t) ; w(r, 0) = w0(r)≡ W0(r) r , wt(r, 0) = w1(r) = W1(r) r , a < r < b . (15) In (15)3 above, set k2 = 1 b, (16)
then we can eliminate the term η(k2−(1/b))w(b, t).
We further make the change of variable
r = b− (b − a)x , 0 ≤ x ≤ 1 . Then (15) becomes 1 (b− a)2wxx(x, t)− 1 c2wtt(x, t) = 0, 0 < x < 1, t > 0 , wt(0, t) =− η b− awx(0, t), t > 0 , wx(1, t) = α(b− a)wt(1, t)− β(b− a) a2 · w3 t(1, t)− k1+ 1 a (b− a)w(1, t), t > 0 ; w(x, 0) = w0(b− (b − a)x) , wt(x, 0) = w1(b− (b − a)x) , 0 < x < 1 . (17) Further, setting b− a = c , η b− a = ˜η , α(b− a) = ˜α , β(b− a) a2 = ˜β , k1+ 1 a (b− a) = ˜γ . Then (17) is converted exactly to the form of (1)–(4).
Note that, even though we are able to elim-inate the w(b, t) term in (15)3 by (16), we
can-not eliminate the w(a, t) term in (15)2 at the same
time because its coefficient,−(k1+ (1/a)), is always
The statements we have made in Remark 1.1(2) above actually opens the door for the investigation of chaotic vibration of the wave equation on a mul-tidimensional domain.
The presence of the γw(1, t) term in the bound-ary condition (3) has added significant technical dif-ficulty to the study of chaos for the system (1)–(4). Most of us will agree that there is not yet available a universally accepted definition of chaos for time-dependent partial differential equations. In the case when γ = 0 in (3), using the method of characteris-tics for hyperbolic systems one can extract clearly defined interval maps [Chen et al., 1998a–1998c, 2001; Huang & Chen, 2001], which come from wave reflection relations totally characterizing the sys-tem, and use them as the natural Poincar´e sec-tion for the system. Since the definisec-tion of chaos for interval maps is more or less standard (see e.g. [Devaney, 1989]), it is thus possible to classify whether the system is chaotic or not when γ = 0. But when γ6= 0 in (3), fixed interval maps no longer exist. What we have instead is a nonlinear integro-differential equation with respect to the t variable on the boundary at x = 1; see (70) below. The presence of the integral term signifies a memory ef-fect. Because the integral term tends to cause the drift of the states out of the invariant region, espe-cially when the time horizon is long, this becomes the most technically challenging part of the paper. It has taken us a long time to analyze this complex-ity and treat it to a desired degree of satisfaction, fruitless until now.
The way we regard that chaos occurs in the sys-tem is from the view of unbounded growth of total variations of snapshots developed by us in [Chen et al., 2001; Huang & Chen, 2001]. If a system starts out from some initial data (at t = 0) whose total variations over the spatial span is finite, as-sume that whatever prescribed forcing term(s) in the boundary data has bounded total variations over the entire time horizon t : 0 < t <∞. If the total variations of the snapshots of the state tend to infinity as t→ ∞, this intuitively speaks for the fact that the system becomes more and more oscil-latory, without any limitation and, thus, is chaotic. This point of view is summarized in Sec. 2.
In Secs. 3 and 4, we actually prove that the to-tal variations of the snapshots grow unbounded, for a certain set of initial data. In Sec. 3, we first regard the γw term in (3) as a prescribed function εf (t); see (26)3. Thus this εf (t) becomes a forcing term
in an open-loop, nonlinear boundary condition. We
then use a perturbation argument and properties of homoclinic orbits to derive the desired unbounded growth of total variations.
In Sec. 4, we then use the equivalence between an open loop system and a closed loop one to prove that total variations of snapshots do go unbounded as t→ ∞, for sufficiently small γ > 0. Graphics for an example are also illustrated.
2. Chaotic Vibration as
Characterized by Unbounded
Growth of Total Variations
of Snapshots
Let I denote a closed interval [a, b] and let f : I →R. We use VI(f ) to denote the total variation of the
function f over I.
Let us first consider the system (1)–(4), but with γ = 0 in (3). Making the transformation
wx = u + v , wt= u− v , (18)
we obtain the following first-order symmetric hy-perbolic system ∂ ∂t " u(x, t) v(x, t) # = " 1 0 0 −1 # ∂ ∂x " u(x, t) v(x, t) # , 0 < x < 1, t > 0 , (19)
with the left-end boundary condition
v(0, t) = Gη(u(0, t))≡
1 + η
1− ηu(0, t) , t > 0 , (20) and the right-end boundary condition
u(1, t) = Fα,β(v(1, t)) , t > 0 , (21)
where for given x ∈ R, y = Fα,β(x) is the unique
real solution of the cubic equation
β(y− x)3+ (1− α)(y − x) + 2x = 0 . (22) The initial conditions for u and v are
u(x, 0) = u0(x)≡ 1 2[w 0 0(x) + w1(x)] , v(x, 0) = v0(x) = 1 2[w 0 0(x)− w1(x)] , 0 < x < 1 . (23)
Let us fix α and β and write Fα,β briefly as F , in
write Gη briefly as G. Then the solution u and v
of (19)–(23) can be determined completely by the reflection relations G and F . The overall system (19)–(23) is chaotic if the composite reflection rela-tion G◦ F is chaotic (see e.g. [Devaney, 1989]).
Assume that
0 < V[0,1](u0) <∞, 0 < V[0,1](v0) <∞ . (24)
Then work in [Chen et al., 2001] shows that for a parameter range of η when Gη◦ F is chaotic, there
exists a large class of initial conditions u0(·) and
v0(·) satisfying (24) such that
lim
t→∞V[0,1](u(·, t)) = ∞, tlim→∞V[0,1](v(·, t)) = ∞ .
(25) The proof of (25) in [Chen et al., 2001] uses key properties of a chaotic interval map such as the ex-istence of a periodic orbit of period 2k· m, where m is an odd integer, or the existence of a homoclinic orbit.
At a more elementary level, the property (24) for the system (19)–(23) hinges on the irregular be-havior of the iterates of the composite reflection map G◦ F . Even though G ◦ F is defined on the entire real line R, it becomes an interval map if we restrict the domain of definition of G◦ F to an invariant interval; see [Chen et al., 1998a–1998c] or Lemma 3.2 below. For an interval map f , the relationship between the chaotic behavior and the property of unbounded growth of total variations of iterates fn of f can be seen in the following two theorems.
Theorem 2.1 [Huang & Chen, 2001]. Let I be a finite closed interval of R and let f : I → I be
continuous. Assume that f has sensitive depen-dence on initial data on I [Devaney, 1989]. Then limn→∞VJ(fn) =∞ for every closed subinterval J
of I. The converse is also true if f has finitely many extremal points.
Theorem 2.2 [Huang & Chen, 2001]. Let I be a finite closed interval of R and let f : I → I be con-tinuous with finitely many extremal points. Assume that limn→∞VJ(fn) =∞ for every closed
subinter-val J of I. Then the map f has periodic points of prime period 2k for k = 1, 2, 3, . . . .
More recently, [Juang & Shieh, 2001] have shown that under the same conditions as in Theo-rem 2.2, f actually has a periodic point with prime period m· 2k for some integers m and k, where m is odd.
Having rationalized the background of how (24) and (25) may be related to chaos, we proceed to es-tablish them for the system (1)–(4) when γ in (3) lies in a certain range.
3. An Open-Loop Perturbation
Approach to Establish the
Unbounded Growth of Total
Variations of Snapshots
In this section, we will view the γw(1, t) term in (3) as an explicitly given perturbation term εf (t), for some bounded continuous function f , for some small ε∈R. Thus, even though the γw(1, t) term in (3) constitutes part of the feedback boundary con-dition, the substitute term εf (t) becomes a forcing term and the new boundary condition at x = 1 is no longer wholly closed-loop. We consider the model wxx(x, t)− wtt(x, t) = 0, 0 < x < 1, t > 0 ; wx(0, t) =−ηwt(0, t), η > 0, η6= 1, t > 0 ; wx(1, t) = αwt(1, t)− βw3t(1, t) + εf (t) , 0 < α < 1, β > 0 ; w(x, 0) = w0(x), wt(x, 0) = w1(x) , 0 < x < 1 . (26) Using (18) in (26)3, we obtain β[u(1, t)− v(1, t)]3+ (1− α)[u(1, t) − v(1, t)] + 2v(1, t) − εf(t) = 0 . (27) For each given value of v(1, t) and that of f (t), there
exists a unique solution u(1, t) of (27). We denote the correspondence by
u(1, t) = F (v(1, t)) .
(For the unperturbed case εf (t) ≡ 0, we retain our old notation u(1, t) = F (v(1, t)) as in (21).) Further, for t∈ [0, 1], write
Throughout the rest of the discussion, we assume that
f ∈ BC[0, ∞) (BC(I), where I = [0, ∞), is the space of all bounded continuous functions on an interval I) , |f(t)| ≤ M0, for some M0 > 0, for all
t∈ [0, ∞) .
(28) Henceforth, let us abbreviate u(1, t) and v(1, t) simply as u(t) and v(t), respectively, in case no ambiguity arises. For given v(i)(t) and εf(i)(t), t ∈ [0, 1], i = 0, 1, 2, . . . , we denote by Fi(v)(t)
the unique solution of
β[Fi(v)(t)− v(i)(t)]3+ (1− α)[Fi(v)(t)− v(i)(t)]
+ 2v(i)(t)− εf(i)(t) = 0 . (29) It is easy to see that
F (v(k + t)) = Fk(v)(t), for
k = 0, 1, 2, . . . , t∈ [0, 1] .
Lemma 3.1. Let f satisfy (28). For given v(i)(t), t∈ [0, 1], any i = 0, 1, 2, . . ., |Fi(v)(t)− F (v(i)(t))| ≤ M0ε 1− α. (30) Proof. Since β[Fi(v)(t)− v(i)(t)]3+ (1− α)[Fi(v)(t)
− v(i)(t)] + 2v(i)(t)− εf(i)(t) = 0 ,
β[F (v(i)(t))− v(i)(t)]3+ (1− α)[Fi(v(i))(t)
− v(i)(t)] + 2v(i)(t) = 0 ,
by subtraction we have
(Fi(v)(t)− F (v(i)(t))){β[(Fi(v)(t)− v(i)(t))2
+ (Fi(v)(t)− v(i)(t))(F (v(i))(t)− v(i)(t))
+ (F (v(i)(t))− v(i)(t))2] + (1− α)} − εf(i)(t) = 0 .
The terms inside [· · · ] above are non-negative. Thus |Fi(v)(t)− F (v(i)(t))| · (1 − α) ≤ |εf(i)(t)|
and, therefore,
|Fi(v)(t)− F (v(i)(t))| ≤
M0|ε|
1− α.
The solution to (26) can now be written as fol-lows: for t = 2k + τ , k = 0, 1, 2, . . . , 0≤ τ < 2 and 0≤ x ≤ 1, u(x, t) = (Fk−1◦ G) ◦ (Fk−2◦ G) ◦ · · · ◦ (F0◦ G)[u0(x + τ )], τ ≤ 1 − x ; G−1◦ (G ◦Fk)◦ (G ◦Fk−1)◦ · · · ◦ (G ◦F0)[v0(2− x − τ)], 1 − x < τ ≤ 2 − x ; (Fk◦ G) ◦ (Fk−1◦ G) ◦ · · · ◦ (F0◦ G)[u0(τ + x− 2)], 2− x < τ ≤ 2 ; (31) v(x, t) = (G◦Fk−1)◦ (G ◦Fk−1)◦ · · · ◦ (G ◦F0)[v0(x− τ)], τ ≤ x ; G◦ (Fk−1◦ G) ◦ (Fk−2◦ G) ◦ · · · ◦ (F0◦ G)[u0(τ − x)], x < τ ≤ 1 + x ; (G◦Fk)◦ (G ◦Fk−1)◦ · · · ◦ (G ◦F0)[v0(2 + x− τ)], 1 + x < τ ≤ 2 . (32)
From (31) and (31), we see that if
lim
k→∞V[0,1]((Fk◦ G) ◦ · · · ◦ (F0◦ G)(u0)) =∞ ,
then we have
limt→∞V[0,1](u(·, t)) = ∞,
limt→∞V[0,1](v(·, t)) = ∞ .
Lemma 3.2. Let η > 0 satisfy either
(i) 0 < ηH ≡ 1− 1 + α 3√3 1 +1 + α 3√3 −1 < η < η0 < 1 , (33) (ii) 1 < η0 < η < ηH ≡ 1−1 + α 3√3 −1 · 1 +1 + α 3√3 , (34)
where η0: 0 < η0 < 1 and η0: 1 < η0 < ∞ are the unique solution of, respectively, the following equations 1 + η0 1− η0 1 + α 3 s 1 + α 3β = 1 + η0 2η0 s 1 + αη0 βη0 , η0+ 1 η0− 1 1 + α 3 s 1 + α 3β = 1 + ¯η0 2 s α + ¯η0 β . Then M1≡ local maximum of Gη ◦ F = 1 + η 1− η 1 + α3 s 1 + α 3β < 1 + η 2 s α + η β ≡ B1, (35)
and [−B1, B1] is an invariant interval of Gη ◦ F
such that
Gη◦ F (M1) > 0 for (33) , (36)
Gη◦ F (M1) < 0 for (34) . (37)
Proof. See [Chen et al., 1998b, Lemmas 2.4, 2.5 and Theorems 4.1, 4.2].
We wish to emphasize here that for the param-eter ranges of η given in (33) and (34), the map Gη ◦ F has homoclinic orbits in [−B1, B1] for each
such η. This homoclinic property is crucial for the perturbation arguments in the subsequent sections.
Lemma 3.3. Let η > 0 satisfy either (33) or (34). Then
(i) for η satisfying (33),
M2 ≡ local maximum of F ◦ Gη = 1 + α 3 s 1 + α 3β < 1− η 2η s 1 + αη βη ≡ B2; (38)
(ii) for η satisfying (34),
M2 ≡ local maximum of F ◦ Gη = 1 + α 3 s 1 + α 3β < η− 1 2 s α + η β ≡ B2, (39)
and [−B2, B2] is an invariant interval of F ◦ Gη
such that
F ◦ Gη(M2) > 0 for (33) , (40)
F ◦ Gη(M2) < 0 for (34) . (41)
Proof. Same as that for Lemma 3.2.
From Lemmas 3.2 and 3.3, we can further choose η satisfying (33) or (34) such that
Gη◦ F (M1) < M1,
F◦ Gη(M2) < M2, for (33) , (42)
Gη◦ F (M1) >−M1,
F◦ Gη(M2) >−M2, for (34) . (43)
Thus, there exist δ1> 0 and δ2 > 0 such that
(Gη ◦ F )[−M1− δ1, M1+ δ1]⊆ [−M1, M1] , (44)
(F ◦ Gη)[−M2− δ2, M2+ δ2]⊆ [−M2, M2] . (45)
Remark 3.1. Lemmas 3.2 and 3.3 allow us to add a perturbation term in the recursive iterations. Let us explain their significance through Fig. 1, where by choosing α = 0.5, β = 1 and η = 1.59 and by setting δ1 to satisfy Gη ◦ F (M1+ δ1) = M1 we see
the following:
(i) S1 ≡ [−B1, B1]× [−B1, B1] is the large
invari-ant square for the map Gη◦Fα,β; see Fig. 1(a);
(ii) S2 ≡ [−M1, M1]× [−M1, M1] is the small
invariant square for the map Gη ◦ Fα,β; see
Fig. 1(b);
(iii) R ≡ [−(M1+ δ1), M1+ δ1]× [−M1, M1] is an
invariant rectangle for the map Gη ◦ Fα,β; see
Fig. 1(c).
The invariant rectangle in (iii) has width 2(M1+δ1),
which is larger than the height 2M1 by a margin
2δ1 > 0. This extra margin δ1 is crucial, providing
what we need in order to allow small perturbations. We call this the extra margin property.
Lemma 3.4. Assume that η satisfies Lemma 3.3 and (43), and that δ2 satisfies (45). Choose ε such
that |ε| < (1 − α)δ2/M0. Then
|(Fk◦ G) ◦ (Fk−1◦ G) ◦ · · · ◦ (F0◦ G)(u0)(t)|
u v 1 2 1 2 -1 -2 -1 -2 -B B -B B 1 1 1 1 α=0.5, β=1, η=1.59 (a) u v 1 2 -1 -2 -1 -2 1 2 -M 1 M -M 1 M 1 1 α=0.5, β=1.0, η=1.59 (b) u v 1 2 -1 -2 1 2 1 2 -M1 δ M1 δ M1 + α=0.5, β=1.0, η=1.59 1 1 --M1 (c)
Fig. 1. We plot the graph of Gη◦ Fα,β, with α = 1/2, β = 1 and η = 1.59. See Remark 3.1. In (a), the square isS1. In (b),
the square isS2. In (c), the rectangle isR. The map Gη◦ Fα,β is invariant onS1,S2 andR. Thus, we see the extra margin
provided that |u0(t)| ≤ M2 + δ2 for t ∈ [0, 1], for
k = 0, 1, 2, . . . .
Proof. By Lemma 3.1, we have
|F0◦ G(u0)(τ )− F ◦ G(u0(τ ))| ≤ M0|ε| 1− α. Therefore |F0◦ G(u0)(τ )| ≤ |F ◦ G(u0(τ ))| + M0 1− α 1− α M0 δ2 ≤ M2+ δ2 (46)
because for|u0(τ )| ≤ M2+ δ2,|F ◦G(u0(τ ))| ≤ M2.
Inductively, if |(Fk◦ G) ◦ (Fk−1◦ G) ◦ · · · ◦ (F0◦ G)(u0)(τ )| ≤ M2+ δ2, then |(Fk+1◦ G) ◦ (Fk◦ G) ◦ · · · ◦ (F0◦ G)(u0)(τ ) − (F ◦ G) ◦ (Fk◦ G) ◦ · · · ◦ (F0◦ G)(u0)(τ )| ≤ M0 1− α 1− α M0 δ2 = δ2. Because|(Fk◦G)◦(Fk−1◦G)◦· · ·◦(F0◦G)(u0)(τ )| ≤
M2+δ2, using the same argument as in (46) we have
|(Fk+1◦ G) ◦ (Fk◦ G) ◦ · · · ◦ (F0◦ G)(u0)(τ )|
≤ M2+ δ2.
Similarly, we can prove the following.
Lemma 3.5. Assume that η satisfies Lemma 3.2 and (42), and that δ1 satisfies (44). Choose ε such
that |ε| < |[(1 − η)/(1 + η)]|[(1 − α)δ1/M0]. Then
|(G ◦Fk)◦ (G ◦Fk−1)◦ · · · ◦ (G ◦ F0)(v0)(t)|
≤ M1+ δ1, t∈ [0, 1] ,
provided that |v0(t)| ≤ M1 + δ1 for t ∈ [0, 1], for
k = 0, 1, 2, . . . .
Lemma 3.6. Assume the conditions of Lemma 3.3 and (41). Let vc = |[(1 − η)/(1 + η)]|[(2 − α)/3] ·
p
(1 + α)/3β be a critical point such that |F ◦
G(vc)| = M2. Then there exists a δ02 > 0 sufficiently
small such that
0 < (F ◦ G)0(v) < 1, for v∈ [vc− δ20, vc] (47)
and
(F◦ G)(M2− 2x) + 2x < 0, for 0 ≤ x ≤ δ02. (48)
Proof. This follows easily from (41) by a continuity argument.
Similarly, we can prove the following.
Lemma 3.7. Assume the conditions of Lemma 3.3 and (40). Let vc = |[(1 − η)/(1 + α)]|[(2 −
α)/3]p(1 + α)/3β be a critical point such that |F ◦ G(vc)| = M2. Then there exists a δ200> 0 sufficiently
small such that
−1 < (F ◦ G)0(v) < 0, for v∈ [v
c− δ200, vc] (49)
and
(F◦G)(M2− 2x)+ 2x > 0, for 0 ≤ x ≤ δ002. (50)
Lemmas 3.6 and 3.7 deal with the map F ◦ G. For the map G◦ F , the following can be proved in a similar way.
Lemma 3.8. Assume the conditions of Lemma 3.2 and (36) or (37). Let vc= [(2− α)/3]
p
(1 + α)/3β be a critical point such that |G ◦ F (vc)| = M1.
Then
(i) for (36), there exists a δ01 > 0 sufficiently small such that
−1 < (G ◦ F )0(v) < 0, for v∈ [vc− δ10, vc]
and
(G◦ F )(M1− 2x) + 2x > 0, for 0 ≤ x ≤ δ10 ;
(ii) for (37), there exists a δ001 > 0 sufficiently small such that
0 < (G◦ F )0(v) < 1 , for v∈ [vc− δ001, vc]
and
Theorem 3.1. Let η satisfy the assumptions in Lemma 3.4, and let
|ε| < 1− α M0
˜
δ2, ˜δ2 ≡ min(δ2, δ20) (51)
where δ2 and δ02satisfy (45), (47) and (48). Assume
that |u0(t)| ≤ M2, 0≤ t ≤ 1 ; Range u0⊇ [−˜δ2, M2− ˜δ2] . (52) Then lim k→∞V[0,1]((Fk◦ G) ◦ (Fk−1◦ G) ◦ · · · ◦ (F0◦ G)(u0)) =∞ . (53)
Proof. To simplify notation, write gk(τ ) = (Fk◦ G) ◦ (Fk−1◦ G) ◦ · · · ◦
(F0◦ G)(u0)(τ ), τ ∈ [0, 1], k = 0, 1, 2, . . . .
We wish to show
lim
k→∞V[0,1](gk) =∞ .
Let x1 ∈ (0, vc) satisfy x1 = (F◦G)−1(M2). Let t0,
t1, t2∈ [0, 1] be such that
u0(t0) =−˜δ2, u0(t1) = x1, u0(t2) = M2− ˜δ2.
We can choose t1 such that
either t0< t1< t2, or t2 < t1< t0.
Define
J1 = the closed interval with endpoints t0 and t1,
J2 = the closed interval with endpoints t1 and t2.
Then |g0(t1)− (F ◦ G)(u0(t1))| ≤ M0ε 1− α < ˜δ2, |g0(t1)− (F ◦ G)(x1)| = |g0(t1)− M2| < ˜δ2, ∴ g0(t1) > M2− ˜δ2. (54) Also |g0(t2)− (F ◦ G)(u0(t2))| =|g0(t2)− (F ◦ G)(M2− ˜δ2)| ≤ ˜δ2, ∴ g0(t2)≤ (F ◦ G)(M2− ˜δ2) + ˜δ2. (55)
By (48) and the fact that F ◦ G is decreasing on [vc, M2], we have
F ◦ G(M2− ˜δ2) < F ◦ G(M2− 2˜δ2)≤ −2˜δ2. (56)
Combining (55) and (56), we have g0(t2)≤ F ◦ G(M2− 2˜δ2) + ˜δ2 ≤ −2˜δ2+ ˜δ2 =−˜δ2. (57) Also |g0(t0)− (F ◦ G)(u0(t0))| =|g0(t0)− F ◦ G(−˜δ2)| < ˜δ2, ∴ g0(t0) < F ◦ G(−˜δ2) + ˜δ2 <−˜δ2+ ˜δ2= 0 , because (F ◦ G)0(0) > 1 . (58) From (54)–(57), we obtain g0(J2)⊇ [−˜δ2, M2− ˜δ2] , g0(J1)⊇ [0, M2− ˜δ2] . (59)
Using the same ideas as for the above, we can show that J2 has two subintervals J2,1 and J2,2, such that
g1(J2,1) = F1◦ G(g0(J2,1))⊇ [0, M2− ˜δ2] ,
g1(J2,2) = F1◦ G(g0(J2,2))⊇ [−˜δ2, M2− ˜δ2] .
On the other hand for J1, we now show that J1 has
a subinterval J1,2 such that
g1(J1,2)⊇ [−˜δ2, M2− ˜δ2] . (60)
Since 0 < x1 < M2− ε ≤ M2 − ˜δ2, consider [x1,
M2− 2˜δ2]⊇ g1(J1). J1 thus has a subinterval J1,2
such that [x1, M2− 2˜δ2] =g1(J1,2). We have
|F1◦ G(x1)− F ◦ G(x1)| =|F1◦ G(x1)− M2| < M0ε 1− α, ∴F1◦ G(x1)≥ M2− M0ε 1− α ≥ M2− ˜δ2. (61) From (48), F◦ G(M2− 2˜δ2) <−2˜δ2, |F1◦ G(M2− 2˜δ2)− F ◦ G(M2− 2˜δ2)| ≤ M0ε 1− α ≤ ˜δ2, F1◦ G(M2− 2˜δ2)≤ F ◦ G(M2− 2˜δ2) + ˜δ2 ≤ −2˜δ2+ ˜δ2 = ˜δ2. (62)
Continuing this argument inductively, we can construct a sequence of subintervals
Ji0,i1,i2,...,in, ik∈ {1, 2}, for k = 0, 1, . . . , n ,
(63) of Ji0 such that
gk(Ji0,i1i2,...,ik)⊇ [0, M2− δ2] if ik = 1 ,
gk(Ji0,i1i2,...,ik)⊇ [−δ2, M2− δ2] if ik = 2 .
Note that in (63), an index ij = 1 can only follow
ij−1 = 2; the subscript i0, i1, . . . , ik consists of all
combinations of ij ∈ {0, 1} except those when two
adjacent indices ij−1ij are 11. Therefore, by
sum-ming over all such admissible i0, i1, . . . , ik, we have
V[0,1](gk)≥
X
VJi0,i1,...,ik(gk)
≥ 2k−1(M
2− δ2)→ ∞, as k → ∞ ,
and (53) has been proved.
Theorem 3.1 covers just the case under the con-ditions of Lemma 3.4. If, instead, the concon-ditions are those stated in Lemma 3.5, then a proof can be sim-ilarly established that
lim k→∞V[0,1]((G◦Fk)◦ (G ◦Fk−1)◦ · · · ◦ (G◦F0)(v0)) =∞ , (64) if |v0(t)| ≤ M1, 0≤ t ≤ 1 ; Range v0⊇ [−˜δ1, M1− ˜δ1] (65) for |ε| < 1 + η 1− η 1M− α0 ˜δ1, δ˜1≡ min(δ1, δ0 1) . (66)
We omit the details.
4. Chaotic Vibration in the
Sense of Unbounded Growth
of Snapshots for the van der Pol
Boundary Condition Containing
Displacement
We now proceed to study the system (1)–(4). The focus is the boundary condition (3). Write
X(t) = u(t)− v(t) . (67) Then w(1, t) = Z t 0 wt(1, τ )dτ + a0 (a0 ≡ w(1, 0)) = Z t 0 X(τ )dτ + a0. (68)
For each reflection of waves at x = 1, we need only consider 0≤ t ≤ 1. Note that because
an≡ w(1, n) = a0+ Z n 0 X(τ )dτ = an−1+ Z n n−1 X(τ )dτ , (69)
if we know an, u(1, t), v(1, t) for t ∈ [n − 1, n),
then we can determine an+1, u(1, t), v(1, t) for
t∈ [n, n + 1).
Let us now consider the nonlinear wave reflec-tion operator at x = 1. For the first wave reflecreflec-tion at x = 1, i.e. for t∈ [0, 1], using (18) and (68), we rewrite (3) as βX3(t) + (1− α)X(t) + γ Z t 0 X(τ )dτ + a0 + 2v(t) = 0, 0≤ t ≤ 1 . (70)
Lemma 4.1. Let β > 0, 0 < α < 1, γ > 0. Then for any v ∈ BC[0, 1] and a0 ∈ R, Eq. (70) has a
unique solution X ∈ BC[0, 1].
Proof. (i) Uniqueness: Let a∈Rand v∈ BC[0, 1] be given. Assume that
βXi3(t) + (1− α)Xi(t) + γ Z t 0 Xi(τ )dτ + a + 2v(t) = 0 for i = 1, 2 . Then subtraction gives
βY (t){[X12(t) + X1(t)X2(t) + X22(t)] + (1− α)} + γ Z t 0 Y (τ )dτ = 0 , (71) where Y ≡ X1− X2. If X1 6≡ X2 on [0, 1], then P (t)≡ β{[X12(t) + X1(t)X2(t) + X22(t)] + (1− α)} > 0 on [0, 1] . From (71), we have βP (t)Y (t) + γ Z t 0 Y (τ )dτ = 0 , Y (0) = 0 .
By a Gronwall argument, we can show that Y ≡ 0 on [0, 1], a contradiction. Therefore, the uniqueness follows.
(ii) Existence and smoothness: Let us construct the solution of (70) iteratively as follows. Choose X0(t)
to be the (unique) solution of (70) when γ = 0.
Then X0 ∈ BC[0, 1]. Iterate by solving Xj+1 from
βXj+13 (t) + (1− α)Xj+1(t) + γ Z t 0 Xj(τ )dτ + 2v(t) = 0 . (72)
Then Xj+1 is unique and Xj+1 ∈ BC[0, 1]. By
subtracting (72)j=n with (72)j=n−1, we obtain
[Xn+1(t)− Xn(t)]{β[Xn+12 (t) + Xn+1(t)Xn(t) + Xn2(t)] + (1− α)} + γ Z t 0 [Xn(τ )− Xn−1(τ )]dτ = 0 , Xn+1(t)− Xn(t) =− γ β[X2 n+1(t) + Xn+1(t)Xn(t) + Xn2(t)] + (1− α) Z t 0 [Xn(τ )− Xn−1(τ )]dτ , |Xn+1(t)− Xn(t)| ≤ γ 1− α Z t 0 |Xn (τ )− Xn−1(τ )|dτ ≤ γ 1− α 2Z t 0 Z τ1 0 |Xn−1 (τ2)− Xn−1(τ2)|dτ2dτ1 ≤ · · · ≤ γ 1− α nZ t 0 Z τ1 0 Z τ2 0 · · · Z τn−1 0 |X1 (τn)− X0(τn)|dτndτn−1. . . dτ2dτ1 ≤ C1· γ 1− α nZ t 0 Z τ1 0 Z τ2 0 · · · Z τn−1 0 dτndτn−1. . . dτ2dτ1 (C1 ≡ max t∈[0,1]|X1(t)− X0(t)|) = C1· γ 1− α ntn n! → 0 as n → ∞, uniformly on [0, 1] . Therefore lim
n→∞Xn= X∈ BC[0, 1], for a limit function X satisfying (70) .
Note that the uniqueness result in Lemma 4.1 guarantees the uniqueness of the solution w for the system (1)–(4).
The following lemma contains the most needed important information about the boundedness of the γw(1, t) term in (3).
Lemma 4.2(Key Technical Lemma). Let w be the solution of (1)–(4). Let w0 and w1 in (4) be
suffi-ciently smooth and are compatible with the boundary conditions (2)–(3). Assume that u0 and v0 in (23)
satisfy
|v0(x)| ≤ M1, |u0(x)| ≤ M2, x∈ [0, 1] ,
(cf. M1 and M2, respectively, in (73)
Lemmas 3.2 and 3.3) . (74)
Then there exists an M0 > 0 (prescribed in (88)
below) such that if
|a0| ≤ M0, then |an| ≤ M0 for n = 1, 2, . . . , and |v(t)| ≤ M1, |u(t)| ≤ M2 and |w(1, t)| ≤ M0+ M1+ M2, ∀t ∈ [0, ∞)
provided that γ > 0 is sufficiently small (satisfying (90) below).
Proof. Let us consider the outcome of each reflec-tion at x = 1. For t ∈ [n, n + 1], from (69)
and (72), βX(n)3 (t) + (1− α)X(n)(t) + 2v(n)(t) + γ Z t 0 X(n)(τ )dτ + an = 0 , 0≤ t ≤ 1 , (75) where an= w(1, n), u(n)(t) = u(n + t) , v(n)(t) = v(n + t), X(n)(t) = u(n)(t)− v(n)(t) . (76) As in [Chen et al., 1998a, 1998b], for any v ∈ R, let g(v) be the unique real solution of the cubic equation
βg3(v) + (1− α)g(v) + 2v = 0 . The X(n)(t) in (75) satisfies, for 0≤ t ≤ 1, X(n)(t) = g v(n)(t) + γ 2 Z t 0 X(n)(τ )dτ + an = g v(n)(t) + γ 2 Z t 0 X(n)(τ )dτ + an − g(0) (∵g(0) = 0) = g0(h(t)) v(n)(t) +γ 2 Z t 0 X(n)(τ )dτ + an , (77)
where h(t), by the Mean Value Theorem, is a continuous function taking values between 0 and v(n)(t) + (γ/2)[R0tX(n)(τ )dτ + an]. For any x ∈ R,
from [Chen et al., 1998a, (30), p. 4279]
g0(x) =− 2 3βg2(x) + (1− α) < 0 , |g0(x)| ≤ 2 1− α, x→±∞lim |g 0(x)| = 0 . From (77), X(n)(t) + p(t)· γ Z t 0 X(n)(τ )dτ + an =−2p(t)v(n)(t) , (78) where p(t)≡ −1 2g 0(h(t)) > 0 , p(t)≤ 1 1− α.
From (78), by Gronwall’s method, d dt eγ Rt 0p(τ )dτ Z t 0 X(n)(τ )dτ + an =−2p(t)eγ Rt 0p(τ )dτv(n)(t) , eγ Rt 0p(τ )dτ Z t 0 X(n)dτ + an − an =−2 Z t 0 p(s)v(n)(s)eγ Rs 0 p(τ )dτds , Z t 0 X(n)(τ )dτ + an = ane−γ Rt 0p(τ )dτ − 2Z t 0 p(s)v(n)(s)e−γ Rt 0p(τ )dτds , Z0t X(n)(τ )dτ + an ≤ |an|e−γ Rt 0p(τ )dτ + 2 1− α Z t 0 |v(n) (s)|e−γ Rt sp(τ )dτds . (79) Assume that |v(j)(t)| ≤ M1, |u(j)(t)| ≤ M2, |aj| ≤ M0 (80)
are satisfied, for j = 0, 1, . . . , n, where M0 will be determined below in (88). Then
v(n)(t) + γ 2 Z t 0 X(n)(τ )dτ + an ≤ v(n)(t) +γ 2 Z t 0 |X(n) (τ )|dτ + |an| ≤ M1+ γ 2[M1+ M2+ M 0]≡ M 3. (81) Therefore, by letting M00≡ max x∈[0,M3] g2(x) , (82)
we have M00 = g2(M3) because g is monotone
de-creasing, g(−x) = −g(x) and g(x) < 0 for x < 0. We have 1 3βM00+ (1− α) ≤ p(t) = − 1 2g 0(h(t)) = 1 3βg2(h(t)) + (1− α) ≤ 1 1− α. (83)
From (83), − 1 1− α(t− s) ≤ − Z t s p(τ )dτ ≤ − 1 3βM00+ (1− α)(t− s) . (84) Hence, from (84), |an|e−γ Rt 0p(τ )dτ ≤ |an|e−γ Rt 0 1 3βM 00+(1−α)dτ =|an|e− γ 3βM 00+(1−α)t, (85)
and from (79), (80) and (84), 2 1−α Z t 0 |v(n) (s)|e−γ Rt sp(τ )dτds ≤ 2M1 1−α Z t 0 e− γ(t−s) 3βM 00+(1−α)ds = 2M1 1−α 3βM00+(1−α) γ [1−e − γt 3βM 00+(1−α)] . (86) From (79), (85) and (86), |w(1, n + 1)| =|an+1| = Z01 X(n)(τ )dτ + an ≤ |an|e− γ 3βM 00+(1−α) + 2M1 1− α · 3βM00+ (1− α) γ [1− e − γ 3βM 00+(1−α)] . (87) But 3βM00+ (1− α) γ [1− e − γ 3βM 00+(1−α)] = 1−1 2 γ 3βM00+ (1− α) + 1 3! γ 3βM00+ (1− α) 2 ± · · · ≤ 1 − c0γ 2[3βM00+ (1− α)], for some c0: 0 < c0< 1,
if 0 < γ < c1, for some small c1.
Therefore, if we choose M0 > 0 such that
M0e− γ 3βM 00+(1−α)+2M1 1−α 1− c0γ 2(3βM00+(1−α)) ≤M0, i.e. M0 ≥ [1 − e− γ 3βM 00 +(1−α)]−1 · 2M1 1− α 1− c0γ 2(3βM00+ (1− α)) , (88) then from (87) |an+1| ≤ M0.
(Here let us make a little clarification. M00 is de-fined in (82) depending on M3, while M3 is defined
in (81) depending on M1, M2 and M10. From (88),
M0 in turn depends on M1 and M00. Thus it
ap-pears that we were having a vicious cycle of M00 de-pending on M00 itself. However, note that in (81), γ is very small, making M3 depend mainly on M1.
Therefore (88) can be satisfied without problem, for small γ > 0.) The above estimate also gives
|w(1, n + t)|≤M0+(M1+M2)t , t∈[0, 1] . (89) Since for t∈ [0, 1], v(1, 2 + n + t) = GηFα,β v(1, n + t) +γ 2w(1, n + t) , u(1, 2 + n + t) = Fα,β Gη(u(1, n + t)) + γ 2w(1, 1 + n + t) ,
if γ is sufficiently small such that γ2w(1, n + t) ≤ γ 2(M 0+ M 1+ M2) ≤ ˜δ3 ≡ min(δ1, δ2) ,
then by Lemmas 3.2 and 3.3, and (44), (45), we can use the extra margin property in Remark 3.1 (iii) to conclude that
|v(1, t)| ≤ M1 |u(1, t)| ≤ M2,
for all t∈ [0, ∞) .
Finally, we have laid all the ground work for deriving the chaotic property of vibration of the system (1)–(4).
Theorem 4.1. Consider (1)–(4). Assume that M1,
Let w0 and w1 in (4) be sufficiently smooth and
be compatible with the boundary conditions (2)–(3) such that u0= (1/2)(w00 + w1) and v0 = (1/2)(w00 −
w1) satisfy
|v0(x)| ≤ M1, |u0(x)| ≤ M2, x∈ [0, 1] .
Assume that γ > 0 is sufficiently small such that
γ < 1− α M0+ M1+ M2 ˜ δ, δ˜≡ min(˜δ1, ˜δ2) , cf. (51) and (66), respectively, f or ˜δ1 and ˜δ2, (90) Range v0⊇ [−˜δ, M1− ˜δ] , Range u0⊇ [−˜δ, M2− ˜δ] . Then for u = (1/2)(wx+ wt), v = (1/2)(wx− wt), we have lim t→∞V[0,1](u(·, t)) = ∞ , lim t→∞V[0,1](v(·, t)) = ∞ . (91)
Proof. By Lemma 4.1, the solution w to the sys-tem (1)–(4) is unique. Under the assumptions of Lemma 4.2, we have|w(1, t)| ≤ M0+ M1+ M2 for
all t≥ 0. Therefore, we can denote
f (t)≡ w(1, t), t ∈ [0, ∞), ε ≡ γ ,
and regard the closed loop system (1)–(4) as the open loop system (26). Just let M0 ≡ M0+M1+M2.
Let
ε = γ < 1− α M0
˜
δ, δ = min(˜˜ δ1, ˜δ2) .
Then Theorem 3.1 (53) and (64) are applicable, and we conclude (91).
The consequence in (91) also implies that lim t→∞V[0,1](wx(·, t)) = ∞ , lim t→∞V[0,1](wt(·, t)) = ∞ . (92)
Therefore, the gradient w of (1)–(4) is chaotic in the sense of unbounded growth of total variations of the snapshots. The proof of (92) follows from (91) by a little extra work utilizing (18) so we omit it here.
One may also question whether
lim
t→∞V[0,1](w(·, t)) = ∞ (93)
holds under the assumptions of Theorem 4.1. The answer is negative.
Corollary 4.1. Assume that Theorem 4.1 holds. Then
V[0,1](w(·, t)) ≤ M1+ M2,
f or all t > 0 .
Proof. We know that w(x, t) = w(0, t) + Z x 0 wx(ξ, t)dξ = w(0, t) + Z x 0 [u(ξ, t) + v(ξ, t)]dξ, for 0≤ x ≤ 1 . Therefore V[0,1](w(·, t)) = sup P n X i=0 |w(xi+1, t)− w(xi, t)| ,
where P is an arbitrary partition of [0, 1]: P = {xi|i = 0, 1, . . . , n, 0 = x0 < x1 < · · · < xn = 1},
such that max0≤i≤n|xi+1 − xi| → 0 as n → ∞.
Hence V[0,1](w(·, t)) = sup P nX−1 i=0 Z xi+1 ii (u(ξ, t) + v(ξ, t))dξ ≤ sup P n−1X i=0 Z xi+1 xi |u(ξ, t) + v(ξ, t)|dξ ≤ (M1+ M2) sup P nX−1 i=0 (xi+1− xi) = M1+ M2.
Example 4.1. Let us choose w0(x) = 0.5− 0.95x + 1 2x 2, w1(x) =−0.95x − x, 0≤ x ≤ 1 α = 0.5, β = 1.95, γ = 0.01 , in (2)–(4). Then u0(x) = 0.05, v0(x) = x− 1 ,
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 u(x,30) (a) u(x, t), 0≤ x ≤ 1, t = 30. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1.5 −1 −0.5 0 0.5 1 1.5 2 v(x,30) (b) v(x, t), 0≤ x ≤ 1, t = 30. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 −0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01 w(x,30) (c) w(x, t), 0≤ x ≤ 1, t = 30
Fig. 2. Snapshots of (a) u(·, t), (b) v(·, t) and (c) w(·, t) for t = 30 in Example 4.1. The reader may observe quite chaotic oscillatory behavior of u and v, but the contrary for w, which is a consequence of Corollary 4.1.
in (23). We plot the graphics of u(x, t), v(x, t) and w(x, t) for t = 30 in Figs. 2(a)–2(c). The reader may find that the snapshots of u and v display rather chaotic oscillatory behavior, while that of w does not.
There are some still unresolved questions: (a) We believe that for the parameter range of η
prescribed in Lemmas 3.2 and 3.3, the chaotic property (91) holds for a large class of initial data w0, u0 and v0 without the requirement that
γ > 0 be small. But, how do we analyze the case when γ > 0 is not small?
(b) Even if the map Gη ◦ Fα,β does not have
ho-moclinic orbits when η does not belong to the parameter range as prescribed in Lemmas 3.2 and 3.3 (with α and β being fixed), as long as the map Gη ◦ Fα,β: I → I is chaotic on an
in-variant interval, then property (91) should still hold for a large class of initial data u0 and v0.
This still needs to be proved.
(c) The proof of Corollary 4.1 essentially says that there will be no chaos in the w(·, t) variable as t → ∞. In order to have (93), the nonlinear boundary condition (3) must be replaced by
wx(1, t) = [α− 3βw2(1, t)]wt(1, t)
with everything else remaining unchanged. The analysis of the boundary condition (94) is yet to be carried out.
Acknowledgments
This work was completed while G. Chen was vis-iting the Center for Theoretical Sciences (CTS) of National Tsing Hua University in Hsinchu, Taiwan during July 2001. He wishes to thank CTS for the hospitality and the support for the visit.
Note added in proof: The rate of growth (91) in Theorem 4.1 can actually be strengthened to exponential with respect to the time variable t. This is consistent with recent results obtained by Goong Chen, Tingwen Huang and Yu Huang, who have shown that for an interval map f , if the total variation of the nth iterate fn grows exponentially
and if f is piecewise monotone, then t has a prime period which is not a power of z and therefore f is chaotic.
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