PHYSICAL REVIEW E 89, 032921 (2014)

Finite-time synchronization of tunnel-diode-based chaotic oscillators

Patrick Louodop,1 Hilaire Fotsin,1 Michaux Kountchou,1 Elie B. Megam Ngouonkadi,1

Hilda A. Cerdeira,2 and Samuel Bowong3

1Laboratory of Electronics and Signal Processing Faculty of Science, Department of Physics, University of Dschang,
P.O. Box 67, Dschang, Cameroon

2Instituto de Fı́sica Teórica, UNESP, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz 271,
Bloco II, Barra Funda, 01140-070 São Paulo, Brazil

3Laboratory of Applied Mathematics, Department of Mathematics and Computer Science, Faculty of Science,
University of Douala, P.O. Box 24157, Douala, Cameroon

(Received 3 June 2013; published 26 March 2014)

This paper addresses the problem of finite-time synchronization of tunnel diode based chaotic oscillators.
After a brief investigation of its chaotic dynamics, we propose an active adaptive feedback coupling which
accomplishes the synchronization of tunnel-diode-based chaotic systems with and without the presence of
delay(s), basing ourselves on Lyapunov and on Krasovskii-Lyapunov stability theories. This feedback coupling
could be applied to many other chaotic systems. A finite horizon can be arbitrarily established by ensuring that
chaos synchronization is achieved at a pre-established time. An advantage of the proposed feedback coupling is
that it is simple and easy to implement. Both mathematical investigations and numerical simulations followed by
PSPICE experiment are presented to show the feasibility of the proposed method.

DOI: 10.1103/PhysRevE.89.032921 PACS number(s): 05.45.Xt, 05.45.Gg

I. INTRODUCTION

Since chaotic systems were discovered, considerable
interest arose in developing and analyzing various systems
that exhibit chaos due to their importance in many fields
of sciences [1–3]. In biology, epidemiology, or climatology,
investigating mathematical models is a part of the strategies
used to better comprehend the phenomenon [1,2,4–8].
However, there exist difficulties to understand some of them
because of the lack of data or the time taken to produce
reliable data. For example, in epidemiology, some deceases
have to be observed for a long period before they can be
modeled [4]. To turn around these problems, electrical circuits
(analog computers) are built to allow the observation of the
effect of different parameters. Nevertheless, these electrical
circuits have to be synchronized with the time history of data
as much as possible to be reliable. Hence, synchronization
becomes an important property to be studied.

Another characteristic of synchronization is its applications
in chaos based cryptography [9–11]. Even if this problem
can be found in a number of papers, most of them deal with
infinite settling time [9–11]. If we consider, for example, the
application of synchronization in secure communications, the
range of time during which the chaotic oscillators are not
synchronized corresponds to the range of time during which
the encoded message can unfortunately not be recovered or
sent. More than a difficulty, this is a catastrophe in digital
telecommunications, since the first bits of standardized bit
strings always contain signalization data, i.e., the “identity
card” of the message. Hence, it clearly appears that the
synchronization time has to be known and minimized, so
that the chaotic oscillators synchronize as fast as possible.
In this context, it is fair to say that there is a need to
study finite-time chaos synchronization. This characteristic
is also helpful because of its advantages in its applications in
engineering. Hence, it is important to investigate the finite-time
stability of nonlinear systems [12–15].

Furthermore, even if delay complicates synchronization,
dynamical systems with delay(s) are abundant in nature. They
occur in a wide variety of physical, chemical, engineering,
economics, and biological systems and their networks. There
are many examples where delay plays an important role. Some
of these examples are listed and presented by M. Lakshmanan
and D. V. Senthilkumar in [16]. The mathematical description
of delay dynamical systems will naturally involve the delay
parameter in some specified way. This can be in the form of
differential equations with delay or difference equations with
delay and so on [16]. Delay differential equations are given
in many ways. Again, we refer the reader to [16] for more
information.

In telecommunications, the delay notion is inescapable,
for example, because of the relative distance between the
transmitter and the receiver. Thus, the use of fast signals is
indicated. A solution is provided by electronic components
such as tunnel diodes. Developed in 1956 by Léo Esaki, the
tunnel diode is a nonlinear device used in very high frequency
amplifiers with low noise and in microwave conception. The
tunnel diode based sources can provide rf signals above
500 GHz. Fundamental oscillations at 712 GHz from a reso-
nant tunnel diode oscillator were demonstrated by Brown et al.
in 1991 [17]. Recently, fundamental frequency oscillations of
a resonant tunnel diode oscillator close to 831 and 915 GHz
and 1.04 THz at room temperature were reported by Suzuki
et al. [18]—more details can be found in the Ph.D. thesis of
L. Wang (2011) of the University of Glasgow [19], and
references therein. Considering the frequency capability of
the tunnel diode, it can be used as an efficient source for
chaotic signals in wireless communications; in particular, H.-P.
Ren et al. [20] show that, for some particular chaotic signals,
the Lyapunov exponents remain unaltered through multipath
propagation, thus making these systems able to sustain
communications with chaotic signals. However, experimental
observations of generalized synchronization phenomena in
microwave oscillators have been done by B. S. Dmitriev

1539-3755/2014/89(3)/032921(11) 032921-1 ©2014 American Physical Society

http://dx.doi.org/10.1103/PhysRevE.89.032921


PATRICK LOUODOP et al. PHYSICAL REVIEW E 89, 032921 (2014)

et al. [21], and their application to secure communications has
been carried out by A. A. Koronovskii et al. [22]. Nevertheless,
the synchronization schemes used in [20–22] are dealing with
asymptotical synchronization and can only guarantee that two
systems achieve synchronization as time tends to infinity, while
in real-world applications one always expects that two systems
achieve synchronization within a finite and/or predetermined
time. Besides, finite-time synchronization has been confirmed
to have better robustness against parametric and external
disturbances than the asymptotical one [23,24]—thus, the
importance of studying finite-time synchronization of chaotic
systems. The use of tunnel diodes to construct chaotic systems
was also reported by A. Pikovskii [25], A. Fradkov [26],
and others. For the purpose of this work, we base ourselves
on the circuit proposed by A. Y. Markov et al. [26], where
we replace the used tunnel diode by the one with reference
number 1N3858, for which the PSPICE model is available on
the Internet.

In this paper we study an active adaptive finite-time
synchronization of tunnel diodes based chaotic oscillators
with (the first case) and without (the second and third cases)
time delay in both the drive and response systems. Therefore,
we will investigate the adaptive synchronization of chaotic
systems using just one master state variable as output to
construct the nonlinear feedback coupling. A robust adaptive
response system will be therefore designed to always globally
synchronize the given driven tunnel diode based oscillator.
These results are of significant interest to infer relationships
between nonlinear feedback coupling, time delay, and finite-
time synchronization. To the best of our knowledge, this active
adaptive finite-time synchronization of chaotic systems using
a nonlinear feedback coupling has not yet been reported in the
literature.

The manuscript is organized as follows: In Sec. II, the circuit
model and its chaotic dynamics are investigated. Section III
deals with the adaptive synchronization problem. In this
section the active controller is designed and the theoretical
settling time in the case of systems without delay is developed.
Numerical simulations are performed to show the effectiveness
of the synchronization scheme. In Sec. IV, we investigate
the cases of delayed systems, considering two cases. The
theoretical settling is also obtained and the numerical results
are presented graphically to prove the effectiveness of the
scheme. In the last section we present the conclusions.

II. THE MODEL AND ITS PROPERTIES

Before starting to analyze adaptive synchronization and
delay, we introduce the elements of our circuit. Its electronic
diagram is given in Fig. 1(a), where the tunnel diode model
is the 1N3858 with the current-voltage characteristic given in
Fig. 1(b). The chaotic circuit is a RLC oscillator, where E is
a continuous voltage source.

The circuit diagram presents two capacitors C1 and C2 and
an inductor L. The nonlinearity is introduced by the tunnel
diode, which has its current-voltage characteristic depicted
in Fig. 1(b). As an approximation of this current-voltage
characteristic, we use the following relation [26]:

i(V1) = a1(V1 − b)3 − a2(V1 − b) + a3, (1)

FIG. 1. (a) Circuit diagram and (b) current voltage characteristic
of tunnel diode 1N3858.

where a1, a2, and a3 are positive constants. The circuit can be
described by the following system of differential equations:

ẋ1(τ ) = α[x2 − x1 − rf (x1)]

ẋ2(τ ) = β(x1 − x2 + rx3) (2)

ẋ3(τ ) = γ (E − x2),

where

c = C2

C1
, r = R1, q = r

Lw
, τ = wt,

� = 1 V, I = 1 A, x1 = V1

�
, x2 = V2

�
,

x3 = i

I
, α = c

q
, β = 1

q
, γ = q

r
, and

f (x1) = a1(x1 − b)3 − a2(x1 − b) + a3,

where V1 and V2 are the voltages at landmarks of C1 and C2,
respectively; i is the current which flows through the inductor;
and w is a constant. Note that Eq. (2) is similar to the so-called
modified Chua’s circuit [27].

Before entering into the description of the system under
study we present the chaotic behavior of the circuit due
to the influence of parameter α on the evolution of the
one-dimensional Lyapunov exponent [Fig. 2(a)] and on the
bifurcation diagram [Fig. 2(b)]. These two graphs allow us
to determine the value of the control parameter, leading to
a chaotic behavior of the system through period doubling.
Figure 2(a) shows that for some values of the parameter α the
system Eq. (2) has a positive Lyapunov exponent.

Figures 3(a) and 3(b) show the graphs x2 vs x1 and
x1 vs x3 obtained from PSPICE simulations of the circuit
of Fig. 1(a). An approximate behavior is displayed with
the parameters c = 8.4, q = 3.35, E = 0.25, r = 16, a1 =
1.3242872, a2 = 0.06922314, a3 = 0.005388, and b = 0.167
and initial conditions x1(0) = 0.15, x2(0) = 0.27, and x3(0) =
0.008.

In Fig. 4 the space (c,q) for which the system behaves
chaotically is shown in black. With α = 2.507462687 and the
values of the parameters given above, basing ourselves on
Fig. 4, it follows that the Lyapunov exponent is positive and
the system shows chaotic behavior (Fig. 5).

032921-2



FINITE-TIME SYNCHRONIZATION OF TUNNEL-DIODE- . . . PHYSICAL REVIEW E 89, 032921 (2014)

FIG. 2. (a) One-dimensional Lyapunov exponent and (b) bifurca-
tion diagram when E = 0.253208 and q = 3.6.

The graphs of Fig. 6 show the Poincaré sections which help
to prove the chaotic behavior of the model [Eq. (2)].

III. ADAPTIVE SYNCHRONIZATION

The starting point of the control law design is always a
mathematical model of the real system which describes the
system response to various control inputs. It is usually assumed
that such a model is known rather completely. However, in
practice the model describes the real system with some degree
of uncertainty. To cope with this uncertainty in such cases,
methods of the adaptive control theory can be employed. The
concept of chaos synchronization emerged much later, not
until the gradual realization of the usefulness of chaos by
scientists and engineers. Chaotic signals are usually broadband
and noiselike. Because of these properties, synchronized

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28
0.2

0.22

0.24

0.26

0.28

x
1

x 2

(a )

−0.01 −0.005 0 0.005 0.01 0.015

0.1

0.15

0.2

0.25

x
3

x 1

(b )

FIG. 3. Chaotic attractor obtained through PSPICE simulation of
the circuit of Fig. 1(a).

7 8 9 10 11 12 13
2

2.5

3

3.5

4

c

q

FIG. 4. Pair (c,q) for which chaos occurs with E = 0.25.

chaotic systems can be used as cipher generators for secure
communication.

In the following section, we investigate two identical
systems of the type described in the previous section, where
the system drive xi(τ ),i = 1,2,3 drives the response system
yi(τ ),i = 1,2,3.

A. Design of the nonlinear controller

Here we investigate the finite-time synchronization of
tunnel diode based chaotic oscillators described by Eq. (2). For
this purpose we consider the corresponding response system
described by

ẏ1 = α[y2 − y1 − rf (y1)] − ζksign(y1 − x1) − ζu(τ )

ẏ2 = β[y1 − y2 + ry3 − 2(y1 − x1)] (3)

ẏ3 = γ (E − y2),

where k and ζ are positive constant gains to be de-
fined by the designer. The term −2(y1 − x1) added in the

0.05 0.1 0.15 0.2 0.25
0.22

0.23

0.24

0.25

0.26

0.27

0.28

x
1

x 2

0.05 0.1 0.15 0.2 0.25

0

5

10

15

x 10
−3

x
1

x 3

(a)

(b)

FIG. 5. Chaotic attractor obtained through numerical simulation
of Eq. (2).

032921-3



PATRICK LOUODOP et al. PHYSICAL REVIEW E 89, 032921 (2014)

0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17
−0.04

−0.02

0

0.02

x
1

dx
1/d

τ

0.24 0.245 0.25 0.255 0.26 0.265

−5

0

5

10

x 10
−3

x
2

dx
2/d

τ

4 5 6 7 8 9 10 11 12
x 10

−3

−4

−3

−2

−1

0

1

2

3x 10
−3

x
3

dx
3/d

τ

(a)

(b)

(c)

FIG. 6. Poincaré sections with E = 0.253205, c = 8.846, and
q = 3.6.

y2 axis stabilizes the slave system. The nonlinear con-
troller is expressed as −ζksign(y1 − x1) − ζu(τ ), where
u(τ ) is the adaptive feedback coupling designed to achieve
finite-time synchronization. All parts of the used controller
contribute to synchronize both drive-response systems at
an established time and also to stabilize the response
dynamics.

Let us define the synchronization error as follows:

ei(τ ) = yi(τ ) − xi(τ ), with i = 1,2,3. (4)

Subtracting Eq. (2) from Eq. (3) and using Eq. (4), we
obtain

ė1 = α[e2 − e1 − φ0(x1,y1)] − ζksign(e1) − ζu(τ )

ė2 = β[−e1 − e2 + re3] (5)

ė3 = −γ e2,

where φ0(x1,y1) = r [f (y1) − f (x1)].
The synchronization problem can be stated as follows:

considering the transmitter Eq. (2) and the receiver Eq. (3)
with any initial conditions, e1(0) = y1(0) − x1(0), e2(0) =
y2(0) − x2(0), e3(0) = y3(0) − x3(0), it is our aim to design
the form of the function u(τ ) which synchronizes the orbits of
both the transmitter and the response systems and thus provides

the stabilization of Eq. (5) at an established finite-time τs , i.e.,

lim
t→τs

xi(τ ) = yi(τ ), i = 1,2,3,

(6)
lim
τ→τs

‖e(τ )‖ = 0.

where τs is the settling time. We consider that the function
φ0(x1,y1) respects the Lipschitz condition; i.e., there exists a
positive constant χ0 such that |φ0(x1,y1)| � χ0|e1(τ )|.

B. Main results

The global finite-time synchronization is achieved when
the Lyapunov function candidate is proper [15], namely, the
time derivative of the used Lyapunov function is bounded by
a negative constant.

Let us consider the following candidate for the Lyapunov
function [28]:

V = 1

2

(
e2

1

α
+ e2

2

β
+ re2

3

γ

)
+ |u(τ )|. (7)

The time derivative along the trajectories of the system of
Eqs. (5) yields

V̇ = −e2
1 − e2

2 − φ0(x1,y1)e1 − ζk

α
|e1| − ζu(τ )

α
e1

+ sign(u)u̇(τ ),

V̇ � −e2
1 − e2

2 + |φ0(x1,y1)||e1| − ζk

α
|e1| − ζu(τ )

α
e1

+ sign(u)u̇(τ ), (8)

V̇ � − (1 − χ0) e2
1 − e2

2 − ζk

α
|e1| − ζu(τ )

α
e1

+ sign(u)u̇(τ ).

Letting χ0 < 1 and ζ = α,

V̇ � −k|e1| − u(τ )e1 + sign(u)u̇(τ ). (9)

Considering the finite-time stability theory applied in [15],
it follows that there exists a certain positive constant p such
that the following relation holds:

V̇ � −p. (10)

If the objective is reached, the theoretical finite time for
synchronization is obtained by integrating Eq. (10) from zero
to τs . Thus, one has

τs � 1

2p

(
e2

1(0)

α
+ e2

2(0)

β
+ re2

3(0)

γ

)
+ |u(0)|

p
. (11)

Therefore, in accordance with Eq. (10), the controller u(τ )
is designed through the following relation:

u̇ = sign(u) (k|e1| + ue1 − p) , (12)

and we construct the slave system Eq. (3) as follows:

ẏ1 = α[y2 − y1 − rf (y1)] − ζksign(y1 − x1) − ζu

ẏ2 = β[y1 − y2 − ry3 − 2(y1 − x1)]
(13)

ẏ3 = γ [E − y2]

u̇ = sign(u)(k|e1| + ue1 − p).

032921-4



FINITE-TIME SYNCHRONIZATION OF TUNNEL-DIODE- . . . PHYSICAL REVIEW E 89, 032921 (2014)

(a)

(b)

(c)

0 50 100 150 200 250 300
−0.1

0

0.1

0.2

0.3

0.4

τ

x
1
(τ) y

1
(τ)

(d)

0 50 100 150 200 250 300

0

0.2

0.4

0.6

τ

x
2
(τ) y

2
(τ)

(e)

0 50 100 150 200 250 300

−0.05

0

0.05

τ

x
3
(τ) y

3
(τ)

(f)

FIG. 7. Time histories of state variables: the solid line depicts the
drive system, and the dashed line depicts the response system.

C. Numerical results

To illustrate the effectiveness of the proposed scheme
we present some numerical results for the initial condi-
tions given by (x1(0),x2(0),x3(0)) = (0.15,0.27,0.008) and

FIG. 8. Time histories of (a) e1(τ ) and (b) the controller u(τ ).

(y1(0),y2(0),y3(0)) = (−0.15,−0.27,−0.008); the parame-
ters p = 0.001, k = 0.005; and the initial condition of con-
troller u(0) = 0.0001. The integration time step was taken as
10−4. For such values, the theoretical settling time is deter-
mined to be τsTH = 521.0579. The graphs of Figs. 7(a)–7(c)
show the time dependence of the master system state variables
(solid lines) and the corresponding slave system state variables
(dashed lines), while those in Figs. 7(d)–7(f) show the same
graphs for another set of initial conditions, for which the
controller was turned on at τ = 50 to show the dynamics
before and after synchronization. From the graphs in Fig. 8
we confirm that the synchronization is reached in finite time.
It is observed that synchronization is reached at time τsNU �
245.3, which respects the finite-time condition τsNU � τsTH

[Fig. 8(b)].
The value of the finite time of stability depends strongly

on the values of parameters p. Considering k � 0.005 the
following graphs represent the behavior of the error state e1(τ )
and of the adaptive parameter u(τ ) for two given values of
p [Fig. 9(a)]. In principle p can take any values, but to deal
with finite-time stability it should be small. The graphs in
Fig. 9 represent the behavior of both e1(τ ) and u(τ ) for p =
0.0017 and 0.002 [Figs. 9(a) and 9(b) and Figs. 9(c) and 9(d),
respectively). In this paper we define the numerical finite time
of convergence as the end time of the transitory phase of the
time evolution of e1(τ ). For the graphs in Figs. 9(a) and 9(b)
the numerical finite time is τsNU � 100 and the theoretical
finite time is τsTH � 306.5047, while for the ones in Figs. 9(c)
and 9(d) the theoretical finite time is τsTH � 260.5290 and
the numerical finite time is τsNU > τsTH. Thus we see that the
finite-time condition τsNU � τsTH is fulfilled in both cases.

IV. FINITE-TIME TIME-DELAY SYNCHRONIZATION OF
TUNNEL DIODE BASED OSCILLATORS

A. Systems with internal delay

1. Synchronization analysis

In this section, we investigate the finite-time synchroniza-
tion of delayed tunnel-diode-based chaotic oscillators. We first
consider the presence of one delay affecting the nonlinearity

032921-5



PATRICK LOUODOP et al. PHYSICAL REVIEW E 89, 032921 (2014)

0 50 100 150 200 250 300

−0.2

−0.1

0

0.1

0.2

e 1(τ
)

τ (a)

0 100 200 300 400 500 600 700
−5

0

5
x 10

−3

U
(τ

)

τ (b)

0 50 100 150 200 250 300
−0.4

−0.2

0

0.2

0.4

e 1(τ
)

τ (c)

0 100 200 300 400 500 600 700
−5

0

5
x 10

−3

U
(τ

)

τ (d)

FIG. 9. Behaviors of e1(τ ) and u(τ ). (a and b) p = 0.0017. (c and
d) p = 0.02.

of each drive system and response system. Thus, the master
and slave systems become, respectively,

ẋ1(τ ) = α{x2 − x1 − rf [x1(τ − θ )]}
ẋ2(τ ) = β(x1 − x2 + rx3) (14)

ẋ3(τ ) = γ (E − x2)

and

ẏ1 = α{y2 − y1 − rf [y1(τ − θ )]}
− ζksign(y1 − x1) − ζu(τ )

(15)
ẏ2 = β[y1 − y2 + ry3 − 2(y1 − x1)]

ẏ3 = γ [E − y2],

where θ is the time delay.

For such a case, we define the functions φ1(x1(τ −
θ ),y1(τ − θ )) = r {f [y1(τ − θ )] − f [x1(τ − θ )]} and e1(τ −
θ ) = y1(τ − θ ) − x1(τ − θ ). Thus the error state is given by
the following set of equations:

ė1 = α[e2 − e1 − φ1(x1(τ − θ ),y1(τ − θ ))]

− ζksign(e1) − ζu(τ )
(16)

ė2 = β[−e1 − e2 + re3]

ė3 = −γ e2.

From here, using the same controller as in the previous sec-
tion, we choose the following Krasovskii-Lyapunov function
candidate [28,31]:

V = 1

2

(
e2

1

α
+ e2

2

β
+ re2

3

γ

)
+ |u(τ )|

+ η

∫ 0

−θ

e2
1 (τ + s) ds, (17)

where η is a positive constant to be determined.
The time derivative along the trajectories of the system

Eq. (16) yields

V̇ = −(1 − η)e2
1 − e2

2 − φ1(τ − θ )e1 − ζk

α
|e1|

− ζu(τ )

α
e1 + sign(u)u̇(τ ) − ηe2

1(τ − θ ),
(18)

V̇ � −(1 − η)e2
1 − e2

2 + |φ1(τ − θ )||e1| − ζk

α
|e1|

− ζu(τ )

α
e1 + sign(u)u̇(τ ) − ηe2

1(τ − θ ).

Let us assume that |φ1(τ − θ )| � χ1|e1(τ − θ )|, where χ1 is a
positive constant. It follows from here that

V̇ � −e2
1 − e2

2 + χ1|e1(τ − θ )||e1| − ζk

α
|e1| − ζu(τ )

α
e1

+ sign(u)u̇(τ ) + ηe2
1 − ηe2

1(τ − θ ),

V̇ � −
(

1 − χ1

2
− η

)
e2

1 − e2
2 +

(χ1

2
− η

)
e2

1(τ − θ )

− ζk

α
|e1| − ζu(τ )

α
e1 + sign(u)u̇(τ ). (19)

Let η = χ1

2
, χ1 < 1, and ζ = α. It follows that

V̇ � −k|e1| − u(τ )e1 + sign(u)u̇(τ ). (20)

Thus, using the controller

u̇ = sign(u) (k|e1| + ue1 − p) , (21)

it follows that

V̇ � −p. (22)

Hence, global finite-time stability is achieved [15]. For any
time τ contained in the interval 0 < τ < θ both the drive and
response system do not oscillate at the considered regime.
Therefore, to determine the theoretical finite settling time, we
integrate Eq. (22) from θ to τs and we obtain

V (τs) − V (θ ) � −p (τs − θ ) . (23)

032921-6



FINITE-TIME SYNCHRONIZATION OF TUNNEL-DIODE- . . . PHYSICAL REVIEW E 89, 032921 (2014)

Using the fact that V (τs) → 0 when τ → τs , V > 0∀τ and,
taking into account Eq. (22), we can see that the Lyapunov
function V is a monotonous and decreasing function. It follows
that

τs = θ + 1

2p

(
1

α
e2

1(θ ) + 1

β
e2

2(θ ) + R

γ
e2

3(θ )

)
+ |u(θ )|

p

+ η

∫ 0

−θ

ε2
1 (θ + s) ds,

τs � θ + 1

2p

(
1

α
e2

1(θ ) + 1

β
e2

2(θ ) + R

γ
e2

3(θ )

)
+ |u(θ )|

p
.

(24)

Equation (24) gives the maximum settling time for synchro-
nization. Hence, the finite-time synchronization is reached
when the numerical settling time τsNU satisfies the rela-
tion τsNU < τsTH, where τsTH = θ + 1

2p
( 1
α
e2

1(θ ) + 1
β
e2

2(θ ) +
R
γ
e2

3(θ )) + |u(θ)|
p

.

2. Numerical results

In this section we investigate the finite-time synchronization
of two delayed tunnel-diode-based chaotic systems basing
ourselves on the obtained numerical results of the established
theory in the previous subsection. The integration time step
taken was 10−4. Figure 10 helps to confirm the synchronization
behavior of both the delayed master system [Eq. (14)] and
the slave system [Eq. (15)] when p = 0.001 and k = 0.005.
With the selected values of parameters p and k, the theoretical
settling time result is τsTH � 127.5448, while the numerical
settling time corresponding to the condition given before is
τsNU � 99.4.

When varying the time lag θ , just the theoretical settling
time is modified when synchronization occurs and the numer-
ical finite time is determined as τsNU � 99.4.

0 50 100 150 200 250 300
−0.2

−0.1

0

0.1

0.2

e 1(τ
)

τ (a)

0 50 100 150 200 250 300
0

0.1

0.2

0.3

0.4

e n
(τ

)

τ (b)

FIG. 10. Time history of (a) e1(τ ) and (b) en(τ ) =√
e2

1(τ ) + e2
2(τ ) + e2

3(τ ).

B. Multidelayed systems

1. Synchronization analysis

In this section we consider that there exist two delays:
θ1 represents the time lag taken for the introduction of the
nonlinearity, and θ2 represents the time delay between the
master state and the slave state. Thus, the systems become

ẋ1(τ − θ2) = α{x2(τ − θ2) − x1(τ − θ2)

− rf [x1(τ − θ1 − θ2)]}
ẋ2(τ − θ2) = β[x1(τ − θ2) − x2(τ − θ2)

(25)
+ rx3(τ − θ2)]

ẋ3(τ − θ2) = γ [E − x2(τ − θ2)]

and

ẏ1(τ ) = α{y2(τ ) − y1(τ ) − rf [y1(τ − θ1)]}
− ζksign[y1(τ ) − x1(τ − θ2)] − ζu(τ )

ẏ2(τ ) = β{y1(τ ) − y2(τ ) + ry3
(26)

− 2[y1(τ ) − x1(τ − θ2)]}
ẏ3(τ ) = γ [E − y2(τ )].

In this case, we define the functions φ2(τ,θ1,θ2) =
r {f [y1(τ − θ1)] − f [x1(τ − θ1 − θ2)]} and e1(τ − θ1 −
θ2) = y1(τ − θ1) − x1(τ − θ1 − θ2). Thus the error state is
given by the following set of equations:

ė1 = α[e2 − e1 − φ2(τ,θ1,θ2)]

− ζksign(e1) − ζu(τ )
(27)

ė2 = β[−e1 − e2 + re3]

ė3 = −γ e2.

Let us now select the Krasovskii-Lyapunov function as [28,31]

V = 1

2

(
e2

1

α
+ e2

2

β
+ re2

3

γ

)
+ |u(τ )|

+ λ

∫ 0

−θ1

e2
1(τ − θ2 + s)ds, (28)

where λ is a positive constant to be determined.
The time derivative along the trajectories of the system

Eq. (27) yields

V̇ = −(1 − λ)e2
1 − e2

2 − φ2(τ,θ1,θ2)e1 − ζk

α
|e1|

− ζu(τ )

α
e1 + sign(u)u̇(τ ) − λe2

1(τ − θ1 − θ2),
(29)

V̇ � −(1 − λ)e2
1 − e2

2 + |φ2(τ,θ1,θ2)||e1| − ζk

α
|e1|

− ζu(τ )

α
e1 + sign(u)u̇(τ ) − λe2

1(τ − θ1 − θ2).

Let us assume that |φ2(τ,θ1,θ2)| � χ2|e1(τ − θ1 − θ2)|,
where χ2 is a positive constant. It follows from

032921-7



PATRICK LOUODOP et al. PHYSICAL REVIEW E 89, 032921 (2014)

here that

V̇ � −e2
1 − e2

2 + χ2|e1(τ − θ1 − θ2)||e1| − ζk

α
|e1|

− ζu(τ )

α
e1 + sign(u)u̇(τ ) + λe2

1 − λe2
1(τ − θ1 − θ2),

V̇ � −
(

1 − χ2

2
− λ

)
e2

1 +
(χ2

2
− λ

)
e2

1(τ − θ1 − θ2)

− ζk

α
|e1| − e2

2 − ζu(τ )

α
e1 + sign(u)u̇(τ ). (30)

Let λ = χ2

2
, χ2 < 1, and ζ = α. It follows that

V̇ � −k|e1| − u(τ )e1 + sign(u)u̇(τ ). (31)

Thus, using the controller

u̇ = sign(u) (k|e1| + ue1 − p) , (32)

it follows that

V̇ � −p. (33)

Hence, the global finite-time stability is achieved [15]. When
0 < τ < θ1 + θ2 the system cannot oscillate at the considered
regime. In order to determine the theoretical finite settling
time, we integrate Eq. (33) from θ1 + θ2 to τs . Thus, the finite
settling time is given by

τs = 1

2p

(
1

α
e2

1(θ1 + θ2) + 1

β
e2

2(θ1 + θ2) + R

γ
e2

3(θ1 + θ2)

)

+ |u(θ1 + θ2)|
p

+ λ

∫ 0

−θ1+θ2

ε2
1 (θ1 + θ2 + s) ds + θ1 + θ2,

τs � 1

2p

(
1

α
e2

1(θ1 + θ2) + 1

β
e2

2(θ1 + θ2) + R

γ
e2

3(θ1 + θ2)

)

+ |u(θ1 + θ2)|
p

+ θ1 + θ2. (34)

At this moment, as in the previous section, the finite-time
synchronization is reached when the numerical settling time
τsNU satisfies the relation τsNU < τsTH, where

τsTH = 1

2p

(
1

α
e2

1(θ1 + θ2) + 1

β
e2

2(θ1 + θ2) + R

γ
e2

3(θ1 + θ2)

)

+ |u(θ1 + θ2)|
p

+ θ1 + θ2.

0 50 100 150
−0.2

−0.1

0

0.1

0.2

e 1(τ
)

τ

FIG. 11. Time history of e1(τ ).

0 50 100 150
−0.2

−0.1

0

0.1

0.2

e 1(τ
)

τ (a)

0 50 100 150
0

0.1

0.2

0.3

0.4

0.5

e n
(τ

)

τ (b)

FIG. 12. Time history of (a) e1(τ ) and (b) en(τ ) when θ2 = 0.0003.

2. Numerical results

From our investigations, it comes out that the finite-time
synchronization is hardly reached when we consider that
both dynamics of the drive system and response system are
subjected to delay θ2. The integration time step used in this
part was 10−4. The graphs in Fig. 11 show that the goal can be
achieved only if θ2 is relatively small. For the values k = 0.005,
p = 0.001, θ1 = 0.1, and θ2 = 0.0003, one has the following
behavior.

Considering the behavior of the synchronization error e1(τ )
and en(τ ) (defined in Fig. 10), one can observe the destruction
of the synchronization when we increase the value of θ2

(Figs. 12–14).

0 50 100 150 200 250 300
−0.2

−0.1

0

0.1

0.2

e 1(τ
)

τ (a)

0 50 100 150 200 250 300
0

0.1

0.2

0.3

0.4

0.5

e n
(τ

)

τ (b)

FIG. 13. Time history of (a) e1(τ ) and (b) en(τ ) when θ2 = 0.004.

032921-8



FINITE-TIME SYNCHRONIZATION OF TUNNEL-DIODE- . . . PHYSICAL REVIEW E 89, 032921 (2014)

0 50 100 150
−0.2

−0.1

0

0.1

0.2

e 1(τ
)

τ (a)

(b)0 50 100 150
0

0.1

0.2

0.3

0.4

0.5

e n
(τ

)

τ

FIG. 14. Time history of (a) e1(τ ) and (b) en(τ ) when θ2 = 0.08.

V. CONCLUSIONS

The target of this paper was to investigate the possibility to
achieve synchronization in finite time of two tunnel-diode-
based chaotic oscillators. We have considered the case of
chaotic systems without and with delay (internal delay and
multiple delay). The controller was built basing ourselves
on the absolute stability theory [28] and on the Krasovskii-

FIG. 15. Circuit diagram of the analog Tunnel Diode based
Oscillator system.

FIG. 16. Chaotic attractor from the circuit in Fig. 15.

Lyapunov stability theory [31]. Later, the expression of the
settling finite time was investigated in all considered cases.
Numerical simulations were performed and given to confirm
our theoretical analysis. We observe that the finite-time

FIG. 17. Circuit diagram of the drive system.

032921-9



PATRICK LOUODOP et al. PHYSICAL REVIEW E 89, 032921 (2014)

FIG. 18. Circuit diagram of the response system.

synchronization is reached when some conditions, which are
given and proven by numerical results, are filled.

APPENDIX: CIRCUIT DESIGN AND SIMULATIONS

Here we investigate the electrical circuit of the system. First
we build the analog computer of the chaotic oscillator, which
is shown in Fig. 15. The use of the analog computer here is
helpful because we can modify the frequency of the oscillator
and thus facilitate the design of the controller.

FIG. 19. (Color online) Circuit diagram of the controller.

FIG. 20. Time histories of ei(t), i = 1,2,3 and of the systems
variables.

This circuit is obtained through the following relations:
α = 1

χR1Cx1
= 1

χR5Cx1
, αra1 = 1

χR3Cx1
, αra2 = 1

χR2Cx1
, αra3 =

1
χR4Cx1

, β = 1
χR7Cx2

= 1
χR6Cx2

, βr = 1
χR8Cx2

, γ = 1
χR9Cx3

, and

γE = 1
χR10Cx3

, where χ is a parameter chosen as χ =
104. Thus, considering that R = 10k� and all capacitors
are C = 10nF , we obtain the following component values:

032921-10



FINITE-TIME SYNCHRONIZATION OF TUNNEL-DIODE- . . . PHYSICAL REVIEW E 89, 032921 (2014)

R1 = R5 = 3988.095 238�, R3 = 188.218 9546�, R2 =
3.48k�, R4 = 46 261.312 62�, R6 = R7 = 33.5k�, R8 =
2093.75�, R9 = 47 761.194 03�, and R10 = 191 044.7761�.
The chaotic behavior of the above circuit is given by the graphs
in Fig. 16.

Later, the drive (Fig. 17) and response (Fig. 18) systems
are designed as follows. The controller (Fig. 19) is designed
using the following parameters: 2β = 1

χRC1 Cy2
, ζk = 1

χRC2 Cy1
,

ζ = 1
χRC3 Cy1

, k = 1
χRC4 Cu1

, 1 = 1
χRCu1

, and p = 0.001
χRCu1

, with

which we obtain the component values: RC1 = 16.75k�,
RC2 = 797 619.0476�, RC3 = 3988.095 238�, RC4 = 2M�,
RS1 = 1k�, and RS2 = 13.508k�. The part of the controller
circuit in the red box, according to J. C. Sprott [29], simulates
the absolute value function, while the ones in the green
domains simulate the sign functions [30]. The results obtained
from these circuits are given in Fig. 20.

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