Contents
1 One Dimensional Dynamical Systems in Discrete Time 5
1.1 One Dimensional Linear Autonomous Systems . . . 5
1.2 Qualitative and Graphical analysis of One Dimensional Nonlinear Model . . . 7
1.3 Local Bifurcations in One-Dimensional Discrete Dynamical Systems 12 1.4 The Logistic Map and the Creation of Chaos . . . 14
1.5 The Bifurcation Diagram of the Logistic Map . . . 18
1.6 Basins of Attraction in One Dimensional Dynamical Systems . . . . 19
2 Two Dimensional Discrete Dynamical Systems 25 2.1 Linear Systems . . . 26
2.2 Solutions of a Linear System . . . 28
2.3 Nonlinear Discrete Dynamical Systems in Two Dimensions . . . 31
2.4 The Hénon Map . . . 32
2.5 Stable and Unstable Manifolds . . . 35
3 Chaos in Multidimensional Maps 39 3.1 Lyapunov Exponents in the One Dimensional Case . . . 39
3.2 Asymptotically Periodic Orbits . . . 41
3.3 Chaotic Orbits . . . 42
3.4 Matrix Times Circle Equals Ellipse . . . 44
3.6 Chaotic Orbits in Rm . . . . 48
3.7 The Chaotic Orbit of The Hénon Map . . . 49
3.8 Chaotic Attractors . . . 50
3.9 Forward Limit Sets . . . 52
4 The Study of Nonlinear Time Series 55 4.1 The Concept of Chaotic Attractor as Fractal . . . 55
4.2 The Determination of Fractal Dimension: The Box Counting Di-mension . . . 56
4.3 Delay Plots from Time Series . . . 58
4.4 Delay Coordinates . . . 59
4.5 Embedology . . . 61
4.6 Lorenz System’s Time Series Analysis . . . 62
4.7 The Case of the Human Electrocardiogram . . . 75
4.8 The Analysis of the Logistic Time Series . . . 80
Abstract
The paper aims to describe and implement some mathematical techniques for the analysis of historical series generated by both empirical observations and non-linear dynamic models. In particular, after introducing some concepts related to the theory of dynamic systems, the thesis concentrates on examining the presence of chaos within the historical series through algorithms introduced in recent decades1.
Chapter 1
One Dimensional Dynamical
Systems in Discrete Time
1.1
One Dimensional Linear Autonomous
Sys-tems
Consider the system
xt+1= T (xt); t2 N
where xt represents the state of the system at a given time t. The state of the
system at time t + 1 is then obtained by the application of the map T : M > M , de…ned in the phase space M R , into itself. The evolution of the system states through time can be represented as
x1 = T (x0); ::::; xt = Tt(x0)
where Tt(x) is the result of applying T to the initial state t times; in general, we de…ne a trajectory this composition of the map into itself.
Consider now the simple linear iterate map xt+1= axt
with initial condition x(0) = x0. The map represents a transformation of the
This kind of system is de…ned Autonomous, as it doesn’t depend explicitly on the variable t but only through the e¤ect that t has on the state variable xt ;
in addition, systems of this kind, characterized by the absence of any additive coe¢ cient, are de…ned Homogeneous.
Solving the previous map by induction, we obtain the expression xt= x0at; t2 N:
which allows us to distinguish among two possible scenarios:
1. If jaj < 1 the sequence converges to the unique asymptotic equilibrium x = 0 as the term t tends to in…nity; in particular, it displays exponential convergence torwards the equilibrium if a 2 [0; 1] , while it converges to the equilibrium through oscillatory movements if a 2 ( 1; 0) ;
2. If jaj > 1 the sequence diverges from the equilibrium value, respectively, monotonically if a > 1 and through oscillations if a < 1:
Two additional cases are represented by the limiting values a = 1 and a = 1 :
1. If a = 1 the constant sequence xt= x0, 8t 2 N is obtained;
2. If a = 1we observe the oscillating sequence described by xt= ( 1)tx0.
Consider now the nonhomogeneous linear map xt+1= axt+ b
where the unique steady state is represented by the value x = 1 ab , provided that a 6= 1 . We can transform this linear non homogeneous recurrence into a linear homogeneous one by changing its starting coordinates, such that
Xt= xt x = xt
b
1 a
By replacing the last condition into the initial equation we obtain the homo-geneous system
Xt+1= aXt
whose general solution is Xt = X0at; substituting Xt into the equation, we are
…nally able to get
xt = x0 b 1 a a t + b 1 a :
Such solution converges to x = 1 ab for jaj < 1, oscillates between values ( x0)
and (x0) for a = 1 and becomes the arithmetic sequence xt+1 = xt+b for[a = 1].
1.2
Qualitative and Graphical analysis of One
Dimensional Nonlinear Model
Consider now the generic discrete time dynamical system xt+1 = f (xt)
characterized by the initial condition x(0) = x0 . The equilibrium points x
are de…ned by the …xed point condition xt+1 = xt:
and are obtained as solutions of the equation
f (x) = x
Accordingly, given the equilibrium point x , we de…ne the linear approximation of the initial system in a neighborood of x as
f (x) = f (x ) + f0(x )(x x ) + o(x x ) which leads to the equation
that, in turn, reduces to the linear homogeneous case Xt+1 = aXt after the
translation Xt = xt x , which measures the displacement from the equilibrium
point.
Proposition 1 (Stability) Let x be an equilibrium point of xt+1= f (xt) , then
1) if jf0(x )j < 1 the equilibrium point x is stable and is de…ned a Sink.
2) if jf0(x )j > 1 the equilibrium point x is unstable and is de…ned a Source.
Given an equilibrium point x , we de…ne the neighborood N"(x ) , we can
formally the neighborood of x as
N"(x ) =fx 2 R : jx x j < "; " > 0g
This means that x will be a locally asymptotically stable equilbrium if there exists one value " > 0 such that for all x in the neighborood N"(x ) we have
limt!1ft(x) = x ; viceversa, x is de…ned an unstable equilibrium if all points
su¢ ciently close to x are repelled from it .
The stability analysis of the …xed points can also be conducted graphically: starting from the location of initial value x0 on the horizontal axis, the successive
value of the recurrence x1 is obtained by moving upward to the graph and then
horizontally torwards the diagonal; once we have done this, we can obtain in a similar way all the following values assumed by x by moving from the digonal to the graph of the function and then again from the graph to the diagonal. This graphical representation of the recurrence takes the name of staircase diagram and allows to visualize the behavior of the trajectories in a neighborood of the equilibrium point.
Consider …gure (1.1)(a): as we can notice, the trajectories of the function xt+1 =pxt which start from an initial point x0 2 (0; 1] display convergence from
the left to the equilibrium point x = 1 . We obtain a similar behavior for any point x0 2 [1; +1) , with convergence displayed from the right.
0 0 . 5 1 1 . 5 2 0 0 . 5 1 1 . 5 2 (a) f (x) =pxt - 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2 - 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2 (b) f (x) = x2 t 1 (1.1) In …gure (1.1) (b) the iterated quadratic map f (x) = x2 1 is considered. In
this case, given the two equilibrium points x0 and x1 with x0 < x1 , whatever initial condition x0 < x0 < x1 is chosen, we are not able to observe any process
of convergence of the trajectories torwards one of these two values. Rather, the trajectories, start assuming the same couple of values 1 and 0 at each iteration, actually repeating the same set inde…nitely. Structures of this kind are de…ned periodic cycles.
De…nition 2 (Periodic Cycle)
Let f be a map on R, we de…ne c a periodic point of period k if fk(c) = cand k
is the smallest such positive integer; the trajectory with initial point c (consisting of k points) is then called a Periodic Cycle of period k. More formally, we de…ne periodic cycle of period k the set of points
Ck =fc1; c2; :::; ckg such that 8 < : ci 6= c1 with i = 2; :::; k f (ci) = ci+1 with i = 1; :::; k 1 f (ck) = c1
where c1 represents a …xed point of the composite function fk(x). Since the
of fk, such that fk(c
i) = ci for each i = 1; :::; k : after k iterations of f , all the
points of the cycle are observed and the initial point is reached again. It is worth notice that every periodic point ci of a k-cycle Ck is a …xed point of fk(x) but it
is not a …xed point of any fj(x)with j < k ; in addition notice that a …xed point
x of f (x) is also a …xed point of any composite function fj(x) for any j > 1, as
the condition f (x ) = x implies f2(x ) = f (f (x )) = f (x ) = x and so on.
Accordingly, the k-periodic points are all and only the …xed points of fk(x)which
are not …xed …xed ponts of fj(x) for any j < k.
Consider now the map
f (xt) = xt+1 = xt(1 xt); > 0 (1.2)
which is graphically represented by a concave parabola with …xed x0 = 0 and
x1 = 1 1; this kind of map is commonly known as logistic map. We analyze the
case described by the parameter value = 3:3 and characterized by …xed points x0 = 0 and x1 = 0:69 , which are both repellers; for almost every …xed initial
condition x0, the resulting trajectory settles into a pattern of alternating values
c1 = 0:4794 and c2 = 0:8236. In …gure (1.3) we represent the behavior of the
trajectory starting in the initial point x0 = 0:3
- 0 . 4 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 - 0 . 5 0 0 . 5 1 1 . 5 f (x) = 3:3x(t)(1 x(t)) (1.3) As we can notice, the …gure shows the typical evolution of an orbit converging to a period-2 attracting orbit fc1; c2g : it is attracted to c1 every two iterates, and
to c2 on alternate iterate. We observe:
1. f (c1) = c2 and f (c2) = c1 , or, equivalently, f2(c1) = c1 , that is, c1 is a
…xed point of the function f2 (the same holds for c 2);
2. The periodic oscillation between c1 and c2 is stable and attracts trajectories.
Finally, the evolution of some trajectories starting from diverse initial points is reported in table (1.4) t ft 1(x) f2t(x) f3t(x) 0 0:2 0:5000 0:9500 1 0:5280 0:8250 0:1568 2 0:8224 0:4764 0:4362 3 0:4820 0:8232 0:8116 4 0:8239 0:4804 0:5047 5 0:4787 0:8237 0:8249 6 0:8235 0:4792 0:4766 7 0:4796 0:8236 0:8232 8 0:8236 0:4795 0:4803 9 0:4794 0:8236 0:8237 10 0:8236 0:4794 0:4792 (1.4)
and this clearly shows that the pair fc1 = 0:4794 ; c2 = 0:8236g is an example
of a stable (invariant over time) periodic orbit.
De…nition 3 (Stability of the Periodic Cycle) Let f be a map and assume that p is a period-k point., then
1) The period-k cycle of c is a periodic sink if c is a sink for the map fk;
2) The orbit of c is a periodic source if c is a source for the map fk.
In fact, the stability of a k-cycle can be determined by the study of the stability of its periodic points ci as …xed points of fk(x), i.e. by the condition df
k
dx(ci) < 1,
where, for the chain rule for the derivation of composite function, we have dfk dx(ci) = f 0(c 1) f0(c2) ::: f0(ck) = k Y i=1 f0(ci)
Given the map f (x) = 3:3x(t)(1 x(t)) , the periodic orbit f0:4794; 0:8236g will be a sink as long as the derivative (g2)0(c1) = g0(c1)g0(c2) = (g2)0(c2)is smaller
1.3
Local Bifurcations in One-Dimensional
Dis-crete Dynamical Systems
If the …rst derivative of the function, computed into one of the equilibrium points, assumes one of the limiting values for the stability condition, +1 and 1, then even a slight modi…cation of the shape of the function f (x) can lead to changes in the stability and in the number of equilibrium points of the system. The changes induced in the model after a variaton of some parameters i 2 R are called
struc-tural modi…cations, as such changes modify the shape of the function and the properties of the trajectories. We denote as bifurcations the changes of the model’s parameters leading to qualitatively di¤erent dynamic scenarios.
Let us consider the discrete system
xt+1 = f (xt; ); 2 R
and let x ( ) be a …xed point de…ned by the equation f (x; ) = x. Given the stability condition jf0(x ( ))j < 1, as the parameter varies, the …xed point can lose its stability if one of the two limiting values of the stability conditions, f0(x ( )) = +1 and f0(x ( )) = 1 respectively, is crossed ; if this happens, the
model is said to be structurally unstable and the values of the parameter for which we observe these variations are de…ned bifurcation values.
We can now proceed to classify the di¤erent types of bifurcations that result in such qualitative changes; to this purpose, we start considering the case f0(x ( )) = +1.
1. Fold Bifurcation: This type of bifurcation is characterized by the creation of two equilibrium points, one stable and one unstable, following a change in the system parameters; at the same time, a variation of the parameters in the opposite direction causes the two equilibria to merge and then to disappear in correspondence of the same bifurcation point.
2. Transcritical Bifurcation: This kind of bifurcation is de…ned by the be-havior of two equilibrium points, one stable and one unstable, that merge when the parameter reaches the bifurcation value and then continue to exist with reversed stability properties even after the bifurcation has occurred.
3. Pitchfork Bifurcation: It is characterized by the transition from a single equilibrium point to three distinct equilibria. The original point changes its stability property as the bifurcation parameter crosses the bifurcation value, leading to the creation of two additional equilibria. If the same parameter varies in the contrary direction, then the two equilibria merge and disap-pear, and only one equilibrium point is left, changing its stability property. In addition, we can distinguish among two peculiar cases: the supercritical pitchfork bifurcation and the subcritical pitchfork bifurcation, the last one de…ned by a unique unstable equilibrium which becomes stable at the bifurcation value with the simultaneous creation of two unstable equi-librium points located at opposite sides and constitutes the upper and lower boundary of the basin of attraction of the central stable one.
Considering instead the case f0(x ( )) = 1we have
4. Flip Bifurcation: The …xed point changes its oscillatory stability into oscil-latory instability; however, while in the linear case the osciloscil-latory expansion has generally no limits, in the nonlinear one we observe the creation of a 2-periodic cycle, that may be stable if it is created around the unstable …xed point (supercritical case) or unstable if exists around the stable …xed point(subcritical case). Speci…cally, in the …rst case the periodic cycle at-tracts all the trajectories diverging from the repulsive point x , whereas in the second case, it forms a boundary for the basin of attraction of the …xed point x . As the ‡ip bifurcation in the one-dimensional case leads to the creation of two periodic points of a 2-cycle, we can observe the creation of two new …xed points in the map f2(x) = f (f (x)), in addition to the original to the original …xed point x , as any …xed point for f (x) is a …xed point for the composite function f2(x): this means that a ‡ip bifurcation of f (x) is
always associated to a pitchfork bifurcation of f2(x).
It is important to notice that all the cases of local bifurcations previously described, can also be observed in the perspective of periodic cycles. Considering a periodic cycle Ck =fc1; c2; :::; ckg and its associated multiplier
(Ck) = k Y i=1 f0(ci) = dfk dx(ci)
if, after a variation in the parameters of the system, the multiplier exits the stability range 1 < (Ck) < 1 through the value +1, then a bifurcation of the
cycle is observed, that may be of the types previously analyzed. In particular, we observe the creation or the destruction, through merging, of a couple of k-cycles,
one stable and one unstable, in the case of a fold bifurcation; the merging of two k-periodic cycles of opposite stability and subsequent switch in their stability properties, in the case of a transcritical bifurcation; the creation of two further k-cycles if a pitchfork bifurcation occurs.
1.4
The Logistic Map and the Creation of Chaos
Let us consider again the logistic model
xt+1= xt(1 xt); > 0
where the …xed points are represented by values x0 = 0 and x1 = 1 1.
Substituting the two equilibrium points into the function’s …rst derivative f0(x) =
(1 2x), we repectively obtain the two values f0(x0) = and f0(x1) = 2 and then, by considering the local stability condition jf0(x
i)j < 1, i = 0; 1, we have
jf0(x0)j < 1 () < 1 jf0(x
1)j < 1 () 1 < < 3
In …gure (1.5) the staircase diagrams for the two bifurcation parameters = 1 and = 3 are represented
- 0 . 4 - 0 . 2 0 0 . 2 0 . 4 0 . 6 - 0 . 4- 0 . 2 0 0 . 2 0 . 4 0 . 6 f (x) = (1 x) - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 f (x) = 3(1 x) (1.5)
At = 1, a transcritical bifurcation occurs and the two …xed points merge and exchange their stability properties: in fact, x1( ) < 0 and unstable for 0 < < 1, whereas x1 > 0 and stable as crosses the bifurcation value = 1 at which
x1 = x0 = 0 . For = 2 we have f0(x
1) = 0 and x1 is said to be a superstable
…xed point; for 2 < < 3, instead, we have f0(x
1) < 0, giving us oscillatory
convergence
Finally, at = 3, a ‡ip bifurcation of x1 occurs and a stable cycle of period two C2 = f ; g is created around the unstable …xed point: the two periodic
points describing the cycle, and , can be directly computed as …xed points of the following compound function
F (x) = f2(x) = f ( x(1 x)) = ( x(1 x)(1 x(1 x)): We can easily …nd the …xed points of F (x) solving the equation
fF (x) = xg =) x 2(1 x)(1 x(1 x)) 1 = 0 (1.6)
Being the points x0 = 0 and x1 = 1
…xed points for f (x), they are …xed points for the compound function F (x) as well; we can therefore factorize equation (1.6) as follows
x 1 x2+ + 1 x + + 12 = 0
in order to obtain the two …xed points of F (x) that are not …xed points of f (x), that is, the two periodic points f ; g
8 < : = +1+ p ( 3)( +1) 2 = +1 p ( 3)( +1) 2
which are both de…ned for 3and coincide with the bifurcating equilibrium for x1 at = 3; this represents the pitchfork bifurcation value for F (x) and leads to the creation of two new …xed stable points of F (x), corresponding to the periodic points of a stable cycle of period 2.
We can check the stability of the cycle computing the derivatives F0( ) = F0( ) = f0( )f0( ) . This value is slightly below one just after crosses the
bifurcation value, it is decreasing as increases beyond the value = 3 and eventually assumes value 1at = 1 +p6.
Summing up:
f0( )f0( )2 ( 1; 0) for 2 3; 1 +p6 f0( )f0( ) = 1 for = 1 +p6
In particular, the value of the parameter = 1 +p6 corresponds to a second ‡ip bifurcation , this time of F (x), at which the cycle C2 loses its stability and a
stable cycle of period four is created; as the value of the parameter is further increased, even the cycle of period four becomes unstable and a new stable cycle of period eight is created and this progresses inde…nitely hereafter.
Thus, for increasing values of we observe the creation of in…nitely many stable cycles of period 2n
, n 2 N, which become unstable as is increased. This sequence of period doubling bifurcations is de…ned period doubling cascade and occurs in a …nite range of the parameter . If we de…ne the distance between two successive bifurcation points as 4n= n+1 n, we have
lim
! 1
[4n = n+1 n] = 0
where the limit point is 1= 3:56994571869(:::)
In general, as increases, the bifurcations become more and more frequent and tend to accumulate at the limit point 1. After 1, all the cycles of period 2nhave
been created and have become unstable, periodic trajectories of any period can appear as well as aperiodic trajectories (bounded trajectories generated by an in…nite iteration of the map which never hit an already visited point ); these aperiodic trajectories are de…ned chaotic trajectories.
De…nition 4 (Chaotic Trajectories) Chaotic trajectories …ll an invariant in-terval satisfying the following properties
1) In…nitely many periodic points exist and they are dense inside the invariant set.
2) An aperiodic trajectory exists in the set and it is dense inside it. 3) Sensitivity to initial conditions: two trajectories starting from dif-ferent, although arbitrarily close, initial conditions, remain bounded, but their reciprocal distance exponentially increases and, in a …nite time, becomes as large as the state variables.
An invariant set for which the …rst two properties hold is said to be chaotic; once they are are satis…ed, the third property consequentially hold.
In particular, the …rst property tells us about the existence of dense and re-pelling periodic points inside the invariant set where chaotic dynamics occurs; the second one, instead, states the existence of a trajectory moving erratically inside the invariant set …lling it completely, while trajectories starting outside the inter-val, are not allowed to escape once have entered inside it. In addition, as the dense
trajectory is aperiodic, it will never reach an already visited point after which the countable set of the points will be repeated periodically: this clearly implies that the trajectory …lls all the set in the long run.
0 0 . 2 0 . 4 0 . 6 0 . 8 1 0 2 0 4 0 6 0 8 0 1 0 0 = 4; x0 = 0:1 0 0 . 2 0 . 4 0 . 6 0 . 8 1 0 2 0 4 0 6 0 8 0 1 0 0 = 4; x0 = 0:100001 (1.7)
Figure (1.7) clearly shows the e¤ect of the sensitivity to initial conditions prop-erties on logistic model’s time series; in particular, two distinct time series are drawn for the same value of the parameter = 4, respectively, for initial condition x0 = 0:1and for x00 = 0:00001. It is clearly shown how also in…nitesimal variations
in the starting values can cause the series to signifcantly di¤er from each other. It is important to notice that the existence of deterministic chaos can signify that at the basis of the behaviour of some time series which appear to have a quite irregu-lar and random path, there could actually be a deterministic, although nonlinear, law of motion. - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 f (x) = 3:61(1 x(t)) (1.8)
Figure (1.8) depicts the trapping interval for the parameter value = 3:61. As we can notice, the interval is characterized by both an upper bound, corresponding to the vertex of the parabula c and by a lower bound given by its image f (c) = c1.
This means that any chaotic motion de…ned inside the interval is trapped inbetween these two extremal values. The knowledge of maximum and minimum values (i.e. the foldings of the graph of the iterated function) as well as their images, show us feasible regions for asymptotic dynamics: in the example, the vertex of the parabola and its images ci = fi(c) , i = 1; :::; 3 bound a trapping region with a
hole inside and even if the dynamics is chaotic, no iterated points are allowed to enter in the space between the two intervals.
1.5
The Bifurcation Diagram of the Logistic Map
In general, we can summarize all the di¤erent dynamic scenarios generated by the model thanks to the use of the bifurcation diagram, reported in …gure (1.9) and depicted for an initial value x0 = 0:3 and de…ned over values ranged [2; 4] .
(1.9) Each vertical slice of the …gure represents the attractor of the system for a given value of the parameter . In particular, for values of smaller than 3, whenever for a particular value of the chosen initial point lies in the basin of a …xed point attractor, only one point will appear in the vertical slice corresponding to that value of the parameter. In the same way, when the initial value is in the basin of a periodic attrractor, isolated points in the attracting orbit will appear in the vertical slices; this occurs when reaches value 3, when we observe the creation of a period two attracting orbit, which later becomes a period-four attractor, then a period-eight attractor, etc., as the value of the parameter is increased. For values of higher than 3:57 we observe an entire interval of plotted point, clearly displaying the creation of chaos.
Finally, we can observe, for a certain range of the bifurcation parameter, the presence of white vertical strips in which chaos seem to disappear and the overall
dynamics are captured by an attracting periodic cycle: these strips are called periodic windows; such periodic windows are in…nitely many: quite evident is the periodic window of a 3-cycle obtained for values of around 3:85.
1.6
Basins of Attraction in One Dimensional
Dy-namical Systems
Consider the generic discrete dynamical system
x(t + 1) = T (x(t)); x2 R we can give some important de…nitions
De…nition 5 (Trapping Set) A generic set A M is trapping if x(t0) 2 A
implies x(t) = T (t; x(t0))2 A for each t > t0, or, equivalently, G (t; A) A and
T (t; A) =fx 2 M : 9t > t0 and x (t0)2 A so that x = T (t; x (t0))g
In other words, any trajectory starting inside the set cannot escape from it.
De…nition 6 (Invariant Set) A closed set A M is invariant if T (t; A) = A, i.e., each subset A0 A is not trapping.
This means that any trajectory starting inside an invariant set remains there, and all the points of the invariant set can be reached by a trajectory starting inside.
Let us now consider an invariant attracting set A R (where A is also trapping, i.e. if x 2 A then Tt(x)2 A for any t > 0 ); the basin of attraction of A is the set of all the points that generate trajectories converging to A, such that
B(A) = x Tt(x)! A as t ! +1
Let U (A) be a neighborood of A whose points converge to A : clearly U (A) B(A) ; in addition, it is implied that also the points of the phase space which are mapped inside U after a …nite number of iterations belong to B(A). Accordingly, the Total Basin of A is given by
B(A) =
1
[
n=0
T t(U (A))
where T 1(x) represents the preimages of x and T t(x) represents the set of
points that are mapped in T after t iterations of the map T . Suppose f : I ! I is a continuous and increasing function: the only possible invariant sets are rep-resented by the …xed points; when many …xed points exist, they are alternatingly stable and unstable, with the unstable ones representing the separating boundaries of the basins of the stable points.
Let us now consider the increasing function f (x) = arctan (x 1) for in-creasing values of : when < 2:5 , a unique …xed points exists and it is globally asymptotically stable. By way of example, we report the case for = 1 in …gure (1.10)(a). In addition, if = 2:5 we have the creation of a fold bifurcation and a pair of …xed point, one stable and one unstable, is created and the unstable equi-librium is the boundary separating the two basins of attractions; this situation is described in …gure (1.10)(b). - 4 - 3 - 2 - 1 0 1 2 3 4 - 4 - 3 - 2 - 1 0 1 2 3 4 (a) f (x) = arctan(x 1) - 4 - 2 0 2 4 - 4 - 2 0 2 4 (b) f (x) = 2:5 arctan (x 1) (1.10)
If f : I ! I is a continuous and decreasing map, the only possible invariant set are represented by the unique stable …xed point and the cycles of period 2,
alternatingly stable and unstable, where the latters determine the boundaries of the basins of attraction of the stable cycles.
Let us consider the map
f (x) = 1 ax3
the unique …xed point x is bounded by the periodic points fc1; c2g of an
unstable cycle of period 2 ; progressively increasing the value of the parameter a a ‡ip bifurcation occurs and the …xed point x becomes unstable.
In this case, a stable 2 cycle f 1; 2g is created around x and its basin is
bounded by the unstable cycle fc1; c2g and we observe the divergence of the
tra-jectories outside this interval ; we give a graphical representation of this stable cycle in …gure (1.11)(b). In general, in the case of one-dimensional invertible maps the only kind of attractor is represented by …xed points and cycles of period 2 : in the …rst case, the basin is an open interval which includes the …xed point, while in the second case, the basin is the union of two open intervals, each one including an attracting periodic point; moreover, if the map is invertible, then the basins of the attracting sets are always intervals that include the attractors.
- 1 0 1 2 3 4 - 1 0 1 2 3 4 (a) f (x) = 1 0:7x3 - 1 0 1 2 3 4 - 1 0 1 2 3 4 (b) f (x) = 1 0:25x3 (1.11)
This is no longer necessarily true in the case of noninvertible maps, as there may exist non connected portions of the basins that are far from the attractor; this is due to the "unfolding action" of the inverses, creating preimages of a neighborhood of the the attractor far away from this one.
The logistic map is an example of a Noninvertible Map: given the equation f (x) = (1 x), the unique image x0 is associated at each x in the domain, while
x1 = f1 1(x0) = 1 2 p ( 4x0) 2 x2 = f2 1(x0) = 1 2 + p ( 4x0) 2 Clearly, if x0 >
4, where 4 is the value assumed by the image in the vertex of
the parabola, no real preimages are obtained; accordingly, we can use the critical point c = 4 to separate the real line into two distinct subsets:
Z0 = (c; +1) , where no inverses are de…ned
Z2 = ( 1; c) , whose points havetwo rank-1 preimages
accordingly, we de…ne the logistic map as a Z0 Z2 noninvertible map. If the
image x0 2 Z2, its two rank-1 preimages are symetrically located with respect to
the point c 1 = 1=2 = f1 1( =4) = f 1
2 ( =4) ; hence c 1 is the point where two
merging preimages of c are located. Notice that, as the logistic map is di¤eren-tiable, its …rst derivative is null at c 1. Geometrically, we can also describe the
action of the noninvertible map by saying that it " Folds and Pleads" its domain, mapping distinct points in the domain into the same point of the codomain; as the map is partially increasing (for c < c 1) and partially decreasing (for c > c 1),
it is orientation preserving for x < c 1 and orientation reversing for x > c 1.
Ac-cordingly, a nonlinear map with a relative maximum or minimum, will "Fold and Stretch" any segment of points of the domain including c 1: as the point x in
the domain varies along the set of values [0; 1] , the corresponding image moves up and down and the sum of the two segments drawn by the image’s values is greater than one.
This, in particular, represents the folding and the stretching action of the map: the repeated application of the map, represents the repeated geometric application of stretching and folding actions: the compound function ft(x) represents the
tth application of these actions. To conclude, we can say that in the case of
nonlinear maps even small initial segments of the domain (i.e. sets of points in the domain which are initially very close ) will appear to be quite dispersed after many applications of stretching and folding actions: this is another way to state our third property, i.e. sensitivity dependence on initial conditions.
As far as < 4, every initial condition x0 2 [0; 1] generates bounded sequences,
converging to a unique attractor A, which may be the …xed point x1 = 1
or a more complex attractor; initial conditions outside of the interval [0; 1] generate sequences diverging to 1 . The boundary that separates the basin of attraction
B(A) of the attractor A , from the basin B(1) is formed by the unstable …xed point x0 = 0and its rank-1 preimage (di¤erent from itself), 0 1 = 1. Observe that,
of course, a …xed point is always preimage of itself, but in this case also another preimage exists because x0 2 Z2 . If < 4 then the maximum value (vertex)
c = =4 < 0 1 = 1 , where c is the critical point (maximum) that separates Z0
and Z2 : the basin’s boundary 0 1 = 1 2 Z . When = 4 we have 0 1 = 1 = c
and a contact between the critical point and the basin boundary occurs. This represents a global bifurcation, introducing radical changes in the structure of the basin. In fact, for > 4 we have 0 1 < c and the portion (0 1; c) of B (1) enters
Z2 and new images of that portion are created. Now almost every point belongs
to the basin of divergent trajectories, the only points which are left on the interval I are the points belonging to a chaotic invariant set , a subset of zero measure on which the restriction of the map is still chaotic, a chaotic repellor.
Chapter 2
Two Dimensional Discrete
Dynamical Systems
A discrete dynamical system with two state variables, respectively, x1(t) and x2(t)
, with t 2 R, assumes the form
x1(t + 1) = T1(x1(t); x2(t))
x2(t + 1) = T2(x1(t); x2(t))
(2.1)
and generates a unique trajectory in the two-dimensional phase space for each initial condition (x1(0); x2(0)); the stationary equilibrium points of the dynamical
system are the …xed points of the map T : R2
! R2 and are de…ned by the system
T1(x1; x2) = x1
T2(x1; x2) = x2 (2.2)
In order to simplify notation, we sometimes refer to (2.1) as follows
x0
1 = T1(x1; x2)
x0
2 = T2(x1; x2)
where 0 denotes the unit-time advancement operator, that is, if the right-hand side variables are productions of period t then the left-hand ones represent pro-ductions of period (t + 1).
x0i = Ti(x1; x2; :::; xm); i = 1::m
Accordingly, we can generalize some of the concepts that have been introduced in the description of the properties of one dimensional systems
De…nition 7 Let f be a map on Rm and let x be a …xed point, such that f (x ) =
x ; if there is an > 0 such that for all vectors v 2 Rm in the -neighborood
N (p), limt!1ft(v) = x , then x is de…ned a sink or attractor. Viceversa, if there is an -neighborood N (p) such that each v in N p except for x itself eventually maps outside of N (p) then x is a Source or Repeller.
In the following sections, however, we will mainly focus on the two-dimensional case, since it allows to obtain some clear geometrical intuitions from our analylitical results. First of all, we start from the case in which Ti is linear.
2.1
Linear Systems
De…nition 8 (Linear Map) A map A (v) from Rm
to Rm is linear if for
each a; b 2 R and v; w 2 Rm , A (av + bw) = aA (v) + bA (w). Equivalently, a linear map A (v) can be represented as multiplication by an m m matrix.
Linear maps de…ned on R2
are commonly exempli…ed by the simple form v 7! Av, where A is a 2 2 matrix; in our particular case
x(t + 1) = Ax(t) where A = a11 a12 a21 a22 and x(t) = x1(t) x2(t) :
so obtaining the following linear and homogeneous 2-dimensional system x1(t + 1) = a11x1(t) + a12x2(t)
In some cases the dynamics for a two dimensional map resemble one-dimensional dynamics: we recall that the scalar is an eigenvalue of the matrix A if there is a nonzero vector v such that
Av = v
and the vector v is called eigenvector of the matrix A. Also notice that if v0 is
an eigenvector with eigenvalue , we can write down a special trajectory
vt+1= Avt satisfying v1 = Av0 = v0 v2 = A v0 = Av0 = 2v0 and in general vt= tv0
Hence the map behaves like the one dimensional map xt+1 = xt
Example 9 (Distinct Real Eigenvalues)
Let v(x; y) denote a two-dimensional vector and let A(v) be the map on R2
de…ned by
A (x; y) = (ax; by) (2.4)
and since each linear map can be represented by multiplication by a matrix, we have
A (v) = Av = a 0
0 b
v1
v2
The eigenvalues of the matrix A are a and b , with associated eigenvectors (1; 0) and (0; 1) respectively and we assume a 6= b .
The result of the iteration of the map t times is
At= a
t 0
0 bt
The unit disk is mapped into an ellipse with semi-major axes of length jajt along the x-axis and jbjt along the y-axis. For example, suppose that a and b are smaller than 1 in absolute value; the ellipse shrinks toward the origin as t ! 1 , so (0; 0) is a sink for A. If jaj ; jbj > 1 , then the origin is a source. On the other hand, if jaj > 1 > jbj as t is increased, the ellipse grows in the x-direction and shrinks in the y-direction and the origin is neither a sink nor a source; …nally, if the ellipses formed by successive iterates of the map grow without bound along one direction and shrink to zero along another, we will call the origin a saddle.
We can also study the same behavior for the iteration of generic points: at this purpose, consider the matrix
A = 2 0
0 1=2
The generic point (x0; y0)maps to 2x0;12y0 and then to 4x0;14y0 and so on;
as can be noticed, the product between the x- and y- coordinates is the constant quantity x0y0 so that the orbits traverse the hyperbola xy = k = x0y0 . More
generally, for a linear map A on R2 , the origin is a saddle if and only if iteration
of the unit disk results in ellipses whose two axis lengths converge to zero and in…nity, respectively.
2.2
Solutions of a Linear System
The linear case is of great importance in the study of dynamic systems, since it gives the opportunity to obtain explicitly a solution or set of solutions. Their importance is also related to the fact that, as we will see later in more detail, the properties of linear systems allow us to have, under appropriate hypothesis, information on the behavior of nonlinear systems.
In general, the set of all the solutions is obtained by the linear combination of two linearly independent solutions: accordingly, we solve the system using what we can de…ne as a trial solution, characterised as follows
Replacing it into the original system, we get
t+1v
1 = a11 tv1+ a12 tv2 t+1v
2 = a21 tv1+ a22 tv2
and then dividing for t we get the following eigenvalue problem
(a11 )v1+ a12v2 = 0
a21v1 + (a22 )v2 = 0
that has non trivial solutions if is a solution of the characteristic
P ( ) = 2 T r(A) + Det(A) = 0 (2.5)
where T r(A) = a11+ a22 and Det(A) = a11a22 a12a21:
Let us now de…ne the measure 4 = T r(A)2 4Det(A) , we have:
1) If 4 > 0 we have two real and distinct eigenvalues and we can write the general solution in the form
xt= c1v1 t1+ c2v2 t2
where v1 and v2 are the corresponding eigenvectors and c1; c2 are real
constants, uniquely determined imposing initial conditions xi0 , i = 1; 2 .
2) If 4 = 0 then we have real and coincident eigenvalues 1 = 2 = and
the general solution has the form
xt= c1v t+ c2vt t
3) If 4 < 0 then we have two complex conjugate eigenvalues
1;2 = T r(A) 2 i p 4 2 =j j (cos i sin ) where
j j =pRe( )2 + Im( )2 =pDet(A) and = arctan Im( )
Re( ) The general real solution is obtained as
xt= t [(c1v1 c2v2) sin( t) + (c1v1+ c2v2) cos( t)]
where v = v1+ iv2 is acomplex eigenvector associated with 1 2 C.
We can now notice that in any of the cases above mentioned, the general solu-tion of the dynamical system converges dynamically to the the trivial equilibrium x = 0 if and only if j ij < 1; i = 1; 2, that is, both the eigenvalues belong to the
unit circle drescribed in the complex plane, de…ned by Re( )2+ Im( )2 < 1
We note that we are excluding the case where some eigenvalues are on the unit circle, since, even if in this case solutions are always available in closed form, the information on such solutions is not helpful in understanding the behavior of nonlinear systems.
We can summarize the necessary and su¢ cient conditions that allow us to have eigenvalues less than one in modulus as follows
Theorem 10 (Sinks and Sources) Let A(v) be a linear map on Rm, which is represented by the matrix A; then
1) The origin is a sink if all eigenvalues of A are smaller than one in absolute value;
2) The origin is a source if all eigenvalues of A are larger than one in absolute value.
In addition, we can give the following de…nition
De…nition 11 (Hyperbolic Map) A is hyperbolic if it has no eigenvalues of absolute value one. If a hyperbolic map A has at least one eigenvalue of absolute value greater than one and at least one eigenvalue of absolute value smaller than one, then the origin is called a saddle.
2.3
Nonlinear Discrete Dynamical Systems in Two
Dimensions
In order to analyse the stability properties of …xed points in the case of nonlinear maps we need …rst to introduce the concept of linearization of a nonlinear map in a neighborood of a …xed point:
De…nition 12 (Jacobian Matrix) Let f (f1; f2; :::; fm) be a map on Rm, and
let x0 2 Rm; the Jacobian matrix of f (:) at x0 , denoted by J (x0) , is the
matrix J (x) = 2 4 @f1 @x1 (x0) ::: @f1 @xm(x0) ::: ::: ::: @fm @x1 (x0) ::: @fm @xm(x0) 3 5 (2.6)
Given a vector x0 and a small vector h, the increment in f due to h is
approximated by the Jacobian matrix times the vector h : f (x0+ h) f (x0) = J (x0) h
where the error in the approximation is proportional to jhj2 for a small h . If
we assume that f (x0) = x0 , then for a small change h the map moves x0 + h
approximately J (x0) haway from x0 ; as long as this deviation remains small (so
that jhj2 is negligible and our approximation is valid), the action of the map near x0 is essentially the same as the linear map h ! Ah, where A = J (x) , with …xed
point h = 0 .
Accordingly, given the map f : R2
! R2 we can de…ne the two dimensional
Jacobian matrix J (:) as follows
J (x) = @f1=@x1 @f1=@x2 @f2=@x1 @f2=@x2
and given the …xed point (x1; x2) , the linear approximation of the system around itis given by the equation
The necessary and su¢ cient conditions for the asymptotic stability of 2-dimensional linear systems, can be extended also to the case of 2-2-dimensional nonlinear systems to determine local stability. In particular, these conditions result to be su¢ cient (but not necessary) for the study of the local stability of an equilibrium point of the nonlinear system; in this way we can compute eigenvalues by using the Jacobian Matrix evaluated at the …xed point J (x ) as it was the co-e¢ cient matrix of the system. This means that if all the eigenvalues are less than one in absolute value (i.e. they lie inside the unit circle), then the …xed point is locally stable; on the contrary, when at least one of the eigenvalues is in absolute value greater than one, then the …xed point is unstable.
Theorem 13 Let f be a map on Rm and assume f (x ) = x
1) If the magnitude of each eigenvalue of J (x ) is less than one, then x is a sink.
2) If the magnitude of each eigenvalue of J (x ) is greater than one, then x is a source.
Moreover, like in the case of linear maps of Rm , for m > 1 , nonlinear maps
can have directions in which orbits converge to the …xed points and others along which the orbits are repelled.
De…nition 14 Let f be a map on Rm , m 1 and assume f (x ) = x , then the …xed point x is called hyperbolic if none of the eigenvalues of J (x ) has magnitude one . If x is hyperbolic, at least one eigenvalue of J (x ) has magnitude greater than 1 and at least one eigenvalue has magnitude less than 1, then x is called a saddle. (In order to express the same result for a periodic point of period t , we simply need to replace f by ft).
2.4
The Hénon Map
Let us consider the two dimensional map
Ha;b(x; y) = (a x2+ by; x) (2.7)
This kind of map is also commonly known as Hénon Map; in particular, it can be rewritten in terms of the two dimensional system
xt+1 = a x2t + byt
yt+1= xt
where a and b are the system parameters. It should be observed that if b = 0 we are back to the case of the 1-D quadratic map xt+1 = 1 axt ; this clearly
shows that the Hénon map is a 2-D generalization of the 1-D quadratic map. The steady states of the map must satisfy the equations
a x2+ by = x
x = y
Substituting the …rst equation into the second one, we obtain the condition x2+ (1 b)x a = 0
Finally, by using the quadratic formula we are able to …nd the solutions of the equation, respectively
x1;2 =
(b 1) p(1 b)2+ 4a
2 :
Accordingly, the steady states are:
(x1; y1) = (b 1) +p(1 b)2+ 4a 2 ; x1 ! (x2; y2) = (b 1) p(1 b)2+ 4a 2 ; x2 !
implying that two distinct …xed points exist as long as 4a > (1 b)2 . In addition, we can …nd the period-two points of the map by imposing the condition (x; y) = f2(x; y) ; we obtain
x = a (a x2+ by)2+ bx
y = (a x2+ by)
Solving the second equation for y and substituting the result into the …rst, we get an expression for the x-coordinate of a period-two point:
0 = x2 a 2+ (1 b)3 (1 b)2a (2.8)
= x2 (1 b) x a + (1 b)2 x2+ (1 b) x a
The zeros of the previous equation correspond to the …xed points of f ; as we know these points also represent …xed points for f2 . In order to investigate
the stability of these points, we have to consider the Jacobian matrix and its eigenvalues at the steady states.
The Jacobian matrix of the Hénon map is given by J Ha;b(x; y) =
2x b
1 0
and its characteristic polynomial is
2+ 2 x
i b = 0
so that, the eigenvalues of the Jacobian matrix are
i = xi
p
(xi)2+ b
Let us for instance set values a = 0 and b = 0:4; in this case f has the two …xed points (0; 0) and ( 0:6; 0:6). The Jacobian matrix is modi…ed as follows
J = 2x b
1 0
The eigenvalues for the matrix J (0; 0) are 1;2 =
p
0:4 ; thus (0; 0) is a sink. The eigenvalues for the matrix J ( 0:6; 0:6) are 1 = 1:472and 2 = 0:272
; thus ( 0:6; 0:6) is a saddle.
Consider another numerical example involving the Hénon Map and pose a = 0:43, b = 0:4 : in this case, the map generates the period-two orbit f(0:7; 0:1); (0:1; 0:7)g . In order to check the stability of this orbit, we need to compute the Jacobian matrix of f2 evaluated at the point (0:7; 0:1). Posing J (x) = Df (x) , we have
Df2(x) = Df (f (x)) Df (x) and then
Df2((0:7; 0:1)) = Df ((0:1; 0:7)) Df ((0:7; 0:1)) =
= 0:12 0:08
1:4 0:4
The eigenvalues of this Jacobian matrix are approximately 0:26 0:30i, which are complex numbers of magnitude 0:4 and so the period-two orbit is a sink; also nNotice that the same eigenvalues can be obtained by evaluating
Df2((0:1; 0:7)) = Df ((0:7; 0:1)) Df ( (0:1; 0:7))
implying that stability is a property of the periodic orbit as a whole, not of the individual points of the orbit. This is true because the eigenvalues of a product AB of two matrices are identical to the eigenvalues of BA.
(2.9) Figure (2.9) shows a bifurcation diagram for the Hénon map for the case b = 0:4. For each …xed value 0 a 1:25 along the horizontal axis, the x-coordinates of the attracting set are plotted vertically; at a = 0:27 we observe the occurrence of a period-doubling bifurcation as the …xed point loses stability and a period-two orbit is born. The period two orbit is a sink until a 0:85, when we observe its period doubling. For a > 0:85, the attractors of the Hénon map become more complicated: as soon as the period-two orbit becomes unstable, it is immediately replaced with an attracting period-four orbit, then a period-eight orbit and so on, inde…nitely.
2.5
Stable and Unstable Manifolds
A saddle …xed point, despite being unstable, is characterized by a set of of initial values which converge torwards itself: this set of values is de…ned stable manifold of the saddle. Consider the linear map
characterised by a saddle …xed point in the origin, and its associated matrix form f (v) = Av = 2 0 0 1=2 x y
The eigenvectors computed for the map are represented by v1 =
1
0 ;
associated to 1 = 2 and v2 =
0
1 associated to 2 = 1=2.
It is possible to prove that the stable manifold of the origin is represented by eigenspace generated by the eigenvector v2 and associated to the stretching
eigenvalue 2 = 1=2 ; conversely, we de…ne unstable manifold the direction
given by v1 and associated with the shrinking eigenvalue 1 = 2. We will observe
convergence to the origin for all the trajectories generated by points located along the vertical axis and divergence for all the others .
De…nition 15 Let f be a smooth one to one map on R2 and let x be a saddle
…xed point or periodic saddle point for f .
The stable manifold of x , denoted by S (x ), is the set of points v such that ft(v) ft(x ) ! 0 as t ! 1:
The unstable manifold of x , denoted by U (x ) , is the set of points u such that f t(v) f t(x ) ! 0 as t ! 1
.
Example 16 (Linear Maps) Consider the linear map
f (x; y) = 2x + 5
2y; 5x + 11
2 y
and the associated system
xt+1= 2xt+52yt
The map has a saddle …xed point at O = (0; 0) with eigenvalues 1 = 1=2
and 2 = 3 . The corresponding eigenvectors are v1 =
1 1 and v2 = 1 2 , respectively. In particular:
1. The line y = x represents the stable manifold of the system and all the points lying on it follow the dynamics v ! 12 v on each iteration of the map.
2. The line y = 2x ( the direction of the eigenvector (1; 2)) represents the unstable manifold of the map and its points follow v ! 3v under the map.
Example 17 (Nonlinear Maps) Consider the invertible nonlinear map
f (x; y) = x=2; 2y 7x2 with associated system
xt+1 = x2t
yt+1 = 2yt 7x2t
characterized by a …xed point at O = (0; 0) . We evaluate the Jacobian matrix in the …xed point and we get
J (0; 0) = 1=2 0
0 2
We derive that the origin is a saddle …xed point and the two eigenvectors v1 =
1=2
0 and v2 =
0
2 lie on the coordinate axes. Accordingly, we can state:
1. With respect to linear maps, stable and unstable manifolds coincide with the eigenvector directions.
2. In the case of nonlinear maps, the stable manifold is tangent to the shrinking eigenvector direction while unstable manifold is tangent to the one of the stretching eigenvector.
Since f (0; y) = (0; 2y) , the y axis can be seen as part of the unstable man-ifold; in particular, it coincides with the entire unstable manifold. The stable manifold of O, on the contrary, is described by the parabola y = 4x2, or ,
S(O) =f(x; 4x2) : x
2 Rg. In general, when a map is linear, the stable and unsta-ble manifolds of a saddle are always linear subspaces, this is not necessarily true in the nonlinear case.
Chapter 3
Chaos in Multidimensional Maps
3.1
Lyapunov Exponents in the One Dimensional
Case
We know that for …xed points of discrete dynamical systems, stability can be determined through the study of the …rst derivative of the map: if x is a …xed point of a one dimensional map and f0(x ) = a > 1 , then the trajectory of any
point x near to x will separate from x at a multiplicative rate of approximately a per iteration, until the orbit of x moves signi…cantly far away from x . This means that the distance between fk(x) and fk(x
1) = x1 will be magni…ed by
approximately a > 1 for each iteration of f . For a periodic point of period k, we have to look at the derivative of the kth iterate of the map, which, by the chain
rule, is the product of the derivatives at the k points of the orbit. Suppose this product of derivatives is A > 1 , then the orbit of each neighbor x of the periodic point x1 separates from x1 at a rate of approximately A after each k iterates: this
means that it takes k iterations to separate by a distance A , as A1=k can be seen
as the average rate of separation per iterate. The concept of Lyapunov number is introduced in order to quantify this average multiplicative rate of separation of points x very close to x1 (the Lyapunov exponent will be simply the natural
logarithm of the Lyapunov number), for example, a Lyapunov number equal to 2 (or equivalently, a Lyapunov exponent equal to ln 2) for the orbit of x1 implies
that the distance between the orbit of x1 and the orbit of a nearby point x doubles
each iteration, on average. Similarly, for a periodic point x1of period k this is
equivalent to say that
In the same way, the concept of Lyapunov number can be adopted for the analysis of nonperiodic orbits, and in particular, for the analysis of chaotic orbits; as we will specify later, a chaotic orbit is characterised by a Lyapunov number greater than one as the orbit does not tend torward periodicity.
Let us consider the discrete time, one dimensional dynamical system xt+1 =
f (xt) and de…ne nominal trajectory the trajectory fx0; x1; x2:::g . We also
de…ne perturbed trajectory the trajectory fex0;ex2; :::g starting from the point
ex0 = x0+ @x0 near to x0. We have, from the Taylor series expansion
ex1 x1 = f (ex0) f (x0) = f0(x0) (ex0 x0) + :::
where jf0(x
0)j is the expansion/contraction rate of initial di¤erence @x0
be-tween the two trajectories. After t iterates we shall obtain
ext xt= ft(ex0) ft(x0) = @ft @x x 0 (ex0 x0) + ::: = =ff0(xt 1) f0(xt 2) :::f0(x0)g (ex0 x0) + :::
Then, the Average Rate of Separation of trajectories in the long run (a long run geometric mean) will be
L (x0) = lim t!1jf
0(x
t 1) f0(xt 2) :::f0(x0)j 1=t
and is referred to as Lyapunov Number. Then, h (x0) = ln L (x0) is the
Lya-punov Exponent of the trajectory starting from x0; in fact, the Lyapunov Exponent
is given by h(x0) = lim t!1 lnjf0(x t 1)j + ln jf0(xt 2)j + ::: + ln jf0(x0)j t = limt!1 1 t t 1 X k=0 lnjf0(xk)j
It follows from the de…nition that the Lyapunov number of a …xed point x1 for
a one-dimensional map f is jf0(x1)j , or equivalently, the Lyapunov exponent of
the orbit is h = ln jf0(x
1)j. If x1 is a periodic point of period k, then it follows
that the Lyapunov exponent is h (x1) =
lnjf0(x
1)j + ::: + ln jf0(xk)j
Finally, if @x0 is in…nitesimal, for t ! 1 we will have j@xtj ! (L (x0)) t j@x0j or j@xtj ! etj@x0j
and its unity of measure is the inverse time. Summing up our …ndings, we have
1) If h (x0) > 0, starting from x0, near trajectories tend to move away, on average.
2) If h (x0) < 0, starting from x0, near trajectories tend to get nearer, on average
3.2
Asymptotically Periodic Orbits
We give now a de…nition for asymptotically periodic orbits
De…nition 18 (Asymptotically Periodic Orbit) Let f be a smooth map. An orbit fx1; x2; :::xt; :::g is called asymptotically periodic if it converges to a
peri-odic orbit as t ! 1 ; this means that there exists a periperi-odic orbit fy1; y2; :::; yk; y1; y2; :::g
such that
lim
t!1jxt ytj = 0
From the previous de…nition, it is therefore easy to deduce that any orbit which happens to be attracted to a sink is asymptotically periodic.
Theorem 19 (Asymptotic Periodicity and Lyapunov Exponents) Let f be a map of the real line R . If the orbit fx1; x2; :::g satis…es the condition f0(xi)6= 0
for all i and is asymptotically periodic to the periodic orbit fy1; y2; :::g, then the
3.3
Chaotic Orbits
A chaotic orbit is a bounded, non-periodic orbit that displays sensitive dependence with respect to initial conditions. In this sense, chaotic orbits separate exponen-tially fast from their neighbors after some iterations and identi…ed by positive Lyapunov exponents. More formally:
De…nition 20 (Chaotic Orbit) Let f be a map of the real line R and let fx1; x2; :::g
be a bounded orbit of f . The orbit is chaotic if
1) fx1; x2; :::g is neither periodic nor asymptotically periodic;
2) The Lyapunov Exponent h(x1) is greater than zero.
Example 21 (Logistic Map)
We report the graph of the maximum Lyapunov exponent of the logistic map in …gure (3.1)
(3.1)
The maximum Lyapunov exponent is represented for an entire set of values of the system parameter, in our case ranging from = 3 to = 4 , as depicted on the horizontal axis. The maximum Lyapunov appears to be persistently positive after value = 3:57, in such a way con…rming that all the trajectories observed for values of the parameter above the threshold are chaotic.
Let us consider the one dimensional map
T (x) = ax
a(1 x)
if x 1=2
if x 1=2 ; a > 0
where a is the parameter characterizing the slope of the map for di¤erent values of the independent variable. The map is clearly continuous but non smooth, as it displays a corner at x = 1=2.
In the case 0 < a < 1, the tent map has a single attracting …xed point in 0 to which all initial conditions. For a = 1, the complement of the unit interval I = [0; 1] maps to the complement of I : a point x that is mapped outside I, will not return to the same set even after a great number of iterations . In addition, when 1 a 2, the points of I stay within I . Finally, for parameter values a > 2 most of the points I leave the interval after a certain number of iterations, never to return. - 0 . 5 0 0 . 5 1 1 . 5 - 0 . 5 0 0 . 5 1 1 . 5 T2(x) ; a = 2 (3.2) Consider the case of the slope-2 tent map (a = 2)
T2(x) =
2x
2(1 x)
if x 1=2
if x 1=2
shown in …gure (3.2). It is easy to demonstrate that T2(x), which maps the unit
interval I = [0; 1] onto itself , is characterised by positive Lyapunov Exponents: since the absolute value of the slope of T2(x) is 2 whenever T2(x) exists (for
x 6= 1=2), the Lyapunov Exponent of any orbit of T2(x) is equal to ln 2 . This
means that any orbit that avoids x = 1=2 and is not asymptotically periodic, is indeed a chaotic orbit. In particular, there are in…nitely many chaotic orbits.
(3.3) Finally,fFigure (3.3) represents the value of the maximum Lyapunov exponent of the map for increasing values of the system parameter a
3.4
Matrix Times Circle Equals Ellipse
We have previously shown that, in the proximity of a …xed point v0 , the dynamics
of the system essentially reduces to a single linear map A = Df (v0) and the
magnitude of the eigenvalues of A is decisive for classifying suc point. The same holds true for a period-k orbit, with the di¤erence that the matrix A which de…nes the dynamics is a product of k other matrices. However, if the studied orbit is non periodic there is no such matrix A and the local dynamics around the orbit is determined by an in…nite product of usually nonrepeating Df (v0). In this case,
the role of the eigenvalues of A is assumed by Lyapunov numbers, which, as we know, measure contraction and expansion rates along trajectories. We can have an example of this by calculating the image of a disk from the matrix representing a linear map. In this sense, we can consider the disk of radius one centered at the origin and a square matrix: the resulting image will be an ellipse. In particular, the image of the unit disk N under the linear map A is determined by the eigenvectors and eigenvalues of AAT, where AT denotes the transpose matrix of A . For a well
known theorem, the eigenvalues of AAT are nonnegative for any matrix A.
Theorem 23 (The Ellipse from a Unit Disk) Let N be the unit disk in Rm,
and let A be an m m matrix. Let s2
1; :::; s2m be the eigenvalues and u1; :::; um
the unit eigenvectors of the m m matrix AAT. Then: 1) u1; :::; um are mutually orthogonal unit vectors
Example 24 (Symmetric Matrix) Consider again the linear map
A (x; y) = (ax; by) and the associated matrix form
Ax = a 0
0 b
x y
The eigenvalues of the matrix are s1 = a and s2 = b , while v1 and v2 are
the x and y unit vectors, respectively . Accordingly, a and b are the lengths of the axes of the ellipse AN . For the t-th iterate of A, represented by the matrix At, we …nd ellipse axes of length at and bt for AtN, the t-th image of the unit disk. Accordingly, each axis is an eigenvector not only of AAtbut also of A, whose
length is the corresponding eigenvalue of A.
This particular kind of map, however, due to the symmetricity of its matrix A, represents a very peculiar case. Infact, in the case of nonsymmetric matrices the eigenvectors of A do not generally coincide with the directions along which the ellipse lies and we need to use theorem .
Example 25 (Nonsymmetric Matrix) Consider the map
A(x; y) = (0:8x + 0:5y; 1:3y) and its associated matrix form
Ax = 0:8 0:5
0 1:3
x y
The eigenvalues of the matrix A are 1 = 0:8 and 2 = 1:3, with eigenvectors
1
0 and
1
1 ; respectively . The …xed point at the origin is a saddle point and the two eigenvectors give us the directions along which the …xed point attracts and repels. Speci…cally, the attracting direction corresponds to
At 1 0 = (0:8) t 1 0 = (0:8)t 0 while the repelling direction corresponds to
At 1 1 = (1:3) t 1 1 = (1:3)t (1:3)t
The stable manifold of the origin saddle point is given by y = 0, while the unstable manifold is represented by the line y = x : points along the x-axis move toward the origin under iteration by A, while points along the line y = x move toward in…nity. As we know, the t-th iterate of the unit circle is an ellipse with one growing direction and one shrinking direction, as in the limit the ellipse becomes longer and thinner. Acoordingly, the ellipses AtN representing higher iterates of
the unit disk gradually line up along the dominant eigenvector 1
1 of A . The
…rst few images of the unit disk under the map A can be found using the theorem; as an example, we consider the …rst iterate of the unit disk under A . Given the matrix multiplication AAt= 0:8 0:5 0 1:3 0:8 0 0:5 1:3 = 0:89 0:65 0:65 1:69
the unit eigenvectors of AAt are approximately 0:873
0:488 and
0:488
0:873 ,
with eigenvalues 0:527 and 2:053:respectively. Taking square roots, we are able to show that the ellipse AN has principal
axes of lengths 0:527 0:726 and 2:053 1:433.
3.5
Lyapunov Exponents in
R
mWhen studying a map on Rm, each orbit has m Lyapunov numbers representing
the m average multiplicative expansion rates of the m orthogonal axes or, equiv-alently, giving us the rates of separation from the current orbit point along m orthogonal directions. In particular
1. The …rst of the directions we are going to consider, will be the one along which the separation between nearby points is greater or at least, less contracting
2. The second direction, among all the directions perpendicular to the …rst, will be that for which we observe the greatest separation.
3. The third direction will be that perpendicular to the …rst two having the largest stretching of all directions. And so on.
The Lyapunov Exponents of the orbit represent the stretching factors in each of these directions: they quantify the amount of the stretching and shrinking of the image due to the map’s iterations near the orbit, starting from a point v0.
Consider for instance a sphere of small radius centered on the …rst point v0of the
orbit. When we examine the image f (S) of the small sphere under one iteration of the map, we see it approaching an ellipsoidal shape, with long axes appearing along the expanding directions for f and short axes along contracting directions. After t iterates of the map f , the small sphere will have thus evolved into a longer and thinner ellipsoidal object. The Lyapunov numbers represent the per-iterate changes of the axes of this image ellipsoid : they quantify the amount of stretching and shrinking due to the dynamics near the orbit beginning at v0. For
the formal de…nition, replace the small sphere about v0 and the map f by the
unit sphere N and the …rst derivative matrix Df (v0), which allows us to study
the in…nitesimal behavior of the orbit around v0. As we have seen, for whatever
matrix A, the image AN is necessarily an ellipsoid. Accordingly, let Jt= Dft(v0)
denote the …rst derivative matrix of the t-th iterate of f , then JtN will be an
ellipsoid with m orthogonal axes. In particular, the axes of the resulting ellipsoid will be longer than 1 in expanding directions of ft(v
0) and shorter than 1 in the
contracting directions. Again, the m average multiplicative expansion rates of the m orthogonal axes will be the Lyapunov numbers.
De…nition 26 Let f be a smooth map on Rm, let Jt = Dft(v0) and for k =
1; :::; m let rt
k be the length of the k-th longest orthogonal axis of the ellipsoid
JtN for an orbit with initial point v0 . Then rtk measures the contraction or the
expansion near the orbit of v0 during the …rst t iterations. The kth Lyapunov
Number of v0 is de…ned by Lk = lim t!1 r t k 1=t
If the previous limit exists, we are also able to de…ne the kth Lyapunov Exponent of v0 by taking hk = ln Lk:
Finally, it is important to notice that we have built into de…nition the property that L1 L2 :::: Lm and h1 h2 :::: hm .
3.6
Chaotic Orbits in
R
mWe are …nally able to extend the concept of chaotic orbits to the case of higher dimensional maps.
De…nition 27 (Chaotic Orbit in Rm)
Let f be a map of Rm, where m 1 ,
and let fv0; v1; v2;:::g be a bounded orbit of f . Then, the orbit is chaotic if
1) It is not asymptotically periodic 2) No Lyapunov Exponent is exactly one 3) L1(v0) > 1 or, equivalently, h1(v0) > 0
If N is the unit sphere in Rm and A is an m m matrix, then the orthogonal
axes of the ellipsoid AN can be computed according to theorem (23). The lengths of the axes are the square roots of the m eigenvalues of the matrix AAt, and the axis directions are given by the m corresponding orthonormal eigenvectors.
Example 28 (Skinny Baker Map) Consider the map
B (x1; x2) = 1 3x1; 2x2 if 0 x2 1 2 1 3x1+ 2 3; 2x2 1 if 1 2 < x2 1
commonly known as Skinny Baker map on R2. In particular, it represents a
map that, if de…ned upon the unit square [0; 1] [0; 1]in the plane and exhibits the properties of stretching in one direction and shrinking in the other that are typical for chaotic two-dimensional maps. Notice that the map is discontinuous, as points (x; y)where x2is less than 1=2 are mapped to the left side of the unit square, while
points with x2 greater than 1=2 are mapped to the right. In this sense, there are
pairs of points on either side of the line y = 1=2 that are arbitrarily close but are mapped at least 1=3 unit apart.
We can write the map in matrix form as follows
B (x1; x2) = 8 > > < > > : 1 3 0 0 2 x1 x2 if 0 x2 12 1 3 0 0 2 x1 x2 + 2 3 1 if 1 2 < x2 1
The Jacobian matrix is constant for each point in the unit square for which it is de…ned; in particular we have
J B (v) = 1=3 0
0 2
for all v except along the discontinuity line x2 = 1=2. After one iteration of
the map, a generic circle of radius r and centered at a unit square is transformed into an ellipse with axes of lenght 13r in the horizontal direction and 2r in the vertical; after t iterates, the ellipse 13 tr by 2tr , provided that the ellipse never maps across the line x2 = 12. We conclude that the Lyapunov numbers of B are
1
3 and 2 or, equivalently, the Lyapunov Exponents are ln 3 and ln 2 for every
orbit. Since ln 2 > 0 , every orbit that is not asymptotically periodic is chaotic.
3.7
The Chaotic Orbit of The Hénon Map
We report in …gure (3.4) the shape of the Hénon Map orbit, plotted for parameters values a = 1:4 and b = 0:3 , obtained by iterating a certain set of initial values. As can be noticed, after many iterates the orbit represented into the phase space ends up repeatedly assuming the same set of values, appearing to be bounded within a …nite set of values, neither displaying a periodic nor an asypmtotically periodic behavior.
(3.4)
In addition, in …gure (3.5) we plot the maximum Lyapunov exponent of the system as a function of the system parameter a . In particular, the maximum Lyapunov exponent appear to be persistently positive for parameter values a > 1 , thus implying the presence of chaos.
