X leaves the point unchanged or, alternatively, the point xD is invariant under the mapping /). By a periodic point of / [6, 10, 31], we mean that after some specific number of iterations of a mapping / of a particular point x0, we return to the point x0. That is, for some specified integer n, n = 1,2,3,..., (fn)(xD) = x0. It is seen immediately that a fixed point is simply a periodic point of period n = 1. For all periodic points, the smallest n > 0 with ( / n ) ( £ o ) = x0 is called the prime period of x0. Now, letting A\" be a subset of the real line, by definition [13, 6, 30], a periodic point x0 of prime period n (and from this point forth, all periodic points will be referred to by their prime period, unless otherwise indicated) is called hyperbolic if | {fn)'(x0) [^ 1. For | (fn)'(x0) |> 1, x0 is called a repelling periodic point, and for | (fn)'(x0) |< 1, x0 is called an attractive periodic point. The case of | (fTl)'(x0) |= 1 for a periodic point x0 defines an elliptic periodic point, and shall not be considered an}7 further here. Some important properties of hyperbolic fixed points (i.e., for n = 1) are as follows [15, 16, 2, 6, 30]: If x0 is an attracting hyperbolic fixed point, i.e., | f'(xD) |< 1, then 3 an open ball O about xc such that Vx £ O, l im n _ 0 0 fn(x) = xQ. If x0 is a repelling hyperbolic fixed point, i.e., j f'(x0) \\> 1, then 3 an open ball O about x0 such that VIE £ O. i / xOJ Hm n_ + 0 0/ n(x) ^ O. Chapter 2. Dynamical Systems 8 Figure 2.1: Stable and unstable lines induced about a fixed point x0. Hence, 3 an open set of points about an attracting fixed point x0 which all con-verge under forward iteration of / to xQ. Conversely, 3 an open subset of points about a repelling fixed point x0 which move away from their initial neighborhood of xQ, or alternatively, move toward x„ under backward iteration of / . We define the unstable set of points Wu(x0) as being the largest open punctured interval about x0 which has the repelling characteristic. Likewise, we define the stable set of points W(x0) as being the largest open interval about x0 for which the attracting property holds. For dimensions higher than one, the stable and unstable sets of points may be ex-tended naturally from characteristic intervals to characteristic manifolds (see figure 2.1). Chapter 2. Dynamical Systems 9 2.2 General Properties In general terms, a dynamical system may be defined as being a pair (X,tp), where X is a topological space and

Y such that the following diagram commutes: X tpt X i -> i 9 I I 9 I -> I Y A Y That is, tpt — g'1 o ipt c g or g o ipt — ipt o g \\/t. By definition, we see that the homeomorphic mapping g conjugates ipt with ipt-A mapping / : X —> X is said to be topologically transitive if, for open non-empty sets 0 i , 0 2 C A there exists a k > 0 such that (fk)(Ox) fl 02 ± 0. This states that under iteration, / moves points from any open neighborhood to any other. Chapter 2. Dynamical Systems 12 We note that a subset U C X is said to be dense if the closure of it is equal to the entire set. A function g(x) is a CT diffeomorphism if g(x) and g~l(x) are Cr homeomorphisms, where by Cr we mean that both g(x) and g~^[x) are differentiable to degree r. We are now equipped to give a full definition of strange attractors and structural stability, two of the most indicative features in chaotic system analysis. For a dynamical system (X,

0. By definition, we look for the presence of exponential separation of our system under N iterations. Hence is sought. Taking logarithms of both sides and observing as N —> oo, e —»• 0, we get: L(x0) = limx_toolim!:_^o fN(Xo + e)-fN(x0) e Chapter 2. Dynamical Systems 13 L(x0) = limx^oojj log dfN(x0) (2.1) dxD From this definition, we note that if L(x0) is positive, the system will yield exponential separation, even over the long term, from only slight perturbations and hence the sign of L has become one of the most consistently employed methods of assessing a nonlinear system's degree of chaos. Finally, an attractor is said to be strange if it is sensitive (in the Lyapunov sense) with respect to initial conditions, i.e., that small perturbations to initial conditions give rise to positive Lyapunov exponents. We note immediately that homoclinic points would be potential candidates for being strange attractors, and often are [11, 13, 6, 27]. For a space X and a mapping

X,

R such that and This then gives an algorithm for the determination of whether or not a system contains a period doubling bifurcation at certain parametrically dependent fixed or periodic points. 3.3 The Quadratic M a p The most widely accepted law for constrained population growth and dynamics in a bounded region is the logistic or Verhulst equation [27, 24, 6, 13, 14, 5]: = rxn(l - xn) 0 < xn < 1, 0 < r < 4 (3.4) We note that x is restricted to the unit interval as one may always rescale equation 3.4 to retain all dynamics found in larger intervals [24, 27, 8]. Amazingly, this simple quadratic difference equation exhibits very complex and in fact highh* chaotic behavior as the growth rate control parameter r is altered in the interval 0 < r < 4. 3.4 The Renormalization Group It is of interest to note that maps of the form 3.4 contain a single extremum in the unit interval, and under the influence of an increase of r from 0 to 4 a cascade of period doublings (i.e., for n —> oo, a cascade would be 2n period doublings [1, 27]) occurs. Chapter 3. Population Dynamics 18 Xn Figure 3.3: The attractor vs. the control parameter r for the quadratic difference equa-tion. Amazingly, there exists a high degree of self similarity at each level of branching or period doubling of equation 3.4, where the branches occur at the bifurcating periodic points. This self similarity property was first documented by Mitchell Feigenbaum and in fact he demonstrated that for maps such as 3.4, universal constants 8 = 4.6692016... and a = —2.5029078... determine the interval rate of doubling of such a system [8, 27, 1]. The usefulness of the renormalization group is found not only in the Feigenbaum constants 8 and a, but also in the microscopic investigation it affords on a system such as 3.4 [18, 27, 6]. The renormalization group gives what is often called a window [27, 14, 12] for the mapping produced by a particular control parameter value. To see how this renormalization group method works, we define a conjugation oper-ator: Chapter 3. Population Dynamics 19 LT(x) - (JC - qT) 9r - qT (3.5) where qT and qT are seen in figure 3.4. From equation 3.5, we produce the inverse operator (x) - (qT - qT)x + qT and the conjugation of F2, i.e., iterating Fr down to the next period doubling level, gives the renormalization of Fr, producing, for the quadratic map, an exact duplicate of the original period doubling map. Consequently, the renormalization of FT is defined as 3.5 Chaos, Period Doubling, and the Quadratic M a p Whereas multi-leveled period doubling bifurcations in a dynamical system give rise to being more fully inspected under analytic techniques such as the renormalization group, the presence of such cascading bifurcations does not necessarily imply the existence of chaos within a system. To assert the presence of chaos in a bifurcating dynamical system such as 3.4, we find it necessary to return to the topics of strange attractors and structural stability. In par-ticular, we find that by determining the Lyapunov exponents produced by equation 3.4 for different values of the population growth rate parameter r, there exists a direct cor-respondence -between the behavior of the bifurcations and the signs of the corresponding Lyapunov exponents. To see this consider the following: Define FT(x) as in equation 3.4. Then using equation 2.1 we calculate the Lyapunov exponents produced from the mapping Fr for various values of r. That is (3.6) Chapter 3. Population Dynamics Figure 3.4: A renormalization of F, demonstrating the self similarity that the gation operator LT produces. Chapter 3. Population Dynamics 21 1 N L ( T ) = lim — T ln | Fr'(xi) |, 0 < x{ < 1, 0 < r < 4 (3.7) 1 i=Q where F' represents the slope of the map. So, in our particular case, we obtain a spectrum of Lyapunov exponents from 1 N L(r) = Jirr^ — £ l n | r ( l - 2a;,-) |, 0 < X i < 1, 0 < r < 4 (3.8) By inspection of figure 3.6, we see that when the bifurcation diagram produces dark vertical bands (indicating an infinite frequency spectrum [27, 28]) we have a direct correspondence with the sign of the Lyapunov exponents, i.e., the period doublings occur at the points where the Lyapunov exponents vanish, L(r) = 0, and dark bands occur where L(r) > 0. In other words, infinite period doubling type branching in a dynamical system is a clear indication of the presence of chaos, as indicated by the positive, L{r) > 0, Chapter 3. Population Dynamics 22 Lyapunov exponents. Figure 3.6: An alignment of the bifurcation and Lyapunov mappings shows the direct correspondence of the two analytic techniques. Chapter 4 Electricity Demand and the Effect of Population Dynamics 4.1 Introduction To study the population growth model in a setting more appropriate to electric demand forecasting, we need not only look at the inherent dynamical model of constrained popu-lation growth, but also we must consider what kind of per capita demand for electricity is to be associated with our growing population. This is obvious as one can see immediately that a huge and dense population with no need of electricity is hardly a chaotic system as far as electric power supply is concerned. Consequently, we shall only be concerned with populations that are dependent on electricity, but with their per capita demand or dependency being at least partly unknown. We say at least partly unknown because long term trends in per capita demand for electricity are not completely understood [22, 3, 7]. Hence, we cannot construct a model with completely determined variables for electric demand in the same way as was done for the model of population dynamics. Further-more, other relevant issues, such as the elasticity of demand for electricity, the price of electric power substitutions (e.g., gas and/or oil), and the electrical dependency of the population under inspection, all play a significant role in demand forecasting, and hence their influence must be accounted for. As none of the above factors are well understood in a deterministic sense, it becomes apparent that we must somehow construct a relevant probabilistic model of our original quadratic population growth model, equation 3.4. Furthermore, this probabilistic model must incorporate some kind of demand variation 24 Chapter 4. Electricity Demand and the Effect of Population Dynamics 25 to account for the inherent differences in electric demand when considering areas or re-gions of different population densities (i.e., it is only reasonable to expect areas of high, medium, and low population densities to exhibit different levels of variation or fluctuation in their electric power demand). The construction of this model is the primary focus of the next section, following which, a chaotic analysis of it is presented. It is hoped that such as analysis will reveal levels of confidence as to when chaos would most likely become the predominant influence in the pattern of areal electric demand. 4.2 Model Construction We recall from the previous chapter two equations of central importance to our proba-bilistic model construction and analysis. 1. x n + 1 = rxn(l — xn) 0 < xn < 1, 0 < r < 4 , equation 3.4. 2. L(r) = Hmiv-^oo -p ££L 0 l n I K 1 - 2xi) I. 0 < xi < 1. 0 < r < 4 , equation 3.8. It is hoped that the model to be proposed, a probabilistic version of equation 3.4, while somewhat crude (i.e., this model does not attempt to incorporate the demand differences of various income groups, as little of practical use has come from this type of segmented analysis [22], at least in part because of a lack of knowledge of the underlying demand distributions.), will be much more realistic in its appraisal of ensuing chaos, and so be of more practical use in the forecasting of electric demand. In a sense, the above quadratic population model may be seen as having a probability of per capita electric demand of one, i.e., it reflects a situation where every member of the population always demands their maximum amount of electricity. Consequently, to determine equation 3.4 in a probabilistic sense, we would seek to determine the behavior of Chapter 4. Electricity Demand and the Effect of Population Dynamics 26 H{il,xn) =r}Xi(xn)-rxn(l - xn) = vXi(xn) • x n + 1 0 < 7 7 , x n < l , 0 < r < 4 (4.9) where the 7jXi{xn) are randomly distributed. Even though xn+1 has already been well analyzed, it remains to determine the addi-tional behavioral effect of rjXi(xn) on xn+1 in order to understand #(77, £ n ) more fully. As r/Xi(xn) must represent many economic behavioral factors simultaneously, we con-sider an underlying normal or gaussian distribution of per capita demand for each value of Xi, i = 1,... ,n. This distribution assumption is made primarily because of the gen-erally unknown state of the summed cumulative true distributions (i.e., the distributions of all probabilistic factors pertinent to electric demand) as well as the presence of an eventual large number of member representations. As has been previously stated, it is unrealistic to expect the variation in per capita demand of electric power to be equal for areas of low, medium, and high population densities. Consequently, it is necessary to construct an T)Xi(xn) that is not only randomly distributed, but also variable in spreading the distributions. Hence, r\\ will reflect the distribution variance of each randomly distributed X\\. Finally, since the chaotic nature of equation 3.4 has already been demonstrated, we are not so much concerned as to whether or not equation 4.9 will ultimately be chaotic, but rather, because of the probabilistic nature of 4.9. we are more concerned with how and when to expect the onset of chaos. 4.3 The Development of the Distribution Function First, since we are to assume a gaussian distribution as being an intrinsic part of 77^ . (a^), we expect to have, in general, for all x Chapter 4. Electricity Demand and the Effect of Population Dynamics 27 1 H V ) 2 a\\/2ir = v((xt(x)) where the corresponding distribution of £,Xi{x) is =>Xi{x) = / —\\=e J-oo ay I T Ydt where, by the substitution s = —^ , dt = ads, equation 4.11 becomes where (4.10) (4.11) (4.12) = / ~7==e *dt J — oo V £K with corresponding standard normal density function (4.13) 1 _S (4.14) The mean of equation 4.11, i.e., of a normally distributed random variable is defined to be 1 r°° 11 T—U. \\ 2 E(Xi) = — = / x-e~{*)dx CT\\/2TT J-OO (4.15) 1 f°° , N _ i ( £ ^ i ) 2 . = —y= / (x — p)e 2K- \" > dx + (TyJI'K J-oo and using the same substitution as in equation 4.12 (but replacing x for t, and letting x = as + fi), we get E(Xi) = / — / 5 • e 2 ds •+ fi = fi J — oo 2TT (4.16) Chapter 4. Electricity Demand and the Effect of Population Dynamics 28 as the integral —containing an odd function intergrand— evaluates to zero. Hence, we see that E(Xi) = p, where the particular values of p and a are yet to be determined. Now rj is to be determined as a function which yields different coefficients of variations (i.e., the relative distribution spread determined by coeff. of var. = (r/p), for various population densities. The primary justification for this is that very low and very high population densities are expected to be less subject to variability in per capita demand (a floor and ceiling effect, respectively) than will be intermediate population densities. Consequently, 77 should distribute each random variable X; in a manner appropriate to the particular expected behavior of a given density per capita electric demand. Letting Xi be the corresponding normally distributed value for each ajj , i = 1, .. . , n, we may see 77 as a variance function of each Xi, with the variational behavior being as follows: 1. For small values of (xn), i.e., for xn < 0.25, we expect 77 to yield a coefficient of variation of 10 per cent. 2. For intermediate values of (xn), we expect 77 to yield a coefficient of variation of 20 have been achieved, and hence, once again a coefficient of variation of 10 per cent. By the above behavioral description of the 77 function, we see that 77 may be described as follows per cent. 3. For large values of (a;n), i.e., xn > 0.75, we expect a per capita demand ceiling to E(X t)with £ VXiM = I E{X{) with 2 . E(Xi) with ^ 20 10 10 , .25 < xn < .75 xn < .25 xn > .75 where the Xi are normally distributed. Chapter 4. Electricity Demand and the Effect of Population Dynamics 29 Hence, our equation of interest for electric demand forecasting, when considering both population dynamics and per capita electric demand is H(r],xn) = VXi{xn)-rxn(l-xn) 0