Umnouy Ponsukcharoen
Project IV: Mapping Regions for a Double Well System
Mapping Regions for a Double Well System: A
In this project, you will investigate the evolution of phase space regions for the damped driven motion of a
particle in the double well. The idea is to take a whole bunch of points (at least a thousand) bunched into a
fairly small area, propagate them all forward in time using the equations of motion, and see what the
resulting region looks like. Can you think of a way of estimating the area of the resulting region? In any
case, try to determine if the area seems to decrease exponentially as derived in Chapter 7. The rate of
shrinkage of the area of the region after a long time should be independent of the value of F0. However, the
shape of the region does depend on F0. Investigate the evolution of the shapes of the regions in the
different regimes. This project is quite open-ended, and we expect you to do interesting and imaginative
work.
A. Introduction
In Chapter 6, we have studied the equation of motion of an object in a double well, which
is
d2x
dV
dx
m 2 

 F cost
dt
dx
dt
1
1
, where V   k1 x 2  k 2 x 4 .
2
2
From this equation, one can treat (x(t), p(t)) as a 2-dimensional object and study its
 In Chapter 7, we use the fact that there is a natural periodicity
behavior in phase space.
given by the periodic driving force in order to construct iterated function:

xi1 : x(t  ), pi1 : p(t  ) with xi : x(t), pi : p(t).
In order to see the chaotic behavior (i.e. period doubling), one may plot x i after number
of iterations as a function of driving force amplitude F. Figure 7.1 in Chapter 7 illustrates
the bifurcate (chaotic) behavior of this double well system reduced to 1 dimension [1].
 in this project, we are going to study
 (x(t),p(t)) as a 2-dimensional object
However,
 to examine the
directly via phase-space plot. First, as a preliminary, we are going
bifurcate behavior of the double well system via the phase space plot for different F.
Then, we will study the behavior of colony of points in phase-space propagating in time
by measuring area and length of the colony. Also, we will study the chaotic behavior by
considering two close points in phase-phase in x-coordinate. Finally, we will discuss how
chaos accounts for the result.
B. Preliminary: Bifurcate behavior of trajectory in phase-space
In Chapter 6 and 7, we have used ‘Oscillator.java’ to represent the motion of the object
by graphs x(t) and p(t). Now, we are going to represent the motion of the object by phasespace plot, i.e. p(x). For simple harmonic motion, for example, the phase-space plot is
just an ellipse. Here, we will do the phase-space plot for our system in the same way, but
we will start plotting after some time in order to eliminate the initial values dependence
(this is just same as what we did with bifurcation plot). This method yields “attractor” for
each case. The revised ‘Oscillator.java’ is used to produce figures below. More details
and comments are given in the program.
Initial condition: x(0) = -1.2, p(0) = 0.0,   1.0 .
Other variables are set as same as ones in default, i.e., dt = 0.01, A =1.0,    , mass
=1.0. These will be fixed throughout the project unless I note a change in particular
section.

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F =0.5
F= 1.5
F=1.75
F=2.0
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F=2.2
F=2.25
F=2.3
F=2.35
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F=2.375
F=2.4
F=2.5
F=4.0
Figure 1: Phase space plots for different F’s.
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These phase space plots clearly illustrate bifurcate behavior of the system. If one draws a
straight-line p = 0 on the graph, the number of intersections between p=0 line and
object’s trajectory in phase-space are just doubling from 2, 4, 8, … as F increases. Under
some specific values of parameters, we can draw a conclusion
0.5 < F < 1.5
2 distinct local x maximum/minimum (i.e. at p = 0)
1.75 < F < 2.2
4 distinct local x maximum/minimum
2.25 < F < 2.35
8 distinct local x maximum/minimum
F = 2.375
16 distinct local x maximum/minimum
2.4 < F < 2.5
16 or more distinct local x maximum/minimum
F = 4.0
very large number of distinct local x maximum/minimum
(equivalent to graph beyond r in one dimensional bifurcation plot – chaos appears
here).
From this information, we can estimate roughly what F is. The value of F’s
corresponding to “bifurcations” are F2 1.75 , F3  2.25 , F4  2.375. Hence, similar to
F  F3 2.375  2.25

 0.25 . The value of F is about
analysis to Project II,   4
F3  F2
2.25 1.75

F  1.75  (2.25 1.75)  (2.25 1.75)(0.25)1  ....



1
n
 1.75  (2.25 1.75) 0.25  1.75  0.5
 2.42 
1  0.25


This is not the only way to see the chaotic behavior of the double-well system. Next, we
will show how chaos appears in a colony of points in phase-space, not just a trajectory of
a single point in phase-space.
C. Behavior of colony of points in phase-space propagating in time: Area
Suppose there is a colony or a blob of points in phase-space. Since we know from
previous section that there is chaos in the system, we might suspect that colony of points
after propagating in time behaves chaotically. Still, what kind of behavior is counted as
“chaotic”? One of properties that might relate to the chaos is area of the colony. As time
propagates, area of the colony may increase in order to create unpredictability, which is a
main feature of chaos. That is, if the area increases as time forwards, it will be more and
more difficult to predict the position of object in the next time step. However, as we will
show by both analytically and numerically, this is not the case. Area of the colony shrinks
as time increases without dependence of F.
In order to prove analytically that the area of colony in phase-space decreases as time
increases, one may refer to the proof of Liouville’s Theorem or the proof in Chapter 7. I
will just quote the result. It is found that,
dA
 A or A(t )  A0 exp( t )
dt
,where  is the damping factor in the dissipative force term [1]. Notice that the
derivation is given in general, so there is no chaos behavior involved here.
Now, in order to show that the area decreases as time forwards independence of chaos
numerically, we may write a program to plot the colony and to compute the colony’s area
at different time. In ‘Blobplot.java’, we start with simplest colony of initial points:
101x101 points distributes uniformly on the 0.1x0.1 square unit in phase-space with
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lower-left coordinate (-1.2,0.0). Then, we use same algorithm as one in ‘Oscillator.java’
to find positions of those points in phase-space. Then, we can sum the area of
parallelogram in which the corner of each parallelogram are consecutive points in the
blob. More details and comments are given in the program ‘Blobplot.java’. First, let’s test
the relation between area and time
Figure 2: The plot of natural log-value of area versus time in seconds for different  ’s. All other
parameters are fixed. F = 4.0.
The plot above agrees with the relation between area of colony and time propagated that
has been proposed. Note that at F = 4.0 and  =1.0 is in chaos as we have shown in
Section B. Now, let’s make another plot for F = 2.0.
Figure 3: The plot of natural log-value of area versus time in seconds for different  ’s. All other
parameters are fixed. F = 2.0.
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The case also yields the same law of area shrinkage. Observe that on Figure 2, setup F =
4.0 and  =1.0 represents chaotic case as illustrated in Section B, while on Figure 3, setup
F = 2.0 and  =1.0 represents non-chaotic case. The exponential of area shrinkage applies
to all cases here.
Remark: more deviation from the expected value of small  ’s occurs because the method

to find the area is accurate up to a certain size of blob. When the points in colony disperse

without shrinking the area (  =0.0), the deviation becomes significant.

So far, we have shown that, no matter it is chaotic case or not, as time forwards, the area
of colony of points in phase-phase shrinks as long as the dissipative factor exists. The
 how does unpredictability appear if the area shrinks down to zero?
next question is then,
The answer is quite fascinating: zero area does not mean zero information. We will see
how the unpredictability arises more closely in the next section.
D. Behavior of colony of points in phase-space propagating in time: Length
Let’s revisit the phase space plot from the previous section setup
1) F = 4.0 and  =1.0
2) F = 1.5 and  =1.0
Here ‘Blobplot.java’ is used in order to make the plot of blob evolved in time. Notice that
the plot is made every 0.20 s time step.


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Figure 4: The evolution of colony of size 0.1x0.1 unit in phase space. The first graph corresponds to F = 4.0
and  =1.0 and the second graph corresponds to F = 1.5 and  =1.0

First, two graphs confirm the area shrinkage in both chaotic and non-chaotic case.
However, what distinguishes two graphs is the perimeter of the colony. As time passes,
 into a line. The line tends to grow longer for
colony of points in both case are squeezed
chaotic case, while the line tends to shrink shorter for non-chaotic case. Let’s investigate
this effect more thoroughly by constructing a new kind of colony – line-like instead of
square blob. For simplicity, we start with initial points: 10001 points distributes
uniformly on a circle with center (-1.2,0.0) and diameter 0.1 unit in phase-space. Then,
we use same algorithm as one in ‘Oscillator.java’ to find positions of those points in
phase-space. Then, we can sum the length of line connecting consecutive points in the
colony. More details and comments are given in the program ‘Lineplot.java’. First, let’s
find the relation between length and time under different F (and same  ). In order to
avoid effect of initial setup, we do same analysis to new circle with new center (-0.6,0.8)
but same diameter.

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Figure 5: The plot of natural log-value of length versus time in seconds for different  ’s. All other
parameters are fixed.  =1.0. The first graph is for a circle with center (-1.2,0.0) and diameter of 1 unit, and
the second graph with center (-0.6,0.8) and diameter of 1 unit.
Note that data in the range t=0 to t=10 s is neglected since it does not behave linearly
 to Lyapunov spectrum, which I will explain it in the discussion). Here one can
(this is due
see that, for extreme cases of chaos (F =3.0, 4.0), the slope is definitely positive. On the
other hand, for extreme cases of non-chaos (F= 0.0,0.5,1.0) the slope is definitely
negative. Hence, the rate of the length between points in phase-space, or, in other words,
the rate of the separation of close trajectories is an indicator of chaos. One way to
understand the picture is that, the points on circumference of circle in phase-space can be
spread out or squeeze in as time increases as they moves in the phase-space. If we
connect the line of each point in time, that line is just the trajectory of point. Therefore,
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even though the length defined in phase-space sounds unphysical unlike area in phasespace, it can still be used as indicator of chaos.
Now, can one find F from this analysis? Let’s make a plot of slope in each line versus
the corresponding driving force.

Figure 6: The relation between the driving force strength F and the slope of plot between log L vs. t. The
data is taken from graphs in Figure 5.
Here one can see two regimes separated by x-axis. The regime far above x-axis
represents chaotic system, while the regime far below x-axis represents non-chaotic
system. From this graph, one may find the x-intercept and claim that it is F . However,
two difficulties arise. First, we do not know exactly what the relation actually look like. It
may be linear, polynomial, or even not smooth at x-intercept. Secondly, the determination
of slope near the critical point becomes poor. As we have seen in logistic map before, it is
 If the number of
difficult to distinguish between F and FN where N is very large.
iterations (i.e. time to run the program) is not large enough, one might get the slope of
graph log L vs. greater than 0 and misunderstand that the critical has already passed. For
example, in our plot, F = 2.3 and F = 2.35 gives the slope value greater than 0. However,
they are still less than 
F >2.4as we know from Section B. Therefore, in practice, the
rate of perimeter change in time in phase-space fits in some extension but it is a good
way to extract value of F .

E. The study
of two close points in phase-space: x-coordinate perspective
In this section, instead of length in phase-space, we are going to investigate in only one
 length measurement in x-coordinate only corresponds to direct
coordinate. This
measurement in real life. First, let set up two points, which are very close in position, and
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both of them are at rest. Then, let the system evolves. Similar to previous section, we
might expect the chaotic behavior when F is large enough. The distance in x-coordinate
between two close points should grow larger and larger. The program “testchaos.java” is
written in order to demonstrate this phenomenon. More details and comments are given
in the program.
F=2.35
F=2.375
F=2.4
F=2.5
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F=3.0
F=4.0
Figure 7: The plot of distance in x-coordinate between 2 points and time under different F’s. The initial
positions of two points are x= -1.2 and x= -1.2001.  =1.0.
For F= 2.35 and F=2.375, the non-chaotic behavior becomes clear. The distance between
two points does not grow; it even shrinks. However, for F=2.5 or up, the distance

between two points grows from 0.0001
to the order of 1. Before it reaches the “saturation
point” where the distance is bound by the overall size of trajectory [2], the distance
between two points grows exponentially over time. One might check this by plotting
Log(delta x) vs. time during this period.
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Figure 8: The plot of Log(delta x) vs. t for F= 2.5, F= 3.0 and F=4.0 respectively. The estimate slope of
each plot is 0.125, 0.25 and 0.375 respectively.
Overall, the plot of Log(delta x) vs. t shows linear trend even though there are some
periodic shift. If we consider only a specific time in period of driving force (i.e. consider
in Poincare section), we will have clearer linear trend. From this graph, one can estimate
the slope of this relation. The values obtained here is same order as in Figure 6 in Section
E. This is not surprising since the distance of two points in x-coordinate is just the project
of a certain length in phase-space. Chaos appears in both phase-space and x-coordinates,
which we can observe directly in real life.
F. Discussion and Conclusion
At this point, we have seen chaotic behavior in the double well system in many ways:
shape of attractor, length in phase-space and distance in x-coordinates. Is there any theory
to explain these phenomena systematically and quantitatively? Here Lyapunov exponent
is introduced in order to quantify the dynamic system systematically.
Remark: this analysis turns out to be similar to discussion in Chapter 7a [1].
The Lyapunov exponent of a dynamical system is quantity that characterizes the
rate of separation of infinitesimally close trajectories:
v
v
Z (t)  et Z0
For n-dimensional phase space, there is a set of n Lyapunov exponents called “spectrum
of Lyapunov exponents” [3]. Now, what can we say about Lyapunov spectrum in our
system?

In the double well problem
here, there are 2 Lyapunov exponents since the phasespace is 2 dimensional. Let’s define it to be 1 and 2 . Each exponent is responsible to
the separation in each direction in phase-space. Now, the separation in phase-space is be
written as
v
v
v
Z (t  )  e1Z(t)1  e2Z (t)2


,where  is small time-step. After many iterations, the contribution in the direction with
smaller Lyapunov exponent will be suppressed by the contribution in the direction with
bigger Lyapunov exponents due to the exponential factor. Therefore, in Section D Figure
t=10 s is removed from the analysis since the contribution from small
5, data from t=0 to
Lyapunov exponent is still significant. Also, if those two Lyapunov exponents are not
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
much different, it took longer time to diminish the effect from the smaller Lyapunov
exponent. That might also be the reason why the data at the boundary between chaotic
and non-chaotic case (Figure 6) is not quite consistent with our prediction. At this point,
one can see that the maximal Lyapunov exponent is just the quantitative measure of
chaos. For chaotic case, Lyapunov is positive, and for non-chaotic, it is negative. Section
D and E are just how to measure the maximal Lyapunov exponent in different projection.
After all, the length of perimeter of close loop in phase-space or the distance in xcoordinate reflects the same rate of separation. However, near the regime where the
maximal Lyapunov exponent changes from negative to positive, the methods to find it
becomes less accurate as we have already seen.
Other than length, one can understand
the
v
v Lyapunov spectrum via the area. The
area of parallelogram formed by Z (t) 1 and Z (t) 2 develops in time proportional to
e 1 t e 2 t  e( 1  2 )t . According to the derivation in Chapter 7a [1], we obtain the relation of
Lyapunov exponents as
1  2  

 exponent,  , is positive, the minimal Lyapunov
Therefore, when the maximal
Lyapunov
m
exponent is negative with even higher magnitude, m   . In other words, the separation
shrinks in one direction faster than the separation grows in another direction. However,
when the maximal Lyapunovexponent is negative, the magnitudes of two exponents are

bound by  . Figure 4 agrees with this argument.
Under the same  and same point of

time, the blob in chaotic case tends to be thinner than the blob in non-chaotic case. Here
the notion of Lyapunov exponent is quite powerful in order to explain chaotic behavior in
phase-space of double well system.


G. Bibliography
[1] Benjamin Blander, Marko Kleine Berkenbusch, Susan Coppersmith, Leo P. Kadanoff,
Amy J. Kolan, Marcelo Magnasco, Michael J. Vinson, Thomas A. Witten, and Scott
Wunsch. (2010) Lecture notes 2010 Phys 251/CS 279/Math 292 Chaos, Complexity and
Computers.
[2] Hilborn, Robert C. (2000). Chaos and Nonlinear Dynamics, An Introduction for
Scientists and Engineers. Oxford University Press.
[3] Lyapunov Exponent in Wikipedia. Retrieved March 11, 2010, from
http://en.wikipedia.org/wiki/Lyapunov_exponent
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