lecture 25 nonlinear dynamics in r (cont’d)links.uwaterloo.ca/pmath370docs/week9.pdflecture 25...
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Lecture 25
Nonlinear dynamics in R2 (cont’d)
A very brief review:
The previous lecture was concerned with nonlinear dynamics – as opposed to linear dynamics – of the
form,
xn+1 = f(xn) , n ≥ 0 , (1)
where f : R2 → R2, having the general form,
f(x, y) = (f1(x, y), f2(x, y)) . (2)
In particular, we were concerned with the behaviour of iterates near a fixed point x of f , i.e.,
f(x) = x . (3)
With this in mind, we employed the linear approximation to f(x) at x on the RHS of Eq. (1),
xn+1 = f(xn) ≈ f(x) +Df(x)(xn − x)
= x+Df(x)(xn − x) . (4)
Now subtract x from both sides of the above result to obtain,
xn+1 − x ≈ Df(x)(xn − x) , (5)
which may be written as
un+1 ≈ Aun , (6)
where un is the displacement of the iterates from the fixed point, i.e.,
un = xn − x , (7)
and
A = Df(x) = Jf (x) (8)
is the Jacobian matrix associated with the mapping f evaluated at the fixed point x.
This leads to the main result of the previous lecture:
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Locally, i.e., near the fixed point x of a nonlinear mapping f : R2 → R2, the
iteration sequence,
xn+1 = f(xn) , (9)
is well approximated by the linear dynamical system,
un+1 = Aun , (10)
where
A = Jf (x) (11)
is the Jacobian matrix of f at x and
un = xn − x . (12)
This implies that the (local) nature of the fixed point x, e.g., attractive, repulsive, can be determined
from the eigenvalue/eigenvector properties of the Jacobian matrix Jf (x) evaluated at x. If λ1 and λ2
are the eigenvalues of A, then from the results of our previous analysis,
1. If |λ1| < 1 and |λ2| < 1, then the fixed point x is locally attractive.
2. If |λ1| > 1 and |λ2| > 1, then the fixed point x is locally repulsive.
The above conditions include the case of complex conjugate eigenvalues, in which case |λ1| = |λ2|.
Note: The above discussion, which involves an approximation sign in Eq. (10), may appear to be
rather nonrigorous, as compared to the following result we obtained in one dimension:
Let f : R → R be a C1 mapping with fixed point x i.e., f(x) = x. Furthermore suppose
that |f ′(x)| < 1. Then there exists an interval I ⊃ x such that
|f(x)− x| < K|x− x| for some 0 ≤ K < 1 . (13)
This, in turn, implies that x is an attractive fixed point: For x ∈ J , fn(x) → x as n → ∞.
The proof of the above result was easily accomplished by means of the Mean Value Theorem for
functions f : R → R.
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Stronger, more rigorous results do exist for the higher-dimensional case. However, they are a bit more
complicated because of the lack of a Mean Value Theorem for functions of several variables. One must
resort to a “Generalized Mean Value Theorem.” Here we shall simply state one result that assumes
only that the Jacobian operator Df(x, y) is continuous:
Theorem: Let f : Rn → Rn with fixed point x, i.e, f(x) = x. Furthermore suppose that Df (x, y) is
continuous in a neighbourhood of x and all eigenvalues of Df (x) satisfy ‖λ‖ < 1. Then there exists a
neighbourhood N(x) of x such that
‖f(x)− x‖ < K‖x− x‖ for some 0 ≤ K < 1 . (14)
This, in turn, implies that x is an attractive fixed point: For x ∈ N(x), fn(x) → x as n → ∞.
A return to the example introduced in the previous lecture:
f(x, y) =
(
1
2(x2 + y2), xy
)
. (15)
We found the following four fixed points:
x1 = (0, 0) , x2 = (2, 0) , x3 = (−1, 1) , x4 = (1, 1) . (16)
It now remains to analyze the Jacobian matrix A = Jf at each of these fixed points. First of all, we
compute the Jacobian at (x, y) to be
A(x, y) =
∂f1∂x
∂f1∂y
∂f2∂x
∂f2∂y
=
x y
y x
. (17)
1. Fixed point x1 = (0, 0): Here, the Jacobian matrix is
A =
0 0
0 0
= 0 . (18)
The eigenvalues of A are λ1 = λ2 = 0. Clearly, |λi| = 0 < 1¡ so the fixed point (0, 0) is
attractive. We can choose any linearly independent set of vectors to be the eigenfunctions – it
doesn’t really matter here since λ1 = λ2 = 0. We can choose v1 = (1, 0) and v2 = (0, 1).
Locally, i.e., near the fixed point (0, 0), the dynamics will be as follows,
287
O
2. Fixed point x2 = (2, 0): Here, the Jacobian matrix is
A =
2 0
0 2
. (19)
The eigenvalues of A are λ1 = λ2 = 2. Since |λi| > 1, this fixed point is repulsive. Eigenvectors:
v1 = (1, 0) and v2 = (0, 1).
Locally, i.e., near the fixed point (2, 0), the dynamics will be as follows,
2
3. Fixed point x3 = (1,−1): Here, the Jacobian matrix is
A =
1 −1
−1 1
. (20)
Compute eigenvalues:
(1− λ)2 − 1 = 0 =⇒ λ1 = 2 , λ2 = 0 . (21)
This fixed point is a saddle point. Now compute eigenvectors:
Case 1: λ1 = 2
A =
1 −1
−1 1
v1
v2
= 2
v1
v2
(22)
yields equations,
v1 − v2 = 2v1 =⇒ v1 = −v2 . (23)
−v1 + v2 = 2v2 =⇒ v1 = −v2 . (24)
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Let v1 = (1,−1). Eigenspace E1 = spanv1 is repulsive.
Case 2: λ2 = 0
A =
1 −1
−1 1
v1
v2
= 0
v1
v2
(25)
yields equations,
v1 − v2 = 0 =⇒ v1 = v2 . (26)
−v1 + v2 = 0 =⇒ v1 = v2 . (27)
Let v2 = (1, 1). Eigenspace E2 = spanv2 is attractive.
Locally, i.e., near the fixed point (1,-1), the dynamics will be as follows,
E1
E2
(1,−1)
4. Fixed point x4 = (1, 1): Here, the Jacobian matrix is
A =
1 1
1 1
. (28)
Compute eigenvalues:
(1− λ)2 − 1 = 0 =⇒ λ1 = 2 , λ2 = 0 . (29)
This fixed point is also a saddle point. Now compute eigenvectors:
Case 1: λ1 = 2
A =
1 1
1 1
v1
v2
= 2
v1
v2
(30)
yields equations,
v1 + v2 = 2v1 =⇒ v1 = v2 . (31)
v1 + v2 = 2v2 =⇒ v1 = v2 . (32)
Let v1 = (1, 1). Eigenspace E1 = spanv2 is repulsive.
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Case 2: λ2 = 0
A =
1 1
1 1
v1
v2
= 0
v1
v2
(33)
yields equations,
v1 + v2 = 0 =⇒ v1 = −v2 . (34)
v1 + v2 = 0 =⇒ v1 = −v2 . (35)
Let v2 = (1,−1). Eigenspace E2 = spanv2 is attractive.
Locally, i.e., near the fixed point (1,-1), the dynamics will be as follows,
(1, 1)
E1
E2
We may now put all of the results together on one plot to get a rough idea of how the dynamics of
the iteration of f are operating in the region occupied by the four fixed points, as sketched below.
x
y
1
1
-1
2
Of course, this is not a complete picture of the dynamics. For example, the point (−1,−1) is mapped
to (1, 1), jumping completely over the origin. (This is reminiscent of how the point x = −1 is mapped
290
to x = 1 under the map f(x) = x2.)
The fixed point (2, 0) is repulsive. As such, no nearby points can come close to it. The fixed
points (1,−1) and (1, 1) are “mostly repulsive” – there are particular lines of approach, namely,
the eigenspaces corresponding to eigenvalues with magnitudes less than 1. The only fixed point that
is attractive is (0, 0). There are two options for further study:
1. Computer experiments,
2. Further analysis.
Of course, it’s usually easier to do some experiments on the computer. And they may lead to some
further insight. This is, indeed, the case for the above example. If we perform an experiment to
see what points in the plane are attracted to (0, 0), in other words, the basin of attraction of
(0, 0), a very interesting result is obtained. We find, experimentally, that all points lying inside the
diamond-shaped region (but not on the boundary) shown in the figure below are attracted to (0, 0).
x
y
2-2
2
-2
.
.
..
x2
x3
x4
x0
Some understanding of this experimental result can be obtained. For example, the line y = 2− x
in the first quadrant coincides with the eigenspace of the fixed point x4, which was found to be locally
attractive: Points on this line near (1,1) are attracted to it. In the case of x2 = (2, 0), points on this
line are repelled away from it along the line. We can go further and establish theoretically that the
line y = 2−x is invariant under the action of f : If (x, y) is a point on this line, then the point f(x, y)
is also a point on this line. We prove this as follows:
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The point (x, y) = (x, 2 − x) lies on this line. We substitute this into the formula for f(x, y), i.e.,
f(x, y) =
(
1
2(x2 + y2), xy
)
. (36)
Then
f(x, y) = f(x, 2− x)
=
(
1
2(x2 + (2− x)2), x(2 − x)
)
=(
x2 − 2x+ 2, 2x − x2)
= (x′, y′) . (37)
The point (x′, y′) lies on the line y = 2− x since
2− x′ = 2− (x2 − 2x+ 2) = 2x− x2 = y′ . (38)
The invariance of the other lines can be shown in the same way.
This illustrates that with a little additional work, some more knowledge of the iteration dynamics of
a map may be obtained.
292
Appendix: Basin of attraction of a fixed point of a nonlinear dynamical system
with attractive fixed point
(This material is considered to be supplementary.)
We once again return to the idea of determining the behaviour of iterates of the nonlinear dy-
namical system,
xn+1 = f(xn) , n ≥ 0 , (39)
near a fixed point x of f , i.e., f(x) = x. Earlier, we showed that for iterates xn sufficiently close to x,
i.e.,
xn = x+ un , (40)
where the un are small displacements, then the behaviour of these displacements are well approximated
by the linear dynamical system,
un+1 = Aun , n ≥ 0 , (41)
where
A = Df(x) , (42)
the Jacobian matrix of f at x. If |λn| < 1 for all of the eigenvalues of Df(x), then the fixed point x is
locally attractive.
Note: This result is local – it does not say anything about the structure of the basin of attraction of
an attractive fixed point x i.e., the set of points x ∈ Rn for which the sequence f◦n(x) → x.
Example : Consider the function f : R2 → R2 defined as follows
f(x, y) =
12x+ xy + 1
2y2
x2y
. (43)
After a little work, fixed points of f are found to be:
(x1, y1) = (0, 0), (x2, y2) = (−1, 1), (x3, y3) = (1,−1 +√2), (x4, y4) = (1,−1−
√2). (44)
The Jacobian of f is easily computed to be:
Df(x, y) =
12 + y x+ y
2xy x2
(45)
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An examination of Df(x, y) at each of the fixed points (eigenvalues) shows that (x1, y1) = (0, 0) is the
only fixed point at which the eigenvalues ‖λn‖ < 1. (We omit the details here.) As such, (0, 0) is the
only fixed point of f that can be locally attractive.
The Jacobian of f at (0, 0) is
Df(0, 0) =
12 0
0 0
. Eigenvalues: λ1 =1
2, λ2 = 0. (46)
From the expression for DF (0, 0) above, we see that for (xn, yn) near (0,0),
xn+1 =1
2xn yn+1 = 0 · yn = 0 ,
implying that any iteration sequence sufficiently close to (0, 0) will approach (0, 0) in the limit
n → ∞. We do not expect this to be the case around the other fixed points. This is confirmed by
numerical calculations.
Note the phrase “sufficiently close” in the above paragraph. Neither the Jacobian matrix, in
Eq. (46), nor even the original dynamical system in Eq. (43) can give us an idea of what “sufficiently
close” means, i.e., how large is the region of points that are mapped to the attractive fixed point (0, 0)
in the limit n → ∞, i.e., the basin of attraction of (0, 0).
One might be able to perform some additional analysis in order to come up with some conservative
estimates of the size of the basin of attraction or, most likely, a part of it. Here, we will simply resort
to some computer experiments.
In Fig. 1 is a plot (in black) of the region of points (x, y) near (0, 0) for which the eigenvalues λn of
the the Jacobian Df(x, y) are such that |λn| < 1. From a result stated in a previous lecture involving
the so-called “Generalized Mean Value Theorem” for mappings f : Rn → Rn, we expect the nonlinear
mapping f to be contractive. As such, we expect that all of these points will, under iteration by f ,
approach the fixed point (0, 0).
Note that the above result is a conservative estimate of the size/structure of the basin of attraction
of (0, 0). In order to obtain an idea of the basin of attraction itself, we have to resort to another
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Fig. 1: Region (in black) around fixed point (0, 0) for which ‖DF (x, y)‖ < 1. The entire region
pictured above is −5 ≤ x, y ≤ 5.
computation, namely, testing to see which points are actually mapped to (0, 0). One can do this by
constructing a grid of points around (0, 0) and using each point on the grid as a starting point for the
iteration of f in Eq. (43). The result is an approximation to the basin of attraction.
In Fig. 2 below is plotted an approximation to the actual basin of attraction of the locally
attractive fixed point (0, 0): The set of points x0 = (x0, y0) ∈ R2 for which the sequence xn+1 = f(xn)
converges to (0, 0). Note the that portions of the boundary appear to be quite complicated, perhaps
even “fractal” in nature.
A slight magnification of the lower-right boundary of the basin of attraction is shown in Fig. 3 below.
We can actually state one global result: The x-axis is part of the basin of attraction of (0, 0).
To see this, note that
f(x, 0) =
12x
0
.
The x-axis is an invariant set with respect to f . In other words, if we start on the x-axis, i.e., y = 0,
we remain on the x-axis. And the iterates xn are contracted toward x = 0 geometrically.
A final note: The function f(x, y) in the above example has three other fixed points, all of which
are repulsive. As such, it is highly unlikely that these fixed points would be detected in any numerical
295
Fig. 2: Basin of attraction of the fixed point (0, 0). The entire region pictured above is −5 ≤ x, y ≤ 5.
Fig. 3: Boundary of the basin of attraction of the fixed point (0, 0) in the region 0 ≤ x ≤ 1,
−3 ≤ y ≤ −2.
296
scheme because of roundoff errors: If one is not exactly on a repulsive fixed point, which will most
probably be the case if the fixed point is an irrational number, then the iterates will be repelled away
from the fixed point.
The Henon map
We now introduce a nonlinear vector-valued mapping that has received a great deal of attention by
researchers in dynamical systems theory, for reasons that will be come clear later.
The so-called Henon map, which is actually a two-parameter family of maps, is defined as follows,
Hab
x
y
=
1− ax2 + y
bx
. (47)
For the moment, we simply show that in spite of having a quadratic term in x, the Henon map is
invertible, i.e.,
v
w
= Hab
x
y
(48)
implies that
x
y
= H−1ab
v
w
. (49)
From (48), we can solve for x and y in terms of v and w: Firstly, from the second component,
x =1
bw . (50)
Now substitute this result into the first component,
v = 1− a
b2w2 + y , (51)
and solve for y:
y = v − 1 +a
b2w2 . (52)
Therefore,
x
y
= H−1ab
v
w
=
1bw
v − 1 + ab2w2
. (53)
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We can rewrite the inverse function as a function of x and y:
H−1ab
x
y
=
1by
x− 1 + ab2y2
. (54)
We now examine a few special cases of the parameters, in order to obtain an idea of how the map
is operating in the plane.
Special case: a = b = 0:
H00
x
y
=
1− y
0
. (55)
Not very interesting.
Special case: a = 0, b 6= 0:
H0b
x
y
=
1− y
bx
=
0 −1
b 0
x
y
+
1
0
. (56)
In the special case b = 1, the 2 × 2 matrix is a rotation ofπ
2. For other b values, it is a combination
rotation/shearing. The final term produces a translation by 1 in the positive x direction.
The action of the Henon map on sets of concentric circles centered at (0, 0) is shown on the next few
pages.
298
Circles x2 + y2 = r2, 0.5 ≤ r ≤ 5.0.
Left: a = 0.1, b = 1. Right: a = 0.5, b = 1.
Left: a = 1, b = 1. Right: a = 1, b = 0.5.
299
Lecture 26
The Henon Map in R2
Fixed points of Henon map
We look for solutions to
Hab
x
y
=
1− ax2 + y
bx
=
x
y
. (57)
From second equation,
y = bx . (58)
Substitute this into top equation:
1− ax2 + bx = x =⇒ ax2 + (1− b)x− 1 = 0 , (59)
with solutions,
x =(1− b)±
√
(1− b)2 + 4a
2a. (60)
Recall that
y = bx . (61)
Cases:
1. Two distinct real fixed points if (1− b)2 + 4a > 0.
2. One real fixed point if (1− b)2 + 4a = 0.
3. No real fixed points if (1− b)2 + 4a < 0. In this case, a must be negative.
In order to study the dynamics, we must determine the nature of fixed points via the Jacobian matrix
of Hab(x, y) = (f(x, y), g(x, y)):
DHab(x, y) =
∂f∂x
∂f∂y
∂g∂x
∂g∂y
=
−2ax 1
b 0
. (62)
The Jacobian will be evaluated at the fixed points (x, y). Eigenvalues:
(−2ax− λ)(−λ)− b = 0 , (63)
300
or
λ2 + 2aλ− b = 0 , (64)
so that
λ =−2ax±
√4a2x2 + 4b
2
= −ax±√
a2x2 + b . (65)
After a significant amount of algebra, we obtain the following result: For
0 < b < 1 (66)
and
−1
4(1− b)2 < a <
3
4(1− b)2 , (67)
then the fixed point
x =(1− b) +
√
(b− 1)2 + 4a
2a, y = bx , (68)
is attractive.
Some numerical results:
b = 0.3
a limn→∞
(xn, yn)
-0.1225 (2.85608, 0.85682) (fixed point)
0.3675 (0.97059,0.28020) , (0.93400, 0.29118) (2-cycle)
0.9125 4-cycle
1.0260 8-cycle
1.0510 16-cycle
1.0565 32-cycle
· · · · · ·1.0580459 end of period-doubling
Question: What happens beyond this period-doubling?
Answer: A complex attractive set, called an attractor results:
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Henon attractor a = 1.4, b = 0.3
This Henon attractor is composed of infinitely many lines that lie arbitrarily close to other lines.
For example, if you magnify a portion of the attractor, you will see a set of roughly parallel lines. And
if you iterate the magnification of small regions, you will obtain more lines. This is illustrated in the
computer plots shown in figure on the next page.
In each region, in the limit, the attractor is a union of an infinite number of lines. If you took a cross
section of these lines, you would see a Cantor set! The “dimension” of this attractor is between 1
and 2 and therefore referred to as a strange attractor. Moreover, Hab is one-to-one and onto when
restricted to this attractor. The dynamics is chaotic.
As mentioned briefly in class, a tremendous amount of research activity was devoted by the scientific
and mathematical community to understand “strange attractors.” Simple “toy models” such as “2D
Baker maps” and “horseshoe maps” were developed in order to obtain an understanding of the dynam-
ical processes involved in the existence of a strange attractor – a combination of expansion/stretching
302
From the website, sprott.phys.wisc.edu/chaos/henongp.htm
in one direction and attraction/compression in an orthogonal direction.
Indeed, if we go back to the earlier figures in which the action of the Henon map on sets of concen-
tric circles was shown, one can see, as the parameter a is increased, an increased expansion/stretching
in the x-direction along with a compression in the y-direction. But note that something else is going
on: The circles are not simply being deformed into ellipses that increase in size in the x-direction and
decrease in size in the y-direction. Yes, such ellipses are being produced, but they are then “folded”
over – reminiscent of what is done with pastry. It is as if one end of the ellipse is lifted from the
x-axis upward and then folded leftward in order to produce an object that is symmetric with respect
to the x-axis. It is the combination of such expansion, compression and folding that can give rise
to chaotic behaviour in the plane.
More information on these subjects may be found in the book, Encounters with Chaos and Frac-
tals, by D. Gulick. Section 3.4 is devoted to the Henon map and its strange attractor. Section 3.5 is
devoted to the Horseshoe Map. Both of these subjects are also discussed in the book, An Introduction
to Chaotic Dynamical Systems by R.L Devaney.
These topics could make excellent central points for course projects for PMATH 370.
303
A simple class of chaotic dynamical systems on [0, 1]2
We end this section on dynamical systems in R2 by describing briefly a simple class of chaotic dy-
namical systems on the unit square [0, 1]2, that is, the square region 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 in the
first quadrant of the plane. Each of these dynamical systems can be considered as a product of one-
dimensional chaotic dynamical systems in the x and y coordinates.
First of all, let us recall the following two chaotic dynamical systems on the interval [0, 1]:
Baker Map on [0, 1]
B(x) = 2x mod 1, 0 ≤ x ≤ 1 ,
=
2x , 0 ≤ x < 12 ,
2x− 1 , 12 ≤ x < 1 ,
0 , x = 1 .0 1
x
y
1
Tent Map on [0, 1]
T (x) =
2x, 0 ≤ x ≤ 12 ,
2− 2x, 12 < x ≤ 1 .
1x
y
1
0
Each of these maps was shown to give rise to a chaotic dynamical system on [0, 1]. And recall that
this was done by means of symbolic dynamics. For each map, one may define, for any point x ∈ [0, 1]
an itinerary sequence b ∈ Σ2, where
Σ2 = {b0b1b2 · · · | bk ∈ {0, 1}, k ≥ 0 } , (69)
is the set of all infinite binary sequences. In the case of the Tent Map, for example, given an x ∈ [0, 1],
the elements of its itinerary sequence,
x : x0x1x2 · · · , (70)
are defined as follows,
xn =
0 if T n(x) ∈ I0
1 if T n(x) ∈ I1 ., (71)
304
where
I0 =
[
0,1
2
]
, I1 =
(
1
2, 1
]
. (72)
The action of the Tent Map on a point x ∈ [0, 1] then induces a Bernoulli shift map S on its itinerary
sequence, i.e.,
S : x0x1x2 · · · → x1x2x3 · · · . (73)
Finally, recall that the Bernoulli shift map S possesses the following properties on the space Σ2 which
then allowed us to conclude that the Baker Map and Tent Map are chaotic on [0, 1]:
1. SDIC
2. existence of a dense orbit on [0, 1] which implies transitivity of T ,
3. regularity: the set of period-n points is dense on [0, 1].
Let us now define the following two simple maps in R2 which map the set [0, 1]2 to itself:
1. 2D Baker Map
fB(x, y) = (B(x), B(y)) , (74)
which gives rise to the dynamical system,
(xn+1, yn+1) = fB(xn, xn) = (B(xn), B(yn)) , n ≥ 0 . (75)
2. 2D Tent Map
fT (x, y) = (T (x), T (y)) , (76)
which gives rise to the dynamical system,
(xn+1, yn+1) = fT (xn, yn) = (T (xn), T (yn)) , n ≥ 0 . (77)
In both cases, the dynamics of the xn and yn are uncoupled, i.e.,
1. 2D Baker Map
xn+1 = B(xn) yn+1 = B(yn) n ≥ 0 . (78)
2. 2D Tent Map
xn+1 = T (xn) yn+1 = T (yn) n ≥ 0 . (79)
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In each case, if we start with a seed point (x0, y0) ∈ [0, 1], the coordinate sequences xn and yn evolve
independently of each other. Recalling that a typical sequence {xn} describes a chaotic motion over
the interval [0, 1] in the x-direction, and a typical sequence {yn} will do the same in the y-direction,
then the sequence of ordered pairs (xn, yn) might be expected to behave chaotically over the unit
square [0, 1]2. This is observed numerically, as shown in the figure below.
Plot of 10000 iterates, (xn, yn) ∈ [0, 1], of 2D Tent Map fT (x, y) as defined in Eq. (79).
You may note that the distribution of the iterates (xn, yn) appears to be rather evenly distributed
over the region [0, 1]2, i.e., there are no regions of concentration of the iterates. We recall that for both
the Baker Map and the Tent Map in one-dimension, the density function for the invariant measure
is ρ(x) = 1, implying that (almost all) iteration sequences are uniformly distributed over the interval
[0, 1]. Since the motions of iterates (xn, yn) in the x- and y-directions are uncoupled, we expect that
their distribution over the planar region [0, 1]2 is uniform.
Establishing symbolic dynamics for 2D Baker and Tent Maps
We now show how the chaotic nature of these maps may be established using symbolic dynamics.
For simplicity, we’ll use the 2D Tent Map to illustrate. Our dynamical system of concern is given in
Eq. (79. One way to associate sequences with this dynamical system is to simply consider x and y
306
separately. Associated with a point (x, y) ∈ [0, 1]2, would then be a pair of itinerary sequences in
Σ2, i.e.,
x : s0s1s2 · · · , sk ∈ {0, 1} , k ≥ 0 ,
y : t0t1t2 · · · , tk ∈ {0, 1} , k ≥ 0 . (80)
(Note that we are using sn and tn for the sequence elements instead of xn and yn so that there is no
confusion with the iterates in Eq. (79).) Each of these sequences would be defined in the usual way:
For n ≥ 0,
sn =
0 if T n(x) ∈ I0
1 if T n(x) ∈ I1 .
tn =
0 if T n(y) ∈ I0
1 if T n(y) ∈ I1 .(81)
The action of the 2D Tent Map fT , as defined in Eq. (79) would then involve a simultaneous Bernoulli
shift on both sequences, i.e.,
T (x) : s1s2 · · · , sk ∈ {0, 1} , k ≥ 1 ,
T (y) : t1t2 · · · , tk ∈ {0, 1} , k ≥ 1 . (82)
In general, for n ≥ 0,
T n(x) : snsn+1 · · · , sk ∈ {0, 1} , k ≥ n ,
T n(y) : tntn+1 · · · , tk ∈ {0, 1} , k ≥ n . (83)
The leading sequence elements sn and tn will then tell us where in the square [0, 1]2 the point (xn, yn) =
(T n(x), T n(y)) lies, according to Eq. (81):
1. If sn = 0, then T n(x) ∈ I0. If sn = 1, then T n(x) ∈ I1.
2. If tn = 0, then T n(y) ∈ I0. If tn = 1, then T n(y) ∈ I1.
But we can do this more compactly by associating a single itinerary sequence to the point (x, y) ∈[0, 1]2. Note that for a given pair of leading sequence elements (sn, tn), there are four possibilities, as
summarized above. The net result:
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The partitioning of (i) the x-interval [0, 1] into subintervals I0 and I1 as well as (ii) the
y-interval [0, 1] into subintervals I0 and I1 produces a partitioning of the region R = [0, 1]2
into four subregions Ri, 0 ≤ i ≤ 3, as shown in the figure below.
Note that the numbering of these regions is quite arbitrary – we have chosen a counterclockwise
indexing scheme starting at the region which contains the origin.
R1
R2
x
y
R0
R3
0 1/2 1
1
I0 I1
I0
I1
1/2
As a result, we may replace the two binary sequences s and t with a single quaternary sequence
u which is an element of the sequence space Σ4,
Σ4 = {q0q1q2 · · · | qk ∈ {0, 1, 2, 3}, k ≥ 0 } , (84)
The sequence u associated with a point (x, y) ∈ [0, 1] will be defined as follows,
(x, y) : u = u0u1u2 · · · , (85)
where
un =
0 if fnT (x, y) = (T n(x), T n(y)) ∈ R0
1 if fnT (x, y) = (T n(x), T n(y)) ∈ R1
2 if fnT (x, y) = (T n(x), T n(y)) ∈ R2
3 if fnT (x, y) = (T n(x), T n(y)) ∈ R3 .
(86)
The action of the 2D Baker map fT on a point (x, y) then induces a Bernoulli left-shift map S on the
sequence u ∈ Σ4, i.e.,
fT (x, y) : u1u2u3 · · · , uk ∈ {0, 1, 2, 3} , k ≥ 0 . (87)
and in general, for n ≥ 0,
fnT (x, y) : unun+1un+2 · · · , uk ∈ {0, 1, 2, 3} , k ≥ n . (88)
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In the same way as was done for the Baker and Tent maps on [0,1], the Bernoulli shift map S : Σ4 → Σ4
associated with the 2D Tent Map – and the 2D Baker Map as well – may now be used to establish
that it is chaotic over the region [0, 1]2, i.e., that it possesses the following properties:
1. SDIC
2. existence of a dense orbit on [0, 1]2 which implies transitivity of T ,
3. regularity: the set of period-n points is repulsive and dense on [0, 1]2.
Note that 1) and 2) would now involve some kind of two-dimensional distance function on [0, 1]2. A
convenient distance function is the two-dimension Euclidean distance.
With regard to point 3) above, it may be useful to examine the question of periodic points of the
2D Tent and Baker Maps. Once again, we’ll consider the 2D Tent Map. First of all, recalling that the
map fT (x, y) is “diagonal” in form, i.e.,
fT (x, y) = (T (x), T (y)) , (89)
its Jacobian will have a very simple form,
DfT (x, y) =
T ′(x) 0
0 T ′(y)
=
2 0
0 2
, (90)
for almost all (x, y) ∈ [0, 1]2. Of course, the eigenvalues of this Jacobian are λ1 = λ2 = 2, which
implies that all periodic points (including fixed points) are repulsive, as was perhaps expected. This
is an important component of chaotic dynamics. Let us now examine the question of the location of
periodic points.
Fixed points of the 2D Tent Map: One of the fixed points of the 1D Tent Map is x1 = 0, i.e.,
T (0) = 0. This implies that x1 = (0, 0) is a fixed point of the 2D Tent Map, fT , i.e.,
fT (0, 0) = (T (0), T (0)) = (0, 0) . (91)
In an earlier Problem Set, you showed that the other fixed point of T (x) is x2 = 23 , i.e., T (
23) =
23 .
This implies that x2 = (23 ,23) is a fixed point of fT , i.e.,
fT
(
2
3,2
3
)
=
(
T
(
2
3
)
, T
(
2
3
))
. (92)
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So far, we have found two fixed points, i.e., two period-1 points of fT in [0, 1]2. But there are two
more fixed points, namely,
(x1, x2) , (x2, x1) . (93)
In summary, there are four period-1 points of fT in [0, 1]2.
Two-cycles of the 2D Tent Map: In that same Problem Set, you showed that (p1, p2) = (25 ,45) is
a two-cycle of T (x), i.e., T (25) = (45 ) and T (45 ) = (25 ). This implies that the points p1 = (p1, p1) and
p2 = (p2, p2) comprise a two-cycle for fT :
fT (p1, p1) = (T (p1), T (p1)) = (p2, p2)
fT (p2, p2) = (T (p2), T (p2)) = (p1, p1) . (94)
But the points p3 = (p1, p2) and p4 = (p2, p1) also comprise a two-cycle,
fT (p1, p2) = (T (p1), T (p2)) = (p2, p1)
fT (p2, p1) = (T (p2), T (p1)) = (p1, p2) . (95)
So far, we have found two two-cycles, which means that there are at least four period-2 points of fT
in [0, 1]2. We say “at least”, because there are other two-cycles of fT ! For example,
(x1, p1) , (x1, p2) , (96)
where x1 = 0 is a fixed point of T (x), is also a two-cycle. And so is
(x2, p1) , (x2, p2) . (97)
where x2 = 23 is the other fixed point of T (x). This implies that the reflections of the above points
also form two-cycles, i.e.,
(p1, x1) , (p2, x1) , (98)
and
(p1, x2) , (p2, x2) . (99)
So far, we have found six two-cycles for a total of 12 period-2 points. We’re not finished! There are
four more period-2 points, namely, the four fixed points from the previous section,
(x1, x1) , (x2, x2) , (x1, x2) , (x2, x1) . (100)
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The net result is that there are 16 period-2 points.
At this point, the reader may begin to see the pattern. We’ll simply state the final result:
Recall that T (x) has 2n period-n points on [0, 1]. Let us denote these period-n points as
0 ≤ r1 ≤ r2 ≤ · · · ≤ r2n ≤ 1. Then the following set of (2n)2 = 4n points in [0, 1]2,
(ri, rj) i, j ∈ {1, 2, · · · , 2n} , (101)
are period-n points for fT . Note that these points are situated on a rectangular grid that
lies in the region [0, 1]2. As n increases, the 4n points (ri, rj) become “denser” over [0, 1]
in the same way that the 2n period-n points ri become “denser” over [0, 1]. Here, we will
simply state that in the limit n → ∞, the set of period-n points is dense in [0, 1]2.
There is a lot more going on here if one wishes to extract various k-cycles from the period-n points, but
we’ll leave this for the interested reader. That being said, let’s begin an examination of three-cycles,
just to see how things get more complicated.
Three-cycles of the 2D Tent Map: Let (p1, p2, p3) be a three-cycle of T (x). This implies that the
points
(p1, p1) , (p2, p2) , (p3, p3) , (102)
comprise a three-cycle for fT in [0, 1]2. Another three-cycle is
(p1, p2) , (p2, p3) , (p3, p1) . (103)
And another three-cycle is
(p1, p3) , (p2, p1) , (p3, p2) . (104)
The three-cycle (p1, p2, p3) for T (x) has generated three three-cycles for fT , for a total of nine period-3
points.
But the story does not end here. Recall that T (x) has two three-cycles in [0,1]. So we should be
able to generate another set of three three-cycles for fT , for a total of six three-cycles.
We can now use these two two-cycles, along with the fixed points of T (x) to eventually come up
with 43 = 64 period-3 points.
311
Other possible two-dimensional mappings
One might also wish to consider systems which “mix” the Baker and Tent Maps, e.g.,
fBT (x, y) = (B(x), T (y)) , (105)
which gives rise to the dynamical system,
(xn+1, yn+1 = fBT (xn, yn) = (B(xn), T (yn)) n ≥ 0 . (106)
Since the Baker and Tent Maps produce uniformly distributed iterates, we expect that iterates of this
map will be uniformly distributed over [0, 1]2.
We could also look at simple two-dimensional systems which employ the logistic function, f4(x),
which is known to be chaotic on [0, 1]:
Logistic Map on [0, 1]
f4(x) = 4x(1 − x) , 0 ≤ x ≤ 1 .
1x
y
1
0
Recall that iterates of the Logistic Map are distributed over [0, 1] in a nonuniform manner, with a
much higher concentration at the endpoints of the interval.
For example, consider the following 2D Logistic Map,
fL(x, y) = (f4(x), f4(y)) , (107)
which gives rise to the dynamical system,
(xn+1, yn+1 = fL(xn, yn) = (f4(xn), f4(yn)) n ≥ 0 . (108)
We expect that iterates of this system will concentrate along the four boundary curves of the region
[0, 1]2. This is observed numerically, as shown in the figure below.
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
y(n
)
x(n)
Plot of 50000 iterates (xn, yn) ∈ [0, 1] of 2D Logistic Map fL(x, y) as defined in Eq. (107).
The density function, σ(x, y), associated with the invariant measure of this two-dimensional system is
a simple product of the one-dimensional density functions,
σ(x, y) = ρ(x)ρ(y)
=1
π2√
xy(1− x)(1 − y). (109)
Another possibility is to use the Logistic Map in one direction and a Baker or Tent Map in the other,
e.g.,
fLT (x, y) = (f4(x), T (y)) , (110)
which gives rise to the dynamical system
(xn+1, yn+1 = fLT (xn, yn) = (f4(xn), T (yn)) n ≥ 0 . (111)
How would you expect iterates to be distributed over [0, 1]2?
313
Admittedly, all of the above examples of chaotic dynamical systems in R2 are still rather “tame,”
since they simply involve chaotic maps acting independently in the x and y directions. One may well
wish to consider coupling the maps so that the two component functions of f(x, y) are dependent on
both x and y. An example would be the following coupled Logistic Map system,
f(x, y) = (4xy(1− x), 4xy(1 − y)) . (112)
That being said, some work is involved in order to avoid unwanted situations, e.g., (1) attractive fixed
points or cycles, (2) mappings which do not map the region of concern, e.g., [0, 1]2 into or onto itself.
For example, the above map has (0, 0) as an attractive fixed point.
314
Higher-dimensional dynamics (cont’d)
Complex dynamics: Iteration of complex-valued functions of a com-
plex variable
Here we let C denote the field of complex numbers, i.e.,
C = {z = a+ bi , a, b ∈ R} (113)
where i2 = 1, commonly written as
i =√−1 . (114)
(Just one square root here, no plus and minus square roots.) In books and papers, it is standard
practice to let “z” and “w” denote complex numbers/variables.
We’ll be concerned with the dynamics of the iteration process,
zn+1 = f(zn) , n = 0, 1, 2, · · · , (115)
where is now a complex-valued map of a single complex variable, i.e.,
f : C → C . (116)
For example, we’ll be looking at the one complex-parameter family of quadratic maps having the form,
fc(z) = z2 + c , c ∈ C , (117)
i.e., the iteration process,
zn+1 = z2n + c . (118)
Let us recall some basic facts and properties:
1. As mentioned earlier, a complex number z ∈ C may be expressed in Cartesian form as follows,
z = a+ bi , a, b ∈ R (119)
where a is the real part of z and b its imaginary part, denoted as follows,
a = Re(z) b = Im(z) . (120)
315
Re(z)
Im(z)
.z = a + bi
a
b
2. Cartesian representation: Complex numbers are conveniently plotted on an Argand dia-
gram in which the horizontal axis represents the real part and the vertical axis represents the
imaginary part, as shown below.
The Argand diagram is a representation of the complex plane C in which complex numbers
“live”. (It’s not the only representation. Another one is the so-called Riemann sphere.)
3. The space C is a vector space over the field of complex-valued scalars, i.e., If z1 = a1+ b1i and
z2 = a2 + b2i, then for C1,C2 ∈ C,
C1z1 + C2z2 ∈ C . (121)
Details on multiplication of complex numbers will follow.
4. The magnitude of a complex number z = a+ bi, denoted as |z| or ‖z‖, is
|z| =√
a2 + b2 . (122)
The complex conjugate of a complex number z = a+ bi, denoted as z∗ is
z∗ = a− bi . (123)
Two complex numbers, z1 = a1 + b1i and z2 = a2 + b2i, can be multiplied as follows,
z1z2 = (a1 + b1i)(a2 + b2i)
= a1a2 + a1b2i+ b1a2i+ b1b2i2
= a1a2 + a1b2i+ b1a2i− b1b2
= (a1a2 − b1b2) + (a1b2 + b1a2)i . (124)
316
5. From the above rule, it follows that for z = a+ bi,
zz∗ = (a+ bi)(a − bi)
= a2 − abi+ abi+ b2
= a2 + b2
= |z |2 . (125)
6. This makes it possible to treat division by complex numbers as follows,
z1z2
=z1z2
· z∗2
z∗2
=z1z
∗2
|z2|2
=1
a22 + b22z1z
∗2
... (126)
7. Polar representation of complex numbers:
For z = a+ bi, we let
a = r cos θ , b = r sin θ , θ ∈ [0, 2π) , (127)
so that
r = |z| =√
a2 + b2 , tan θ =b
a. (128)
Re(z)
Im(z)
z = a + bi
a
br
θ
The scalar r is the magnitude of the complex number z. The angle θ is often called the
argument of z, denoted mathematically as follows,
θ = Arg(z) . (129)
317
It follows that
z = a+ bi
= r cos θ + ir sin θ
= r(cos θ + i sin θ)
= “ r cis θ ” . (130)
8. De Moivre’s Theorem: (Easily proved using addition rules for cosine and sine)
[cos θ1 + i sin θ1][cos(θ2) + i sin θ2] = cos(θ1 + θ2) + i sin(θ1 + θ2) . (131)
9. Euler’s equation: (Easily proved using power series expansion of ex))
cos θ + i sin θ = eiθ , also written as exp(iθ) . (132)
These results provide a more transparent picture of what happens when two complex numbers
are multiplied. If z1 = r1eiθ1 and z2 = r2 = eiθ2 , then
z1z2 = r1eiθ1r2e
iθ2
= r1r2ei(θ1+θ2) . (133)
When two complex numbers z1 and z2 are multiplied, the resulting complex number z1z2 has:
(a) magnitude equal to the products of the magnitudes of z1 and z2,
(b) argument equal to the sum of the arguments of z1 and z2.
A number of other consequences follow, e.g.,
z1z2
=r1r2
ei(θ1−θ2) . (134)
10. nth roots of a complex number: Given z = reiθ, we compute the n nth roots z1, · · · , zn so
that znk = z as follows: Let
z = rei(θ+2kπ) 0 ≤ k ≤ n− 1 . (135)
Then
z1/n = r1/nei(θ+2kπ)/n
= r1/neiθ/nei2kπ/n , 0 ≤ k ≤ n− 1 . (136)
318
The n roots are obtained by multiplying the principal nth root of z,
z1/n = r1/neiθ/n , (137)
by the n nth roots of unity,
wk,n = ei2kπ/n , 0 ≤ k ≤ n− 1 , (138)
which are distributed uniformly over the unit circle |z | = 1 and separated from each other by
the angle 2π/n. The principal root z1/n in (137) is effectively rotated n times by angle 2π/n to
produce the n roots.
Aside: The nth roots of unity satisfy the equation,
zn = 1 . (139)
To solve for z, write 1 as ei2kπ, k ≥ 0, so that the above equation becomes,
zn = ei2kπ , k ≥ 0 . (140)
Raising both sides to the 1/nth power,
z = [ei2kπ]1/n = ei2kπ/n = wk,n 0 ≤ k ≤ n− 1 , (141)
where we have used the fact that the case k = n generates the root 1.
319
Lecture 27
Complex dynamics (cont’d)
Functions of complex variables
We shall be working with complex-valued functions of a complex variable, i.e., f : C → C, for example,
polynomials of the form
f(z) = anzn + an−1z
n−1 + · · · a1z + a0 . (142)
In the spirit of this course, we’ll be considering the iteration of such complex-valued functions, i.e.,
zn+1 = f(zn) , n ≥ 0 , zn ∈ C . (143)
And, as before, we’ll be interested in the existence of fixed points of such maps, i.e.,
f(z) = z . (144)
Note: Unfortunately, the bar in z often denotes complex conjugation. In these notes, we’ll use the
star, i.e., z∗ to denote complex conjugation, unless otherwise indicated.
Another note: The functions f : C → C which we consider here are functions only of the complex
variable z and not its complex conjugate. In order words, we shall not be working with functions
such as
f(z, z∗) = Az2 +Bz∗ . (145)
As a result, the functions which we consider are much “nicer”, i.e., “well behaved” in terms of conti-
nuity, derivatives, etc.. (In the parlance of the theory of functions of complex variables, our functions
will generally be analytic.)
We’ll also be interested in period-n points, the components of an n-cycle, {p1, p2, · · · , pn}, where
f(p1) = p2 , f(p2) = p3 , · · · f(pn) = f(p1) . (146)
Here we mention one important difference between complex-valued functions (of a complex vari-
able) and real-valued functions (of a real variable) – the existence of fixed points which is related to
the existence of zeros of a function.
320
To illustrate, suppose that f : R → R is a real-valued function of a real-valued variable. If x is a fixed
point of f , i.e.,
f(x) = x , (147)
then x is a zero of the function g(x) defined as
g(x) = f(x)− x . (148)
It is quite possible that f(x) has no real fixed points because there are no real zeros of g(x). For
example, the function,
f(x) = x2 + 1 , (149)
has no fixed points because
g(x) = x2 − x+ 1 (150)
has no real zeros.
But if we now consider f as a complex-valued mapping of a complex variable, i.e., f : C → C,
with
f(z) = z2 + 1 (151)
then f(z) has fixed points because
g(z) = z2 − z + 1 (152)
has two complex zeros,
z1,2 =1
2±
√3
2i . (153)
Of relevance here is a famous result by C.F. Gauss (1777-1855), namely the Fundamental Theorem
of Algebra:
Fundamental Theorem of Algebra: Let f : C → C be a nonconstant polynomial function. Then
f has a zero, i.e., there is a z0 ∈ C such that f(z0) = 0.
321
Some properties of complex-valued functions
We list some important properties of complex-valued functions. Since the functions that we consider
in this course are, for the most part, “well-behaved,” the following properties will generally hold.
Limit of a function: The existence of a limit of a function f : C → C at a point z0 ∈ C, written as
limz→z0
f(z) = L , (154)
is understood in an “ǫ− δ” sense. We’ll avoid the details here, but only mention that the concept
z → z0 (155)
must be understood in a two-dimensional way. Going back to the Argand diagram, z must be allowed
to approach z0 from all possible angles. For example, we may consider all z ∈ C of the form,
z = z0 + reiθ . (156)
For a fixed θ ∈ [0, 2π), we then let r → 0+. This is then done for all θ ∈ [0, 2π).
Continuous function: We say that a function f : C → C is continuous at z0 ∈ C if
limz→z0
f(z) = f(z0) . (157)
Derivative of a function: We shall also have to work with derivatives of functions f : C → C. In
the cases that we are considering, where f is a function of the variable z only, and not z∗ as well,
things are rather straightforward. We define
d
dzf(z)
∣
∣
∣
∣
z=z0
= f ′(z0) = limz→z0
f(z)− f(z0)
z − z0, (158)
provided that the limit exists. Once again, it is understood that the same limit is obtained regardless
of the way in which z approaches z0.
In general, if derivatives exist, they may be computed using the standard rules of differentiation,
i.e.,
f(z) = z5 + 2 sin 3z =⇒ f ′(z) = 5z4 + 6cos 3z . (159)
322
Nature of a fixed point z of a complex-valued function f(z):
Here we simply state the following result, which is a natural extension of the one-dimensional
result for real-valued functions:
Let z be a fixed point of f : C → C.
1. If |f ′(z)| < 1, then the fixed point z is attractive, i.e., there exists a neighbourhood N(z) ⊂ C
of z such that for z ∈ N(z),
|f(z)− z| < K|z − z0| for some 0 ≤ K < 1 . (160)
This in turn implies that for z ∈ N(z),
fn(z) → z as n → ∞ . (161)
2. If |f ′(z)| > 1, then the fixed point z is repulsive, i.e., there exists a neighbourhood N(z) ⊂ C
of z such that for z ∈ N(z),
|f(z)− z| > L|z − z0| for some L > 1 . (162)
This implies that for a z ∈ N(z), there exists an n > 0 such that
fn(z) /∈ N(z) . (163)
323
Iteration dynamics for some simple complex-valued functions
Linear maps
As mentioned earlier, we’ll be looking at the dynamics associated with the following iteration scheme,
where f : C → C:
zn+1 = f(zn) , n ≥ 0 . (164)
The iterates zn ∈ C will be situated in the complex plane. As such, the dynamics is two-dimensional.
Many of the ideas encountered in our earlier study of two-dimensional real dynamical systems will
apply here.
We first consider one of the simplest functions, the linear function
f(z) = az , a ∈ C , (165)
the complex version of the simple function,
f(x) = ax , (166)
one of the first functions examined in this course. The only fixed point of f(z) is z = 0. The iteration
scheme is
zn+1 = azn . (167)
If we start with a point z0 ∈ C and apply f iteratively, we find, as we did in the real-valued case, that
z1 = az0 z2 = a2z0 , (168)
so that, in general,
zn = anz0 . (169)
If we let
a = reiθ , (170)
Then Eq. (169) becomes
zn = rneinθz0 , n ≥ 0 . (171)
Perhaps its best to look at the first iterate,
z1 = az0 = reiθz0 . (172)
There are two processes operating here:
324
1. Multiplication of z0 by the complex number eiθ rotates it by the angle θ = Argz.
2. Multiplication of eiθz0 by the real number r scales it in a radial direction, i.e., the argument is
kept the same. If r < 1, then the magnitude of the complex number is reduced, i.e., the point
moves toward the origin. If r > 1, then it is increased, i.t., the point moves away from the origin.
If r = 1, the magnitude remains the same, i.e., a pure rotation.
Of course, each additional multiplication by the complex scalar a = reiθ will produce an additional
rotation by the angle θ = Argz along with a radial scaling by r.
In summary, iteration of the function f(z) = az produces a rotation and scaling of iterates, which
is identical to the situation for two-dimensional linear real dynamical systems,
xn+1 = Axn (173)
where the 2× 2 matrix A has complex eigenvalues. In fact, we can make this more precise and state
that iteration of the function f(z) = az is identical to a particular linear real dynamical system
studied in a previous section. If we let
w = az , (174)
and express w, a and z in Cartesian form,
w = u+ vi , a = a1 + a2i , z = x+ yi , (175)
then
az = (a1 + a2i)(x+ yi)
= (a1x− a2y) + (a1y + a2x)i . (176)
Equating real and imaginary parts of w and az yields the following system of equations,
u = a1x− a2y
v = a2x+ a1y , (177)
or, in matrix-vector form,
u
v
=
a1 −a2
a2 a1
x
y
. (178)
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If we let a = a1 and b = a2, then we have the matrix A,
A =
a −b
b a
, (179)
with eigenvalues a ± bi = a1 ± a2i. You will recall that this matrix was encountered in a previous
lecture – it was actually called B in that lecture, being the representation of a matrix with complex
eigenvalues a± ib in the basis {v2,v1}.Also recall that if multiply and divide each component of the matrix A in (179) above by the
scalar√a2 + b2, it can be expressed as
A = r
cos θ − sin θ
sin θ cos θ
, r =√
a2 + b2 , tan θ =b
a. (180)
In other words, A represents a rotation by angle θ along with a scaling by the factor r.
In summary, we easily summarize the dynamics of the complex iteration scheme,
zn+1 = azn , n ≥ 0 , (181)
where a ∈ C, as follows:
1. r = |a| < 1: The origin 0 is attractive.
2. r = |a| > 1: The origin 0 is repulsive.
3. r = |a| = 1: The origin is neither attractive nor repulsive. It is stable because trajectories do
not move away from it.
Note that we didn’t even use a derivative-based analysis of the fixed point z = 0 here. The fact that
f ′(z) = f ′(0) = a , (182)
leads to (1) and (2) above.
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Quadratic complex maps
We first consider the simple quadratic complex map,
f(z) = z2 . (183)
It is a special case of the general one-complex-parameter family of maps, gc : C → C,
gc(z) = z2 + c c ∈ C . (184)
You will recall that we studied this family in the real-valued case. We’ll return to this a little later.
Back to the function f(z) = z2, let’s recall the dynamics associated with the real-valued case,
f(x) = x2, a graph of which is presented below (from Lecture 3).
4y
x
y = x
y = x2
0 1 2−1−2
1
2
3
The phase portrait of this iteration scheme is shown below. Since the real line R is a subset of the
complex plane C, we expect that this phase portrait will somehow be a subset of the phase portrait
of the complex map f(z) = z2, especially since the complex map f(z) maps the real line to itself, i.e.,
if z is real, i.e., z = a+ 0i, then f(z) = z2 = a2 is also real.
-1 0 1
Let us now examine what happens when we start at a point z0 ∈ C and perform the iteration
scheme associated with (183), i.e.,
zn+1 = z2n , n ≥ 0 . (185)
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It will be beneficial to consider the polar representation of z0, i.e.,
z0 = reiθ . (186)
Then
z1 = z20 = r2e2iθ . (187)
Continuing,
z2 = z21 = r4e4iθ . (188)
And once more,
z3 = z22 = r8e8iθ . (189)
The reader should be able to see the pattern: For n ≥ 0,
zn = r2n
ei2nθ . (190)
We see that
1. The magnitude of zn is r2n
.
2. The argument of zn is 2nθ.
Unlike the rotation dynamics associated with the linear problem studied earlier, the angles of the zn
are not growing linearly here, but exponentially. And the magnitudes of the zn are also increasing
(r > 1) or decreasing (r < 1) exponentially.
The one situation in which the magnitudes do not change is when r = 1, i.e.,
|z0 | = 1 . (191)
This implies that the starting point z0 lies on the unit circle in the complex plane which, for reasons
that will be come clear later, we shall denote as J :
J = {z = x+ iy , x2 + y2 = 1 } . (192)
Note that:
z ∈ J =⇒ z2 ∈ J . (193)
If z ∈ J , then z = eiθ for some θ ∈ [0, 2π]. This means that z2 = e2iθ ∈ J .
This implies that the unit circle J is an invariant set under the action of the map f(z) = z2.
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ORe(z)
Im(z)
|z| = 1
The unit circle J ∈ C, |z| = 1.
Fixed points of f(z) = z2
This suggests that if f(z) has any fixed points, they may lie in the invariant set. We know one of these
fixed points, i.e., z1 = 1, and it certainly lies on J . Are there any other fixed points? We perform the
computation:
f(z) = z =⇒ z2 = z =⇒ z(z − 1) = 0 . (194)
The other fixed point is z2 = 0.
The difference between these two fixed points is that x1 = 1 is repulsive and x2 = 0 is attractive,
as we know from the real case:
f(z) = z2 =⇒ f ′(z) = 2z , (195)
so that
|f ′(x1)| = |f ′(1)| = 2 , |f ′(x2)| = |f ′(0)| = 0 . (196)
Note: The two fixed points z1 = 1 and z2 = 0 actually belong to another invariant set of the
map f(z) = z2, namely, the real line, R. In other words, if z ∈ R, then f(z) ∈ R. Of course, we
have already examined the dynamics of the this map on R, having referred to it as f(x) = x2.
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Period-n points of f(z) = z2, n > 1
We now look for period-n points, i.e., fixed points of the composition map fn(z). Note that
f2(x) = f(f(x)) = (z2)2 = z4
f3(x) = f(f2(x))) = (z4)2 = z8
f4(x) = f(f3(x))) = (z8)2 = z16 . (197)
We saw this pattern earlier. In general,
fn(z) = z2n
, n ≥ 0 . (198)
For a given n > 0, period-n points will be roots of the equation,
z2n
= z . (199)
One of the roots is z = 0, a fixed point of f(z). For z 6= 0, the period-n points satisfy
z2n−1 = 1 , (200)
implying that they are the (2n − 1)th roots of unity, i.e.,
wk,2n−1 = ei2kπ/(2n−1), 0 ≤ k ≤ 2n − 2 . (201)
Note that all of these roots lie on the unit circle J .
Let’s check this derivation with a couple of special cases.
1. n = 1 (fixed points): There is 21 − 1 = 1 period-1 point on J :
w0,1 = 1 . (202)
2. n = 2 (period-2 points): There are 22 − 1 = 3 period-2 points on J :
w0,3 = 1 , (203)
the fixed point of f(z),
w1,3 = ei2π/3
= cos
(
2π
3
)
+ i sin
(
2π
3
)
= −1
2+
√3
2i , (204)
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and
w2,3 = ei4π/3
= cos
(
4π
3
)
+ i sin
(
4π
3
)
= −1
2−
√3
2i , (205)
It is easy to verify that {w1,3, w2,3} comprise a two-cycle, and especially so by using polar
representations,
w21,3 = ei4π/3 = w2,3 w2
2,3 = ei8π/3 = ei2π/3 = w1,3 . (206)
Notice that the elements of the two-cycle are complex, i.e., they have nonzero imaginary parts.
This is consistent with our earlier finding that no real two-cycles exist.
Regularity of the period-n points on J: Note that for a given n > 0, the 2n − 1 period-n points
wk,2n−1 are equally spaced over the unit circle J – the angle between consecutive points is
∆θ =2π
2n − 1, (207)
which implies that the distance between consecutive points is (radius times subtended angle)
d = ∆θ =2π
2n − 1. (208)
As n → ∞, ∆θ → 0. It should be easy to see that the set of all period-n points for n > 0 is
dense on the unit circle J. Given any point z ∈ J and a δ > 0, can find an n sufficiently large so
that a period-n point wk,2n−1 lies within a distance δ of z. This follows from that fact that for each
n > 0, the points wk,2n−1, 0 ≤ k ≤ 2n − 1, divide the unit circle J into 2n − 1 arcs. For n sufficiently
large, the point z will be contained in an arc of length less than δ, which implies the the endpoints of
the arc will be distances less than δ from z.
Repulsivity of period-n points on J: Recalling that
fn(z) = z2n
, (209)
it follows that
(fn)′(z) = 2nz2n−1 , (210)
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so that for a given n > 0,
|(fn)′(wk,2n−1)| = 2n|wk,2n−1| = 2n > 1 , 0 ≤ k ≤ 2n − 1 . (211)
It follows that all period-n points on J are repulsive.
All of this sounds familiar: We have two of the three ingredients for chaotic behaviour of the function
f(z) on the unit circle J . All that remains to be shown is transitivity. Let us return to the action of
f(z) on a point z = eiθ ∈ J where θ ∈ [0, 2π].
f(eiθ) = ei2θ . (212)
We can now forget the fact that the points lie on the unit circle J and focus on the arguments of the
points involved. The map f : J → J induces the following map g : [0, 2π] → [0, 2π]:
g(θ) = 2θ mod 2π . (213)
We’ve seen this map, but on the unit interval [0, 1]. We can always rescale θ by defining
φ =θ
2π, (214)
so that φ ∈ [0, 1]. The map g(θ) then becomes
h(φ) = 2φ mod 1 . (215)
We know that this map is chaotic on [0, 1]. It follows that g(θ) is chaotic on [0, 2π] which, in turn,
implies (after only a little work) the following:
The map f(z) = z2 restricted to the unit circle J, i.e., f : J → J is chaotic.
We must now investigate the dynamics of iterating f(z) = z2 at points not on the unit circle J .
Recall, from Eq. (190 that if we start at a point z0 = reiθ, then the iterates zn = fn(z0) are given by
zn = r2n
ei2nθ . (216)
There are two cases that should be readily evident:
1. Case No. 1: r < 1, i.e., points inside the circle J . In this case,
r2n → 0 =⇒ zn → 0 as n → ∞. (217)
The iterates zn spiral inward exponentially rapidly toward the attractive fixed point 0.
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2. Case No. 2: r > 1, i.e., points outside the circle J . In this case,
r2n → ∞ =⇒ |zn| → ∞ as n → ∞. (218)
The iterates zn spiral outward exponentially rapidly.
The entire dynamics of the iteration of the function f(z) = z2 may be summarized in the following
phase portrait in the complex plane.
Re(z)
Im(z)
|z| = 1
O
Phase portrait for f(z) = z2 on C.
From this phase portrait, we can see that the unit circle J can be viewed as a separator or
boundary set – it acts as the boundary between two dynamical processes:
1. |z| < 1: Points travel toward the attractive fixed point z2 = 0. We see that the interior of the
unit circle, i.e., the set
W (0) = {z ∈ C , |z| < 1 }, (219)
is the basin of attraction of the attractive fixed point z2 = 0.
Note that this is consistent with our earlier results for the real-valued mapping f(x) = x2 on
the real line, where we showed that the basin of attraction of the attractive fixed point x = 0
was the open interval (−1, 1). This is simply the intersection of the real line R with the basin of
attraction W (0) ⊂ C given above, i.e., the interior of the unit circle.
2. |z| > 1: Points travel outward, i.e., |zn| → ∞. In complex variable theory, it is “kosher” to
consider the point at infinity, i.e., z = ∞, as a bona fide point. We can then write that iterates
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outside the unit circle behave as zn → ∞, and that z = ∞ is an attractive fixed point. Its basin
of attraction is the set
W (∞) = {z ∈ C , |z| > 1 } . (220)
From our earlier analysis of fixed points, we can also state that the set J contains all of the
repulsive period-n points of f(z) = z2. If we “close” the set of all repulsive period-n points, i.e.
include all limit points of sequences contained in this set, then the result is the unit circle J . As a
result, the set J is a repeller set. For any point z sufficiently close to J , f maps z away from J .
We’ll state this more precisely a little later.
The set J is called the Julia set of the map f(z) = z2 in honour of Gaston Julia, a French mathe-
matician who made many important contributions to the study of complex dynamics. Very shortly,
we shall examine the Julia sets of the maps gc(z) = z2 + c for nonzero values of c ∈ C.
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