- Email: [email protected]
Parameter estimation and synchronization in the uncertain financial network
Physica A 535 (2019) 122418
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Parameter estimation and synchronization in the uncertain financial network ∗
Ling Lü , Qingtao Wei School of Electrical Engineering, Dalian University of Science and Technology, Dalian 116052, China
article
info
Article history: Received 14 July 2019 Available online 16 August 2019 Keywords: Uncertain financial network Parameter estimation Synchronization Lyapunov exponent
a b s t r a c t In this work, we research the parameter estimation and synchronization in the uncertain financial network. First, we introduce the characteristics of a typical financial system. And we design an observer to effectively estimate the uncertain parameter in the system. On this basis, we research the synchronization of the uncertain financial network consisting of several financial systems. The conditions of network synchronization are obtained by calculating the Lyapunov exponent of the network. Finally, we test the network synchronization performance using numerical simulation. This technique does not need to design the network controller or the control input, it is simple and convenient for practical application. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Modern research has found that the financial system is an open and highly complex nonlinear system. Financial crisis and financial overheating belong to the uncertain behavior of the system after instability, which corresponds to the chaotic phenomenon in the nonlinear theory. The existence of chaos in the financial system is first discovered by Stutzer in the economic growth equation [1]. Since then, chaos has been found in the economic cycle, money, finance and savings. Therefore, using chaos theory to study uncertain behavior in the financial system has become a hot topic in the financial field. In fact, the financial system of each country or region does not exist in isolation. It is interrelated with the financial system of other countries or regions. Therefore, these interactive financial systems constitute a complex network. In other words, these interacting financial systems have characteristics of complex networks. Therefore, in the process of global economic development, it is of great practical significance to prevent financial crisis and to achieve synchronous development in countries or regions with different degrees of economic development. In the theory of complex networks, the above problems are equivalent to the synchronization problems of complex networks. Nowadays, the network synchronizations have attracted particular attention [2–8]. After people’s unremitting efforts, a large number of research results have been reported, and a variety of synchronization types have emerged, such as complete synchronization [9– 11], phase synchronization [12,13], projective synchronization [14,15] and cluster synchronization [16,17], etc. Moreover, many effective synchronization strategies have been proposed, including the Master Stability Functions (MSF) criterion [18], connection graph method [19], adaptive control [20,21], pinning technique [22,23], impulsive control [24], coupling technique [25] and so on. ∗ Corresponding author. E-mail address: [email protected] (L. Lü). https://doi.org/10.1016/j.physa.2019.122418 0378-4371/© 2019 Elsevier B.V. All rights reserved.
2
L. Lü and Q. Wei / Physica A 535 (2019) 122418
Fig. 1. Evolution of Lyapunov exponent with parameter.
Fig. 2. Chaotic attractor of the system.
In general, the network synchronization is achieved by applying the controller or the control input in the controlled network. However, the network contains a large number of interactive nodes and the network structure is also very complex. These factors lead to the complexity of the network controller structure or the control input, which is difficult to design and implement. Meanwhile, the added controller or the control input often changes the original dynamics of the network, which is not allowed in practice. In addition, for a large financial network, there are often some uncertain factors leading to the uncertainty of network parameters. Therefore, it is a meaningful work to estimate parameter for the uncertain financial network. In this work, we research the parameter estimation and synchronization in the uncertain financial network. First, we introduce the characteristics of a typical financial system. And we design an observer to effectively estimate the uncertain parameter in the system. On this basis, we research the synchronization of the uncertain financial network consisting of several financial systems. The conditions of network synchronization are obtained by calculating the Lyapunov exponent of the network. Finally, we test the network synchronization performance using numerical simulation. This technique does not need to design the network controller or the control input, it is simple and convenient for practical application. The rest of this work is organized as follows. In Section 2 introduces the characteristics of a typical financial system. And the uncertain parameter of the system is estimated effectively. In Section 3, we research the synchronization of the uncertain financial network. Section 4 is the conclusions of this work.
L. Lü and Q. Wei / Physica A 535 (2019) 122418
3
Fig. 3. Estimation process of uncertain parameter.
Fig. 4. Financial network structure.
2. Characteristics of a financial system and parameter estimation Consider a financial system consisting of factors such as production, currency, securities and labor [26]
⎧ ⎪ dx ⎪ ⎪ = z + (y − a) ⎪ ⎪ ⎪ ⎨ dt dy = 1 − by − x2 ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dz = −x − cz
(1)
dt where x stands for interest rate, y represents investment demand, z denotes price index. a, b and c are parameters, representing the amount of savings, the cost of investment and the elasticity of commodity demand, and a = 0.9, b = 0.2, c = 1. Based on the Lyapunov theorem, when the maximum Lyapunov exponent is greater than zero, the system is in chaotic state. We keep the other parameters in the system unchanged and calculate the evolution of the maximum Lyapunov exponent with the parameter c, as shown in Fig. 1. Fig. 2 is the chaotic attractor when the parameter c = 1. Further, we assume that the parameter b in the system is uncertain, an observer is designed to estimate it. Assuming that b¯ is the estimated value of uncertain parameter b, the observer structure is designed as follows db¯ dt
=
dR(y) dy
¯ y(b − b)
where R(y) is an undetermined gain function.
(2)
4
L. Lü and Q. Wei / Physica A 535 (2019) 122418
Fig. 5. Evolutions of the maximum Lyapunov exponents with the correlation coefficients.
From Eq. (1), we get by = −
dy dt
+ 1 − x2
(3)
By substituting Eq. (3) into Eq. (2), the following relationship can be obtained db¯ dt
=
dR(y) dy
(−
dy dt
+ 1 − x2 ) −
In fact, it is hard to observe
dy , dt
dR(y) dy
yb¯
(4)
so the observer is not applicable. We introduce an auxiliary variable
u = b¯ + R(y)
(5)
then according to Eqs. (4) and (5), we have du dt
=
dR(y) dy
[1 − x2 − y(u − R(y))]
(6)
We choose the form of the gain function as follows R(y) =
1 2
gy2
where g is the arbitrary positive coefficient.
(7)
L. Lü and Q. Wei / Physica A 535 (2019) 122418
5
Finally, the structure of the designed observer is
⎧ ⎪ ⎨ b¯ = u − 1 gy2 2
(8)
du 1 ⎪ ⎩ = gy[1 − x2 − y(u − gy2 )] dt
2
It is easy to see from Eq. (2), b¯ converges to b with exponential rate as t → ∞. We choose the coefficient as g = 0.5, the estimation process of uncertain parameter b is simulated as shown in Fig. 3. It is found that the estimation curve tend gradually to a fixed value 0.2 from arbitrary initial value, indicating that the designed observer is effective. 3. Synchronization in the uncertain financial network Let us assume that we make the financial systems of countries with different levels of economic development or regions with different levels of economic development within a country establish some kind of connection to form a financial network. The node of the network is the financial system of each country or region, and the edge of the network is the correlation between these financial systems. Here, we arbitrarily choose five financial systems as network nodes. Among them, the financial system (FS) of country or region with higher economic development level is linked to the other four financial systems (FS1, FS2, FS3, FS4), the network structure is shown in Fig. 4. Based on Fig. 4, the network node equations are respectively
⎧ ⎪ dx ⎪ ⎪ = z + (y − a) ⎪ ⎪ ⎪ ⎨ dt dy = 1 − by − x2 ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dz = −x − cz ⎧ dt ⎪ dxi ⎪ ⎪ = zi + (yi − a) + ki (x − xi ) ⎪ ⎪ ⎪ dt ⎨ dyi
= 1 − byi − x2i ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dzi = −x − cz i i
(9)
(i = 1, 2, 3, 4)
(10)
dt
where ki is the correlation coefficient. We still calculate the Lyapunov exponents as the criterion of network synchronization. The evolutions of the maximum Lyapunov exponents λmax i (i = 1, 2, · · · , 4) of the related nodes with the correlation coefficients are shown in Fig. 6. In Fig. 5, the critical value of correlation coefficient is about 0.32 when all Lyapunov exponents are negative. Fixed ki = 0.32, the time evolutions of network errors exi = x − xi , eyi = y − yi , ezi = z − zi (i = 1, 2, · · · , 4) are simulated as shown in Fig. 6. It can be seen that the errors tend to zero, that is, all financial systems achieve complete synchronization. The results mean that countries or regions with different levels of economic development can make their economies tend to develop synchronously by interlinking with each other. Therefore, this work has very important application value. 4. Conclusion The parameter estimation and synchronization in the uncertain financial network have been researched. We have designed an observer to effectively estimate the uncertain parameter in the financial system. By calculating the maximum Lyapunov exponent of the financial network, we have obtained the criterion of network synchronization. The research results show that when the correlation coefficients between network nodes are greater than 0.32, the uncertain financial network achieves complete synchronization. This technique does not need to design the network controller or the control input, it is simple and convenient for practical application. Acknowledgment This research was supported by the National Natural Science Foundation of China (Grant No. 11747318).
6
L. Lü and Q. Wei / Physica A 535 (2019) 122418
Fig. 6. Errors between network nodes.
References [1] M.J. Stutzer, Chaotic dynamics and bifurcation in a macro model, J. Econom. Dynam. Control 2 (1980) 353–376. [2] T. Botmart, N. Yotha, P. Niamsup, W. Weera, Hybrid adaptive pinning control for function projective synchronization of delayed neural networks with mixed uncertain couplings, Complexity 2017 (2017) 1–18. [3] A. Diwadkar, U. Vaidya, Synchronization in large-scale nonlinear network systems with uncertain links, Automatica 100 (2019) 194–199. [4] M. Syed Ali, J. Yogambigai, Synchronization criterion of complex dynamical networks with both leakage delay and coupling delay on time scales, Neural Process. Lett. 49 (2019) 453–466. [5] L. Lü, F.L. Zhang, A. Sun, Synchronization between uncertain spatiotemporal networks based on open-loop and closed-loop coupling technology, Physica A 526 (2019) 120712–120717. [6] A. Otto, G. Radons, D. Bachrathy, G. Orosz, Synchronization in networks with heterogeneous coupling delays, Phys. Rev. E 97 (2018) 012311–012313. [7] D. Ibánez-Soria, J. Garcia-Ojalvo, A. Soria-Frisch, G. Ruffini, Detection of generalized synchronization using echo state networks, Chaos 28 (2018) 033118-8. [8] P. DeLellis, F. Garofalo, F. Lo Iudice, The partial pinning control strategy for large complex networks, Automatica 89 (2018) 111–116. [9] A. Arellano-Delgado, R.M. López-Gutiérrez, R. Martínez-Clark, C. Cruz-Hernández, Small-world outer synchronization of small-world chaotic networks, J. Comput. Nonlinear Dyn. 13 (2018) 101008–101111. [10] A. Sun, L. Lü, C.R. Li, Study on synchronization in an uncertain neuronal network with small-world effects, Nonlinear Dynam. 82 (2015) 1905–1912. [11] L. Lü, C.R. Li, G. Li, G.N. Zhao, Cluster synchronization transmission of laser pattern signal in laser network with ring cavity, Sci. Sin. Phys. Mech. Astron. 47 (2017) 080501–080510 (in Chinese). [12] P.S. Skardal, D. Taylor, J. Sun, A. Arenas, Erosion of synchronization in networks of coupled oscillators, Phys. Rev. E 91 (2015) 010802–010805. [13] F. Dörfler, F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica 50 (2014) 1539–1564. [14] G. Al-Mahbashi, M.S.M. Noorani, S.A. Bakar, Projective lag synchronization in drive-response dynamical networks, Internat. J. Modern Phys. C 25 (2014) 771–776. [15] L. Lü, C.R. Li, G. Li, G.N. Zhao, Projective synchronization for uncertain network based on modified sliding mode control technique, Internat. J. Adapt. Control Signal Process. 31 (2017) 429–440. [16] S. Jalan, A. Singh, S. Acharyya, J. Kurths, Impact of a leader on cluster synchronization, Phys. Rev. E 91 (2015) 022901–122905.
L. Lü and Q. Wei / Physica A 535 (2019) 122418 [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
7
A.B. Siddique, L. Pecora, J.D. Hart, F. Sorrentino, Symmetry-and input-cluster synchronization in networks, Phys. Rev. E 97 (2018) 042217–042219. L.M. Pecora, T.L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett. 80 (1998) 2109–2112. V.N. Belykh, I.V. Belykh, M. Hasler, Connection graph stability method for synchronized coupled chaotic systems, Physica D 195 (2004) 159–187. A.A. Selivanov, J. Lehnert, T. Dahms, P. Hövel, A.L. Fradkov, E. Schöll, Adaptive synchronization in delay-coupled networks of Stuart–Landau oscillators, Phys. Rev. E 85 (2012) 016201–016208. A. Selivanov, A. Fradkov, E. Fridman, Passification-based decentralized adaptive synchronization of dynamical networks with time-varying delays, J. Franklin Inst. 352 (2015) 52–72. S. Dharani, R. Rakkiyappan, J.H. Park, Pinning sampled-data synchronization of coupled inertial neural networks with reaction–diffusion terms and time-varying delays, Neurocomputing 227 (2017) 101–107. R. Rakkiyappan, N. Sakthivel, Pinning sampled-data control for synchronization of complex networks with probabilistic time-varying delays using quadratic convex approach, Neurocomputing 162 (2015) 26–40. N. Mahdavi, M.B. Menhaj, J. Kurths, J.Q. Lu, A. Afshar, Pinning impulsive synchronization of complex dynamical networks, Int. J. Bifurcation Chaos 22 (2012) 1250239-14. K. Srinivasan, V.K. Chandrasekar, R. Gladwin Pradeep, K. Murali, M. Lakshmanan, Different types of synchronization in coupled network based chaotic circuits, Commun. Nonlinear Sci. Numer. Simul. 39 (2016) 156–168. J.H. Ma, Y.S. Chen, Study for the bifurcation chaotic topological structure and the global complicated character of a kind of nonlinear finance system (I), Appl. Math. Mech. 22 (2001) 1240–1251 (in Chinese).
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Parameter estimation and synchronization in the uncertain financial network ∗
Ling Lü , Qingtao Wei School of Electrical Engineering, Dalian University of Science and Technology, Dalian 116052, China
article
info
Article history: Received 14 July 2019 Available online 16 August 2019 Keywords: Uncertain financial network Parameter estimation Synchronization Lyapunov exponent
a b s t r a c t In this work, we research the parameter estimation and synchronization in the uncertain financial network. First, we introduce the characteristics of a typical financial system. And we design an observer to effectively estimate the uncertain parameter in the system. On this basis, we research the synchronization of the uncertain financial network consisting of several financial systems. The conditions of network synchronization are obtained by calculating the Lyapunov exponent of the network. Finally, we test the network synchronization performance using numerical simulation. This technique does not need to design the network controller or the control input, it is simple and convenient for practical application. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Modern research has found that the financial system is an open and highly complex nonlinear system. Financial crisis and financial overheating belong to the uncertain behavior of the system after instability, which corresponds to the chaotic phenomenon in the nonlinear theory. The existence of chaos in the financial system is first discovered by Stutzer in the economic growth equation [1]. Since then, chaos has been found in the economic cycle, money, finance and savings. Therefore, using chaos theory to study uncertain behavior in the financial system has become a hot topic in the financial field. In fact, the financial system of each country or region does not exist in isolation. It is interrelated with the financial system of other countries or regions. Therefore, these interactive financial systems constitute a complex network. In other words, these interacting financial systems have characteristics of complex networks. Therefore, in the process of global economic development, it is of great practical significance to prevent financial crisis and to achieve synchronous development in countries or regions with different degrees of economic development. In the theory of complex networks, the above problems are equivalent to the synchronization problems of complex networks. Nowadays, the network synchronizations have attracted particular attention [2–8]. After people’s unremitting efforts, a large number of research results have been reported, and a variety of synchronization types have emerged, such as complete synchronization [9– 11], phase synchronization [12,13], projective synchronization [14,15] and cluster synchronization [16,17], etc. Moreover, many effective synchronization strategies have been proposed, including the Master Stability Functions (MSF) criterion [18], connection graph method [19], adaptive control [20,21], pinning technique [22,23], impulsive control [24], coupling technique [25] and so on. ∗ Corresponding author. E-mail address: [email protected] (L. Lü). https://doi.org/10.1016/j.physa.2019.122418 0378-4371/© 2019 Elsevier B.V. All rights reserved.
2
L. Lü and Q. Wei / Physica A 535 (2019) 122418
Fig. 1. Evolution of Lyapunov exponent with parameter.
Fig. 2. Chaotic attractor of the system.
In general, the network synchronization is achieved by applying the controller or the control input in the controlled network. However, the network contains a large number of interactive nodes and the network structure is also very complex. These factors lead to the complexity of the network controller structure or the control input, which is difficult to design and implement. Meanwhile, the added controller or the control input often changes the original dynamics of the network, which is not allowed in practice. In addition, for a large financial network, there are often some uncertain factors leading to the uncertainty of network parameters. Therefore, it is a meaningful work to estimate parameter for the uncertain financial network. In this work, we research the parameter estimation and synchronization in the uncertain financial network. First, we introduce the characteristics of a typical financial system. And we design an observer to effectively estimate the uncertain parameter in the system. On this basis, we research the synchronization of the uncertain financial network consisting of several financial systems. The conditions of network synchronization are obtained by calculating the Lyapunov exponent of the network. Finally, we test the network synchronization performance using numerical simulation. This technique does not need to design the network controller or the control input, it is simple and convenient for practical application. The rest of this work is organized as follows. In Section 2 introduces the characteristics of a typical financial system. And the uncertain parameter of the system is estimated effectively. In Section 3, we research the synchronization of the uncertain financial network. Section 4 is the conclusions of this work.
L. Lü and Q. Wei / Physica A 535 (2019) 122418
3
Fig. 3. Estimation process of uncertain parameter.
Fig. 4. Financial network structure.
2. Characteristics of a financial system and parameter estimation Consider a financial system consisting of factors such as production, currency, securities and labor [26]
⎧ ⎪ dx ⎪ ⎪ = z + (y − a) ⎪ ⎪ ⎪ ⎨ dt dy = 1 − by − x2 ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dz = −x − cz
(1)
dt where x stands for interest rate, y represents investment demand, z denotes price index. a, b and c are parameters, representing the amount of savings, the cost of investment and the elasticity of commodity demand, and a = 0.9, b = 0.2, c = 1. Based on the Lyapunov theorem, when the maximum Lyapunov exponent is greater than zero, the system is in chaotic state. We keep the other parameters in the system unchanged and calculate the evolution of the maximum Lyapunov exponent with the parameter c, as shown in Fig. 1. Fig. 2 is the chaotic attractor when the parameter c = 1. Further, we assume that the parameter b in the system is uncertain, an observer is designed to estimate it. Assuming that b¯ is the estimated value of uncertain parameter b, the observer structure is designed as follows db¯ dt
=
dR(y) dy
¯ y(b − b)
where R(y) is an undetermined gain function.
(2)
4
L. Lü and Q. Wei / Physica A 535 (2019) 122418
Fig. 5. Evolutions of the maximum Lyapunov exponents with the correlation coefficients.
From Eq. (1), we get by = −
dy dt
+ 1 − x2
(3)
By substituting Eq. (3) into Eq. (2), the following relationship can be obtained db¯ dt
=
dR(y) dy
(−
dy dt
+ 1 − x2 ) −
In fact, it is hard to observe
dy , dt
dR(y) dy
yb¯
(4)
so the observer is not applicable. We introduce an auxiliary variable
u = b¯ + R(y)
(5)
then according to Eqs. (4) and (5), we have du dt
=
dR(y) dy
[1 − x2 − y(u − R(y))]
(6)
We choose the form of the gain function as follows R(y) =
1 2
gy2
where g is the arbitrary positive coefficient.
(7)
L. Lü and Q. Wei / Physica A 535 (2019) 122418
5
Finally, the structure of the designed observer is
⎧ ⎪ ⎨ b¯ = u − 1 gy2 2
(8)
du 1 ⎪ ⎩ = gy[1 − x2 − y(u − gy2 )] dt
2
It is easy to see from Eq. (2), b¯ converges to b with exponential rate as t → ∞. We choose the coefficient as g = 0.5, the estimation process of uncertain parameter b is simulated as shown in Fig. 3. It is found that the estimation curve tend gradually to a fixed value 0.2 from arbitrary initial value, indicating that the designed observer is effective. 3. Synchronization in the uncertain financial network Let us assume that we make the financial systems of countries with different levels of economic development or regions with different levels of economic development within a country establish some kind of connection to form a financial network. The node of the network is the financial system of each country or region, and the edge of the network is the correlation between these financial systems. Here, we arbitrarily choose five financial systems as network nodes. Among them, the financial system (FS) of country or region with higher economic development level is linked to the other four financial systems (FS1, FS2, FS3, FS4), the network structure is shown in Fig. 4. Based on Fig. 4, the network node equations are respectively
⎧ ⎪ dx ⎪ ⎪ = z + (y − a) ⎪ ⎪ ⎪ ⎨ dt dy = 1 − by − x2 ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dz = −x − cz ⎧ dt ⎪ dxi ⎪ ⎪ = zi + (yi − a) + ki (x − xi ) ⎪ ⎪ ⎪ dt ⎨ dyi
= 1 − byi − x2i ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎩ dzi = −x − cz i i
(9)
(i = 1, 2, 3, 4)
(10)
dt
where ki is the correlation coefficient. We still calculate the Lyapunov exponents as the criterion of network synchronization. The evolutions of the maximum Lyapunov exponents λmax i (i = 1, 2, · · · , 4) of the related nodes with the correlation coefficients are shown in Fig. 6. In Fig. 5, the critical value of correlation coefficient is about 0.32 when all Lyapunov exponents are negative. Fixed ki = 0.32, the time evolutions of network errors exi = x − xi , eyi = y − yi , ezi = z − zi (i = 1, 2, · · · , 4) are simulated as shown in Fig. 6. It can be seen that the errors tend to zero, that is, all financial systems achieve complete synchronization. The results mean that countries or regions with different levels of economic development can make their economies tend to develop synchronously by interlinking with each other. Therefore, this work has very important application value. 4. Conclusion The parameter estimation and synchronization in the uncertain financial network have been researched. We have designed an observer to effectively estimate the uncertain parameter in the financial system. By calculating the maximum Lyapunov exponent of the financial network, we have obtained the criterion of network synchronization. The research results show that when the correlation coefficients between network nodes are greater than 0.32, the uncertain financial network achieves complete synchronization. This technique does not need to design the network controller or the control input, it is simple and convenient for practical application. Acknowledgment This research was supported by the National Natural Science Foundation of China (Grant No. 11747318).
6
L. Lü and Q. Wei / Physica A 535 (2019) 122418
Fig. 6. Errors between network nodes.
References [1] M.J. Stutzer, Chaotic dynamics and bifurcation in a macro model, J. Econom. Dynam. Control 2 (1980) 353–376. [2] T. Botmart, N. Yotha, P. Niamsup, W. Weera, Hybrid adaptive pinning control for function projective synchronization of delayed neural networks with mixed uncertain couplings, Complexity 2017 (2017) 1–18. [3] A. Diwadkar, U. Vaidya, Synchronization in large-scale nonlinear network systems with uncertain links, Automatica 100 (2019) 194–199. [4] M. Syed Ali, J. Yogambigai, Synchronization criterion of complex dynamical networks with both leakage delay and coupling delay on time scales, Neural Process. Lett. 49 (2019) 453–466. [5] L. Lü, F.L. Zhang, A. Sun, Synchronization between uncertain spatiotemporal networks based on open-loop and closed-loop coupling technology, Physica A 526 (2019) 120712–120717. [6] A. Otto, G. Radons, D. Bachrathy, G. Orosz, Synchronization in networks with heterogeneous coupling delays, Phys. Rev. E 97 (2018) 012311–012313. [7] D. Ibánez-Soria, J. Garcia-Ojalvo, A. Soria-Frisch, G. Ruffini, Detection of generalized synchronization using echo state networks, Chaos 28 (2018) 033118-8. [8] P. DeLellis, F. Garofalo, F. Lo Iudice, The partial pinning control strategy for large complex networks, Automatica 89 (2018) 111–116. [9] A. Arellano-Delgado, R.M. López-Gutiérrez, R. Martínez-Clark, C. Cruz-Hernández, Small-world outer synchronization of small-world chaotic networks, J. Comput. Nonlinear Dyn. 13 (2018) 101008–101111. [10] A. Sun, L. Lü, C.R. Li, Study on synchronization in an uncertain neuronal network with small-world effects, Nonlinear Dynam. 82 (2015) 1905–1912. [11] L. Lü, C.R. Li, G. Li, G.N. Zhao, Cluster synchronization transmission of laser pattern signal in laser network with ring cavity, Sci. Sin. Phys. Mech. Astron. 47 (2017) 080501–080510 (in Chinese). [12] P.S. Skardal, D. Taylor, J. Sun, A. Arenas, Erosion of synchronization in networks of coupled oscillators, Phys. Rev. E 91 (2015) 010802–010805. [13] F. Dörfler, F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica 50 (2014) 1539–1564. [14] G. Al-Mahbashi, M.S.M. Noorani, S.A. Bakar, Projective lag synchronization in drive-response dynamical networks, Internat. J. Modern Phys. C 25 (2014) 771–776. [15] L. Lü, C.R. Li, G. Li, G.N. Zhao, Projective synchronization for uncertain network based on modified sliding mode control technique, Internat. J. Adapt. Control Signal Process. 31 (2017) 429–440. [16] S. Jalan, A. Singh, S. Acharyya, J. Kurths, Impact of a leader on cluster synchronization, Phys. Rev. E 91 (2015) 022901–122905.
L. Lü and Q. Wei / Physica A 535 (2019) 122418 [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
7
A.B. Siddique, L. Pecora, J.D. Hart, F. Sorrentino, Symmetry-and input-cluster synchronization in networks, Phys. Rev. E 97 (2018) 042217–042219. L.M. Pecora, T.L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett. 80 (1998) 2109–2112. V.N. Belykh, I.V. Belykh, M. Hasler, Connection graph stability method for synchronized coupled chaotic systems, Physica D 195 (2004) 159–187. A.A. Selivanov, J. Lehnert, T. Dahms, P. Hövel, A.L. Fradkov, E. Schöll, Adaptive synchronization in delay-coupled networks of Stuart–Landau oscillators, Phys. Rev. E 85 (2012) 016201–016208. A. Selivanov, A. Fradkov, E. Fridman, Passification-based decentralized adaptive synchronization of dynamical networks with time-varying delays, J. Franklin Inst. 352 (2015) 52–72. S. Dharani, R. Rakkiyappan, J.H. Park, Pinning sampled-data synchronization of coupled inertial neural networks with reaction–diffusion terms and time-varying delays, Neurocomputing 227 (2017) 101–107. R. Rakkiyappan, N. Sakthivel, Pinning sampled-data control for synchronization of complex networks with probabilistic time-varying delays using quadratic convex approach, Neurocomputing 162 (2015) 26–40. N. Mahdavi, M.B. Menhaj, J. Kurths, J.Q. Lu, A. Afshar, Pinning impulsive synchronization of complex dynamical networks, Int. J. Bifurcation Chaos 22 (2012) 1250239-14. K. Srinivasan, V.K. Chandrasekar, R. Gladwin Pradeep, K. Murali, M. Lakshmanan, Different types of synchronization in coupled network based chaotic circuits, Commun. Nonlinear Sci. Numer. Simul. 39 (2016) 156–168. J.H. Ma, Y.S. Chen, Study for the bifurcation chaotic topological structure and the global complicated character of a kind of nonlinear finance system (I), Appl. Math. Mech. 22 (2001) 1240–1251 (in Chinese).