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Snapback Repellers in Chaos Theory and
their Application in Ecology
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Snapback Repellers in Chaos Theory and
their Application in Ecology
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Student: Hsun-Hui Chen
Advisor: Dr. Shu-Ming Chang
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A Thesis
Submitted to Department of Applied Mathematics
College of Science
National Chiao Tung University
in Partial Fulfillment of the Requirements
for the Degree of
Master
in
Applied Mathematices
June 2010
Hsinchu, Taiwan, Republic of China
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dddddddddddd(chaos theory)dddddddddddddd(chaos)dddd dddd(Lyapunov exponent)dddd(topological entropy)dddddd(strange attrac-tor)ddddd(snapback repeller)ddddddddd(Sarkovskii’s theorem)dddddd ddddddddddddddddddddddddddddddddddddd dddddddddddddddddddd(Akiko Satake)dddd(Yoh Iwasa)ddd dddd(Yuji Isagi)ddddddd(resource budget model)dddddddddddd ddd (generalized resource budget model)ddddddddddddddddddd d(depletion coefficient)ddddddddddddddddddddddddddddd dddd(Devaney)ddddddddd(sensitivity)ddddddddddddddddd ddddddddddddddddddddddddddddddddddddddd ddddddddddddddddddddddddddddddddddddddd dddddddddddddd d dddddddddddd-ddddddddddddddddddddddddddddddd ddddSnapback Repellers in Chaos Theory and
their Application in Ecology
Student: Hsun-Hui Chen
Advisors: Dr. Shu-Ming Chang
Department (Institute) of Applied Mathematics
National Chiao Tung University
Abstract
This work briefly reviews the history of chaos theory and elucidates the relationship among chaos, Lyapunov exponent, topological entropy, strange attractor, snapback re-peller, and Sarkovskii’s theorem, connecting them to each other using a relational graph. Mathematical and computer-assisted tools can be used to determine whether maps or systems are chaotic by finding a quantity or sometimes identifying the existence of a property. In ecology, Satake’s generalized resource budget model that modified from Isagi’s resource budget model, Satake and Iwasa proved by computing the positive Lya-punov exponent that if the depletion coefficient k is greater than one, then the system is chaotic. However, a positive Lyapunov exponent means only sensitivity in Devaney’s chaos. Therefore, this work presents mathematical views and a numerical analysis on Satake’s model, using the “snapback repeller method” to prove that the model is chaotic in Devaney’s sense (involving transitivity, density, and sensitivity). Moreover, this work also overcomes the omission of Satake’s paper (Satake & Iwasa, 2000) when the deple-tion coefficient k is a positive integer. Furthermore, the end of this work investigates the difference between odd depletion coefficients and even depletion coefficients.
Keywords: Li-Yorke’s chaos, Davaney’s chaos, topological entropy, snapback repeller, Sarkovskii’s theorem, resource budget model.
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ddd ddddddd 2010d6dContents
dddddd i Abstract ii dddddd iii Contents iv List of Tables v List of Figures vi Symbols vii 1 Introduction 1 1.1 History of Chaos . . . 1 1.2 Definitions of Chaos . . . 2 1.3 Topological Entropy . . . 5 1.4 Lyapunov Exponent . . . 9 1.5 Strange Attractor . . . 10 1.6 Snapback Repeller . . . 12 2 Ecological Model 15 2.1 Isagi’s Resource Budget Model . . . 162.2 Satake’s Generalized Resource Budget Model . . . 17
3 Mathematical Analysis 18
4 Numerical Simulation 25
5 Conclusions 33
Appendix (A, B, and C) 33
Reference 38
List of Tables
1 When k > kp, system (2) is chaotic in Devaney’s sense as determined by
numerical computation in Matlab. The result of Y1 and Y2 in Theorem 3.5
is above dotted line, and the iterative number n greater than two is below the dotted line. . . 23 2 When k > kp, system (2) is chaotic in Devaney’s sense as determined by
numerical computation in Maple. The result of Y1 and Y2 in Theorem 3.5 is
above dotted line, and the iterative number n greater than two is below the dotted line. . . 24 3 Rate of convergence of kp. . . 25
List of Figures
1 Relational graph of chaos and relative checking methods. . . 15 2 Resource budget model of an individual plant [38]. . . 16 3 Time series of Cawith various Rcs. For all plates, Ps, LT, and Cf(0) were 3.0,
6.0, and 2.5, respectively. Results were excluded for the initial 50 years [38]. . 17 4 Bifurcation diagram of a single tree. The horizontal axis represents the
deple-tion coefficient k, and the vertical axis represents Y(t) for many units of time
t. . . 26 5 Iterative number t v.s Y(t) when k is a positive even number. The system
converges to a lower periodic cycle with period k + 1; however, when k is a positive odd number, the dynamics are not lower periodic. (a) k = 2; (b) k = 3; (c) k = 4; (d) k = 5. . . 28 6 Relational graph of chaos and relative checking methods including period three. 35
Symbols
S(t) : the amount of energy reserved at the beginning of year t
PS : the annual amount of photosynthate
LT : energy threshold of a tree
Y(t) : non-dimensionalized variable of (S(t) + P
S− LT)/PS
k : depletion coefficient
kp : critical point with respect to Satake’s generalized resource budget model
under the iterative number 2p
Cf : a cost of flowering
Ca : a cost of fruiting
1
Introduction
In recent decades, chaotic theory has advanced rapidly. This work discusses definitions of chaos and their relationships. Many mathematical tools can be used to measure or describe the “chaotic” condition of systems. This work introduces the commonly utilized Lyapunov exponent and topological entropy, and most importantly connects these tools to chaos. The strange attractor is considered. Like the Lorenz attractor, although it has an attract property, it exhibits “unstable”, meaning that its interior exhibits a “sensitive dependence on initial conditions”, which phenomena are together identified as “strange”. Finally, a computer-assisted method, involving snapback repellers, is introduced: if a system has a snapback repeller, then it is chaotic.
1.1
History of Chaos
In 1960, the meteorologist Edward Lorenz was working on the problem of weather predic-tion. In 1961, he discovered the “butterfly effect” while trying to forecast the weather. This phenomenon, the “butterfly effect” refers to the sensitive dependence on initial conditions in chaos theory. Lorenz called the image he obtained when he graphed the associated equations the “Lorenz attractor”. Lorenz was the discoverer of chaos and its first true experimental researcher.
A decade later (1971), David Ruelle and Floris Takens elucidated a phenomenon they called a “strange attractor”, and in so doing, gave birth to a whole new area of chaos theory. The association of “turbulence” with a strange attractor was revolutionary [77].
The word “chaos” was coined by James A. Yorke [55]. According to one investigation [52], many scientists believe that the “existence” and “uniqueness” of a solution to a ordinary difference equation if the system is smooth is required by the Poincar`e-Bendixson theorem, even when the system has more than over two dimensions. They considered that such a solution was “regular” and that convergence to such an (almost) periodic or quasi-periodic solution is unaffected by noise. Whenever the solution was irregular, they regarded the problem as a computing error. However, in 1974, Li and Yorke solved the logistic model problems of the ecologist Robert May, and an increasing number of scientists no longer thought that these problems involved computing mistakes. In the following year, Li and Yorke revised their draft, which had been written one year earlier and published it in 1975 [55].
At that time, chaos was not a science, or even a cohesive theory. During the 1970s, several brave scientists such as Robert May, Mitchell Feigenbaum, and Benoit Mandelbrot, were so
intrigued with the new concept of chaos that they began to research it.
In 1989, Devaney explicitly defined chaos [21]. Diamond [22, 23], Marotto [23, 57], Wig-gins [98], Robinson [74] and Martelli [59] also provided their definitions of chaos in 1976, 1978, 1991, 1998, and 1999, respectively.
The existence of chaotic dynamics has been discussed in the mathematical literature for many decades with important contributions by Poincar´e, Birkhoff, Cartwright, Littlewood, Levinson, Smale, and Kolmogorov and his students, among others. The field is now undergo-ing explosive growth, and has various applications across a wide range of scientific disciplines, including–ecology [20, 33, 34, 61, 80, 86, 95, 102], economics [82], physics [73], sociology [78], anthropology [70, 71], biology [18, 65], chemistry, engineering, fluid mechanics, and many others. Specific examples of chaotic time-dependence include the convection of a fluid that is heated from below, simple models of the annual variation of insect populations, stirred chemical reactor systems, and determination of the limits on the length of reliable weather forecasting. The number of such examples continues to increase [31].
Many studies have applied these definitions, and some have emphasized their mutual relationships [7, 8, 30, 35, 46, 47, 60, 97].
1.2
Definitions of Chaos
The chaos of a map has been defined in several ways [46]. Although the comment “so many au-thors, so many definitions,” is true, a basic component of all definitions is the unpredictability of the behavior of the trajectory which is determined with some certain error. (The associated phenomenon is usually described in terms of sensitive dependence on initial conditions.)
The definitions of chaos of Devaney and of Li and Yorke are considered herein because they are fundamental and widely accepted:
Definition 1.1 (Devaney [21]). Let X be a metric space. A continuous map f : X → X is
said to be chaotic on X if
(D1) f is topologically transitive. That is, for any pair of nonempty open sets U, V ⊂ X there
exists k > 0 such that fk(U)∩ V 6= ∅;
(D2) periodic points are dense in X;
(D3) f has sensitive dependence on initial conditions,meaning that, there exists δ > 0 such
that, for any x∈ X and any neighborhood N of x, there exists y ∈ N and n ∈ N such that |fn(x)− fn(y)| > δ.
Intuitively, a topologically transitive map has points that eventually move under iteration from one arbitrarily small neighborhood to any other. Consequently, a dynamical system cannot be decomposed into two disjoint open sets that are invariant under the map. Notably, if a map possesses a dense orbit, then it is necessarily topologically transitive. The converse is also true (for compact subsets of R or S1).
A map possesses sensitive dependence on initial conditions if there exist points arbitrarily close to x that are eventually separated from x by at least δ under iteration of f . Importantly, not all points close to x need eventually to separate from x under iteration, but at least one such point must exist in every neighborhood of x. If a map possesses sensitive dependence on initial conditions, then for all practical purposes, the dynamics of the map defy numerical computation. Small rounding errors in the computation become magnified upon iteration. The results of numerical computation of an orbit, no matter how accurate, may bear no resemblance whatsoever to the real orbit.
In summary, a chaotic map possesses three ingredients, which are: indecomposability, an element of regularity, and unpredictability. A chaotic system cannot be broken down or decomposed into two subsystems (two invariant open subsets) that do not interact under f because of topologically transitivity. In the midst of this random behavior, however, is an element of regularity, which is exhibited by the periodic points that are dense. The system is unpredictable because of the sensitive dependence on initial conditions [21].
Banks, Brooks, Cairns, Davis, and Stacey proved that (D1) and (D2) imply (D3). Theorem 1.2 ([8]). If f : X → X is transitive and has dense periodic points, then f has
sensitive dependence on initial conditions.
Vellekoop and Berglund proved that (D1) alone can imply (D2) and (D3) on intervals. Theorem 1.3 ([97]). Let I be a (not necessarily finite) interval and f : I → I a continuous
function and topologically transitive map. Then (1) the periodic points of f are dense in I and (2) f has sensitive dependence on initial conditions.
Remark 1.4 ([7]). Assaf and Gadbois demonstrated that (D1) and (D3) do not imply (D2)
by giving the example X = S1 \ei2πp/q|p, q ∈ Z, q 6= 0 equipped with the usual arclength
metric d and f : X → X defined by f(eiθ) = ei2θ. They also demonstrated that (D2)
and (D3) do not imply (D1) by giving the examples X = S1× [0, 1], where S1 is the unit
circle and [0, 1] is the unit interval with the standard metrics and f : X → X defined by
The definition of chaos in the sense of Li and Yorke is now introduced.
Definition 1.5 (Li and Yorke [55]). Let I be an interval of the real line and f : I → I a
continuous function. f is chaotic if f has an uncountable scrambled set S ⊂ I which satisfies the following condition:
(i) for every p, q ∈ S with p 6= q, lim sup
n→∞ |f
n(p)− fn(q)| > 0 and lim inf n→∞ |f
n(p)− fn(q)| = 0;
(ii) for every p∈ S and periodic point q ∈ I, lim sup
n→∞ |f
n(p)− fn(q)| > 0.
Li and Yorke defined only “chaos” on an interval, Huang and Ye extended the interval to general metric space.
Definition 1.6 ([35]). Let (X, f ) be a dynamical system, where X is a compact metric space
with metric d, and f : X → X is continuous and surjective. A subset S ⊂ X is a scrambled set (for f ), if any different points x and y from S are proximal and not asymptotic:
lim sup
n→∞
d(fn(x), fn(y)) > 0 and lim inf n→∞ d(f
n(x), fn(y)) = 0.
The function f is said to be chaotic in the sense of Li-Yorke, if there exists an uncountable scrambled set.
Li and Yorke proved that if a map has a periodic point with period 3, then the map has Li-Yorke chaos and implies the existence of all other periods. (Detailed discussions see Appendix A)
Theorem 1.7 ([21, 55]). Let f : R→ R be continuous. If f has a periodic point with period 3,
then
(1) for every k = 1, 2, . . ., there is a periodic point having period k; (2) f is chaotic in the sense of Li-Yorke.
Chaos in the sense of Devaney is stronger than that in the sense of Li-Yorke. The rela-tionship is given by the following two theorems.
Theorem 1.8 ([46]). Suppose f : I → I is continuous, where I is an interval on the real line.
Theorem 1.9 ([35, 90]). Let V be a compact set of a metric space (X, d), containing infinitely
many points. If a map f : V → V is continuous, surjective, and chaotic in the sense of
Devaney on V , then it is chaotic in the sense of Li-Yorke.
Remark 1.10. The converse statement is incorrect [46].
Various versions of “chaos”, such as those defined by Diamond, Marotto, Martelli, Robin-son, Wiggins and many others, also exist. Detailed discussions have been presented else-where [22, 23, 57, 59, 74, 98].
1.3
Topological Entropy
Definition 1.11 ([53]). Let X be a sample space where X contains n events w1, w2, ..., wnwith
probability p1, p2, ..., pn, respectively. The Shannon’s entropy with respect to X is defined
by H(p1, ..., pn) =− n X i=1 pilog(pi).
The following holds.
Theorem 1.12 ([53]). In the above definition, H 1 n, ..., 1 n = log n = max ( H(p1, ..., pn) : pi ≥ 0, n X i=1 pi = 1 ) .
Based on studies [53, 54], conducted in 1948, Shannon introduced a definition of entropy for use in information theory to establish the concept of “uncertain quantities” [83]. Later, in 1958, Kolmogorov used Shannon’s entropy to formalize the entropy on dynamical systems and opened the door to established the foundation of ergodic theory. This “Kolmogorov’s entropy” was improved and revised by his student Sinai in 1959. The revised version is called “Kolmogorov-Sinai entropy” or “Measure-theoretic entropy”.
Definition 1.13 ([53, 75]). Suppose (X, Σ, µ) is a probability measure space, that is, X is
a set, Σ is a σ-algebra of subsets of X, and µ is a probability on Σ. Let f : X → X be a measurable function. Arbitrarily choose a finite partition ¯A = {A1, ..., An}, that is, Ai ∈ Σ,
Ai∩ Aj =∅, and n
S
i=1
Ai = X. The entropy of f with respect to ¯A is defined by
hµ f, ¯A = lim sup n→∞ H f−n( ¯A)| n−1 _ i=0 f−i( ¯A) ! ,
where ¯A∨ ¯B =A∩ B : A ∈ ¯A, B ∈ ¯B , and ¯A and ¯B are finite partitions. The Kolmogorov-Sinai entropy of f is defined by
hµ(f ) = sup
hµ(f, ¯A) : ¯A is the finite partition of X
. Remark 1.14. If f is measure-preserving transformation, then
hµ(f, ¯A) = lim sup n→∞ H f−n( ¯A)| n−1 _ i=0 f−i( ¯A) ! = lim n→∞H f −n A¯| n−1 _ i=0 f−i A¯ ! = lim n→∞ 1 nH n−1 _ i=0 f−i( ¯A) !
for any finite partition ¯A of X. Although this equation is not more easily understood or
interpreted than the foregoing definition, its equivalent condition can be more conveniently computed [53].
Modeled after Kolmogorov-Sinai entropy, “topological entropy” was defined by Adler, Konhein, and McAndrew for topologically conjugate invariance in 1965 [2].
Definition 1.15 ([2, 53, 54]). Suppose (X, T ) is a dynamical system, where X is a compact
Hausdorff space and T : X → X is a continuous map. For an open cover ¯A of X, let N( ¯A)
be defined as the smallest cardinality of a subcover of ¯A. If ¯A and ¯B are open covers of X,
then their common refinement is defined as ¯A∨ ¯B ={A ∩ B : A ∈ ¯A, B ∈ ¯B}. Let
n
_
i=0
T−i( ¯A) = ¯A∨ T−1( ¯A)∨ · · · ∨ T−(n−1)( ¯A),
where T−k( ¯A) ={T−k(A) : A∈ ¯A}, and define
htop(T, ¯A) = lim n→∞ 1 nlog N n _ i=0 T−i( ¯A) ! .
The topological entropy of (X, T ) is defined as the supremum
htop(T ) = sup
htop(T, ¯A) : ¯A is the finite open cover of X
.
Remark 1.16. N( ¯A) is finite because of compactness, and the limit exists for any open cover ¯
A.
If the space is compact metric, then the following definition is equivalent to the above notion [12] and it is more useful [4].
Definition 1.17 ([12, 15, 74]). Let f : X → X be a continuous map on the space X with
metric d. A set S ⊂ X is called (n, ǫ)-separated for f for n a positive integer and ǫ > 0
provided that for every pair of distinct points x, y ∈ S, x 6= y, there is at least one k with
0 ≤ k < n such that d(fk(x), fk(y)) > ǫ. The number of different orbits of length n (as
measured by ǫ) is defined by
r(n, ǫ, f ) = {#(S) : S ⊂ X is a (n, ǫ)-separated set for f },
where #(S) is the cardinality of elements in S. Let
htop(ǫ, f ) = lim sup n→∞
log(r(n, ǫ, f )) n ,
and define the topological entropy of f as
htop(f ) = lim
ǫ→0,ǫ>0htop(ǫ, f ).
Remark 1.18. Here, only two definitions are introduced. See other references for topological
entropy [9, 13, 29, 96].
Example 1.19 ([74]). Let f : S1 → S1 have a covering map F : R→ R given by F (x) = 2x.
This map is called the doubling map. The distance on S1 is the one inherited from R by taking
x to x mod 1. Therefore, points near 1 are close to points near 0. Two points x and y stay
within ǫ of each other for n−1 iterations of f if and only if |x−y| ≤ ǫ2−(n−1) (as points in R).
If points are separated by a distance of exactly ǫ2−(n−1) in [0, 1), then the maximum number
of points is [ǫ−12n−1]. However, the last point to the right is close to the first point on the left
when considered modulo 1 in S1, so there are [ǫ−12n−1]− 1 points in S1. These points can
be spread apart slightly to make them (n, ǫ)-separated. Therefore, r (n, ǫ, f ) = [ǫ−12n−1]− 1,
where [a] is the integer part of a. Then,
htop(ǫ, f ) = lim sup n→∞ log ([ǫ−12n−1]− 1) n = lim sup n→∞ log (ǫ−1) + (n− 1) log(2) n = log(2)
for any ǫ > 0, so htop(f ) = log(2).
The following theorem relates the topological entropy to the Kolmogorov-Sinai entropy: Theorem 1.20 ([24]). If K is compact and f : K → K is continuous, then htop(f ) =
Remark 1.21. An invariant measure µ satisfies the equation µ(f−t(K)) = µ(K), t > 0.
An invariant probability measure µ may be decomposable into several pieces, each of which is again invariant. If it is not so decomposable, then it is said to be ergodic.
Once the topological entropy has been computed, whether systems are chaotic or not is known.
Theorem 1.22 ([51]). If a continuous map of the interval has positive topological entropy,
then it is chaotic according to the definition of Li and Yorke.
Remark 1.23 ([28, 51, 92]). The converse of this theorem is incorrect. Xiong [101] and
Sm´ıtal [92] given counterexamples.
In general metric space, a similar result is obtained:
Theorem 1.24 ([10]). If the dynamical system (X, T ) has positive topological entropy, where
X is compact metric space and T is surjective and continuous, then it is chaotic in the sense of Li and Yorke.
Consider the continuous map on the compact interval, the relationship between positive topological entropy and Devaney’s chaos is equivalent:
Theorem 1.25 ([48, 50, 51, 63]). Let f be a continuous map of a compact interval I to itself.
f has positive topological entropy if and only if f is chaotic in the sense of Devaney.
Let f : X → X and g : Y → Y be two maps. f and g are said to be topologically conjugate if there exists a homeomorphism h : X → Y such that h ◦ f = g ◦ h. The homeomorphism h is called a topological conjugacy. Mappings that are topologically conjugate have completely equivalent dynamics [21].
Adler, Konhein, and McAndrew established “topological entropy” for invariant of topo-logically conjugate, such that if two maps are topotopo-logically conjugate, then their topological entropies are equal as follows:
Theorem 1.26 ([2, 54, 74]). Let f : X → X and g : Y → Y be two continuous maps, where X and Y are invariant compact sets under f and g, respectively. That is, f (X) ⊂ X and g(Y )⊂ Y . If f and g are topologically conjugate, then htop(f ) = htop(g).
This basic result that is used to help calculate the entropy, relates the entropy of a map f to a power fk of f.
Theorem 1.27 ([74]). Assume f : X → X is uniformly continuous or X is compact, and k
1.4
Lyapunov Exponent
The Lyapunov exponent is usually computed to measure the exponential rate at which nearby orbits are moving apart. It is used in a diagnostic method that has been proven to be the most useful dynamical diagnostic method for chaotic systems in chaos theory [1, 74, 99, 100]. Therefore, an expression for the growth rate of the derivative of a function f : R→ R as the number of iterations increases is sought. Let x0be an initial value and δ > 0 (δ can sufficiently
small). By Taylor’s expansion [74], |fn(x
0 + δ)− fn(x0)| ≈ |(fn)′(x0)|δ. If |(fn)′(x0)| ∼ Ln,
where L is a linear function, then 1
nlog(|(f
n)′
(x0)|) ∼ log(L).
Definition 1.28 ([74]). Let f : R → R be a C1 function. For each point x
0, define the
Lyapunov exponent of x0, λ(x0), as follows:
λ(x0) = lim sup n→∞ 1 nlog(|(f n)′ (x0)|) = lim sup n→∞ 1 n n−1 X k=0 log(|f′ (xk)|), where xj = fj(x0).
Remark 1.29. Oseledec [68] showed that the limit exists for almost all points. Remark 1.30. Assume λ(x0) > 0, which implies that
log(|(fn)′
(x0)|) ≈ nλ(x0) or |(fn)′(x0)| ≈ enλ(x0) = L(x0)n,
where L(x0) = eλ(x0)> 1, and
|fn(x0 + δ)− fn(x0)| ≈ |(fn)′(x0)|δ ≈ L(x0)nδ → ∞
as n → ∞. Therefore, a positive Lyapunov exponent means sensitive dependence on initial conditions, this result is very important and useful since it enables a single quantity to be computed to determine whether a chaotic process is highly sensitive to initial conditions [74, 100].
In the case of m variables with x ∈ Rm, the derivative df
dx is replaced by the Jacobian matrix, which is evaluated at x: Dxf = (
∂fi
∂xj
), and is called the spectrum of Lyapunov exponents [24]. In a deterministic system, the positivity of the largest Lyapunov exponent is a necessary but not sufficient condition to guarantee the existence of chaos. For example, suppose f : R→ R is defined by f(x) = 2x, it is sensitive dependence on initial conditions, but it is not transitive. However, fulfillment of this condition is often sufficient [100].
Katok proved the following theorem, which relates the Lyapunov exponent to topological entropy:
Theorem 1.31 ([41]). If a C1+α-diffeomorphish f , α > 0, of a compact manifold has a
Borel probability invariant continuous ergodic measure with non-zero Lyapunov exponents, then htop(f ) > 0. (That is, the map is chaotic in the sense of Devaney and of Li and Yorke).
Remark 1.32.
(1) X is called a compact manifold if it is a manifold that is compact as a topological space,
where the manifold is a second countable Hausdorff space that is locally homeomorphic to Euclidean space;
(2) f : X → X is a Cr-diffeomorphish if f is a Cr-homeomorphism such that f−1 is also Cr.
Pesin also proved the relationship in other situations:
Theorem 1.33 ([24]). If ρ is an invariant ergodic measure with compact support under the
diffeomorphism f of a finite-dimensional manifold and ρ has smooth density with respect to the Lebesgue measure, then htop(ρ) =P positive λi.
1.5
Strange Attractor
Roughly, an attractor is an invariant set to which all nearby orbits converge. Hence, attractors are the sets that one “sees” when a dynamical system is iterated on a computer [21]. First, the following is defined:
Definition 1.34 ([21, 74]). A compact region N ⊂ M is called a trapping region for f if f (N) ⊂ int(N), where M is a (smooth) compact manifold and f : M → M is a
diffeomor-phism.
Since f (N) is compact and f (N) ⊂ int(N), the sets fn(N) are all compact and nested
for k ≥ 0. Therefore
Λ = \
n≥0
fn(N)
is a compact, nonempty set. Λ is the set of points whose full orbits, both forward and backward, remain in N for all time [21].
Definition 1.35 ([21, 74]). A set Λ is called an attracting set for f if there exists a trapping
region N such that Λ = T
k≥0
Remark 1.36. Λ is an invariant set [21].
Definition 1.37 ([21, 74]). An ǫ-chain or a pseudo-orbit of length n from x to y for a map f is a sequence {x = x0, ..., xn= y} such that for all 1 ≤ j ≤ n, d(f(xj−1), xj) < ǫ. Let X be
a set. The ǫ-chain limit set of X for f is the set
Ω+ǫ(f ) ={x ∈ X : ∀ n ≥ 1, ∃ y ∈ X and ǫ-chain from y to x of length greater than n}.
Then, the forward chain limit set of X is the set
Ω+(f ) = \
ǫ>0
Ω+ǫ (f ).
Finally, the chain recurrent set of f is the set
R(f ) ={x : there exists an ǫ-chain from x to x for all ǫ > 0}.
Definition 1.38 ([74]). The following relation ∼ on R(f) is defined: x ∼ y if y ∈ Ω+(x) and
x ∈ Ω+(y). Two such points are called chain equivalent, and the relation is an equivalence
relation. The equivalence classes are called the chain components of R(f ). If f has a single chain component on an invariant set Λ, then f is said to be chain transitive on Λ.
Definition 1.39 ([74]). A set Λ is called an attractor if it is an attracting set and f|Λ is
chain transitive, so Λ⊂ R(f).
Remark 1.40. In the above definition,
(1) Sometimes, f|Λ may wish to be assumed to be topologically transitive [74].
(2) Other definitions of attractors are in common use [21, 24, 62, 64, 76].
The omega limit set ω(x) of a point x ∈ M is the collection of all accumulation points for the sequence {x, f(x), f2(x), ...} of successive images of x. If some metric for the topological
space M is chosen, then ω(x) can also be described as the smallest closed set S such that the distance from fn(x) to the nearest point of S tends to zero as n → ∞. A set Λ is said to
be the likely limit set if it is the smallest closed of M with ω(x) ⊂ Λ for every point x ∈ M outside of a set of measure zero [28, 62]. Interestingly, the following properties are identified: Theorem 1.41 ([62]). This likely limit set Λ is well defined and is an attractor for f . In
fact, Λ is the unique maximal attractor, which contains all others.
Theorem 1.42 ([62]). If S is a compact set of positive measure with the property with f (S)⊂ S, then S necessarily contains at least one attractor.
Now, the definition can be presented:
Definition 1.43 ([74]). An invariant set Λ is called a strange attractor if it is an attractor
and f has positive Lyapunov exponent on Λ (meaning that it has sensitive dependence on initial conditions).
1.6
Snapback Repeller
Generally, proving that a dynamical system has chaotic behavior is difficult. Most techniques for making such a determination involve computer simulations, which apply the arithmetic of the Lyapunov exponent, find a period doubling bifurcation, and perform other tasks that are associated with numerical dynamical systems. However, obtaining such results by rigorous mathematical proofs is difficult.
A dynamical system with diffeomorphism has chaotic behavior that can be proved by using known methods, such as the existence of Smale horseshoe, transversal homoclinic orbits, or heteroclinic orbits. Noninvertible maps have chaotic behavior that can be identified by the existence of snapback repellers. However, for general focus problems, applying the above methods without computer-assistance is difficult. In most cases, the verification must be carried out with the aid of a computer and many investigations on the verification of chaotic behavior using computer-assisted techniques for continuous dynamical systems and discrete dynamical systems have been published [69].
In 1978, Marotto defined the snapback repeller. The existence of snapback repellers is adopted to determine whether a system is chaotic or not.
Definition 1.44 ([57]). Let p∗ ∈ Rn, B
r(p∗) be the open ball in Rn of radius r centered at
the point p∗, and Df (p∗) be the Jacobi matrix of f evaluated at p∗. Suppose f : Rn → Rn
be differentiable in Br(p∗), the point p∗ is an expanding fixed point of f in Br(p∗) if
f (p∗) = p∗ and all eigenvalues of Df (x) exceed 1 in norm for all x
∈ Br(p∗).
Definition 1.45 ([57]). Suppose p∗ is an expanding fixed point of f in B
r(p∗) for some r > 0.
p∗ is said to be a snapback repeller of f if there exists a point x
0 ∈ Br(p∗) with x0 6= p∗
and some positive integer m such that fm(x
0) = p∗ and det(Dfm(x0))6= 0.
The surprising main related result [57] is the following theorem, proven formerly by Marotto in 1978.
Theorem 1.46 ([57]). If f : Rn→ Rn has a snapback repeller, then f is chaotic in the sense
The above theorem states that if snapback repellers can be found, then the system ex-hibits Li-Yorke’s chaos. However, there exists an error in Marotto’s paper [57]. Proving Theorem 1.46, Marotto used the statement that if
(A): f satisfies that “expanding fixed point” condition in Definition 1.44, then
(B): there exists a constant s > 1 such that kf(x) − f(y)k > skx − yk for all x, y ∈ Br(p∗).
But, (A) does not imply (B) (the converse is true). Chen, Hsu & Zhou [16] and Li & Chen [49] gave two counterexamples to point out that the above statement is not true. As a result, the proof of Theorem 1.46 is in error.
In order to modify Marotto’s definition and theorem, many authors like Shi, Chen, and Yu [87, 89, 90] proposed their improved version of Definition 1.44, Definition 1.45, and Theo-rem 1.46. Furthermore, Marotto also redefined the definition of snapback repeller as follows. Definition 1.47 ([58]). Let f : Rn → Rn be differentiable in B
r(p∗) and p∗ be a fixed point of
f with all eigenvalues of Df (p∗) exceeding 1 in norm, and there exists a constant s > 1 such
that kf(x) − f(y)k > skx − yk for all x, y ∈ Br(p∗). Suppose there exists a point x0 ∈ Br(p∗)
with x0 6= p∗ and some positive integer m such that fm(x0) = p∗ and det(Dfm(x0)) 6= 0.
Then p∗ is called a snapback repeller of f .
Remark 1.48.
(1) The result of Theorem 1.46 remains essentially unchanged, as long as one refers to the
above revised definition [58].
(2) Actually, Definition 1.45 and Definition 1.47 are all true and have no problems in above
discussions in one-dimensional space R which can be verified by applying the mean value theorem [87]. In principle this work still uses original definition of snapback repeller (Definition 1.45).
(3) In one-dimensional space R, the Jacobi matrix Df (p∗) = f′(p∗) and
det(Dfm(x0)) = (fm)′(x0) = f′ (fm−1(x0))· f′(fm−2(x0))· · · f′(f (x0))· f′(x0) = f′ (xm−1)· f′(xm−2)· · · f′(x1)· f′(x0), where xj = fj(x0), 1≤ j ≤ m − 1.
Definition 1.49 ([87, 89]). Let snapback repeller p∗, f , m, and x
0 be the same as
Defini-tion 1.47. p∗ is said to be a nondegenerate snapback repeller of f if there exist positive
constants µ and δ0 such that Bδ0(x0) ⊂ Br0(p ∗) and
kfm(x)− fm(y)k ≥ µkx − yk for all
x, y ∈ Bδ0(x0); p
∗ is called a regular snapback repeller of f if f (B r0(p
∗)) is open and
there exists a positive constant δ∗
0 such that Bδ∗
0(x0)⊂ Br0(p
∗) and p∗ is an interior point of
fm(B
δ(x0)) for any positive constant δ ≤ δ0∗.
The snapback repeller in Marotto’s theorem is nondegenerate and regular. If f is contin-uously differentiable in some neighborhood of xj and det(Df (xj))6= 0 with 0 ≤ j ≤ m − 1,
then f is chaotic in the sense of Devaney.
Theorem 1.50 ([56, 87, 88, 89, 90]). Let snapback repeller p∗, f , m, and x
0 be the same as
Definition 1.47. If f is C1 in some neighborhood of x
j, det(Df (xj))6= 0, 0 ≤ j ≤ m − 1, and
f has a snapback repeller p∗, then f is chaotic in the sense of Devaney.
Remark 1.51. Chen, Hsu and Zhou [16] proved that if f : Rn → Rn is C1 and it has a
snapback repeller, then f is chaotic in the sense of Devaney. The ecological model in this paper is not C1, hence we prefer using Theorem 1.50 to using the theorem of Chen, Hsu and
Zhou.
Based on Devaney’s chaos and Li-Yorke’s chaos in Section 1.2, this work establishes a relational graph of chaos and relative checking methods (Figure 6). In Theorem 1.22, Theo-rem 1.24, and TheoTheo-rem 1.25. A continuous map on a compact interval (or a surjective and continuous map on a compact matric space) has positive topological entropy implying that the map has both Devaney’s chaos and Li-Yorke’s chaos. At the same time, in Theorem 1.46 and Theorem 1.50, if a map could find a snapback repeller, then it also exhibits Devaney’s chaos and Li-Yorke’s chaos.
In Remark 1.30, a positive Lyapunov exponent exhibits only sensitivity but transitivity, but in Theorem 1.31, if a C1+α-diffeomorphism map (α > 0) of a compact manifold has a
Borel probability invariant continuous ergodic measure and a positive Lyapunov exponent, then its topological entropy is positive. Furthermore, it exhibits both Devaney’s chaos and Li-Yorke’s chaos. On the other hand, in the hypothesis of Theorem 1.33, it has a positive Lyapunov exponent when the map has a positive topological entropy. Furthermore, it is a strange attractor when the map has a positive Lyapunov exponent and an attractor.
S t r a n g e a t t r a c t o r s D e v a n e y ’ s c h a o s L i - Y o r k e ’ s c h a o s - t r a n s i t i v i t y - p e r i o d i c p o i n t s a r e d e n s e - s e n s i t i v i t y L y a p u n o v e x p o n e n t T o p o l o g i c a l e n t r o p y K o l m o g o r o v - S i n a i e n t r o p y ( 3 ) , ( 4 ) ( 6 ) , ( 7 ) ( 5 ) (9), (10) (11) ( 8 ) (12) S n a p b a c k r e p e l l e r s ( 2 ) ( 1 ) (1) Theorem 1.50 (7) Theorem 1.24 (2) Theorem 1.46 (8) Theorem 1.20 (3) Theorem 1.8 (9) Theorem 1.31 (4) Theorem 1.9 (10) Theorem 1.33 (5) Theorem 1.25 (11) Definition 1.28 (6) Theorem 1.22 (12) Definition 1.43
Figure 1: Relational graph of chaos and relative checking methods.
2
Ecological Model
Many trees in forests reproduce intermittently, rather than at a constant rate [79]. A number of flowers and fruits are produced in a particular year (called a mast year) but very little reproductive activity occurs during several subsequent years until the next mast year. This “synchronous production of highly variable amount of seeds from year to years by a population of plants” is called masting [79]. Perfect periodicity in reproduction is rarely observed, and the intervals between masting are rather irregular.
Several explanations of the masting phenomenon have been proposed [3, 5, 6, 17, 19, 25, 26, 27, 32, 36, 37, 39, 40, 43, 44, 45, 67, 72, 85, 91, 93, 94]. They involve environmental fluctuations, weather conditions, swamping predators, the weight of young deer, bird popu-lations, the reproductive success of bears, increased efficiency of wind pollination, attraction to seed distributions, cue masting, and the dispersing animals. However, most of these hy-potheses explain neither the mechanism of masting nor the mechanism by which the timing of reproduction varies among individuals [79].
2.1
Isagi’s Resource Budget Model
Figure 2: Resource budget model of an individual plant [38].
Isagi, Sugimura, Sumidaa and Ito proposed a simple model of the mechanism of masting that was based on the resource budget of an individual tree [38]. They assumed that a constant amount of photosynthate is produced by each tree annually, given that the environmental conditions are constant from year to year. In Figure 2, photosynthate (PS) is consumed for
the growth and the maintenance of the tree; any that is not used by the plant is stored in a pool within the tree. The amount of PS was constant from year to year. In one year when
the accumulated PS exceeded a threshold (LT), the amount of accumulated PS minus LT was
used for flowering, and is regarded as the cost of flowering Cf. Hence, whenever the amount
of photosynthate accumulated in preceding years was large, the tree was inclined to flower more, and the amount of flowering in a year also depended on the amount of photosynthetic products that had accumulated in the previous years. The amount of accumulated PS was
decreased to LT after the flowering. The flowers were pollinated and bore fruits at a cost of
Ca. The ratio Ca/Cf was assumed to be constant RC. After the fruiting had been completed,
the amount accumulated was LT− Ca = LT− RCCf. In the model, PS accumulates annually,
until the tree flowers again when the amount exceeds LT.
Isagi et al. performed several numerical simulations [38], which RC was an important role
in their model. In Figure 3, when RC < 1, the amount of seed production was constant every
intervals between mast years, and therefore, a higher rate of crop failure. When RC = 1, the
autocorrelation values between seed production and lagged values of the prior seed production reciprocated between two specific values. In this case, the mast and non-mast years are predictable because the interval is regular. As RC increases, the model is less able to predict
masting.
Figure 3: Time series of Ca with various Rcs. For all plates, Ps, LT, and Cf(0) were 3.0, 6.0,
and 2.5, respectively. Results were excluded for the initial 50 years [38].
2.2
Satake’s Generalized Resource Budget Model
Let S(t) be the amount of energy reserved at the beginning of year t. If the sum S(t)+ P S is
below the threshold LT, then the tree does not reproduce and saves all of its reserved energy
for the following year. If the sum exceeds LT, then the tree uses energy for flowering.
Isagi et al. assumed that the energy expenditure for flowering exactly equals the excess, S(t)+ P
S−LT. Satake and Iwasa generalized Isagi’s model, the amount of energy expenditure
for flowering is proportional to the excess, a(S(t) + P
S− LT), where a is a positive constant.
be proportional to the cost of flowers, and is expressed as RCa(S(t) + PS − LT). After the
reproductive stage, the energy reserves of the tree have fallen to
S(t)+ PS− a(S(t)+ PS− LT)− RCa(S(t)+ PS− LT) = S(t)+ PS− a(RC+ 1)(S(t)+ PS− LT).
Therefore, S(t+1) = S(t)+ P S, if S(t)+ PS ≤ LT, S(t)+ P S− a(RC+ 1)(S(t) + PS− LT), if S(t)+ PS > LT. (1)
Define the non-dimensionalized variable Y(t) = (S(t)+ P
S − LT)/PS, equation (1) is now rewritten as Y(t+1) = Y(t)+ 1, if Y(t) ≤ 0, −kY(t) + 1, if Y(t) > 0, (2)
where k = a(RC + 1)− 1. The parameter k denotes the degree of resource depletion after a
reproductive year divided by the excess amount of energy in reserve before that year, and is called the depletion coefficient [79]. Notably, the quantity Y(t) is positive if and only if the
tree exhibits some reproductive activity in year t.
After rescaling, the dynamics (2) include only a single parameter k. Other parameters such as PS or LT do not affect the essential features of the dynamics if k remains the same.
In Isagi’s model, a = 1 is assumed and the depletion coefficient is the same as the ratio of the fruiting cost to the flowering cost, k = RC. Since the maximum value of Y(t+1) equals one
when Y(t) = 0, the minimum value of Y(t+1) equals −k + 1. Thus, the possible range of Y(t)
contained in [−k + 1, 1].
3
Mathematical Analysis
The following definitions are present for convenience later:
Definition 3.1. The composition of two functions is denoted by f ◦ g(x) = f(g(x)). The
n-fold composition of f with itself recurs repeatedly in the sequel. The function f is denoted by fn(x) = f ◦ · · · ◦ f(x), where n is an iterative number.
Now, Satake’s model (2) is analyed mathematically. Proposition 3.2. If k ≤ −1, then Y(t) tends to infinity.
Proof. By definition of Y , if Y(t) ≤ 0, then there exists a ¯t > 0 such that Y(¯t) ∈ (0, 1]. Let
Y0 = Y(¯t) and Ym = Y(¯t+m). When k =−1, Ym = Y0+ m for all m > 0 is to be shown. When
m = 1, Y1 =−kY0+ 1 = Y0+ 1. Suppose Yj = Y0+ j, since Yj > 0,
Yj+1 =−kYj + 1 = (Y0+ j) + 1 = Y0+ (j + 1).
By mathematical induction, Ym = Y0+ m for all m > 0. Then,
lim
m→∞Ym = limm→∞Y0+ m =∞.
When k < −1, since k < −1 and −k > 1, Ym > 0 for all m > 0. Next, Ym = (−k)mY0+
(−k)m−1+· · ·+(−k)+1 for all m > 0 is to be shown. Now, Y
1 =−kY0+ 1 is known. Suppose
Yj = (−k)jY0+ (−k)j−1+· · · + (−k) + 1, since Yj > 0, Yj+1=−kYj+ 1 =−k(−k)jY 0+ (−k)j−1+· · · + (−k) + 1 + 1 = (−k)j+1+ (−k)j+· · · + (−k)2+ (−k) + 1.
Again, by mathematical induction, Ym = (−k)mY0+ (−k)m−1+· · · + (−k) + 1 for all m > 0,
and Ym = (−k)mY0+ (−k)m−1 +· · · + (−k) + 1 = (−k)mY0+ 1− (−k)m 1− (−k) . (3) Hence, lim m→∞Ym = limm→∞ (−k)mY0+ 1− (−k)m 1 + k =∞.
Proposition 3.3. If −1 < k < 1, then Y(t) converges to the stable equilibrium 1
k + 1.
Proof. When 0 < k < 1, since Y0 ∈ (0, 1], Y1 = −kY0 + 1 ∈ (0, 1). Assume Yj ∈ (0, 1),
Yj+1 = −kYj + 1 is also contained in (0, 1). By mathematical induction, Ym ∈ (0, 1) for all
m > 0. Since
Ym = (−k)mY0+ (−k)m−1+· · · + (−k) + 1 = (−k)mY0+
1− (−k)m
1− (−k) and 0 < k < 1,| − k| < 1, let m tend to infinity,
lim
m→∞Ym =
1 1 + k. If k = 0, Ym = 1 for all m > 0, then
lim
m→∞Ym = 1 =
1 1 + k.
When−1 < k < 0, 0 < −k < 1, Y1 =−kY0+1 > 0. Suppose Yj > 0, now Yj+1 =−kYj+1 > 0
because −k > 0 and Yj > 0. Hence, Ym > 0 for all m > 0. From (3),
lim m→∞Ym = limm→∞ (−k)mY0+ 1− (−k)m 1− (−k) = 1 1 + k.
Since k is the depletion coefficient, k > 0 can be assumed. Therefore, from Proposition 3.3, if 0 < k < 1, then the tree reproduces every year at a constant rate.
Proposition 3.4. If k = 1, then there exists a number of periodic points with period 2
corresponding to different initial conditions.
Proof. By hypothesis Y0 ∈ (0, 1],
Y1 =−kY0+ 1 =−Y0+ 1 ∈ [0, 1).
If Y1 = 0, then Y2 = 1 and Y3 = 0 = Y1. If Y1 ∈ (0, 1), then Y2 =−Y1+ 1 =−(−Y0+ 1) + 1 =
Y0.
Hence, if k = 1, there are a number of two-point cycles corresponding to different initial conditions. (Isagi’s model satisfies Proposition 3.3 and Proposition 3.4.)
In one study [79], Satake and Iwasa identified chaos by finding a positive Lyapunov ex-ponent if k > 1. Of course, some authors regard the positive Lyapunov exex-ponent as the definition of chaos because sensitivity is the most important property of chaotic systems and is easily observed. However, a positive Lyapunov exponent means only that the model is sen-sitive dependence on initial conditions. The goal here is to prove chaos by identifying dense periodic subsets and transitivity rather than sensitivity (as in the chaos of Devaney). In this work, the model is proven to exhibit Devaney’s chaos by identifying snapback repellers. Theorem 3.5. If k > 1 2 + r 23 108 !1/3 + 1 2− r 23 108 !1/3
≈ 1.3247, then the system is
chaotic in Devaney’s sense.
Proof. First, p∗ = 1
1 + k is a fixed point of Y ; let g = Y
−1. Since |g′(p∗)| < 1, there exists
r > 0 with U = (p∗
− r, p∗+ r), U
⊂ (0, 1) such that limm→∞gm(x) = p∗ if x
∈ U. Choose g(p∗ ) = −k 1 + k < 0 and g 2(p∗ ) = 2k + 1 k2 + k > 0. Let g(p∗) >
−k + 1 and check g2(p∗) < 1, then k2− k − 1 > 0. Solve the inequality,
k > 1 + √ 5 2 ≈ 1.6180 or k < 1−√5 2 ≈ −0.6180.
Choosing k > 1 + √
5
2 allows j to be found such that gj(p∗) > 0 for all j
≥ 3 by the definition of Y . Computing |gj(p∗)
− p∗ |, yield |g(p∗) − p∗ | = 1 = 1 k0, and |gj(p∗ )− p∗ | = 1 kj−1
for all j ≥ 2 by mathematical induction. Hence, |gj(p∗)
− p∗
| decreases to 0 as j tends to infinity. That is, for this r, there exists a natural number J > 0 such that
gj(p∗
)∈ U as j ≥ J.
Fix J and let x0 = gJ(p∗), then x0 ∈ U and YJ(x0) = p∗. Since |Y′(p)| = k > 1 for all
p∈ U, and (YJ)′(x
0)6= 0, p∗ is a snapback repeller of Y .
Next, choose the (upper-right) fixed point p∗∗
= 2 1 + k of Y 2. Suppose h = (Y2)−1 and choose h(p∗∗) < 1 k and h 2(p∗∗) > 1 k. Let h(p∗∗) > −k + 2 and h2(p∗∗) < 1, yielding k4− k3− k2+ 1 = (k− 1)(k3− k − 1) > 0.
In Appendix B, the cubic equation is solved exactly. Hence, solving this inequality,
k > 1 2+ r 23 108 !1/3 + 1 2 − r 23 108 !1/3 ≈ 1.3247 or k < 1. Selecting k > 1 2 + r 23 108 !1/3 + 1 2 − r 23 108 !1/3
, yields j such that
hj(p∗∗
) > 1
k for all j ≥ 3 by the definition of Y2. By mathematical induction,
|hj(p∗∗
)− p∗∗
| = k− 1 kj+1
for all j≥ 1. Therefore, |hj(p∗∗)− p∗∗| is decreasing and
lim j→∞|h j(p∗∗ )− p∗∗ | = lim j→∞ k− 1 kj+1 = 0. (4)
Since |h′(p∗∗)| < 1, there exists r > 0 with V = (p∗∗− r, p∗∗+ r), V ⊂ (1 k, 1) such that lim m→∞h m(x) = p∗∗ if x∈ V.
For this r, from (4), there exists a natural number J′ > 0 such that
hj(p∗∗
)∈ V as j ≥ J′
. Fix this J′, let y
0 = hJ ′ (p∗∗), then y 0 ∈ V and (Y2)J ′ (y0) = p∗∗. Since |(Y2)′(p)| = k > 1
for all p∈ V , and (Y2)J′′
(y0)6= 0, p∗∗ is a snapback repeller of Y2.
Finally, Y2 has an (upper-right) snapback repeller 2
1 + k as k > 1 2+ r 23 108 !1/3 + 1 2 − r 23 108 !1/3 .
By Theorem 1.50, Y2is chaotic in the Devaney sense. Then, from Theorem 1.25, htop(Y2) > 0.
Since htop(Y2) = 2· htop(Y ) by Theorem 1.27, htop(Y ) > 0. From Theorem 1.25, Y is chaotic
in Devaney’s sense as k > 1 2+ r 23 108 !1/3 + 1 2 − r 23 108 !1/3 . Remark 3.6. Let k0 = 1 +√5 2 ≈ 1.6180 and k1 = 1 2+ r 23 108 !1/3 + 1 2 − r 23 108 !1/3 ≈ 1.3247. The term kp will be used later; it denotes the critical point with respect to the
sys-tem (2) under the iterative number 2p.
In the above theorem, the “snapback repeller method” is used when the iterative number is two to find k1 such that the system has Devaney’s chaos when k > k1. Next, when k
is between 1 and k1, a snapback repeller is still required as the iterative number increases.
However, when iterative number of Y is odd and 1 < k ≤ k1, the system has only one fixed
point, and the “snapback repeller method” fails, but when iterative number of Y is even but not two to the power of any natural number, Theorem 3.5 can not be improved upon. Therefore, the last case to be considered is that in which the iterative number is two to the power of any natural number in system (2).
Proof. When the iterative number is two to the power of any natural number, the general form of Y2p can be represented to Y2p(x) = L2p(x), x∈ Cp−3 1 k , Cp−2 1 k , R2p(x), x ∈ Cp−2 1 k , 1 , where p∈ N, L2p(x) = −kR2p(x) + k + 1, p is odd, −R2p(x) + k + 1 k , p is even, R2p (x) = L2p−1R2p−1(x), and j ∈ N Cj = Cj−1AACj−1, j is odd, Cj−1BCj−1, j is even
with C0 = C−1BC−1, where C−1(x)≡ x and C−2 ≡ 0. It is similar to Theorem 3.5 in which
the snapback repeller of system (2) can be found by numerical computation for p ≥ 2.
p kp 0 1.618033988749895 (≡ k0) 1 1.324717957244745 (≡ k1) 2 1.134724138401520 (≡ k2) 3 1.068297188920740 (≡ k3) 4 1.032770966453956 (≡ k4) 5 1.016443864419055 (≡ k5) 6 1.008140050503278 (≡ k6) 7 1.004160992268882 (≡ k7) 8 1.003664292317828 (C1) 9 1.003795792338565 (C2)
Table 1: When k > kp, system (2) is chaotic in Devaney’s sense as determined by numerical
computation in Matlab. The result of Y1 and Y2 in Theorem 3.5 is above dotted line, and
the iterative number n greater than two is below the dotted line.
p kp 0 1.618033988749894848204586834365638117720309179805762862135448622705261 1 1.324717957244746025960908854478097340734404056901733364534015050302828 2 1.134724138401519492605446054506472840279667226382801485925149551668237 3 1.068297188920841276369429588323878282093631016920833444507611946647007 4 1.032770966441042909329492888334744856652058371140403253917031540208661 5 1.016443864059417072092280201941787277910662321454134609733959043245535 6 1.008140032021166342336675311408118208893644908964048997902342844304787 7 1.004073666388692740274952354135845754211121309836120298287534443071976 8 1.002031776333416997088893271971142972647918937489170894541068546238239 9 1.001016116350239987853959635630193675245706270323947596435520337219342 10 1.000507743074500114948189347177723859179135821018512700930688524462566 11 1.000253885799306497646948038000941319259507014651397354037337963961327
Table 2: When k > kp, system (2) is chaotic in Devaney’s sense as determined by numerical
computation in Maple. The result of Y1 and Y2 in Theorem 3.5 is above dotted line, and
the iterative number n greater than two is below the dotted line.
to perform numerical computation and establish the following table to determine the iteration and the regions of k where the system is chaotic in Devaney’s sense. Including the result of Y1 and Y2 in Theorem 3.5, we have Table 1 and Table 2 (see Appendix C).
The k’s regions of Y2p
are found by determining the roots of polynomial with degree 2p+1. However, in Table 1, since the limitation of computer’s binary representation only four
bytes in Matlab, the results of (C1) and (C2) have large errors. Hence, in Table 2, the representation extended to 100 digits to reveal more accurate results in Maple.
In the numerical analysis, the speed at which a convergent sequence approaches its limit is called the rate of convergence. A sequence kp converges linearly to L if there exists a
number M ∈ (0, 1) such that lim
p→∞
kp+1− L
kp− L
= M, and the number M is called the rate of
convergence [81]. The sequence kp is provided in Table 2, and the value
kp+1− L
kp− L
is presented
in Table 3. Decreasing trend point of view of k1, k1,. . . , kp in Table 2 and the result in Table 3
demonstrate that the sequence kp converges linearly to the greatest lower bound 1 at a rate of
p (kp+1− 1)/(kp− 1) 0 0.52540469157943422769003322478664651351376649047830696763492605159518883 1 0.41489586699997431299925477792765572121310751072840368371184368616710443 2 0.50694099610638878113952349805380330694715113829015656080921537994903761 3 0.47982892061671168115785962262331467488250827627624491318606863118921204 4 0.50178148053706772234152456827894503210609893138022208365602874183801158 5 0.49501941829205944983109846713194236053127665283014625698323313050315463 6 0.50044844763510476499969515141640000606898362052434282373474249083429693 7 0.49875864627908427712294912377763749255500672526772548623019825923679742 8 0.50011230740694058357812570761070978087582752392863114119242456520421925 9 0.49968989710695575574326476020722865297776557761856147868459186532735834 10 0.50002808912057385358698942088743772381022001594295705733873847542234191
Table 3: Rate of convergence of kp.
that can manipulate a number with more digits and that has a larger memory can yield more accurate result.
Therefore, Theorem 3.5 and Theorem 3.7 prove that Satake’s generalized resource budget model is chaotic in the sense of Devaney when the depletion coefficient k > 1. This section mathematically interprets the dynamics of system (2) when k > 1. The next section will analyze Satake’s model by calculation for k > 1.
4
Numerical Simulation
The bifurcation diagram (Figure 4) of system (2) with iterations given by the same random initial values that the theoretical results of Proposition 3.2–3.4 satisfy for k ≤ 1. For k > 1, Theorem 3.5 and Theorem 3.7 yield rigorous mathematical and numerical results that show that system (2) is chaotic in Devaney’s sense. However, the system (2) eventually converges to periodic points when the initial value is a rational number and the depletion coefficient is a natural number.
Theorem 4.1. For any initial value x∈ Q and k ∈ N, Y(t)(x) is eventually periodic.
Proof. Without loss of generality, x∈ Q ∩ [−k + 1, 1]. Let x = q
−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 −4 −3 −2 −1 0 1 2 3 Y (t ) Depletion coefficient k
Figure 4: Bifurcation diagram of a single tree. The horizontal axis represents the depletion coefficient k, and the vertical axis represents Y(t) for many units of time t.
Suppose S = j p ∈ [−k + 1, 1] j ∈ Z , then x = q
p ∈ S. The cardinality of S is denoted by |S|, and |S| = j p ∈ [−k + 1, 1] j ∈ Z = k j p ∈ (0, 1] j ∈ Z + 1 = kp + 1. For any x = j p ∈ S, Y (x) = j p+ 1 = j + p p , if x∈ [−k + 1, 0], (−k)j p + 1 = (−k)j + p p , if x∈ (0, 1],
then Y (x)∈ S, that implies Y (S) ⊆ S. Hence, Yn(x)∈ S for all n ∈ N as x = j
p, and S1 ≡ {Y1(x), Y2(x), . . . , Ykp+2(x)} ⊆ S.
Yi(x) = Ykp+2(x) derived from the Pigeonhole Principle. It implies that the system has a
periodic solution with the period at most kp + 2− i.
According to Theorem 4.1, when the initial value x is a rational number and k is a natural number, the initial value eventually converges to a periodic point independently of k. It is no doubt that x only can be expressed using finite digits in binary representation in the computer. Therefore, for any simulation in the computer the initial value is always a rational number such that system (2) eventually goes to a periodic solution under k ∈ N.
Satake et al. [79] used the stochastic variable and the probability distribution density to elucidate in which situation k is a natural number. They assumed a stable distribution to show that the system converges to the periodic cycle {−k + 1, . . . , 0, 1} for all k is an odd number or an even number.
In Figure 5 (Figure 4), the system indeed converges to a periodic point with period k + 1 and the periodic cycle is{−k + 1, . . . , 0, 1} when k is a positive even number (see Figure 5 (a) & (c)). This means that the even number under the computer’s binary representation lets initial value x to carry that it converges to a “lower” period. However, the behavior is not like “lower” periodic when k is a positive odd number (see Figure 5 (b) & (d)). Moreover, whether the distribution that was proposed by Satake et al. is in fact stable is herein unknown. This section proposes an well explanation.
The following theorem show that why system (2) converges to a periodic point with period k + 1, where the periodic cycle is{−k + 1, −k + 2, . . . , 0, 1} when k is a positive even number. Theorem 4.2. Under a binary representation with finite digits, if k is a positive even number,
then Y(t) converges to the periodic cycle{−k + 1, −k + 2, . . . , 0, 1} with period k + 1.
Proof. Assume k = 2n, n∈ N. Let x = 0.x1x2· · · xp ∈ (0, 1) with xi ∈ {0, 1}. Since
−kx + 1 = −(2n)x + 1 = −2(nx) + 1 = −2y + 1,
0 100 200 300 400 500 600 700 800 900 1000 −2 −1.5 −1 −0.5 0 0.5 1 (a) t Y ( t) 0 500 1000 1500 2000 2500 3000 −2 −1.5 −1 −0.5 0 0.5 1 (b) t Y ( t) 0 500 1000 1500 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 (c) t Y ( t) 0 500 1000 1500 2000 2500 3000 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 (d) t Y ( t)
Figure 5: Iterative number t v.s Y(t) when k is a positive even number. The system converges
to a lower periodic cycle with period k + 1; however, when k is a positive odd number, the dynamics are not lower periodic. (a) k = 2; (b) k = 3; (c) k = 4; (d) k = 5.
used, only k = 2 can be considered. Hence, Y (x) = −x1.x2x3· · · xp0 + 1 = −0.x2x3· · · xp0 + 1, if x1 = 0, −1.x2x3· · · xp0 + 1, if x1 = 1 = 0.(1− x2)· · · (1 − xp−1)(2− xp) 0 |{z} 1 zero , if x1 = 0, −0.x2x3· · · xp0, if x1 = 1. Therefore, Y2(x) = −(1 − x2).(1− x3)· · · (1 − xp−1)(2− xp)00 + 1, if x1 = 0, 0.(1− x2)(1− x3)· · · (1 − xp−1)(2− xp)0, if x1 = 1 = −0.(1 − x3)· · · (1 − xp−1)(2− xp)00, if x1 = 0 and x2 = 0, 0.x3x4· · · xp 00 |{z} 2 zeros , if x1 = 0 and x2 = 1, 0.(1− x2)· · · (1 − xp−1)(2− xp) 0 |{z} 1 zero , if x1 = 1,
Y3(x) = 0.x3x4· · · xp00, if x1 = 0 and x2 = 0, −(1 − x2).(1− x3)· · · (1 − xp−1)(2− xp)00 + 1, if x1 = 1 = 0.x3x4· · · xp 00 |{z} 2 zeros , if x1 = 0, x2 = 0 or x1 = 1, x2 = 1, −0.(1 − x3)· · · (1 − xp−1)(2− xp)00, if x1 = 1 and x2 = 0, and Y4(x) = 0.x 3x4· · · xp 00 |{z} 2 zeros , if x1 = 1 and x2 = 0.
Hence, there exists positive integers n1 and n2 such that
Yn1(x) = 0.(1 − x2)· · · (1 − xp−1)(2− xp) 0 |{z} 1 zeros and Yn2(x) = 0.x 3x4· · · xp 00 |{z} 2 zeros . Suppose Ynk(x) = 0. (1− xk+1)· · · (1 − xp−1)(2− xp) | {z } (p−k) terms 00· · · 0 | {z } k zeros , if k ∈ 2N − 1, 0. xk+1xk+2· · · xp | {z } (p−k) terms 00· · · 0 | {z } k zeros , if k ∈ 2N. Then Ynk+1(x) = Y (Ynk(x)) = −1. (1 − xk+2)· · · (1 − xp−1)(2− xp) | {z } (p−k−1) terms 00· · · 0 | {z } (k+1) zeros +1, if k∈ 2N − 1 and xk+1 = 0, −0. (1 − xk+2)· · · (1 − xp−1)(2− xp) | {z } (p−k−1) terms 00· · · 0 | {z } (k+1) zeros +1, if k∈ 2N − 1 and xk+1 = 1, −0. xk+2xk+3· · · xp | {z } (p−k−1) terms 00· · · 0 | {z } (k+1) zeros +1, if k∈ 2N and xk+1 = 0, −1. xk+2xk+3· · · xp | {z } (p−k−1) terms 00· · · 0 | {z } (k+1) zeros +1, if k∈ 2N and xk+1 = 1 = −0. (1 − xk+2)· · · (1 − xp−1)(2− xp) | {z } (p−k−1) terms 00· · · 0 | {z } (k+1) zeros , if k ∈ 2N − 1 and xk+1 = 0, 0. xk+2xk+3· · · xp | {z } (p−k−1) terms 00· · · 0 | {z } (k+1) zeros , if k ∈ 2N − 1 and xk+1 = 1, 0. (1− xk+2)· · · (1 − xp−1)(2− xp) | {z } (p−k−1) terms 00· · · 0 | {z } (k+1) zeros , if k ∈ 2N and xk+1 = 0, −0. xk+2xk+3· · · xp | {z } (p−k−1) terms 00· · · 0 | {z } (k+1) zeros , if k ∈ 2N and xk+1 = 1,
Ynk+2(x) = Y (Ynk+1(x)) = 0. xk+2xk+3· · · xp | {z } (p−k−1) terms 00· · · 0 | {z } (k+1) zeros , if k ∈ 2N − 1 and xk+1 = 0, 0. (1− xk+2)· · · (1 − xp−1)(2− xp) | {z } (p−k−1) terms 00· · · 0 | {z } (k+1) zeros , if k∈ 2N and xk+1 = 1.
Hence, there is a nk+1 ∈ N such that
Ynk+1(x) = 0. xk+2xk+3· · · xp | {z } (p−k−1) terms 00· · · 0 | {z } (k+1) zeros , if k ∈ 2N − 1, 0. (1− xk+2)· · · (1 − xp−1)(2− xp) | {z } (p−k−1) terms 00· · · 0 | {z } (k+1) zeros , if k ∈ 2N. By mathematical induction, Ynj(x) = 0. (1− xj+1)· · · (1 − xp−1)(2− xp) | {z } (p−j) terms 00· · · 0 | {z } j zeros , if j ∈ 2N − 1, 0. xj+1xj+2· · · xp | {z } (p−j) terms 00· · · 0 | {z } j zeros if j ∈ 2N
for all j ∈ N with j ≤ p. When j = p, Ynp(x) = 0. 00· · · 0
| {z }
p zeros
. Therefore, Y (x) converges to the periodic cycle {−k + 1, −k + 2, . . . , 0, 1}.
Moreover, when the initial x could be represented in base-β with a finite number of digits, system (2) converges to periodic cycle{−k + 1, . . . , 0, 1} under k = βm with m ∈ N (β ∈ N). However, when k is a odd number, the following theorem explains that system (2) cannot converge to the periodic point with periodic cycle {−k + 1, −k + 2, . . . , 0, 1}.
Theorem 4.3. Under the binary representation with a finite number of digits, if k is a positive
odd number, then Y(t) can not converge to the periodic cycle S ≡ {−k + 1, −k + 2, . . . , 0, 1}
as the initial value x /∈ S.
Proof. Let x be the same as in Theorem 4.2 and consider only k = 3. Since x 6= 0, let the
first nonzero digits be xm, 1≤ m ≤ p. That is,
x = 0.x1· · · xm−1xm 0· · · 0 | {z } (p−m) zeros = 0.x1· · · xm−11 0· · · 0 | {z } (p−m) zeros .
![Figure 2: Resource budget model of an individual plant [38].](https://thumb-ap.123doks.com/thumbv2/9libinfo/8581395.189367/25.892.128.806.168.785/figure-resource-budget-model-individual-plant.webp)
