Luo A.C.J. (Ed.) Dynamical Systems: Discontinuity, Stochasticity and Time-Delay
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17 Impulsive Control Induced Effects on Dynamics of Complex Networks 215
−80
−60
−40
−20
0
20
40
60
80
−200
−150
−100
−50
0
50
100
150
200
−150
−100
−50
0
50
100
150
200
x
11
x
12
x
13
Fig. 17.6 The dynamical behavior of the first node of networks for D 3:5
17.4 Conclusion
Control and synchronization of complex networks have been extensively
investigated in many research and application fields. Previous works focused upon
realizing synchronization by vary methods. There has been little research on the
dynamics of synchronization manifold of complex networks by now. In this chapter,
impulsive control induced effects on dynamics of complex networks are investi-
gated. And the stability of complex networks with different dynamical nodes by
such impulsive control is discussed too,where the impulse effects have outer input
signals. Complex networks realize synchronization with such special impulsive
control. The dynamics of complex networks are affected by changes of outer input
signals. Complex networks can realize synchronization and its synchronization
manifold can be changed with such impulsive control signals. The synchronization
orbit may be an equilibrium point, a periodic orbit,or a chaotic orbit, which is
determined by a parameter in outer impulse single. Strict theories are given, and
some numerical simulations confirm the correctness of theoretical results.
Acknowledgments This work was supported by the National Natural Science Foundation of
China (Grant No. 10871120), the China Tianyuan Youth Founding of Mathematics (Grant No.
10826029), the Science & Technology Development Funds of Shandong Education Committee
(Grant No. J08LI10).
References
1. BarabKai AL, Albert R, Jeong, H (1999) Mean-field theory for scale-free random networks.
Physica A 272:173–187
2. Fu XL, Yan BQ, Liu YS (2005) Introduction of impulsive differential system. Academic Press
of China, Beijing
3. Goto S, Nishioka T, Kanehisa M (1998) LIGAND: Chemical database for enzyme reactions.
Bioinformatics 14:591–599
216 X. Han and X. Fu
4. Han XP, Lu JA (2010) Impulsive control induced effects on dynamics of single and coupled
ODE systems. Nonlinear Dyn 59:101–111
5. Horne AB, Hodgman TC, Spence HD, Dalby AR (2004) Constructing an enzyme-centric view
of metabolism. Bioinformatics 20:2050–2055
6. Itoh M, Yamamoto N, Yang T, Chua LO (1999) Performance analysis of impulsive synchro-
nization. In: Proceedings of the 1999 European conference on circuit theory and design, Stresa,
Italy, pp 353–356
7. Itoh M, Yang T, Chua LO (2001) Experimental study of impulsive synchronization of chaotic
and hyperchaotic circuits. Int J Bifurcat Chaos 11:1393–1424
8. Jeong H, Tombor B, Albert R, Oltvai ZN, BarabKai AL (2000) The large-scale organization of
metabolic networks. Nat Lond 407:651–654
9. Khadra A, Liu XZ, Shen X (2005) Impulsively synchronizing chaotic systems with delay and
application to secure communication. Automatica 41:1491–1502
10. Lakshmikantham V, Bainov D, Simeonov P (1989) Theory of impulsive differential equations.
World Scientific, Singapore
11. Li C, Chen G (2004) Synchronization in general complex dynamical networks with coupling
delays. Phys A 343:263–278
12. Li ZG, Wen CY, Soh YC (2001) Analysis and design of impulsive control systems. IEEE Trans
Automat Control 46:894–897
13. Li C, Xu H, Liao X, Yu J (2004) Synchronization in small-world oscillator networks with
coupling delays. Phys A 335:359–364
14. Lin W (2002) Some problems in chaotic systems and their applications. Fudan University’s
doctoral dissertation
15. Liu XZ, Willms AR (1996) Impulsive controllability of linear dynamical systems with appli-
cations to maneuvers of spacecraft. Math Prob Eng 2:277–299
16. Liu B, Liu XZ, Chen GR, Wang HY (2005) Robust impulsive synchronization of uncertain
dynamical networks. IEEE Trans Circuits Syst I 52:1431–1440
17. LRu, J, Yu X, Chen G (2004) Chaos synchronization of general complex dynamical networks.
Phys A 334:281–302
18. Ruan J, Lin W (1999) Chaos in a class of impulsive differential equation. Commun Nonlinear
Sci Numer Simul 4:166–169
19. Strogatz SH (2001) Exploring complex networks. Nature 410:268–276
20. Wang X, Chen G (2002) Synchronization in scale-free dynamical networks: Robustness and
fragility. IEEE Trans Circuits Syst I 49:54–62
21. Watts DJ, Strogatz SH (1998) Collective dynamics of small-world. Nature 393:440–442
22. Yang T (2001) Impulsive control theory. Springer, Berlin
23. Yang T, Chua LO (1997) Impulsive control and synchronization of nonlinear dynamical sys-
tems and application to secure communication. Int J Bifurcat Chaos 7:645–664
24. Yang T, Chua LO (1997) Impulsive stabilization for control and synchronization of chaotic sys-
tems: theory and application to secure communication. IEEE Trans Circuits Syst I 44:976–988
25. Yang T, Yang CM, Yang LB (1997) Control of Rossler systems to periodic motions using
control method. Phys Lett A 232:356–361
26. Zhou J, Lu J, LRu, J (2006) Adaptive synchronization of an uncertain complex dynamical net-
work. IEEE Trans Automat Control 51:652–656
Chapter 18
Study on Synchronization of Two Identical
Uncoupled Neurons Induced by Noise
Ying Wu, Ling Hong, Jun Jiang, and Wuyin Jin
Abstract In this paper, noise-induced synchronization between two identical
uncoupled neurons is investigated by using Hodgkin–Huxley .H H/ and FHN
models with sinusoidal stimulations. The numerical results show that the value of
the critical noise intensity for synchronizing two neurons is much less than the
magnitude of mean size of the attractor of the original system, the deterministic
dynamics structure of the original attractor is not swamped under noise, and the
deterministic feature of the attractor of the original system is affected by noise
slightly for FHN neurons. This finding is significantly different from the previous
work (Phys Rev E 67:027201, 2003).
18.1 Introduction
Noise-induced synchronization is widely studied [1–9]. Theoretical results have in-
spired experimental work since the noise-induced synchronization was observed in a
pair of uncoupled sensory neurons of biological system [10]. It was confirmed that
phase synchronization could be achieved between two coupled non-identical HR
neurons when the intensity of common noise exceeds a critical value [9]. The same,
two uncoupled identical HR or HH neurons are able to achieve complete syn-
chronization [6, 8], and two uncoupled non-identical HR neurons or HH neurons
are able to achieve generalized synchronization or phase synchronization respec-
tively [7,8]. Specially, the results of Ref. [6] show that the value of the critical noise
intensity for synchronizing two identical uncoupled neurons is roughly equal to the
magnitude of mean size of the attractor of the original system, and the randomicity
of noise swamps the deterministic dynamics structure of the original attractor.
Y. Wu (
)
School of Science, Xi’an University of Technology, Xi’an, Shaanxi 710048, People’s Republic of
China
e-mail: wying36@xaut.edu.cn
A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay,
DOI 10.1007/978-1-4419-5754-2
18,
c
Springer Science+Business Media, LLC 2010
217
218 Y. Wu et al.
In this paper, noise-induced synchronization between two identical uncoupled
HH neurons and FHN neurons is numerically investigated. The results show that
the value of the critical noise intensity for synchronizing two neurons is much less
than the mean size of the attractor of the original system, and the deterministic
dynamics structure of the original attractor is not swamped under noise, but the de-
terministic dynamics structure of the original attractor in FHN models is changed
slightly by noise. This result is completely different from the conclusion of Ref. [6],
which means that the value of critical noise intensity reaching or exceeding the mag-
nitude of size of the attractor of the original system for synchronizing two identical
uncoupled neurons would not be necessary.
18.2 Numerical Results and Discussion
In Ref. [6], the equations of two identical uncoupled HR neurons are given as
follows:
8
<
:
Px
1;2
D y
1;2
ax
3
1;2
C bx
2
1;2
z
1;2
C I C k.t/
Py
1;2
D c dx
2
1;2
y
1;2
Pz
1;2
D rŒS.x
1;2
/ z
1;2
(18.1)
the parameter I is external input current, .t/ is white noise with zero mean, and
h
.t/.t /
i
D ı./; k is noise intensity. Other parameters can be obtained in [6].
The result is given in Fig. 18.1,Fig.18.1a corresponds to attractors without noise,
and Fig. 18.1b to synchronized attractors with noise intensity k D 2:4.
Obviously, in the noiseless conditions, neurons is chaotic firing for ID3:3.After
noise-induced synchronization, the deterministic dynamics structure of the original
attractors is swamped by noise.
x
I=3.3
a
zz
x
I=3.3
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
3.25 3.3 3.35
−4
−3
−2
−1
0
1
2
3
4
5
b
3.4 3.45
2.4 2.5 2.6 2.7 2.8 2.9
Fig. 18.1 The projections of attractors of HR neurons in z x plane for I D 3:3, the projections
of noiseless attractors (a); the projections of attractors synchronized by noise, k D 2:4 (b)
18 Study on Synchronization of Two Identical Uncoupled Neurons Induced by Noise 219
The differential equations of single H H model are given as follows:
8
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
:
C
dV
dt
Dg
Na
m
3
h.V V
Na
/ g
k
n
4
.V V
K
/ g
L
.V V
L
/ C I
ext
dm
dt
D ˛
m
.V /.1 m/ ˇ
m
.V /m
dn
dt
D ˛
n
.V /.1 n/ ˇ
n
.V /n
dh
dt
D ˛
h
.V /.1 h/ ˇ
h
.V /h
(18.2)
where V is the membrane potential, the definitions of the other variables could be
found in [11], and, the parameters C D 1F=cm
2
;g
Na
D120 mS=cm
2
;g
K
D36
mS=cm
2
;g
L
D0:3 mS=cm
2
;V
Na
D50 mV;V
K
D77 mV;V
L
D54:4 mV;I
ext
is an external periodic signal current, I
ext
DI
shift
C sin.2ft/,whereI
shift
D10 A
=cm
2
, is the amplitude of current shift, and f Df
0
is the stimulus frequency,
f
0
D20 Hz is basic stimulus frequency. In this work, the double precision fourth-
order Runge Kutta method is used with integration time step 0.05, and the thresh-
old V
th
D40. As parameter varies, there are a lot of firing types, such as
quasi-periodic, bursting, chaotic, and periodic motions.
Adding white noise .t/ with zero mean to external input current, the differential
equations of two uncoupled H H models are given as follows:
8
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
:
C
dV
1;2
dt
Dg
Na
m
1;2
3
h
1;2
.V
1;2
V
Na
/ g
k
n
1;2
4
.V
1;2
V
K
/ g
L
.V
1;2
V
L
/
CI
ext
C k.t/
dm
1;2
dt
D ˛
m
.V
1;2
/.1 m
1;2
/ ˇ
m
.V
1;2
/m
1;2
dn
1;2
dt
D ˛
n
.V
1;2
/.1 n
1;2
/ ˇ
n
.V
1;2
/n
1;2
dh
1;2
dt
D ˛
h
.V
1;2
/.1 h
1;2
/ ˇ
h
.V
1;2
/h
1;2
(18.3)
where k is noise intensity. In numerical simulation, the initial conditions are
.V
1
;m
1
;n
1
;h
1
/
0
D.60; 0:2; 0:4; 0:45/ and .V
2
;m
2
;n
2
;h
2
/
0
D.65; 0:1; 0:4;
0:45/, the data of n 10
4
are ignored to avoid transients.
According to the definition in [6], the mean size of attractor of H H model
without noise corresponding to variable V is S
V
106:8, and the mean size is not
changed as stimulus frequency varies. In Fig. 18.2,wegiveV h projections of the
attractors for f D 4:7f
0
. The critical noise intensity for synchronizing two H H
neurons is k D 4:3. The critical noise intensity are much less than the magnitude
of the mean size of the original attractor .S
V
106:8/. Figure 18.2a corresponds
to the attractors without noise which is chaotic firing, Fig. 18.2b to synchronized
attractors. Obviously, after synchronizing the original attractors is not swamped by
noise, which is completely different from the conclusion of Ref. [6].
220 Y. Wu et al.
0 0.1 0.2 0.3 0.4 0.5
−80
−60
−40
−20
0
20
40
h
V
η=4.7 η=4.7
0 0.1 0.2 0.3 0.4 0.5
−80
−60
−40
−20
0
20
40
h
V
a b
Fig. 18.2 The projections of attractors of HH neurons in V h plane for D 4:7, the projection
of noiseless attractors (a) the projection of attractors synchronized by noise (b)
−0.2 0 0.2 0.4 0.6
b=0.2404
w
v
a b
b=0.2404
w
v
−0.1
−0.05
0
0.05
0.1
0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.8 −0.5 0 0.5 1
Fig. 18.3 The projections of attractors of FHN neurons in v w plane for b D 0:2404,the
projection of noiseless attractors (a) the projection of attractors synchronized by noise (b)
The differential equations of two uncoupled FHN models with noise are given as
follows:
8
ˆ
<
ˆ
:
"
dv
i
dt
D v
i
.v
i
a/.1 v
i
/ w
i
C k.t/
dw
i
dt
D v
i
dw b C r sin.ˇt/
(18.4)
in which, i D 1, 2 and other parameters are " D 0:02; d D 0:78; a D 0:5; r D 0:27,
and ˇ D 14:0 respectively, single FHN model can show a lot of firing types with
parameter b changing.
Figure 18.3 gives the projections of attractors of FHN neurons in v w plane
for b D 0:2404. Figure 18.3a corresponds to the attractors without noise which is
chaotic firing, Fig. 18.3b to synchronized attractors. Obviously, after synchronizing
the original attractor is not swamped by noise, but are changed obviously, which
18 Study on Synchronization of Two Identical Uncoupled Neurons Induced by Noise 221
are disturbed by noise slightly. The result is different from the conclusion of HH
neurons. The values of critical noise intensity for synchronizing two FHN neurons
is 0.06, which is much less than the magnitude of the mean size of the original
attractor .S
w
D 0:43/. That is same as the result of HH neurons.
18.3 Conclusions
By numerically investigating noise-induced synchronization between two uncoupled
FHN and HH models, we find that the value of the critical noise intensity reaching
or exceeding the magnitude of the mean size of the attractor of the original system
for synchronizing two models is not necessary. This result is completely different
from conclusion of Ref. [6]. It is interesting that the deterministic dynamics struc-
ture of the original attractor of neurons is swamped by noise for HR model, but that
of the original attractor of neuron is remained for HHmodel, and the original attrac-
tor of neuron is slightly changed by noise for FHN models. These differences mean
that the mechanism of noise-induced synchronization is worthy to further detected.
Acknowledgments The project is grateful for the financial support by the National Natural Sci-
ence Foundation of China under Grant Nos. 10772140, 10872155, and the first author is grateful
for the financial support by the Foundation of Xi’an University of Technology under Grant Nos.
108-210808, 108-210809.
References
1. Maritan A, Banavar JR (1994) Chaos, noise, and synchronization [J]. Phys Rev Lett 72:1451
2. Pikovsky AS (1994) Chaos, noise, and synchronization [J]. Phys Rev Lett 73:2931
3. Lai CH, Zhou CS (1998) Synchronization of chaotic maps by symmetric common noise [J].
Europhys Lett 43:376
4. Zhou CS, Kurths J (2002) Noise-enhanced phase synchronization of chaotic oscillators [J].
Phys Rev Lett 88:230602
5. Toral R, et al (2001) Analytical and numerical studies of noise-induced synchronization of
chaotic systems [J]. Chaos 11:665
6. He DH, Shi PL, Stone L (2003) Noiseinduced synchronization in realistic models [J]. Phys
Rev E 67:027201
7. Wu Y, Xu JX, He DH, et al (2005) Generalized synchronization induced by noise and parameter
mismatching in Hindmarsh–Rose neurons [J]. Chaos Solitons Fractals 23:1605
8. Zhou CS, Kurths J (2003) Noise-induced synchronization and coherence resonance of a
Hodgkin–Huxley model of thermally sensitive neurons [J]. Chaos 13:401
9. Shuai JW, Durand DM (1999) Phase synchronization in two coupled chaotic neurons [J]. Phys
Lett A 264:289
10. Neiman AB, Russell DF (2002) Synchronization of noise-induced bursts in noncoupled sen-
sory neurons [J]. Phys Rev Lett 88:138103
11. Jin WY, Xu JX, Wu Y, et al (2004) An alternating periodic-chaotic ISI sequence of HH neuron
under external sinusoidal stimulus. Chinese Phys 13:335
Chapter 19
Non-equilibrium Phase Transitions
in a Single-Mode Laser Model Driven
by Non-Gaussian Noise
Yanfei Jin
Abstract The non-equilibrium phase transition of a single-mode laser model driven
by non-Gaussian noise is studied in this paper. The stationary probability distribu-
tion (SPD) and its extremal equation are derived by using the path integral approach
and the unified colored noise approximation. It is found that there is a critical rela-
tion between the noise intensity and the correlation time so that there is a transition
line separating the mono-stable region and the bi-stable region. Given the noise
intensity and the correlation time, the single-mode laser system undergoes a suc-
cessive phase transition by varying the departure of the non-Gaussian noise from
the Gaussian noise. Meanwhile, as indicated in the phase diagram, when the noise
intensity and the correlation time are varied, the system undergoes a reentrance
phenomenon.
19.1 Introduction
Noise may play a constructive role and induce new ordering phenomena in
some non-equilibrium systems [1–4] even though it is usually considered as a
source of disorder and chaos. This phenomenon is the so-called noise induced
phase transition. The noise induced phase transitions in systems far away from
thermal equilibrium have been widely investigated in many fields [5–11], such
as physics, chemistry, and biology. Gudyma [7] examined the mechanisms of ac-
tion of colored multiplicative noise in a positionally disordered semiconductor
with Moss–Burstein shift. It is found that the action of multiplicative noise causes
non-equilibrium first-order phase transition of the disorder-order-type in electron
subsystem of semiconductor. Van den Broeck et al. [8] found that the lattice model
with multiplicative noise could undergo a non-equilibrium phase transition to a
symmetry-breaking state. Zaikin et al. [9] showed the non-equilibrium first-order
Y. J i n (
)
Department of Mechanics, Beijing Institute of Technology, Beijing 100081,
People’s Republic of China
e-mail: jinyf@bit.edu.cn
A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay,
DOI 10.1007/978-1-4419-5754-2
19,
c
Springer Science+Business Media, LLC 2010
223
224 Y. J i n
phase transition in a nonlinear lattice of over-damped oscillators with both additive
and multiplicative noise terms, which is induced by additive noise. Castro et al.
[10] reported the reentrance phenomena induced by colored multiplicative noise
and correlated additive and multiplicative white noises. Jia et al. [11] found that
the cross-correlation between noises could induce the reentrance phenomena in a
bi-stable kinetic model. The above studies proved that the noise acts an important
role in the study of non-equilibrium phase transitions.
The single-mode laser model with random fluctuation has received much at-
tention from both theorists and experimentalists [12–18] and can be regarded
as a particular prototype of a nonlinear problem in non-equilibrium statistical
mechanics. Zhu [12] investigated the transient properties of two different single-
mode laser models with loss noise and gain noise. Cao and Liang et al. [13–15]
studied the effects of white and colored cross-correlation between additive and mul-
tiplicative noises on the phase transition in the dye laser system respectively. Luo
et al. [16] developed a two-dimensional decoupling theory, which applied to single-
mode dye laser system with colored noise. Xie and Mei [17] studied a single-mode
laser model with cross-correlated additive and multiplicative noises term and found
that nearby threshold the presence of negatively correlated noises slows down the
decay of fluctuation. Jin et al. [18] investigated the relaxation time of a single-mode
dye laser system driven by cross-correlated additive and multiplicative noises and
found that the correlation intensity speeds up the intensity fluctuation decay of laser
slightly above threshold.
Most of existing theoretical works assume that the noise source yields a Gaussian
distribution (either white or colored). Recently, some experiments in sensory, bio-
logical, and physical systems [19, 20] have indicated that the noise sources may
yield non-Gaussian distributions. Fuentes and Horacio et al. [21–23] investigated
the effects of non-Gaussian noises on the mean first-passage time and stochas-
tic resonance in a bi-stable system. Meanwhile, the effect of non-Gaussian noise
sources in the noise-induced transition for a genetic model is studied by Horacio and
Toral [24]. Goswami et al. [25] studied the barrier crossing dynamics in presence
of additive and multiplicative non-Gaussian noise. This included the multiplica-
tive colored non-Gaussian noise, which can induce resonant activation. To the best
knowledge of the author, no attention has been paid to the noise-induced phenom-
ena in the laser model subjected to non-Gaussian noises because of mathematical
difficulties. It is desirable, hence, to gain an insight into the non-equilibrium phase
transitions of a single-mode laser model subjected to the non-Gaussian noise.
The paper is organized as follows: In Sect. 19.2, the stationary probability dis-
tribution (SPD) and its extremal equation are derived by using the path integral
approach and the unified colored noise approximation. It is found that there is a crit-
ical relation between the noise intensity and the correlation time so that there is a
transition line separating the mono-stable region and the bi-stable region. Analyzing
the roots of the extreme equation of SPD, the effects of non-Gaussian noise on non-
equilibrium phase transition is discussed in Sect. 19.3. Finally, several concluding
remarks are drawn in Sect. 19.4.
19 Non-equilibrium Phase Transitions in a Single-Mode Laser Model 225
19.2 Approximate Stationary Probability Distribution
Consider a single-mode laser loss model with the non-Gaussian noise described by
the following Langevin equation
P
I D 2. I/I C 2I .t/; (19.1)
where I is the laser intensity, is the pump parameter, .t/ is the noise term
with the non-Gaussian distribution, and can be modeled as a Markov process by
the following Langevin equation [21]
P D
1
d
d
V
q
./ C
1
.t/; (19.2)
where .t/ is the Gaussian white noise and can be characterized by the following
mean and variance
h
.t/
i
D 0;
˝
.t/.t
0
/
˛
D 2Dı.t t
0
/ (19.3)
and
V
q
./ D
D
.q 1/
ln
1 C
D
.q 1/
2
2
; (19.4)
where parameter q denotes the departure of the non-Gaussian noise .t/ from the
Gaussian distribution, the parameters and D represent the noise correlation time
and the noise intensity respectively. When q ! 1; .t/ degenerates to Gaussian col-
ored noise. From [21–23], the stationary probability distribution of .t/ is given by
P
st
q
./ D
1
Z
q
1 C
D
.q 1/
2
2
1=.q1/
; (19.5)
here Z
q
is the normalization factor. And the mean and variance of .t/ are derived
as follows
h
.t/
i
D 0;
˝
2
.t/
˛
D
(
2D
.53q/
;q2 .1; 5=3/
1;q2 Œ5=3; 3/
: (19.6)
The multiplicative noise .t/ in (19.1) turns to the additive one by setting x D ln I
Px D 2. e
x
/ C 2.t/: (19.7)