Untangling Complex Systems, page 25
1
λ 2
across the bifurcation points a and b.
Third, a system, whose dynamical evolution involves bifurcation points, can be sensitive to fluc-
tuations. Although a system obeys to deterministic laws, when it is close to a bifurcation point, its
evolution can be remarkably influenced by random fluctuations, determining the path it will follow.
TRY EXERCISE 4.3
7 The root-mean-square of a set of N values ( x , ,
) is given by the following equation:
1 x 2 …, xN
1 (
2
2
2
1
2
)
x
x
x
xN
=
+
+ … +
.
rms
N
For a continuous function f( t) defined over the time interval [ t , ], the root-mean-square is: 0 tf
t f
1
2
f ( t) =
[ f ( t)] dt .
0
∫
rms
( t − )
f
t
t 0
8 For more information about microtubules in cells, see Chapter 9.
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Untangling Complex Systems
α
f
b
i
a
λ 1
λ 2
λ
FIGURE 4.13 Hypothetic system with many bifurcations.
4.2.4 hoPf bifurcaTions
So far, we have studied bifurcations for one-dimensional systems. As we move up from one-
dimensional to two-dimensional systems, we still find that fixed points can be created, destroyed, or
destabilized by varying the “contour conditions.” In the case of two-dimensional systems, we can
encounter also limit cycles. The ensemble of all possible solutions for a two-dimensional system
is illustrated in Figure 3.18, and a synthetic view is also reported in the Figure 4.14. It is evident that if the contour conditions determine a negative trace and a positive determinant of the Jacobian
(see paragraph 3.8.2 in Chapter 3), the system is in a stable fixed point. In fact, the fixed point is located in the right bottom part of the stability graph (covered by a white patch in Figure 4.14).
If we change the contour conditions so much that either tr( J) changes from negative to positive
or det( J) changes from positive to negative values or both variations occur, the fixed point is not
any more stable. Imagine starting from a fixed point that is a stable spiral. If tr( J) becomes posi-
tive, the fixed point transforms from a stable spiral into an unstable one. If the unstable fixed point
is surrounded by a stable limit cycle, we say that we have found a supercritical Hopf bifurcation
(see Figure 4.15a). On the other hand, a subcritical Hopf bifurcation occurs when a stable fixed point is encompassed by an unstable limit cycle (see Figure 4.15b). As we reach the bifurcation point,
the fixed point becomes unstable. The trajectories must jump to a distant attractor, which may be
another fixed point or another limit cycle or goes to infinity. In three or more dimensions, this dis-
tant attractor can be “strange” because it can be chaotic, as we will see for the Lorenz’s equations in
tr( J)
Instability
det( J)
Stability
FIGURE 4.14 Plot of tr( J) versus det( J) distinguishing the stable and unstable solutions for a two- dimensional system.
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111
y
y
λ
λ
x
x
(a)
(b)
FIGURE 4.15 Representation of the supercritical (a) and subcritical (b) Hopf bifurcations for a two-
dimensional system whose variables are x and y, and the control parameter is λ. A dashed line represents an unstable set of fixed points, whereas a continuous line represents a set of stable fixed points.
Chapter 10. Examples of supercritical and subcritical Hopf bifurcations may be found in the oscillating chemical reactions presented in Chapter 8.9
TRY EXERCISE 4.4
In Chapter 9, we will encounter a further type of bifurcation in the two-dimensional system: the
Turing bifurcation. The Turing bifurcation occurs when a stable fixed-point transforms in a saddle
point because the modification of the contour conditions determines the change of the sign for det(J)
from positive to negative.
In the next chapters, we are going to discover how out-of-equilibrium systems give rise to
ordered phenomena in time. This is true in ecology (Chapter 5), in the economy (Chapter 6), in biology (Chapter 7), and bio/chemical laboratories (Chapter 8). The order in space and time emerging far-from-equilibrium will be presented in Chapter 9. In Chapter 10, we will know that in non-linear regime even chaotic dynamics can be observed. Chaotic dynamics produce structures that cannot
be described with the traditional Euclidean geometry, but with a new geometry: Fractal geometry
(see Chapter 11). In Chapter 12, we will learn the features of Complex Systems, and finally, in
Chapter 13, we will discover the promising strategies for facing the Complexity Challenges.
4.3 KEY QUESTIONS
• Which are the criteria of evolution for out-of-equilibrium systems?
• What is a bifurcation?
• Present the model that describes the transition from a lamp to a laser.
• What is a plausible hypothesis about the origin of biomolecular asymmetry?
4.4 KEY WORDS
Static and dynamic self-assembly; Self-organization; Dissipative structures; Saddle-node bifurca-
tions; Trans-critical bifurcation; Supercritical and Subcritical Pitchfork bifurcation; Supercritical
and Subcritical Hopf bifurcations.
4.5 HINT FOR FURTHER READING
Other examples of bifurcations can be found in the enjoyable book by Strogatz (1994).
9 In the presence of a subcritical Hopf bifurcation, a system may exhibit hysteresis. An example is given by Strogatz (1994) in his delightful book, on page 252. In the range of coexistence of stable fixed point and limit cycle, the system will remain stationary or will oscillate periodically, depending upon its history. Small perturbations to either the steady or the oscillatory state decay, but larger ones can cause a transition from stationary to periodic behavior or vice versa.
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Untangling Complex Systems
4.6 EXERCISES
4.1. For the differential equation dx dt = λ + x 2, repeat the linear stability analysis presented in paragraph 4.2.1 and build its bifurcation diagram. Which are the differences if compared
to the diagram of Figure 4.4?
4.2. Find the fixed points for the differential equation dx dt = λ x + x 3.
4.3. Verify graphically that the differential equation [4.29] has just one fixed point when
(λ −λ
(
c )
c ) ≤ 0, whereas it has three solutions when λ − λ
> 0.
4.4. The Brusselator is a hypothetical mechanism for an oscillating chemical reaction. It does
not describe any particular reaction. It was proposed by Ilya Prigogine and his coworkers
at the Université Libre de Bruxelles (Prigogine and Lefever 1968). The mechanism is the
following:
A k 1
→
X
B + X
k 2
→
Y + D
2 X + Y
k 3
→
3 X
X
k 4
→
E
If we assume that A and B are held constant, D and E do not participate in any further reactions; the only variables are X and Y. If we write the differential equations for X and Y and we transform in dimensionless form, we get:
dx = a− bx + x 2 y − x
τ
d
dy = bx − x 2 y
dτ
where:
x = X k 3 k
/ 4
y = Y k 3 k
/ 4
τ = k 4 t
k 1
a =
A
k 3 k
/
k
4
4
b = k 2 B k
/ 4
Find out the steady state solution and discuss its stability as a function of the values of a
and b. Do you find a Hopf bifurcation? Is it supercritical or subcritical?
4.7 SOLUTIONS TO THE EXERCISES
4.1. The solutions of fixed point are x 0 = ± −λ . Real solutions are those with λ ≤ 0. According
to the linear stability analysis, x x
0
2 0
0
e f ( x t)
x 0 e+ x t
=
+
=
+
. The solution x
0 = 0 and
x 0 = + −λ are unstable, whereas the solutions x 0 = − −λ are stable. It derives that the
bifurcation diagram is that shown in Figure 4.16.
4.2. The fixed points are: x′0 = 0, x′′0 = + −λ , and x′′′0= − −λ . When λ ≤ 0, all the three fixed points are real solutions. When λ is positive, only x′0 = 0 is a possible real solution. Through
the linear stability analysis, we define the properties of the fixed points.
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113
x
λ
0
FIGURE 4.16 Saddle-node bifurcation diagram.
dx
∫
f x
2
0
dt (λ 3 x 0 )
(
=
+
dt
x − x 0 ) = ( )∫
∫
( +3 2
λ
0
0 )
After integrating, we obtain: x x 0 e f ( x t)
x 0 e
x t
=
+
=
+
. From the latter equation, it is
evident that x′0 = 0 is stable when λ is negative, whereas it becomes unstable when λ ≥ 0. The
other two solutions, x′′0 = + −λ and x′′′0 = − −λ , which are possible only when λ is nega-
tive, are unstable. The corresponding bifurcation diagram looks like the plot in Figure 4.17.
If we compare this graph with that of Figure 4.9, we notice that now the pitchfork is
inverted, and two backward-bending branches of unstable fixed points bifurcate from the
origin when λ = 0. When our system reaches the bifurcation point after following the stable
branch defined by the solution x′
( )
0 = 0, for a further increase of λ, it blows-up: x t → ±∞ .
The diagram of this exercise is called subcritical pitchfork bifurcation, whereas that
shown in Figure 4.9 is named as supercritical pitchfork bifurcation.
4.3. To find the fixed points of the equation dα dt = − Aα 3 + B(λ − )
c
λ α + Cg graphically, we
plot the functions y
3
(λ λ )
1 = B
− c α − Aα and y 2 = C
− g to find the intersections, which rep-
resent the fixed points. When (λ − λ c ) ≤ 0, the cubic function, y 1, is monotonically decreas-
ing and intersects the horizontal line at one point at positive α. When (λ − λ c) > 0, the
intersection points can be one, two or three depending on the value of y 2 = C
− g. We obtain
two solutions when the horizontal line ( y 2 = C
− g) is tangent to the minimum of the cubic
function. In this case, we are at the bifurcation point. To determine the values of α at the
bifurcation point, we search for the minimum of the cubic function. The derivative of y 1
is y
2
(λ λ )
B(
c )
1 = B
− c − 3 Aα ; it is null when α = ±
λ − λ 3 A. The negative solution cor-
responds to the minimum, whereas the positive solution corresponds to the maximum of
the cubic function (Figure 4.18).
x
Unstable
Stable
Unstable
λ
Unstable
FIGURE 4.17 Subcritical pitchfork bifurcation diagram.
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Untangling Complex Systems
y
y
B( λ − λc) − Aα 3
B( λ − λc) − Aα 3
α
α
− Cg
(− Cg)3
(− Cg)2
(− Cg)1
( λ − λc) ≤ 0
( λ − λc) > 0
FIGURE 4.18 Graphical determination of the fixed points for the differential equation [4.29], for (λ − λ c ) ≤ 0
on the left, and for (λ − λ c ) > 0 on the right.
4.4. The fixed point is ( x
)
)
0 , y 0
= ( a, b a
/ . The Jacobian J is equal to
b −1
a 2
J =
.
−
b
− a 2
The trace is tr ( J ) = b −1− a 2, and the determinant is det( J ) = a 2. The determinant is always positive. The characteristic equation is: λ2 + λ 2
2
( a +1− b) + a = 0. Therefore, the
eigenvalues are: λ
2
2
2
2
( a
b
a
b
a )
1,2 = − (
+1− ) ± ( +1− ) − 4
2.
Whenever
b < 1 + a2, the trace is negative, and if the discriminant is negative, the
eigenvalues are complex numbers with a negative real part. In this case, the fixed point is
a stable focus. If we change the contour conditions and we increase the value of b, when it
results b = 1 + a2, the trace becomes null, and the eigenvalues become imaginary numbers.
The stable focus transforms in a limit cycle. Whenever b > 1 + a2 the trace is positive, and the fixed point becomes unstable. This result means that we have found a Hopf bifurcation.
To decide if it is supercritical or subcritical, we must perform simple numerical experi-
ments. In Figure 4.19, there are three examples by fixing a equal to 1 and changing b from 1.9 in (a), to 2 in (b) and 2.1 in (c).
3
3
2
2
2
y
y
y
1
1
1
0.8
1.2
1.6
0.5
1.0
1.5
0.5 1.0 1.5 2.0
x
x
x
(a)
(b)
(c)
FIGURE 4.19 Dynamics of the Brusselator for a = 1 and b = 1.9 in (a), b = 2 in (b), and b = 2.1 in (c).
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115
The dynamics shown in Figure 4.19a demonstrates that the fixed point is a stable focus.
In Figure 4.19b, when b = 1 + a2, the system converges to a small limit cycle. By increasing the value of b and maintaining constant the value of a, it is evident that the limit cycle
expands. The Hopf bifurcation is supercritical. To do the numerical experiment, you may use
MATLAB and a function file like the one reported as follows.
function dy = Brusselator(t, y)
dy = zeros(2,1)
a = 1
b = 2.1
dy(1) = a–b*y(1) + (y(1)^2)*y(2)–y(1)
dy(2) = b*y(1)–(y(1)^2)*y(2)
The script should look like:
[ t, y] = ode 45(“Brusselator,” [0 750], [0.9 0.
9])
The Emergence of Temporal
5 Order in the Ecosystems
The first law of ecology is that everything is related to everything else.
Barry Commoner (1917–2012 AD)
5.1 INTRODUCTION
Both terrestrial and aquatic ecosystems are networks involving interactions among living beings
and their environment. The reductionist approach brings us to isolate interactions between pairs
of elements. For instance, a recurrent type of simplified interaction in ecosystems is that involving
one kind of predator and one kind of prey. The predator is an organism that eats the prey. Some
examples are lynx and hare in North America; lion and zebra, leopard and impala in the African
Savannah; bear and fish, cat and rat, least weasel and bank vole in Europe; largemouth bass and
mosquito fish in the freshwaters of North America, and many more. Predator and prey make efforts
to survive in the same ecosystem. The dynamical relationship between predator and prey is one of
the dominant themes in ecology.
5.2 PREDATOR-PREY RELATIONSHIP: THE LOTKA-VOLTERRA MODEL
The first plausible model to describe the predator-prey interaction was proposed independently by Lotka
(1925) and Volterra (1926).1 The Lotka-Volterra model involves three elementary steps reported in [5.1], wherein H is the prey, C is the predator, F is the food of prey, and D is the carcass of the predator.
F + H
k
1→
2 H
H + C
k
2→
C
2
[5.1]
C
k
