Chinese Physics B

Chinese Physics B ( First published in 1992 ) Published monthly in hard copy by the Chinese Physical Society and online by IOP Publishing, Temple Circus, Temple Way, Bristol BS1 6HG, UK Institutional subscription information: 2013 volume For all countries, except the United States, Canada and Central and South America, the subscription rate is £977 per annual volume. Single-issue price £97. Delivery is by air-speeded mail from the United Kingdom. Orders to: Journals Subscription Fulfilment, IOP Publishing, Temple Circus, Temple Way, Bristol BS1 6HG, UK For the United States, Canada and Central and South America, the subscription rate is US$1930 per annual volume. Single-issue price US$194. Delivery is by transatlantic airfreight and onward mailing. Orders to: IOP Publishing, PO Box 320, Congers, NY 10920-0320, USA c 2013 Chinese Physical Society and IOP Publishing Ltd ⃝ All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the copyright owner. Supported by the National Natural Science Foundation of China, the China Association for Science and Technology, and the Science Publication Foundation, Chinese Academy of Sciences Editorial Office: Institute of Physics, Chinese Academy of Sciences, PO Box 603, Beijing 100190, China Tel: (86 – 10) 82649026 or 82649519, Fax: (86 – 10) 82649027, E-mail: [email protected] 主管单位: 中国科学院国内统一刊号: CN 11–5639/O4 主办单位: 中国物理学会和中国科学院物理研究所广告经营许可证：京海工商广字第0335号承办单位: 中国科学院物理研究所编辑部地址: 北京中关村中国科学院物理研究所内主编：欧阳钟灿通讯地址: 100190 北京 603 信箱出版：中国物理学会 Chinese Physics B 编辑部印刷装订：北京科信印刷厂电话: (010) 82649026, 82649519 编辑: Chinese Physics B 编辑部传真: (010) 82649027 国内发行: Chinese Physics B 出版发行部 E-mail: [email protected] 国外发行: IOP Publishing Ltd “Chinese Physics B”网址: 发行范围: 公开发行 http://cpb.iphy.ac.cn(编辑部) 国际统一刊号: ISSN 1674–1056 http://iopscience.iop.org/cpb (IOP) Published by the Chinese Physical Society 顾问

Advisory Board

陈佳洱

教授, 院士北京大学物理学院, 北京 100871 教授, 院士南京大学物理系, 南京 210093 教授, 院士北京师范大学低能核物理研究所, 北京 100875 教授, 院士

冯

端

黄祖洽

李政道李荫远丁肇中

研究员, 院士中国科学院物理研究所, 北京 100190 教授, 院士

杨振宁

教授, 院士

杨福家

教授, 院士复旦大学物理二系, 上海 200433 研究员, 院士中国科学技术协会, 北京 100863 研究员, 院士中国原子能科学研究院, 北京 102413 研究员, 院士中国科学院物理研究所, 北京 100190

周光召王乃彦梁敬魁

Prof. Academician Chen Jia-Er School of Physics, Peking University, Beijing 100871, China Prof. Academician Feng Duan Department of Physics, Nanjing University, Nanjing 210093, China Prof. Academician Huang Zu-Qia Institute of Low Energy Nuclear Physics, Beijing Normal University, Beijing 100875, China Prof. Academician T. D. Lee Department of Physics, Columbia University, New York, NY 10027, USA Prof. Academician Li Yin-Yuan Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Prof. Academician Samuel C. C. Ting LEP3, CERN, CH-1211, Geneva 23, Switzerland Prof. Academician C. N. Yang Institute for Theoretical Physics, State University of New York, USA Prof. Academician Yang Fu-Jia Department of Nuclear Physics, Fudan University, Shanghai 200433, China Prof. Academician Zhou Guang-Zhao (Chou Kuang-Chao) China Association for Science and Technology, Beijing 100863, China Prof. Academician Wang Nai-Yan China Institute of Atomic Energy, Beijing 102413, China Prof. Academician Liang Jing-Kui Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

(Continued )

Chin. Phys. B Vol. 22, No. 10 (2013) 100502

Reliability of linear coupling synchronization of hyperchaotic systems with unknown parameters∗ Li Fan(李凡), Wang Chun-Ni(王春妮), and Ma Jun(马军)† Department of Physics, Lanzhou University of Technology, Lanzhou 730050, China (Received 19 February 2013; revised manuscript received 1 April 2013)

Complete synchronization could be reached between some chaotic and/or hyperchaotic systems under linear coupling. More generally, the conditional Lyapunov exponents are often calculated to confirm the stability of synchronization and reliability of linear controllers. In this paper, detailed proof and measurement of the reliability of linear controllers are given by constructing a Lyapunov function in the exponential form. It is confirmed that two hyperchaotic systems can reach complete synchronization when two linear controllers are imposed on the driven system unidirectionally and the unknown parameters in the driving systems are estimated completely. Finally, it gives the general guidance to reach complete synchronization under linear coupling for other chaotic and hyperchaotic systems with unknown parameters.

Keywords: parameter estimation, exponential Lyapunov function, parameter observer, linear coupling PACS: 05.45.–a

DOI: 10.1088/1674-1056/22/10/100502

1. Introduction Hyperchaos is an important subject in the field of nonlinear dynamics, which is often observed in the physical, chemical, and biological systems. [1–8] For example, Rössler [3] presented a four-variable nonlinear equation and generated hyperchaos in this scheme. Yang et al. [7] confirmed the chaos occurrence of the time series of an human electrocardiogram by using nonlinear analysis. Krese et al. [8] studied the transition of chaos to chaos in the laser droplet generation. In a general chaotic/ hyperchaotic system there often emerge finite strange attractors, and it is confirmed that the multiscroll hyperchaotic attractors have much better properties and complex dynamics than the general hyperchaotic attractors. For example, Lü and Chen [9] and Yu et al. [10] even gave an instructive guidance to generate multiscroll chaotic attractors in a theoretical way and investigated its potential applications. One potential application is that time series generated from chaotic or hyperchaotic systems could be used as a secure key or carrier wave for secure communication. [11–18] For example, Kanso et al. [11] presented a keyed hash function from a chaotic map. Li et al. [12] proposed another keyed hash function from a modified coupled chaotic map lattice for encryption. More interesting, Huang [13] designed a class of parallel keyed hash functions from the neural network. On the other hand, secure communication based on the synchronization between hyperchaotic and/or spatiotemporal systems has also received much attention. [19–26] Complex systems often contain a large number of oscillators (or cells), for example, the neuronal system is composed of a large number of oscillators, and these oscillators can communicate with other oscillators in a feasible way such

as coupling. [27–31] Phase synchronization is often reached under appropriate coupling intensity, and it is confirmed that phase synchronization is conducible to memory processes. [30] Chaotic systems can also reach synchronization due to coupling, for example, Rosenblum et al. [32] investigated the phase synchronization between two chaotic Rössler systems and it was also confirmed that phase synchronization could be approached between two chaotic oscillators when the coupling intensity exceeded a certain threshold even if the mismatch of parameters in the two systems occurred. Furthermore, the transition from phase synchronization to lag synchronization was also investigated in two coupled chaotic oscillators, and the synchronization state was dependent on the coupling intensity. [33] More often, much work has discussed the adaptive synchronization between chaotic or hyperchaotic systems, [34–41] the controllers and parameter observers could be approached analytically by using the Lyapunov function control scheme, particularly, parameters with different orders of magnitudes could be estimated completely. [37] However, the controllers could be in the complex form by using the original Lyapunov function scheme. Therefore, it is important to find a simpler controller such as a linear controller to realize synchronization and parameter estimation. It is confirmed that some chaotic or hyperchaotic systems [42,43] can reach complete or phase synchronization under the linear coupling scheme, [44,45] and its reliability of linear coupling on realizing synchronization is often checked by calculating the conditional Lyapunov exponents, the threshold for coupling intensity could be approached so that the conditional Lyapunov exponents of the controlled systems should be negative.

∗ Project

supported partially by the National Natural Science Foundation of China (Grant No. 11265008). author. E-mail: [email protected] © 2013 Chinese Physical Society and IOP Publishing Ltd http://iopscience.iop.org/cpb http://cpb.iphy.ac.cn † Corresponding

100502-1

Chin. Phys. B Vol. 22, No. 10 (2013) 100502 On the other hand, the topic about the complex networks [46–48] has received increasing attention from various fields of science and engineering because various synchronization phenomena are found in the network. For example, in Ref. [46] some synchronization phenomenon were investigated and several network synchronization theorems were proved in a time-varying complex dynamical network model. The complex network shows better application than a single oscillator, for example, Squartini et al. [47] introduced a neural paradigm for complex systems and the new algorithms and discussed its applications in detail. As a result, the synchronization and control of complex networks has become more interesting and challenging. [48] It is also important to discuss the effectiveness of this scheme. In the case of a ring network with a nearest-neighbor connection (similar to the regular network type), a similar Lyapunov function could be designed and the error function could be marked with a subscript (i). In the case of a two-dimensional regular network, in which each node (oscillator) is connected with the four nearest-neighbor nodes (oscillators), the error function in the Lyapunov function will be marked with two subscripts (i, j), and this scheme could be extended to realize the synchronization in network under linear coupling. However, it could encounter much difficulty in the case of a network with a small-world-type connection because the long-range connection with a certain probability should be considered. Based on the Lyapunov function scheme and stability theory, the scheme of adaptive synchronization [49–54] between chaotic or hyperchaotic systems is often used to estimate unknown parameters in the driving system and control chaos, and it sounds reasonable in the analytical and numerical way. In fact, this scheme and most of the previous similar schemes of adaptive synchronization deserve further investigation and improvement for some realistic nonlinear systems. In Refs. [55]– [59] the parmaeter identification in chaotic or hyperchaotic systems was also discussed, and these results sound attractive. However, for realistic nonlinear systems, the problem of parameter estimation is often challenging and is worth further investigation by considering the external noise, finite signal series, diversity of order magnitudes of parameters, sudden shifts in parameters, and other uncertain factors. Synchronization between fractional order chaotic systems is also an interesting problem, for example, Huang and Qi [60] suggested that adaptive sliding mode control could be used to realize the synchronization between two different fractional order chaotic systems. It is expected that this continuous linear coupling-induced synchronization and parameter estimation scheme could be improved, and can be used in some realistic nonlinear systems. In this paper, the four-variable Rössler equations with four unknown parameters will be investigated. The aim of this paper is to prove that linear coupling can realize the com-

plete synchronization between two hyperchaotic Rössler systems under linear coupling, and the unknown parameters in the drive system will be estimated completely. The proof will be given in a theoretical way and then corresponding parameter observers will also be approached analytically. And the design of the exponential Lypunov function could be useful to approach the parameter observer and linear controller for other hyperchaotic or chaotic systems, and thus the control target could be approached completely.

2. Model and scheme At first, the principle of this scheme is introduced. For any given nonlinear dynamical systems in chaotic or hyperchaotic state, the driving and response systems are defined as follows: x˙1 = f1 (x1 , x2 , x3 , . . . xn , a, b, c, d, ...), x˙2 = f2 (x1 , x2 , x3 , . . . xn , a, b, c, d, ...), x˙3 = f3 (x1 , x2 , x3 , . . . xn , a, b, c, d, ...), x˙4 = f4 (x1 , x2 , x3 , . . . xn , a, b, c, d, ...), ··· , x˙n = fn (x1 , x2 , x3 , . . . xn , a, b, c, d, ...); ˆ c, ˆ ...), y˙1 = g1 (y1 , y2 , y3 . . . yn , a, ˆ b, ˆ d,

(1)

ˆ c, ˆ ...), y˙2 = g2 (y1 , y2 , y3 . . . yn , a, ˆ b, ˆ d, ˆ c, ˆ ...), y˙3 = g3 (y1 , y2 , y3 . . . yn , a, ˆ b, ˆ d, ˆ c, ˆ ...), y˙4 = g4 (y1 , y2 , y3 . . . yn , a, ˆ b, ˆ d, ··· , ˆ c, ˆ ...), y˙n = gn (y1 , y2 , y3 . . . yn , a, ˆ b, ˆ d,

(2)

where xi (i = 1, 2, 3, . . . , n), a, b, c, d, . . . , are variables and parameters and f1 , f2 , f3 , . . . are linear or nonlinear functions in the driving system (1), which determine the dynamics of the ˆ c, systems; yi (i = 1, 2, 3, . . . , n), a, ˆ b, ˆ d,ˆ . . . are variables and parameters and g1 , g2 , g3 , . . . are linear or nonlinear functions in the response system (2), more generally, linear controllers could be imposed on the response system (2) as follows: ˆ c, ˆ ...) + k1 (x1 − y1 ), y˙1 = g1 (y1 , y2 , y3 . . . yn , a, ˆ b, ˆ d, ˆ c, ˆ ...) + k2 (x2 − y2 ), y˙2 = g2 (y1 , y2 , y3 . . . yn , a, ˆ b, ˆ d, ˆ c, ˆ ...) + k3 (x3 − y3 ), y˙3 = g3 (y1 , y2 , y3 . . . yn , a, ˆ b, ˆ d, ˆ c, ˆ ...) + k4 (x4 − y4 ), y˙4 = g4 (y1 , y2 , y3 . . . yn , a, ˆ b, ˆ d, ··· , ˆ c, ˆ ...) + kn (xn − yn ), (3) y˙n = gn (y1 , y2 , y3 . . . yn , a, ˆ b, ˆ d, where k1 , k2 , k3 , k4 , . . . , are the coupling intensities, and for simplicity, these coupling intensities could be selected to be the same value, i.e., k1 = k2 = k3 = k4 = · · · = k. The output error series for the locked variables in the drive and driven systems are marked as follows:

100502-2

ei = xi − yi , i = 1, 2, 3, . . . , n,

(4)

Chin. Phys. B Vol. 22, No. 10 (2013) 100502 ˆ ea = a − a, ˆ eb = b − b,

ec = c − c, ˆ

ˆ . . . . (5) ed = d − d,

By calculating the Lyapunov exponent spectra for the driving system (1) and driven system (3), the threshold for coupling intensity k = kc could be approached when the largest conditional Lyapunov exponent keeps negative by increasing the value of k. According to the Lyapunov stability theory, a general positive Lyapunov function in the exponential form could be constructed as follows: n V = exp A1 (e21 + e22 + e23 + · · · + e2n ) + ∑ Ai j ei e j i=1

−

ˆ c, h(ei , ea , eb , . . . , a, ˆ b, ˆ ..., Ai , B, kc )dt

a˙ˆ = −e˙a = −A1 Bga (xi , yi , ei ), b˙ˆ = −e˙b = −A1 Bgb (xi , yi , ei ), c˙ˆ = −e˙c = −A1 Bgc (xi , yi , ei ), ··· .

n

V˙ = V {[ea (e˙a /B − A1 ga (xi , yi , ei ))] + [eb (e˙b /B − A1 gb (xi , yi , ei ))] + [ec (e˙c /B − A1 gc (xi , yi , ei ))] + · · · n

∑ (ki − kc )e2i + f (∗∗)

i=1

ˆ c, − h(ei , ea , eb . . . , a, ˆ b, ˆ . . . , A1 , B, kc )},

(10)

Equation (10) becomes negative completely under ki > kc , so the Lyapunov stability criterion is satisfied, and the controlled (driven) system will keep pace with the driving system and the unknown parameters in driving system are estimated according to Eq. (9c). Clearly, the Lyapunov function in exponential type shown in Eq. (6) keeps positive completely, and the differential coefficient of the Lyapunov function versus time becomes negative when linear controllers are imposed on the response system. The parameters are estimated completely when the two hyperchaotic systems reach the complete syschronization. For further understanding, this scheme is used to realize the parameter estimation based on an improved linear coupling-induced synchronization between two hyperchaotic systems. The dynamical equations for the four-variable Rössler model [3] is often described by

(7)

where f (∗∗) is a superfluous term, which is often used to offˆ c, set the function h(ei , ea , eb , . . . , a, ˆ b, ˆ . . . , Ai , B, kc ) in Eqs. (6) and (7). As a result, equation (7) is redefined as ˙ V = V [ea (e˙a /B − A1 ga (xi , yi , ei ))] + [eb (e˙b /B − A1 gb (xi , yi , ei ))]

(8)

i=1

The parameter observers are approached from Eq. (8) as follows:

x˙ = −y − z, y˙ = x + ay + w, z˙ = b + xz, w˙ = −cz + dw,

xˆ˙ = −yˆ − zˆ, yˆ˙ = xˆ + aˆyˆ + wˆ + u3 , zˆ˙ = bˆ + xˆ ˆz, w˙ˆ = −cˆ ˆz + dˆwˆ + u4 ,

eb (e˙b /B − A1 gb (xi , yi , ei )) = 0, ec (e˙c /B − A1 gc (xi , yi , ei )) = 0, (9a)

(11)

where x, y, z, w are output variables; a, b, c, d are control parameters in system (11). Hyperchaotic state will be observed in this system when parameters are selected as a = 0.25, b = 3.0, c = 0.5, d = 0.05. More generally, controllers are often imposed on the second and fourth formula to change the dynamics of the nonlinear system as shown in Eq. (11). The driven system with nonidentical parameters is often given as

ea (e˙a /B − A1 ga (xi , yi , ei )) = 0,

e˙a /B − A1 ga (xi , yi , ei ) = 0,

∑ (ki − kc )e2i .

i=1

where Ai j (i, j = 1, 2, 3, . . . , n) is certain coefficient, A1 , B are gain coefficients, and kc is the critical coupling intensity for realizing the complete synchronization. The nonlinˆ c, ear function h(ei , ea , eb , . . . , a, ˆ b, ˆ . . . , Ai , B, kc ) could be approached by calculating the differential coefficient of the Lyapunov function in Eq. (6) on the condition that the differential coefficient of the Lyapunov function versus time should be negative

··· ,

(9c)

V˙ = −V (6)

+ [ec (e˙c /B − A1 gc (xi , yi , ei ))] + · · · n − ∑ (ki − kc )e2i .

(9b)

The parameters in the driving system are often constants, and thus the parameter observers are found as follows:

> 0,

−

··· .

As a result, equation (8) is redefined as follows:

+ B(e2a + e2b + e2c + · · · ) Z

e˙c /B − A1 gc (xi , yi , ei ) = 0,

(12)

ˆ c, where x, ˆ y, ˆ zˆ are output variables and a, ˆ b, ˆ dˆ are parameters in Eq. (12). The linear controllers are defined as follows: u3 = k3 (y − y) ˆ and u4 = k4 (w − w), ˆ

e˙b /B − A1 gb (xi , yi , ei ) = 0, 100502-3

(13)

Chin. Phys. B Vol. 22, No. 10 (2013) 100502 where k3 and k4 are coupling intensities. The two nonlinear systems will become identical when the parameters in the two systems are the same and then the two identical systems could reach the complete synchronization only when the intensities of coupling exceed certain thresholds. It is important to exactly prove that the controllers, as shown in Eq. (13), are effective to realize the complete synchronization and the parameter observers for the parameters in Eq. (12) will also be approached analytically. According to Eq. (6), an appropriate positive Lyapunov function is determined as follows:

a result, it is required that the condition as shown in Eq. (17) should be satisfied, i.e., 0 > ea [e˙a /g2 + g1 y(ey − ex )] + eb [e˙b /g2 + g1 (ez − ex ) − k3 ey /g2 ] + ec [e˙c /g2 − g1 z(ew − ex )] + ed [e˙d /g2 + g1 w(ew − ex )] − g1 e2x − k3 g1 e2y − k4 g1 e2w .

A general but simple solution for inequality (17) could be given as ea [e˙a /g2 + g1 y(ey − ex )] = 0, eb [e˙b /g2 + g1 (ez − ex ) − k3 ey /g2 ] = 0,

V = exp[g1 (e2x + e2y + e2z + e2w − 2ex ey − 2ex ez − 2ex ew )/2

0 = ec [e˙c /g2 − g1 z(ew − ex )],

+ (e2a + e2b + e2c + e2d )/2g2 − − − − −

Z Z Z Z Z

0 = ed [e˙d /g2 + g1 w(ew − ex )].

g1 (ex − ey − ez − ew )(−ey − ez )dt g1 (ey − ex )(ae ˆ y + ew )dt −

g1 (k3 + 1)ey ex dt

V˙ = V (−g1 e2x − k3 g1 e2y − k4 g1 e2w ) < 0.

Z

(14) a˙ˆ = −e˙a = g1 g2 y(ey − ex ), b˙ˆ = −e˙b = g1 g2 (ez − ex ) − k3 ey , c˙ˆ = −e˙c = −g1 g2 z(ew − ex ), dˆ˙ = −e˙ = g g w(e − e ).

where g1 and g2 are positive gain coefficients to be selected, and the errors for corresponding variables, parameters in the two systems are determined as follows:

d

ex = x − x, ˆ ey = y − y, ˆ ez = z − zˆ, ew = w − w, ˆ q θ (ex , ey , ez , ew ) = (e2x + e2y + e2z + e2w ).

(19)

When the parameters in the driving system (11) are not timevarying, and the parameter observers are approached as follows:

ˆ w )dt g1 (ew − ex )(−ce ˆ z + de k3 ey eb /g2 dt] > 0,

(18)

And the condition as shown in Eq. (17) is replaced as follows:

Z

g1 (ez − ex )(xz − xˆ ˆz)dt

g1 k4 ex ew dt −

(17)

(15a) (15b)

Then the differential coefficient of the Lyapunov function as shown in Eq. (14) versus time is approached by V˙ = V [(ex e˙x + ey e˙y + ez e˙z + ew e˙w − e˙x ey − ex e˙y − e˙x ez − ex e˙z − e˙x ew − ex e˙w )g1 + (ea e˙a + eb e˙b + ec e˙c + ed e˙d )/g2 − (ex − ey − ez − ew )(−ey − ez )g1

1 2

w

x

(20)

Inequality (19) is satisfied when appropriate gain coefficients and coupling intensity are used, and thus the parameter observers in Eq. (20) will become stable to estimate all the unknown parameters in the driving system as shown in Eq. (11). According to Eqs. (14) and (19), the Lyapunov function decreases with time gradually only when the gain coefficients and coupling intensity are selected to be positive values because (−g1 e2x − k3 g1 e2y − k4 g1 e2w ) < 0, and the convergence rate of error function is dependent on the selection of gain coefficients and coupling intensity.

− (ey − ex )(ae ˆ y + ew )g1 − (k3 + 1)ey ex g1

3. Numerical results and discussion

− (ez − ex )(xz − xˆ ˆz)g1 ˆ w )g1 − g1 k4 ex ew − (ew − ex )(−ce ˆ z + de − k3 ey eb /g2 ] = {ea [e˙a /g2 + g1 y(ey − ex )] + eb [e˙b /g2 + g1 (ez − ex ) − k3 ey /g2 ] + ec [e˙c /g2 − g1 z(ew − ex )] − ed [e˙d /g2 + g1 w(ew − ex )] − g1 e2x − k3 g1 e2y − k4 g1 e2w }V.

(16)

According to the Lyapunov stability theory, the Lyapunov function in Eq. (14) will decrease to zero with a long transient period and thus the two systems become synchronized only when the result in Eq. (16) is negative completely. As

In the numerical studies, the fourth Runge–Kutta algorithm is used to calculate the nonlinear dynamical equations, in time steps of h = 0.001. The initial values for the drive system and driven system are used as (−20, 0, 0, 15), (−25, 0, 0, 20), respectively. The transient period is about 500 time units. The results in Figs. 1 and 2 confirm that the two hyperchaotic Rössler systems could reach the complete synchronization and all the unknown parameters are identified completely under appropriate gain coefficients and coupling intensity. Extensive numerical results confirm that the transient period decreases when bigger gain coefficients and coupling intensity are used. Furthermore, it is also investigated that

100502-4

Chin. Phys. B Vol. 22, No. 10 (2013) 100502 some parameters in the drive system jump suddenly, for example, parameter shift could occur when some electric elements (resistance, capacitor, inductance) in the circuit are damaged. The numerical results confirm that the unknown parameters in the drive system are estimated completely. In Fig. 3, the error function series are plotted, and the estimated results for the unknown parameters are plotted in Fig. 4 when parameter b is jumped from 3 to 4 at t = 250 time units. 2.0

θ(ex, ey, ez, ew)

1.5 1.0 0.5 0 0

100

200

300 400 500 t Fig. 1. Time evolution of the error function as defined in Eq. (15b), with the two gain coefficients being g1 = 5.0 and g2 = 0.05, and the coupling intensity for controllers being k3 = k4 = 5.0.

The results in Fig. 3 confirm that the error function decreases to a stable value quickly, and a small fluctuation occurs when parameter b changes from 3 to 4 at t = 250 time units, then the error function still decreases to zero and a complete synchronization state is reached. The results in Fig. 4 confirm that the unknown parameters still could be identified exactly even though one parameter is

jumped from 3 to 4 at t=250 time units. It indicates that the scheme could be effective to estimate the unknown parameters when some parameters in the drive system jump suddenly. We also check the case where the gain coefficient and coupling intensity are selected to be other values. It is also confirmed that the short transient period costs a lot when a group of bigger values are given to the gain coefficients and coupling intensity. For example in which another group of gain coefficients and coupling intensity are selected, the results are shown in Figs. 5 and 6. The results in Figs. 5 and 6 confirm that the two nonlinear hyperchaotic systems can still reach synchronization completely, and the unknown parameters could be estimated exactly even if one of the unknown parameters in the drive system decreases from b = 3 to b = 2.0 at t = 250 time units. Furthermore, this exponential Lyapunov function scheme is also used to prove the effectiveness of linear coupling on synchronization between the chaotic Chua circuit and hyperchaotic Chen system, and the linear controllers based on linear coupling and the parameter observers could be approached and proved to be stable and effective. In summary, this scheme is helpful to confirm the reliability of linear coupling-induced synchronization of other chaotic or hyperchaotic systems. In some hyperchaotic systems there could emerge multiscroll attractors, and the appearance of multiscroll attractors often depends on the dynamics and parameter region of the system. This scheme is also useful when the dynamics of the system are described by deterministic

2.0 4.5 (a)

(b) 3.0

1.0

1.5 ^ b

a ^

1.5

0

0.5

-1.5 0 -3.0 0

100

200

300

400

0

500

100

200

0.75

300

400

500

t

t 0.10

0.50 0.05

^ d

c^

0.25 0

0 -0.05

(c)

-0.25

(d)

-0.50

-0.10 0

100

200

300

400

500

0

100

200

300

400

500

t

t

Fig. 2. Estimated results for the four unknown parameters in the driven system, with the two gain coefficients being g1 = 5.0 and g2 = 0.05, and the coupling intensity for controllers being k3 = k4 = 5.0. The real values for these unknown parameters in the driving system are a = 0.25, b = 3.0, c = 0.5, d = 0.05.

100502-5

Chin. Phys. B Vol. 22, No. 10 (2013) 100502 2.0

θ(ex, ey, ez, ew)

1.5

1.0

0.5

0 0

100

200

300 400 500 t Fig. 3. Time evolution of the error function as defined in Eq. (15b), with the two gain coefficients being g1 = 4.0 and g2 = 0.04, and the coupling intensity for controllers being k3 = k4 = 5.0.

dynamic equations. It is also important to discuss the potential application of this scheme, and it is also attractive to explain

0.75

whether this scheme is useful in the networks. In our opinion, this scheme confirms the reliability of linear coupling-induced synchronization between hyperchaotic oscillators and that of the parameter estimation. A liner controller could be the simplest controller and it is practical to be realized experimentally. In most of the previous work about adaptive synchronization, the appropriate Lyapunov function is often constructed, and the controllers are often in complex form. It is critical to find the appropriate exponential Lyapunov function when liner coupling is used to realize the synchronization and control of hyperchaotic systems. Furhthermore, this scheme will be investigated in the nonlinear systems with fractional order and time-delay, and it could provide a simpler and practical way to realize the synchronization between more complex hyperchaotic systems.

4.5

(a)

(b)

3.0 ^ b

a ^

0.50

1.5

0.25 0 0 0

100

200

300

400

-1.5

500

1.00

100

200

300

400

500

t (c)

0.4

0.75

(d)

0.2

0.50

^ d

c^

0

t

0 -0.2

0.25

-0.4 0

0 100 200 300 400 500 300 400 500 t t Fig. 4. Estimated results for the four unknown parameters in the drive system, with the two gain coefficients being g1 = 4.0 and g2 = 0.04, and the coupling intensity for controllers being k3 = k4 = 5.0. The real values for these unknown parameters in the drive system are a = 0.25, b = 3.0 (t < 250 time units), b = 4.0 (t > 250 time units), c = 0.5, d = 0.05. 0

100

200

4. Conclusions

50 θ(ex, ey, ez, ew)

40 30 20 10 0 0

200

400

600 800 1000 t Fig. 5. Time evolution of the error function as defined in Eq. (15b), with the two gain coefficients being g1 = 3.0 and g2 = 0.03, and the coupling intensity for controllers being k3 = 4.5 and k4 = 4.0.

In this paper, a kind of exponential Lyapunov function is constructed to prove that linear controllers are very effective to realize the complete synchronization between two hyperchaotic systems, and all the unknown parameters in the driving system could be identified exactly. Our results could give an exact proof about the reliability of linear coupling on realizing the synchronization between chaotic or hyperchaotic systems because most of the previous work focused on checking the possibility of linear coupling by calculating the conditional Lyapunov exponents, while exact analysis was left out. The general Lyapunov function scheme is an 100502-6

Chin. Phys. B Vol. 22, No. 10 (2013) 100502 0.75 (a)

(b)

3

0.50 ^ b

a ^

2 0.25

0

1

0

200

400

600

800

0

1000

t

0.10 (c)

0.6

0

200

400

600

800

1000

t (d)

0.05 0.5 c^

^ d

0

0.4

0.3

-0.05

-0.10

0 200 400 600 800 1000 t t Fig. 6. Estimated results for the four unknown parameters in the drive system, with the two gain coefficients being g1 = 3.0 and g2 = 0.03, and the coupling intensity for controllers being k3 = 4.5 and k4 = 4.0. The real values for these unknown parameters in the drive system are a = 0.25, b = 3.0 (t < 250 time units), b = 2.0 (t > 250 time units), c = 0.5, d = 0.05. 0

200

400

600

800

1000

effective scheme to construct a controller and parameter observer to realize the synchronization, but sometimes the form of controllers is much too complex and these controllers are difficult to realize experimentally. However, the scheme in this paper could give a feasible way to find appropriate linear controllers and parameter observers by constructing an exponential Lyapunov function, and it is also independent of the model selection and dynamics of the system.

References

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100502-8

Chinese Physics B Volume 22

Number 10

October 2013

TOPICAL REVIEW — Magnetism, magnetic materials, and interdisciplinary research 104301

Magnetic microbubble: A biomedical platform co-constructed from magnetics and acoustics Yang Fang, Gu Zhu-Xiao, Jin Xin, Wang Hao-Yao and Gu Ning

107503

Chemical synthesis of magnetic nanocrystals: Recent progress Liu Fei, Zhu Jing-Han, Hou Yang-Long and Gao Song

108104

Magnetic-mediated hyperthermia for cancer treatment: Research progress and clinical trials Zhao Ling-Yun, Liu Jia-Yi, Ouyang Wei-Wei, Li Dan-Ye, Li Li, Li Li-Ya and Tang Jin-Tian RAPID COMMUNICATION

104503

Noether symmetry and conserved quantities of the analytical dynamics of a Cosserat thin elastic rod Wang Peng, Xue Yun and Liu Yu-Lu

106804

Resonant energy transfer from nanocrystal Si to β-FeSi2 in hybrid Si/β-FeSi2 film He Jiu-Yang, Zhang Qi-Zhen, Wu Xing-Long and Chu Paul K.

107801

The design and preparation of a Fabry–Pérot polarizing filter Hou Yong-Qiang, Qi Hong-Ji and Yi Kui GENERAL

100201

A diagrammatic categorification of the fermion algebra Lin Bing-Sheng, Wang Zhi-Xi, Wu Ke and Yang Zi-Feng

100202

Approximate derivative-dependent functional variable separation for quasi-linear diffusion equations with a weak source Ji Fei-Yu and Yang Chun-Xiao

100203

On certain new exact solutions of the Einstein equations for axisymmetric rotating fields Lakhveer Kaur and R. K. Gupta

100204

An element-free Galerkin (EFG) method for numerical solution of the coupled Schrödinger-KdV equations Liu Yong-Qing, Cheng Rong-Jun and Ge Hong-Xia

100301

A geometric phase for superconducting qubits under the decoherence effect S. Abdel-Khalek, K. Berrada, Mohamed A. El-Sayed and M. Abel-Aty

100302

Analytic solutions of the double ring-shaped Coulomb potential in quantum mechanics Chen Chang-Yuan, Lu Fa-Lin, Sun Dong-Sheng and Dong Shi-Hai

100303

Relativistic treatment of the spin-zero particles subject to the second Pöschl Teller-like potential Ekele V. Aguda and Amos S. Idowu

100304

The spin-one Duffin Kemmer Petiau equation in the presence of pseudo-harmonic oscillatory ringshaped potential H. Hassanabadi and M. Kamali (Continued on the Bookbinding Inside Back Cover)

100305

Eigen-spectra in the Dirac-attractive radial problem plus a tensor interaction under pseudospin and spin symmetry with the SUSY approach S. Arbabi Moghadam, H. Mehraban and M. Eshghi

100306

Quantum correlation of a three-particle 𝑊 -class state under quantum decoherence Xu Peng, Wang Dong and Ye Liu

100307

Enhanced electron positron pair creation by the frequency chirped laser pulse Jiang Min, Xie Bai-Song, Sang Hai-Bo and Li Zi-Liang

100308

Generation of steady four-atom decoherence-free states via quantum-jump-based feedback Wu Qi-Cheng and Ji Xin

100309

Bound entanglement and teleportation for arbitrary bipartite systems Fan Jiao and Zhao Hui

100501

Transport dynamics of an interacting binary Bose Einstein condensate in an incommensurate optical lattice Cui Guo-Dong, Sun Jian-Fang, Jiang Bo-Nan, Qian Jun and Wang Yu-Zhu

100502

Reliability of linear coupling synchronization of hyperchaotic systems with unknown parameters Li Fan, Wang Chun-Ni and Ma Jun

100503

Synchronization for complex dynamical Lurie networks Zhang Xiao-Jiao and Cui Bao-Tong

100504

Robust modified projective synchronization of fractional-order chaotic systems with parameters perturbation and external disturbance Wang Dong-Feng, Zhang Jin-Ying and Wang Xiao-Yan

100505

Four-cluster chimera state in non-locally coupled phase oscillator systems with an external potential Zhu Yun, Zheng Zhi-Gang and Yang Jun-Zhong

100506

Chaos synchronization of a chain network based on a sliding mode control Liu Shuang and Chen Li-Qun

100507

Trial function method and exact solutions to the generalized nonlinear Schrödinger equation with timedependent coefficient Cao Rui and Zhang Jian

100508

Phase diagrams of spin-3/2 Ising model in the presence of random crystal field within the effective field theory based on two approximations Ali Yigit and Erhan Albayrak

100701

MEH-PPV/Alq3 -based bulk heterojunction photodetector Zubair Ahmad, Mahdi Hasan Suhail, Issam Ibrahim Muhammad, Wissam Khayer Al-Rawi, Khaulah Sulaiman, Qayyum Zafar, Ahmad Sazali Hamzah and Zurina Shaameri

100702

Multi-rate sensor fusion-based adaptive discrete finite-time synergetic control for flexible-joint mechanical systems Xue Guang-Yue, Ren Xue-Mei and Xia Yuan-Qing

100703

Analysis of influence of RF power and buffer gas pressure on sensitivity of optically pumped cesium magnetometer Shi Rong-Ye and Wang Yan-Hui (Continued on the Bookbinding Inside Back Cover)

ATOMIC AND MOLECULAR PHYSICS 103101

Translational, vibrational, rotational enhancements and alignments of reactions H + ClF (𝑣 = 0 5, 𝑗 = 0, 3, 6, 9) →HCl + F and HF + Cl, at 𝐸rel = 0.5 20 kcal/mol Victor Wei-Keh Chao(Wu)

103102

Further investigations of the low-lying electronic states of AsO+ radical Zhu Zun-Lue, Qiao Hao, Lang Jian-Hua and Sun Jin-Feng

103103

Time-dependent density functional theoretical studies on the photo-induced dynamics of an HCl molecule encapsulated in C60 under femtosecond laser pulses Liu Dan-Dan and Zhang Hong

103301

Control of the photoionization/photodissociation processes of cyclopentanone with trains of femtosecond laser pulses Song Yao-Dong, Chen Zhou, Yang Xue, Sun Chang-Kai, Zhang Cong-Cong and Hu Zhan

103401

The effect of wave function orthogonality on the simultaneous ionization and excitation of helium Liu Li-Juan, Jia Chang-Chun, Zhang Li-Min, Chen Jiao-Jiao and Chen Zhang-Jin

103402

Multiple ionization of atoms and molecules impacted by very high-𝑞 fast projectiles in the strong coupling regime (𝑞/𝑣 > 1) Zhou Man, Zou Xian-Rong, Zhao Lei, Chen Xi-Meng, Wang Shi-Yao, Zhou Wang and Shao Jian-Xiong

103403

Kr L X-ray and Au M X-ray emission for 1.5 MeV–3.9 MeV Kr13+ ions impacting on an Au target Mei Ce-Xiang, Zhang Xiao-An, Zhao Yong-Tao, Zhou Xian-Ming, Ren Jie-Ru, Wang Xing, Lei Yu, Sun YuanBo, Cheng Rui, Wang Yu-Yu, Liang Chang-Hui, Li Yao-Zong and Xiao Guo-Qing

103701

Experiments on trapping ytterbium atoms in optical lattices Zhou Min, Chen Ning, Zhang Xiao-Hang, Huang Liang-Yu, Yao Mao-Fei, Tian Jie, Gao Qi, Jiang Hai-Ling, Tang Hai-Yao and Xu Xin-Ye ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID DYNAMICS

104101

Flat lenses constructed by graded negative index-based photonic crystals with tuned configurations Jin Lei, Zhu Qing-Yi, Fu Yong-Qi and Yu Wei-Xing

104201

The Wigner distribution function of a super Lorentz–Gauss SLG11 beam through a paraxial ABCD optical system Zhou Yi-Min and Zhou Guo-Quan

104202

An improved deconvolution method for X-ray coded imaging in inertial confinement fusion Zhao Zong-Qing, He Wei-Hua, Wang Jian, Hao Yi-Dan, Cao Lei-Feng, Gu Yu-Qiu and Zhang Bao-Han

104203

Phase grating in a doubly degenerate four-level system Liu Yun, Wang Pu and Peng Shuang-Yan

104204

Curved surface effect and emission on silicon nanostructures Huang Wei-Qi, Yin Jun, Zhou Nian-Jie, Huang Zhong-Mei, Miao Xin-Jian, Cheng Han-Qiong, Su Qin, Liu Shi-Rong and Qin Chao-Jian

104205

An all-polarization-maintaining repetition-tunable erbium-doped passively mode-locked fiber laser Zhao Guang-Zhen, Xiao Xiao-Sheng, Meng Fei, Mei Jia-Wei and Yang Chang-Xi (Continued on the Bookbinding Inside Back Cover)

104206

A mode-locked external-cavity quantum-dot laser with a variable repetition rate Wu Jian, Jin Peng, Li Xin-Kun, Wei Heng, Wu Yan-Hua, Wang Fei-Fei, Chen Hong-Mei, Wu Ju and Wang Zhan-Guo

104207

The dependence on optical energy of terahertz emission from air plasma induced by two-color femtosecond laser-pulses Wu Si-Qing, Liu Jin-Song, Wang Sheng-Lie and Hu Bing

104208

Optical bistability induced by quantum coherence in a negative index atomic medium Zhang Hong-Jun, Guo Hong-Ju, Sun Hui, Li Jin-Ping and Yin Bao-Yin

104209

The mobility of nonlocal solitons in fading optical lattices Dai Zhi-Ping, Ling Xiao-Hui, Wang You-Wen and You Kai-Ming

104210

The influence of smoothing by spectral dispersion on the beam characteristics in the near field Fan Xin-Min, Lü Zhi-Wei, Lin Dian-Yang and Wang Yu-Lei

104211

Characteristics of photonic bands generated by quadrangular multiconnected networks Luo Rui-Fang, Yang Xiang-Bo, Lu Jian and Liu Timon Cheng-Yi

104212

Optical phase front control in a metallic grating with equally spaced alternately tapered slits Zheng Gai-Ge, Wu Yi-Gen and Xu Lin-Hua

104213

A high figure of merit localized surface plasmon sensor based on a gold nanograting on the top of a gold planar film Zhang Zu-Yin, Wang Li-Na, Hu Hai-Feng, Li Kang-Wen, Ma Xun-Peng and Song Guo-Feng

104501

A necessary and sufficient condition for transforming autonomous systems into linear autonomous Birkhoffian systems Cui Jin-Chao, Liu Shi-Xing and Song Duan

104502

The dynamic characteristics of harvesting energy from mechanical vibration via piezoelectric conversion Fan Kang-Qi, Ming Zheng-Feng, Xu Chun-Hui and Chao Feng-Bo

104701

Electro–magnetic control of shear flow over a cylinder for drag reduction and lift enhancement Zhang Hui, Fan Bao-Chun, Chen Zhi-Hua, Chen Shuai and Li Hong-Zhi PHYSICS OF GASES, PLASMAS, AND ELECTRIC DISCHARGES

105101

The mechanism of hydrogen plasma passivation for poly-crystalline silicon thin film Li Juan, Luo Chong, Meng Zhi-Guo, Xiong Shao-Zhen and Hoi Sing Kwok CONDENSED MATTER: STRUCTURAL, MECHANICAL, AND THERMAL PROPERTIES

106101

Silicon micro-hemispheres with periodic nanoscale rings produced by the laser ablation of single crystalline silicon Chen Ming, Li Shuang, Cui Qing-Qiang and Liu Xiang-Dong

106102

TiO2 /Ag composite nanowires for a recyclable surface enhanced Raman scattering substrate Deng Chao-Yue, Zhang Gu-Ling, Zou Bin, Shi Hong-Long, Liang Yu-Jie, Li Yong-Chao, Fu Jin-Xiang and Wang Wen-Zhong

106103

A phase-field model for simulating various spherulite morphologies of semi-crystalline polymers Wang Xiao-Dong, Ouyang Jie, Su Jin and Zhou Wen

(Continued on the Bookbinding Inside Back Cover)

106104

Effect of vacancy charge state on positron annihilation in silicon Liu Jian-Dang, Cheng Bin, Kong Wei and Ye Bang-Jiao

106105

Optical and magnetic properties of InFeP layers prepared by Fe+ implantation Zhou Lin, Shang Yan-Xia, Wang Ze-Song, Zhang Rui, Zhang Zao-Di, Vasiliy O. Pelenovich, Fu De-Jun and Kang Tae Won

106106

Effect of In𝑥 Ga1−𝑥 N “continuously graded” buffer layer on InGaN epilayer grown by metalorganic chemical vapor deposition Qian Wei-Ning, Su Shi-Chen, Chen Hong, Ma Zi-Guang, Zhu Ke-Bao, He Miao, Lu Ping-Yuan, Wang Geng, Lu Tai-Ping, Du Chun-Hua, Wang Qiao, Wu Wen-Bo and Zhang Wei-Wei

106107

Electric field modulation technique for high-voltage AlGaN/GaN Schottky barrier diodes Tang Cen, Xie Gang, Zhang Li, Guo Qing, Wang Tao and Sheng Kuang

106108

Accurate measurement and influence on device reliability of defect density of a light-emitting diode Guo Zu-Qiang and Qian Ke-Yuan

106109

Effects of chromium on structure and mechanical properties of vanadium: A first-principles study Gui Li-Jiang, Liu Yue-Lin, Wang Wei-Tian, Zhang Ying, Lü Guang-Hong and Yao Jun-En

106201

Size effect of the elastic modulus of rectangular nanobeams: Surface elasticity effect Yao Hai-Yan, Yun Guo-Hong and Fan Wen-Liang

106401

Thermodynamic properties of 3C SiC B. Y. Thakore, S. G. Khambholja, A. Y. Vahora, N. K. Bhatt and A. R. Jani

106801

Fabrication of pillar-array superhydrophobic silicon surface and thermodynamic analysis on the wetting state transition Liu Si-Si, Zhang Chao-Hui, Zhang Han-Bing, Zhou Jie, He Jian-Guo and Yin Heng-Yang

106802

Fabrication of GaN-based LEDs with 22◦ undercut sidewalls by inductively coupled plasma reactive ion etching Wang Bo, Su Shi-Chen, He Miao, Chen Hong, Wu Wen-Bo, Zhang Wei-Wei, Wang Qiao, Chen Yu-Long, Gao You, Zhang Li, Zhu Ke-Bao and Lei Yan

106803

Influence of Si doping on the structural and optical properties of InGaN epilayers Lu Ping-Yuan, Ma Zi-Guang, Su Shi-Chen, Zhang Li, Chen Hong, Jia Hai-Qiang, Jiang Yang, Qian Wei-Ning, Wang Geng, Lu Tai-Ping and He Miao CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND OPTICAL PROPERTIES

107101

Growth of monodisperse nanospheres of MnFe2 O4 with enhanced magnetic and optical properties M. Yasir Rafique, Pan Li-Qing, Qurat-ul-ain Javed, M. Zubair Iqbal, Qiu Hong-Mei, M. Hassan Farooq, Guo Zhen-Gang and M. Tanveer

107102

First-principles calculations of electronic and magnetic properties of CeN: The LDA + 𝑈 method Hao Ai-Min and Bai Jing

107103

Theoretical optoelectronic analysis of intermediate-band photovoltaic material based on ZnY1−𝑥 O𝑥 (Y = S, Se, Te) semiconductors by first-principles calculations Wu Kong-Ping, Gu Shu-Lin, Ye Jian-Dong, Tang Kun, Zhu Shun-Ming, Zhou Meng-Ran, Huang You-Rui, Zhang Rong and Zheng You-Dou (Continued on the Bookbinding Inside Back Cover)

107104

The effects of strain and surface roughness scattering on the quasi-ballistic characteristics of a Ge nanowire p-channel field-effect transistor Qin Jie-Yu, Du Gang and Liu Xiao-Yan

107105

The structural, elastic, and electronic properties of Zr𝑥 Nb1−𝑥 C alloys from first principle calculations Sun Xiao-Wei, Zhang Xin-Yu, Zhang Su-Hong, Zhu Yan, Wang Li-Min, Zhang Shi-Liang, Ma Ming-Zhen and Liu Ri-Ping

107106

The nonlinear optical properties of a magneto-exciton in a strained Ga0.2 In0.8 As/GaAs quantum dot N. R. Senthil Kumar, A. John Peter and Chang Kyoo Yoo

107201

Structural and electrical properties of laser-crystallized nanocrystalline Ge films and nanocrystalline Ge/SiN𝑥 multilayers Li Cong, Xu Jun, Li Wei, Jiang Xiao-Fan, Sun Sheng-Hua, Xu Ling and Chen Kun-Ji

107202

A shortcut for determining growth mode R. A. Rehman, Cai Yi-Liang, Zhang Han-Jie, Wu Ke, Dou Wei-Dong, Li Hai-Yang, He Pi-Mo and Bao ShiNing

107301

Temperature-dependent

rectifying

and

photovoltaic

characteristics

of

an

oxygen-deficient

Bi2 Sr2 Co2 O𝑦 /Si heterojunction Yan Guo-Ying, Bai Zi-Long, Li Hui-Ling, Fu Guang-Sheng, Liu Fu-Qiang, Yu Wei, Wang Jiang-Long and Wang Shu-Fang 107302

High-mobility germanium p-MOSFETs by using HCl and (NH4 )2 S surface passivation Xue Bai-Qing, Wang Sheng-Kai, Han Le, Chang Hu-Dong, Sun Bing, Zhao Wei and Liu Hong-Gang

107303

The degradation mechanism of an AlGaN/GaN high electron mobility transistor under step-stress Chen Wei-Wei, Ma Xiao-Hua, Hou Bin, Zhu Jie-Jie, Zhang Jin-Cheng and Hao Yue

107401

A new modulated structure in α-Fe2 O3 nanowires Cai Rong-Sheng, Shang Lei, Liu Xue-Hua, Wang Yi-Qian, Yuan Lu and Zhou Guang-Wen

107501

The variation of Mn-dopant distribution state with 𝑥 and its effect on the magnetic coupling mechanism in Zn1−𝑥 Mn𝑥 O nanocrystals Cheng Yan, Hao Wei-Chang, Li Wen-Xian, Xu Huai-Zhe, Chen Rui and Dou Shi-Xue

107502

Transformation behaviors, structural and magnetic characteristics of Ni Mn Ga films on MgO (001) Xie Ren, Tang Shao-Long, Tang Yan-Mei, Liu Xiao-Chen, Tang Tao and Du You-Wei

107701

Dielectric spectroscopy studies of ZnO single crystal Cheng Peng-Fei, Li Sheng-Tao and Wang Hui

107702

Bipolar resistive switching in BiFe0.95 Zn0.05 O3 films Yuan Xue-Yong, Luo Li-Rong, Wu Di and Xu Qing-Yu

107703

Analysis of tensile strain enhancement in Ge nano-belts on an insulator surrounded by dielectrics Lu Wei-Fang, Li Cheng, Huang Shi-Hao, Lin Guang-Yang, Wang Chen, Yan Guang-Ming, Huang Wei, Lai Hong-Kai and Chen Song-Yan

107802

The effect of an optical pump on the absorption coefficient of magnesium-doped near-stoichiometric lithium niobate in terahertz range Zuo Zhi-Gao, Ling Fu-Ri, Ma De-Cai, Wu Liang, Liu Jin-Song and Yao Jian-Quan


107803

In-situ growth of a CdS window layer by vacuum thermal evaporation for CIGS thin film solar cell applications Cao Min, Men Chuan-Ling, Zhu De-Ming, Tian Zi-Ao and An Zheng-Hua

107804

The metamaterial analogue of electromagnetically induced transparency by dual-mode excitation of a symmetric resonator Shao Jian, Li Jie, Li Jia-Qi, Wang Yu-Kun, Dong Zheng-Gao, Lu Wei-Bing and Zhai Ya

107805

Tunable zeroth-order resonator based on a ferrite metamaterial structure Javad Ghalibafan and Nader Komjani

107806

Structural distortions and magnetisms in Fe-doped LaMn1−𝑥 Fe𝑥 O3 (0 < 𝑥 ≤ 0.6) Zheng Long and Wu Xiao-Shan

107901

A comparison of the field emission characteristics of vertically aligned graphene sheets grown on different SiC substrates Chen Lian-Lian, Guo Li-Wei, Liu Yu, Li Zhi-Lin, Huang Jiao and Lu Wei INTERDISCIPLINARY PHYSICS AND RELATED AREAS OF SCIENCE AND TECHNOLOGY

108101

Controllable synthesis, characterization, and growth mechanism of hollow Zn𝑥 Cd1−𝑥 S spheres generated by a one-step thermal evaporation method Yang Zai-Xing, Zhong Wei, Au Chak-Tong and Du You-Wei

108102

The effect of fractional thermoelasticity on a two-dimensional problem of a mode I crack in a rotating fiber-reinforced thermoelastic medium Ahmed E. Abouelregal and Ashraf M. Zenkour

108103

Crystal growth, structural and physical properties of the 5d noncentrosymmetric LaOsSi3 Zhang Xu, Miao Shan-Shan, Wang Pu, Zheng Ping, Yin Wen-Long, Yao Ji-Yong, Jiang Hong-Wei, Wang Hai and Shi You-Guo

108301

A fiber-array probe technique for measuring the viscosity of a substance under shock compression Feng Li-Peng, Liu Fu-Sheng, Ma Xiao-Juan, Zhao Bei-Jing, Zhang Ning-Chao, Wang Wen-Peng and Hao Bin-Bin

108401

A novel slotted helix slow-wave structure for high power Ka-band traveling-wave tubes Liu Lu-Wei, Wei Yan-Yu, Wang Shao-Meng, Hou Yan, Yin Hai-Rong, Zhao Guo-Qing, Duan Zhao-Yun, Xu Jin, Gong Yu-Bin, Wang Wen-Xiang and Yang Ming-Hua

108402

The design and numerical analysis of tandem thermophotovoltaic cells Yang Hao-Yu, Liu Ren-Jun, Wang Lian-Kai, Lü You, Li Tian-Tian, Li Guo-Xing, Zhang Yuan-Tao and Zhang Bao-Lin

108501

Gate-to-body tunneling current model for silicon-on-insulator MOSFETs Wu Qing-Qing, Chen Jing, Luo Jie-Xin, Lü Kai, Yu Tao, Chai Zhan and Wang Xi

108502

Analyses of temperature-dependent interface states, series resistances, and AC electrical conductivities of Al/p Si and Al/Bi4 Ti3 O12 /p Si structures by using the admittance spectroscopy method Mert Yıldırım, Perihan Durmus¸, and S¸emsettin Altındal

108503

Design and fabrication of a high-performance evanescently coupled waveguide photodetector Liu Shao-Qing, Yang Xiao-Hong, Liu Yu, Li Bin and Han Qin


108504

Preliminary results for the design, fabrication, and performance of a backside-illuminated avalanche drift detector Qiao Yun, Liang Kun, Chen Wen-Fei and Han De-Jun

108505

Performance enhancement of an InGaN light-emitting diode with an AlGaN/InGaN superlattice electron-blocking layer Xiong Jian-Yong, Xu Yi-Qin, Zhao Fang, Song Jing-Jing, Ding Bin-Bin, Zheng Shu-Wen, Zhang Tao and Fan Guang-Han

108901

A comparison of coal supply-demand in China and in the US based on a network model Fang Cui-Cui, Sun Mei, Zhang Pei-Pei and Gao An-Na

108902

The effect of moving bottlenecks on a two-lane traffic flow Fang Yuan, Chen Jian-Zhong and Peng Zhi-Yuan

108903

An improvement of the fast uncovering community algorithm Wang Li, Wang Jiang, Shen Hua-Wei and Cheng Xue-Qi

108904

Random walks in generalized delayed recursive trees Sun Wei-Gang, Zhang Jing-Yuan and Chen Guan-Rong GEOPHYSICS, ASTRONOMY, AND ASTROPHYSICS

109501

Modeling and assessing the influence of linear energy transfer on multiple bit upset susceptibility Geng Chao, Liu Jie, Xi Kai, Zhang Zhan-Gang, Gu Song and Liu Tian-Qi