This document discusses two systems that exhibit chaos: the Josephson junction and optical systems. It describes how the Josephson junction, a quantum device made of two superconductors separated by an insulator, can display chaotic behavior when coupled to an RLC circuit. It also explains how feedback in laser systems can induce optical chaos characterized by spectral broadening. The document emphasizes that while chaos is interesting, controlling and harnessing chaos in these systems could enable applications in areas like signal processing and communications.

1. University of Virginia
RF and Optical Realizations of Chaotic Dynamics
Emeka V. Ikpeazu, Jr.
MATH 5250—Dynamical Systems
Professor John Imbrie
December 3, 2016

2. Abstract:
Nonlinear systems have a profound characteristics of exhibiting chaos under very specific
circumstances. What is more profound is there extraordinary sensitivity to small perturbations in
initial conditions. In this paper, I discuss a variety of chaotic systems that show up in electrical
engineering applications, particularly in the GHz and THz ranges of the electromagnetic
spectrum. These include the Josephson junction and optical chaos. Here, I show that chaos is
not only interesting as an effect, but also interesting in the way it can be harnessed in order to
realize efficient methods for problems confronting engineers and applied scientists.
1. The Josephson junction
The Josephson junction is a quantum mechanical device created by sandwiching a
non-superconducting material in between to superconducting devices. This is sort of
similar to a superconducting p-n junction, probably more analogous to a p-i-n junction
wherein the i-layer an intrinsic absorbing region. The device is named for Welsh
physicist Brian Josephson who predicted the tunneling of superconducting electrons
through a superconducting device. The device operates at low temperatures where metals
become superconductors, usually 20 K or lower. At these sort of temperatures, phase
transitions occur in metals and practically diminish any resistance to the flow of electrons
in them.
Figure 1.1. The Josephson junction is operated below a certain “critical temperature” so
the that the electrons on the opposing sides of the junction exhibit an attractive
interaction. This attractive potential creates an energy gap. This energy gap induces a
“critical current” across the junction.
When this critical current is exceeded an AC current is created and the phase
difference between the two junctions 𝜙 = 𝜙1 − 𝜙2 is created. The equation for phase
difference in the Josephson junction is written
𝜙̇ =
2𝑒
ℏ
𝑉
Figure 1.2. An equivalent circuit is created for the Josephson junction wherein there is a
displacement current 𝐶𝑉̇, a regular current 𝑉/𝑅, and a bias current 𝐼𝑐 sin 𝜙. In this way
ℏ𝐶
2𝑒
𝜙̈ +
ℏ
2𝑒𝑅
𝜙̇ + 𝐼𝑐 sin 𝜙 = 𝐼

3. where ℏ is Planck’s constant divided by 2𝜋,
and 𝑒 is the charge of an electron.
The system can be turned into the following two-dimensional system:
𝜙̇ =
2𝑒𝑉
ℏ
𝑉̇ =
𝐼
𝐶
−
𝐼𝑐 sin 𝜙
𝐶
−
𝑉
𝑅𝐶
However, an additional term can be added to make the system three-dimensional. It is by
coupling the JJ to an RLC circuit wherein the system thereby becomes three-dimensional
and chaos is induced [2]. This additional term relates the voltage induced by the inductor
of inductance well and that produced by the current through the resistor of resistance 𝑅.
The term relates the voltage 𝑉 as
𝐿𝐼̇𝑠 + 𝐼𝑠 𝑅 𝑠 = 𝑉
where 𝐼𝑠 is the shunt current. Equivalently, the second equation for 𝑉̇ must be changed to
𝑉̇ =
𝐼 − 𝐼𝑠
𝐶
−
𝐼𝑠 sin 𝜙
𝐶
−
𝑉
𝑅𝐶
Via further normalization we get the system [3],
𝜕𝑥 𝜏 = 𝑦
𝜕𝑦𝜏 =
1
𝛽𝑐
{𝑖 − 𝑔𝑦 − sin 𝑥 − 𝑧}
𝜕𝑧 𝜏 =
𝑦 − 𝑧
𝛽𝐿
where 𝑥 = 𝜙, 𝑦 = 𝑉/𝐼𝑐 𝑅 𝑠, and 𝑧 = 𝐼𝑠 /𝐼𝑐. Time is normalized as 𝜏 = 𝜔𝑐 𝑡 and 𝜔𝑐 =
2𝐼𝑐 𝑅 𝑠
ℏ
.
Voltage is normalized as 𝑣 = 𝑉/𝐼𝑐 𝑅 𝑠. There are other dimensionless parameters such as
the normalized capacitance parameter 𝛽𝑐 = 2𝐼𝑐 𝑅 𝑠
2
𝐶/ℏ, the normalized inductance
parameter 𝛽𝐿 = 2𝑒𝐼𝑐 𝐿/ℏ, and 𝑔 = 𝑅 𝑉/𝑅 𝐿.
Figure 1.3. The figure above is the phase portrait for an RLC-shunted Josephson junction
where 𝛽𝑐 = 1.08, 𝛽𝑙 = 2.57, 𝑖 = 1.11, and 𝑔 = 0.061.
-10
0
10
-10
0
10
-10
-5
0
5
10
xy
z
_x = 0
_y = 0
_z = 0

4. The Jacobian for this system is written
𝐷𝐹𝑋 = 𝐽 = (
0 1 0
−𝛽𝑐
−1
cos 𝑥∗
−𝑔𝛽𝑐
−1
−𝛽𝑐
−1
0 𝛽𝐿
−1
−𝛽𝐿
−1
) = (
0 1 0
−𝛽𝑐
−1√1 − 𝑖2 −𝑔𝛽𝑐
−1
−𝛽𝑐
−1
0 𝛽𝐿
−1
−𝛽𝐿
−1
)
Accordingly, the eigenvalues for the Jacobian linearization are
𝜆1 = −0.6216 + 0.9921𝑖
𝜆2 = 0.1122 − 0.5103𝑖
𝜆3 = −0.4730 − 0.4818𝑖
Figure 1.4. More specifically, a trajectory of the system is shown above
where ( 𝑥0, 𝑦0, 𝑧0) = (1.5,1.5,1.8).
0
50
100
150
200
-2
0
2
4
-0.5
0
0.5
1
1.5
2
xy
z
-5
0
5
10
x 10
-4
-5
0
5
x 10
-4
-2
-1
0
1
2
x 10
-4
 x y
z

5. Figure 1.5. Above is plotted the difference in trajectory of two versions of the system in
Figure 4 separated by 10−4
in each direction for the initial conditions.
Most of the difference is in the x-direction which is plotted below.
Figure 1.6. Quasiperiodicity in the x-direction takes place where multiple periods appear.
The quasi-sinusoidality of 𝛿𝑥(𝑡) shows that the trajectories in the x-direction—and in
fact, all directions—do cross in a semi-periodic interval.
The data above shows the value of Josephson junctions in very high-frequency
applications. More interestingly, however, is vacillation of trajectories that are just
slightly offset in initial directionality. This can be useful in differential RF amplifiers
that rely on detecting phase and frequency differences. Moreover, this behavior allows
for a sort of nonlinear heterodyning scheme wherein slightly offset currents and phases
can be used to generate quasi-periodic signals an encode information.
2. Optical chaos
Optical systems are a different type of dynamical system in regards to chaos. Nonlinear
systems like the Lorenz attractor or the Rӧssler system can easily induce chaos under the
most standard of initial conditions. Optical chaos is more different; it’s a system whose
contributions from nonlinear effects are realized under very intense conditions.
Figure 2.1. Optical chaos begins with this rather simple feedback mechanism with a time
delay 𝜏. Here, the laser diode (LD) sends the optical signal through a circulator where it
goes through the system with varying attenuation. This is to decrease the lasing
threshold, the amount of gain needed for the steady-state condition [7].
0 50 100 150 200 250
-4
-2
0
2
4
6
8
x 10
-4
x
t

6. Moreover, this feedback system induces “coherence collapse” as the spectral linewidth
broadens via the effects of noise and nonlinear effects [5]. The degree to which the
spectral linewidth broadens is a function of the correlation between each cycle in the
feedback system. The electric field induced and the number of carriers in the cavity are a
dynamical system described by the Lang-Kobayashi equations:
𝐸̇ =
1 + 𝑖𝛼
2
𝐺 𝑁 𝑛( 𝑡) 𝐸( 𝑡) + 𝜅𝑒 𝑖𝜔 𝑜 𝜏
𝐸( 𝑡 − 𝜏)
𝑛̇ = 𝑝𝑒𝑥𝑐 𝐽𝑡ℎ − 𝛾𝑛( 𝑡) − [Γ + 𝐺 𝑁 𝑛( 𝑡)]| 𝐸( 𝑡)|2
where 𝛼 is the linewidth enhancement factor,
𝐺 𝑁 is the gain factor,
𝜅 is the coupling between feedback cycles,
𝑝𝑒𝑥𝑐 𝐽𝑡ℎ is the injection rate of the carrier pump,
𝛾 is inverse of the lifetimes of the spontaneous carriers,
and Γ is the cavity loss.
The second term of 𝐸̇ is blanched to denote that this contribution is extrinsic to the cavity
system.
Figure 2.2. The increase in feedback—in this case the parameter 𝜅—leads not only to a
spectral broadening, but also to a spectral shift towards a lower frequency.
What is more is that frequency peaks get lost as Δ𝑓𝑓𝑏, the frequency shift induced by
feedback coupling, approaches 𝑓𝑅𝑂, the relaxation oscillation frequency.
(a) (b)
Figure 2.3. The more widened spectral heat map (a) shows the presence of multiple
frequencies in the GHz range. This is because an increase in the attenuation corresponds
to a decrease in the coupling coefficient 𝜅. The peaks become more pronounced when
the attenuation exceeds 25 dB [6,7].

7. Figure 2.4. At constant 𝜅 values, the effacement of spectral peaks is seen with gradual
increases in input current [7].
In deciphering what is an apparently chaotic time series the appropriate procedure is to
take the autocorrelation function of the signal with itself. When that is done, what is
observed is that the value of the first echo of the autocorrelation varies quasi-sinusoidally
with the change in the attenuation as seen in Figure 2.5. This pattern was further evinced
by doing the same thing with various input photocurrents as shown in Figure 2.6.
Figure 2.5. The maximum correlation Figure 2.6. The maximum correlation
has its first minimum at 18 dB echoes at different photocurrents that
attenuation and its highest around 27.5 are above the threshold photocurrent.
dB attenuation [7]. They all exhibit the same pattern [7].
The correlation pattern is explained but parsing the plot into two regimes: a regime of
‘weak chaos’ and one of ‘strong chaos’. The left side has the highest attenuation and thus
the lowest feedback coupling so correlation comes from the high signal to noise ratios
which increase with the input photocurrent and thus increase the correlation. On the right
side, bistability takes place in that the laser output has two steady-state equilibrium
points. One point is one of chaotic dynamics where feedback cycles experience an
entanglement of sorts and the other is that of stable emission where feedback cycles are
more superposed. Optical chaos has many applications when controlled, that is.
Stability analysis in feedback-coupled dynamical systems is similar to that in
regular dynamical systems save for that fact that it is sub-Lyapunov exponents that are
analyzed instead of Lyapunov exponents [6,7]. In this scheme of analysis, the presence
of weak chaos is characterized by negative sub-Lyapunov exponents while that of strong
chaos is characterized by positive sub-Lyapunov exponents. Characteristic of coupled
M-ary optical systems is the ansatz Lang-Kobayashi scheme:
𝐱̇( 𝑡) = F[ 𝐱( 𝑡)] + 𝜎 ∑ 𝐺𝑖𝑗
𝑀
𝑗=1
𝑯[ 𝐱( 𝑡 − 𝜏)]
Where G is the coupling matrix normalized to unity.

8. In both the RF and optical applications of chaos the common requirement for their
optimal applicability is control. In the context of non-feedback optical chaos, this comes
in the modulation of a parameter, particularly in an oscillatory fashion to achieve an orbit
of some sort [8]. In the context of feedback optical chaos vis-à-vis the Lang-Kobayashi
system control comes via the continuous application of small perturbations using various
methods. The most common of these methods is the OGY (Ott-Grebogi-Yorke) method
wherein a cycle is achieved and small kicks are injected to periodically adjust it.
The case of the Josephson junction is also similar although it seems to maintain a
rather felicitous autonomous synchrony, much like that of a phase-locked loop as
demonstrated in Figure 1.5, thereby eliminating the need for and OGY or Pyragas method
of controlling the ensuing chaos in the circuit. In the limit that the applied current I is
much less than the critical current 𝐼𝑐 and the resistance—assuming this is an RC-shunted
JJ—is large the oscillation will carry on for much longer. However, it almost goes
without saying that the case of the Josephson junction is one that requires much more
similar devices in parallel for any real power to be generated [9]. This, of course, will be
facilitated by feedback coupling between these circuits, making the a more complex
system, whose dynamical incorrigibility will scale with the power it is expected to
generate.
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Synchronization of Chaos in RCL-Shunted Josephson Junction with Noise Disturbance
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378457, 14 pages, 2012. doi:10.1155/2012/378457
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Fischer, I. Kanter, and W. Kinzel, Phys. Rev. E88, 012902 (2013).
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