Let S(t) be the amount of energy reserved at the beginning of year t. If the sum S(t)+ PS 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)+ PS−LT. Satake and Iwasa generalized Isagi’s model, the amount of energy expenditure for flowering is proportional to the excess, a(S(t) + PS− LT), where a is a positive constant.
Flowering trees may be pollinated and set seeds and fruits. The cost for fruits is assumed to
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)+ PS, if S(t)+ PS ≤ LT, S(t)+ PS− a(RC+ 1)(S(t) + PS− LT), if S(t)+ PS > LT.
(1)
Define the non-dimensionalized variable Y(t) = (S(t)+ PS − 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,
m→∞lim Ym = lim
m→∞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, Y1 =−kY0+ 1 is known. Suppose Yj = (−k)jY0+ (−k)j−1+· · · + (−k) + 1, since Yj > 0,
Yj+1=−kYj+ 1
=−k
(−k)jY0+ (−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,
m→∞lim Ym = lim
m→∞
(−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,
m→∞lim Ym = 1 1 + k. If k = 0, Ym = 1 for all m > 0, then
m→∞lim 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),
m→∞lim Ym = lim
m→∞
(−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 g2(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 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)′(x0)6= 0, p∗ is a snapback repeller of Y .
Next, choose the (upper-right) fixed point p∗∗ = 2
1 + k of Y2. Suppose h = (Y2)−1 and
In Appendix B, the cubic equation is solved exactly. Hence, solving this inequality,
k > 1
, 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
Since |h′(p∗∗)| < 1, there exists r > 0 with V = (p∗∗− r, p∗∗+ r), V ⊂ (1
k, 1) such that
m→∞lim hm(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 y0 = hJ′(p∗∗), then y0 ∈ 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
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 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).
Theorem 3.7. For 1 < k≤ k1, the system is chaotic in Devaney’s sense.
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) = the snapback repeller of system (2) can be found by numerical computation for p ≥ 2.
p kp
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.
Therefore, do the same steps as in Theorem 3.5 are used, and Matlab and Maple are used
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 convergence of 0.5. Hence, the system (2) is necessarily chaotic as long as k > 1. A computer
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
p ∈ Q with p ∈ N and q ∈ Z.
−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4
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 ∈ S. The cardinality of S is denoted by
|S|, and
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,
where y = nx = y1.y2· · · ypyp+1 and a binary representation with a number of finite digits is
0 100 200 300 400 500 600 700 800 900 1000
0 500 1000 1500 2000 2500 3000
−2
0 500 1000 1500 2000 2500 3000
−4
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
Y3(x) =
Hence, there exists positive integers n1 and n2 such that Yn1(x) = 0.(1− x2)· · · (1 − xp−1)(2− xp) 0
Ynk+2(x) = Y (Ynk+1(x))
. 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}
Hence,
whose first nonzero digit x[nm1] is also 1. Assume Ynk(x) = 0.x[n1k]x[n2k]· · · x[nm−1k] x[nmk] 0| {z }· · · 0
(p−m) zeros
, x[ni k]∈ {0, 1}, i 6= m,
whose first nonzero digit x[nmk] is also 1, then
Hence, there exists an nk+1 ∈ N such that
Ynk+1(x) = 0.x[n1k+1]x[n2k+1]· · · x[nm−1k+1]x[nmk+1] 0| {z }· · · 0
(p−m) zeros
, x[ni k+1]∈ {0, 1}, i 6= m,
whose first nonzero digit x[nmk+1] is also 1. By mathematical induction, Ynj(x) = 0.x[n1j]x[n2 j]· · · x[nm−1j] x[nmj] 0| {z }· · · 0
(p−m) zeros
, x[ni j] ∈ {0, 1}, i 6= m,
its first nonzero digit x[nmj]is still 1 for all j ∈ N. That is, no n ∈ N exists such that Yn(x) = 0.
Hence, Y can not converge to the periodic cycle {−k + 1, −k + 2, . . . , 0, 1}.
Therefore, from Theorem 4.3, even though x is represented in binary using a finite number of digits in computers, Y does not converge to the periodic cycle {−k + 1, −k + 2, . . . , 0, 1}
when k is an odd number. The above theorem revises the statement of integers k in a investigation [79].
5 Conclusions
The relationship among chaos, Lyapunov exponent, topological entropy, strange attractors, and snapback repellers is elucidated. The conditions under which they are equivalent to each other or imply each other are identified.
Satake and Iwasa proved that the generalized budget resource model is chaotic when k > 1 by computing the Lyapunov exponent, but their proof was not clear. However, the model has no periodic points with period three when k <√
2, so we can not to prove that it is chaotic by the existence of periodic points with period three. Therefore the existence of snapback repellers was used to prove theoretically and numerically that the model exhibits Devaney’s chaos when k > 1.
Any rational number eventually converges to a periodic point when k is a natural number, and the periodic point are found. Under the binary representation with a finite number of digits, if k represents an even number, then Y converges to the periodic cycle {−k + 1, −k + 2, . . . , 0, 1} with period k + 1, but if k represents an odd number, then Y cannot converge to the same periodic cycle even if its points are rational numbers.
Appendix A
Consider the following ordering of the natural numbers:
3 ⊲ 5 ⊲ 7 ⊲· · · ⊲ 2 · 3 ⊲ 2 · 5 ⊲ 2 · 7 ⊲ · · · ⊲ 22· 3 ⊲ 22· 5 ⊲ 22 · 7 ⊲ · · ·
⊲ 23· 3 ⊲ 23· 5 ⊲ 23· 7 ⊲ · · · ⊲ 23⊲ 22⊲ 2 ⊲ 1.
That is, first list all odd numbers except one, followed by 2 times the odds, 22 times the odds, 23 times the odds, etc. This exhausts all the natural numbers with the exception of the powers of two which we list last, in decreasing order. This is the Sarkovskii’s ordering of the natural numbers. This ordering allows us to state Sarkovskii’s Theorem.
Theorem A.1 (Sarkovskii’s Theorem [84]). Let f : R → R be a continuous function, and suppose f has a periodic point of prime period k. If k ⊲ m in the Sarkovskii’s ordering, then f also has a periodic point of period m.
In above theorem, period 3 is the greatest period in the Sarkovskii’s ordering and there-fore implies the existence of all other periods, Devaney, Li and Yorke have the same result.
Furthermore, Li and Yorke proved that if a map has a periodic point with period 3, then the map has Li-Yorke chaos.
Theorem A.2 ([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.
Therefore, from Theorem A.1 and Theorem A.2, if f has only finitely many periodic points, then they all necessarily have periods which are powers of two. Conversely, if f has a periodic point whose period is not a power of two, then f necessarily has infinitely many period points.
Definition A.3 ([21]). Let f (p∗) = p∗ and f′(p∗) > 1. A point x0 is called homoclinic to p∗ if x0 ∈ Wlocu (p∗) = {the maximal such open interval about p∗} and there exists n > 0 such that fn(x0) = p∗.
Remark A.4. If p∗ has a homoclinic point, then p∗ is also called a snapback repeller.
Definition A.5 ([21]). A homoclinic point, together with its backward orbit and forward orbit, is called a homoclinic orbit. A homoclinic orbit is called nondegenerate if f′(x)6= 0, for all points x on the orbit.
Theorem A.6 ([21]). Suppose f admits a nondegenerate homoclinic point to p∗. Then, every neighborhood of p∗ contains infinitely many distinct periodic points.
Comparing Theorem A.2 with Theorem A.6, if f has a periodic point whose period is not a power of two or f has a nondegenerate homoclinic point, then f necessarily has infinitely many period points. The relationship of periodic point whose period is not a power of two and homoclinic point is given by the following theorem.
Theorem A.7 ([11]). f has a periodic point whose period is not a power of two if and only if f has a homoclinic point.
From the above theorem, if f has homoclinic point, then f has a periodic point whose period is not a power of two. Bowen and Franks have shown in [14] that if f has a periodic point whose period is not a power of two, then the topological entropy of f is positive.
Therefore, from Theorem 1.25, f is chaotic in the sense of Devaney. Naturally we have the following theorem.
Theorem A.8. If f has a periodic point with period 3, then f has Devaney’s chaos.
Hence, combining the result of Theorem A.2 and Theorem A.8, we know that if f has a periodic point with period 3, then it is Devaney’s chaotic and Li-Yorke’s chaotic.
Strange attractors
Devaney’s chaos Li-Yorke’s chaos
- transitivity
- periodic points are dense - sensitivity
(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 (13) Theorem A.2 (6) Theorem 1.22 (12) Definition 1.43 (14) Theorem A.8
Figure 6: Relational graph of chaos and relative checking methods including period three.
Appendix B
In mathematics, a cubic function has the form
f (x) = ax3 + bx2 + cx + d,
where a6= 0. If f(x) = 0, then the equation is called a cubic equation. The cubic equation will be solved exactly here [42, 66].
Suppose
Expanding and simplifying (6), yields ay3+
Converting (7) into the form
Substituting this equation into (8),
z6+ f z3− e3 27 = 0.
One more substitution, w = z3, yields the quadratic equation w2+ f w− e3
27 = 0. (10)
Solving the quadratic equation (10),
w = z3 =−f 2 ±
rf2 4 + e3
27. (11)
Six values of z can be determined from (11), since a square root has two possible values (±), and a cubic root has three. The six possible values of z yield the six possible values of y (Equation (9)), but the three values of y will be identical to the other three. Therefore, three values of y, and three values of x (Equation (5)) are obtained. The three roots are
x1 =− b
Appendix C
First, compute Y2p, p = 2, 3, 4, and 5. Choose the polynomial that passes through upper-right periodic point, and select its left polynomial at the same time. Then
Y2(x) =
L22(x) = −R22(x) + k + 1
Therefore, the algorithm yields the general form Y2p in Theorem 3.7, and the “snapback repeller method” can be applied to the iterative number 2p to identify the regions of k in which system (2) exhibits Devaney’s chaos.
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