Appendix A One-Dimensional Maps A.I Introduction The objective of this Appendix is to provide an introduction to the theory of chaos. One-dimensional maps are used as a vehicle to convey the concepts required for study­ ing chaos, firstly because fewer concepts are involved compared to higher dimensional maps and differential equations and secondly because of the fact that one-dimensional maps and their evolutions may easily be visualized, which helps to grasp ideas in­ volved more rapidly. For generalization of these result s to, and additional result s on higher dimensional mappings and differential equations, the reader is referred to the bibliography. Some important theorems are stated in Appendix B without proof. Consider an arbitrary one dimensional, N-th order discrete dynamical system whose time evolution is described by the homogeneous recurrence relation X n+l = f( X n, Xn- l, .•. , X n - N + l ), where f is some function N f: R -t R, and X n is the dynamical system's evolution on the n-th discrete time inst ant , i.e. n = 0,1,2, .... In general f may be nonlinear and is usually at least piecewise differentiable. The function f which describ es some dynamical system is termed a mapping or simply a map. Example A.I Consider the logistic map described by Xn+l = f(x n ), (A.l) where f : I -t R, f(x) := Tx(l - x ), (A.2) 343 344 APPENDIX A. ONE-DIMENSIONAL MAPS 0.9 0.8 0.7 0.6 .-... ~ '-" 0.5 '+-.., 0.4 0.3 0.2 0.1 o x Figure A.I: The logistic map for r = 4 . with 1:= {x I 0 ~ x ~ I} , and r E R+ a so-called bifurcation param eter of the system. From (A.2) we observe that f is of degree 2 in x and hence nonlinear in x. The map given by (A.2) is illustrated in Figure A.I. It is obvious that the logistic map is one-dimensional and of order one. • In th is chapter we consider only one-dimensional, first order map s, that is, maps of the form (A.3) where generally f : R --t R. A.2 Orbits and Their Properties Definition A.2 Consider the map f : X --t X,X c R defined by n E N o (AA ) The set of points A .2. ORBITS AND THEIR PROPERTIES 345 obtained from (A.4) is defined as the forward orbit of the point xo. If f is a homeo­ morphism, we may also define the backward orbit of Xo as the set of points and the full orbit of Xo as the set O(xo) := O-(xo) U O+(xo) = {Xn In E Z} , obtained from (A.4) . • At this point it is worthwhile to mention that another popular notation for represent­ ing the n-th iterate of the point Xo under the map f is by application of the inductive definition f(nl(xo) := f(J(n-ll(xo)), n E N, where f1l(xo) = f(xo). In the case of a causal dynamical system described by (A.4), the forward orbit of the system is its response to the initial condition xo. In general an orbit of a dynamical system is the time evolution of the system coinciding with some predetermined value Xo on iteration instance n = O. Since we classify a dynamical system based on its response to initial conditions (and inputs) it is therefore important to understand all orbits of a map . A particular type of orbit called a periodic orbit plays an important role in the study of dynamical systems. Definition A.3 The point x· is periodic of prime period-n if but f(ml(x') i= x· for I ~ m < n. The set of all iterates of a periodic point form a periodic orbit [50] . We denote the set of all points of prime period-n of the map f by Pern(J). • Definition AA A fixed point (or equilibrium point) x· of the map f is a point of period-l, that is, f(x') = z", The set of all fixed points of the map f will be denoted by Fix(J), i.e. Fix(J) := Perl(J). • 346 APPENDIX A. ONE-DIMENSIONAL MAPS 0.9 0.8 0.7 0.6 "'"""~ '-' 0.5 "+-., 0.4 0.3 0.2 0.1 0 0 0.4 0.6 0.8 x Figure A.2: Fixed points of the logistic map for r = 4. To find the set of fixed points of a map f graphically we may overlay the curve Xn+l = X n onto Xn+l = f(xn ) and look for all points of intersection. Example A.5 Calculate the fixed points of the logistic map, f : I -+ I, f(x) = rx(l - x), From Figure A.2 we see that the logistic map has two fixed points as indicated by the dots. The set of fixed points of f can be shown to be Fix(J) = {O, (r - l)/r}.• Example A.6 Find the fixed points of the map f : I -+ I defined by (refer to Section 6.3) f(x) = 1 - (ax +f3)(mod 1), a,{3 E R+ . (A.5) The graph of f is shown in Figure A.3. The fixed points are indicated by the dots in Figure A.3. To find the fixed points, consider x· = f(x') ~ x· = 1 - (ax' +{3)(mod 1) ~ x• = 1 - (ax' + {3) + n, for some n E Z (see (A.6)) n+1-{3 ~ x· , for some n E Z = l+a A.2. ORBITS AND THEIR PROPERTIES 347 0.9 0.8 0.7 0.6 ,-..., .......,l-: 0.5 "-, 0.4 0.3 0.2 0.1 00 0.2 0.4 0.6 0.8 X Figure A.3: The graph of f with the fixed points indicated by dots. where we have used x = y(mod z) ¢:::::> x - y = kz, for some k E Z. (A.6) Thus the set of fixed points of f is Fix(f) = {x E I Ix= n~ ~ : jJ, nEZ} = { n;~: jJ I nEZ, ff3-ll ::;n ::; Lf3+ aJ}, (A.7) where fYl (respectively LyJ) denotes the least (respectively greatest) integer greater (respectively less) than or equal to the real number y . • Example A.7 Consider the logistic map. Then 5- VS 5+VS} { 8 ' 8 is a period-2 orbit of the logistic map. • Example A.8 The map f : I -+ I given by 2x , if x E [0 ,1/2) f(x) = (A.8) { . 2(1-x), if x E [1/2,1] 348 APPENDIX A. ONE-DIMENSIONAL MAPS is called the tent map. It can easily be verified that is a period-S orbit of the tent map. • In principle the set of points of period-n, Pern(J) of the map j : X ~ X, X C R can be determined analytically by solving the expression for x*. However, because j is usually nonlinear this is sometimes only possible for small values of n (say n = 1,2), while for greater values of n this is generally not pos­ sible. In the latter case the periodic points can only be determined either numerically or graphically. To determine the set Pern(J) graphically, we proceed as follows . First the set of all positive integer divisors for n, say V = {I, nl, n2, ... , np } is found. Then the graphs of the corresponding iterates of j are plotted, that is, the graphs of the set of functions p {j, j(nll .r»,... ,j(n ) }, overlayed onto one another. Next we overlay the graph of y = x onto the aforemen­ tioned graphs. All points of intersection of the above set of iterates of j and the the diagonal (i.e. the graph of y = x) are then removed and then the graph of f n ) is over­ laid onto the resulting graph . The intersections between j(n) and the disconnected segments of the diagonal are all periodic points of period-no Mathematically this can be written as Pern(J) ={x E X Ix == j(n)(x)} '-U {x E X Ix = j(k)(x)}. kEV Example A.9 Find the points of period-3 for the logistic map with r = 3.839. The graphs of j and j(3) are shown in Figure AA. It can be verified that [217] Per3(J) = {al,a2,a3,bl,b2,b3} where al = 0.149888, bl = 0.169040, a2 0.489172, b2 0.539247, a3 = 0.959299, b3 0.953837, up to six decimals. A.2. ORBITS AND THEIR PROPERTIES 349 0.8 ~ ';::::::" 0.6 0 'i-, --'l-( '-' 0.4 'i-, 0.2 o -- . '~------------------------- ----- - 0', "/ o 0.2 0.4 0.6 0.8 x Figure A.4: The graphs of f (dashed line) and f(3) (solid line). Note that the cardinality of Per3(J) is 6. Since by definition a period-3 orbit contains exactly 3 points, this implies that f possesses two independent period-3 orbits . It may easily be verified numerically that the two period-3 orbits are {at, az, a3,at, . .} and {bt, bz , bs , bt, .. .}. • Example A.I0 For the map given by (A.5), determine Perz(J). The two cases a -1= 1 and a = 1 will be considered separately. The case a -1= 1: We have f(Z)(x') = 1 - {af(x') + ,8}(mod 1) = 1- {a[l - (ax' + ,8)(mod 1)] +,8} (mod 1) = 1 - {a +,8 - (azx' + a,8)(mod a)} (mod 1) (A.9) 1 - {a + ,8- aZx' - a,8 - na} + m, for some n, m E Z = aZx' + (a -1)(,8 -1) + an + m, for some n,m E Z, (A.10) where (A.6) was used together with the identity k(y(mod z)) == ky(mod kz).