Definition 2.2. Let f : X → X be a continuous map on the space X which is compact with metric d. A set S ⊂ X is called (n,)-separated for f for n positive integer and > 0 provided that for every pair x, y ∈ S, x 6= y, there is at least one k with 0 ≤ k < n such that d(fk(x), fk(y)) > . The number of different orbits of length n(as measured by ) is defined by
r(n, , f ) = max{(S) : S ⊂ X is (n, ) − separated set f or f },
where#(S) is the cardinality of elements in S. Let
htop(, f ) = lim sup
n→∞
log(r(n, , f )) n and define the topological entropy (46) of f as
htop(f ) = lim
→0,>0htop(, f ).
Consider the continuous map on the compact interval, the relationship be-tween positive topological entropy (htop(f ) > 0) and Devaney’s chaos is equiv-alent.
Theorem 2.1. Let f be a continuous map of a compact interval I to itself, f has positive topological if and only if f is chaotic in the sense of Devaney on a closed invariant set in I [47-50].
The basic result following that is used to help calculate the entropy, and relates the entropy of a map f to a n-fold composition of f , fn.
Theorem 2.2. Assume f : X → X is uniformly continuous or X is compact, and n is an integer with n ≥ 1. Then htop(fn) = n · htop(f ) [46].
There is an another way to calculate topological entropy was defined by Welington and Sebastian [51].
Definition 2.3. Let f : I → I be a continuous piecewise monotone map. The lap number of f , which we defined `(f ) is the number of, maximal intervals on which f is monotone. In other words, `(f ) − 1 is the number of turning points of f [51].
Theorem 2.3. (Misiurewicz and Szlenk). Let f : I → I be a continuous, piecewise monotone map. Then the topological entropy of f is equal to the logarithm of the number s(f )=limn→∞(`(fn))n1 [51].
3 Mathematical analysis
In this section, we will prove that the generalized resource budget model is chaotic in the sense of Devaney on a closed invariant set in [1 − d, 1] when the depletion coefficient d is greater than 1 by using the preliminaries, the topological entropy theory.
Theorem 3.1. f is finitely renormalizable when the depletion coefficient d is greater than 1.
Proof. We have the recursion of slope of equation f algebraically.
xn+1= right part of function f2n in the box, respectively. Hence, by the recursion, we
can conclude that
Now, we want to express fixed point and its preimage of f2n in the box if n is odd or n is even. First, n is odd, solve the fixed point for the following equation:
flef t(2n)(x) = (−d)anx + f(2n)(0),
where flef t(2n)(x) means the equation of f2n’s left part in the box. Thus x∗ =
f(2n )(0)
1−(−d)an is the fixed point of it. Moreover, solve the solution x∗−1 of equation:
(−d)an+1x + f(2n)(0) = x∗,
Continuing the calculation process is like above described. Therefore, y∗ =
f(2n )(0) through the bottom of the box. Furthermore, we want to find the exactly n0
such that f(2n0+1)(0) > x∗−1 or f(2n0+1)(0) < y−1∗ . Define y = log(
log(log(d+1)log(d) +1)
log(2) < 1 . There are three possibilities for choosing n0which is related to y and x.
Hence, f is finitely renormalizable when the depletion coefficient d is greater than 1.
Theorem 3.2. htop(f ) is no less than 2n0+1ln 2 when the depletion coefficient d is greater than 1.
Proof. We want to demonstrate that f2n0+1+n has 2n−1 peaks on [x∗, x∗−1] for n0 in theorem3.1. Define f2n0+1 = h and the proof is by mathematical induction on n. The result is immediate if n = 1. Suppose that the result is true for n = k. That is, hk has 2k−1 peaks. Consider n = k + 1, we denote
htop(f2n0+1|[x∗, x∗−1]) = ln 2 by theorem2.3. Then, htop(f2n0+1) is no less than ln 2 on the compact set [1 − d, 1]. Finally, the result shows that htop(f ) is no less than 2n0+1ln 2 by theorem2.2.
Theorem 3.3. The generalized resource budget model is chaotic in the sense of Devaney on a closed invariant set in [1 − d, 1] when the depletion coefficient d is greater than 1.
Proof. We have htop(f ) > 0 for d > 1 according to the theorem3.2. Therefore, f is chaotic in the sense of Devaney on a closed invariant set in [1 − d, 1] by theorem2.1. Hence, the result shows that the map f can possess Devaney’s chaos when the depletion coefficient d is greater than 1.
4 Finite period doubling route to chaos
Usually the term ”route to chaos” refers to formation of chaotic attractors.
In this section, we consider route from a parameter having no chaos to one with chaotic sets, where the sets are not necessarily attractors. To be more precise, we require only one aspect of chaos: we say that a map has a chaos at a particular µ if there exist infinitely many periodic orbits; otherwise, it is said that the map has no chaos at the particular µ. For example either one of the following conditions is sufficient for a continuous map to have chaos. (i) The positivity of the topological entropy (see e.g. [52]) . (ii) The existence of a horseshoe [53] . (iii) The existence of a nondegenerate homoclinic orbit [54] . (iv) The existence of a periodic point with its period being not the power of two. (v) The map is finitely renormalizable [52]. The concepts of a horseshoe and being finitely renormalizable are to be used through out this section. The definition of the latter, which is more complicated, is to be given at the appropriate place. If I ⊂ R is a closed interval, f : I → R continuous, and a < c < b ∈ I, then we say that [a.b] is a horseshoe for f if [a, b] ⊂ f ([a, c]) ∩ f ([c, b]). The presence of a horseshoe clearly produces a full two-shift as a factor of the restriction of
f to an invariant set. Consequently, f has periodic points of all period and its topological entropy is no less than ln 2.
For smooth dynamical systems that depend on a parameter, one of the basic route to chaos is the period-doubling cascade. For instance, it is well-known that for the quadratic family fµ(x) = µx(1−x), the route to chaos is through period-doubling. A geometric and intuitive answer to the process can be provided as follows.
For 0 ≤ µ ≤ 3, fµ has a globally attracting fixed point. Before fµ can pos-sibly have infinitely many periodic points with distinct periods, it must have periodic points with all periods of the form 2jaccording to Sarkovskii’s Theorem (see e.g., (54)). That leads to the consideration of the graph of fµ2 which re-sembles the graph of the original quadratic map (for a different µ-value). Using graphical analysis of fµ, we may also sketch the graphs of fµ2for various µ-values.
These are depicted in Fig 1. Note that in Fig 1-c, we say that [ˆpµ,0, pµ,0] is a horseshoe for f2. Inside the box, fµ2 has one fixed point pµ,0 at an endpoint of the interval [ˆpµ,0, pµ,0] and a unique critical point with this interval. Note that, as long as fµ0(pµ,0) < 0 (resp., > 0), there exists a ”partner” ˆpµ,0for pµ,0in the sense that fµ(ˆpµ,0) = pµ,0and ˆpµ,0 < pµ,0 (resp., ˆpµ,0> pµ,0). As µ increases, we first expect a new fixed point pµ,1in [ˆpµ,0, pµ,0] for fµ2(i.e., a period 2 point for fµ) to be born. Eventually, this ”fixed point” will itself period-double, just as pµ,0 did for fµ, producing a period 4 point. Continuing the procedure, we may find a small box in which the graphs of fµ4, fµ8, etc., resemble the origi-nal quadratic function. Such ideas can be made precise, by using the so called renormalization techniques. Thus we are led to expect that fµ undergoes a series of period-doublings as µ increases. On the other hand, if one views this process algebraically, then at the bifurcation value µ1 = 3 for the family fµ, the fixed point changes from attracting for 1 < µ < µ1 to repelling for µ > µ1. For µ slightly larger than µ1, the 2-period orbit is born and is attracting. As µ moves past µ2, where the period four orbit is created and is attracting. Again, the original 2-periodic orbit changes from attracting to repelling. Such period
four orbit becomes repelling for µ > µ3 and a new attracting period eight orbit is born. This process repeats itself; at µ > µ+n, the period 2n orbit is added.
This orbit is attracting for µn< µ < µn+1and becomes repelling for µ > µn+1.
Figure 1: The graphs of fµ2(x) for µ = 2.5, µ = 3.4 and µ = 3.8, respectively.
Now, combining the geometric and algebraic view together, we have that for µn < µ < µn+1, the corresponding box containing the graph of fµ2n, i.e., the graph of fµ2n on [pµ,n−1, ˆpµ,n−1] if n is even or on [ˆpµ,n−1, pµ,n−1] if n is odd, is similar to that of in Fig 1-(b). Here pµ,n is the 2n periodic point of f . However, for the same range of µ, the associated box containing the graph of fµ2n+1 is similar to that of in Fig 1-(a). As a result, the parameters in this range yield no new fixed point, and hence, no chaos. It should be mentioned that for µn+1 < µ < µn+2, the graph of fµ2n+1 in the corresponding box is similar to that of in Fig 1-(b). Such sequence {µn} produces a universal constant as the rate of convergence. For µ∞:= lim
n→∞µn, fµ∞ is called infinitely renormalizable.
Geometrically speaking, this means that for any n, the graph of fµ2n∞ on the corresponding box has the following two properties:
(i) there exists a fixed point in (ˆpµ,n−1, pµ,n−1) or (pµ,n−1, ˆpµ,n−1) for fµ2∞n; (ii) the ”hump” will not extend out the box.
Pictorially, this means that for any n ∈ N, the graph of f2nin the corresponding box resembles that in Fig 1-(b). For µ > µ∞, there exists an n such that fµ2n
has a horseshoe. That is to say, the hump of fµ2n protrudes through the bottom or the top of the box, or equivalently f is said to be finitely renormalizable.
Pictorially, the graph of fµ2n in the corresponding box looks like that of in Fig 1-(c). This completes the process of the period-doubling route to chaos.