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2016
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Exponential extinction of discrete Nicholson's blowflies systems with patch structure and mortality terms
Exponential extinction of discrete Nicholson's blowflies systems with patch structure and mortality terms
en
en
Discrete Nicholson's blowflies systems with patch structure and mortality terms are considered in this
paper. Based on certain discrete inequalities, we prove the boundedness of the systems. Using this result,
sufficient conditions are then established to guarantee the exponential extinction for the systems. We provide
numerical examples verified by illustrative figures to demonstrate the validity of the proposed results.
298
307
J.
Alzabut
S.
Obaidat
Z.
Yao
Extinction
discrete Nicholson's blowflies model
patch structure
mortality terms.
Article.1.pdf
[
[1]
J. O. Alzabut, Almost periodic solutions for an impulsive delay Nicholson's blowflies model, J. Comput. Appl. Math., 234 (2010), 233-239
##[2]
J. O. Alzabut, Existence and exponential convergence of almost periodic solutions for a discrete Nicholson's blowflies model with a nonlinear harvesting term , Math. Sci. Lett., 2 (2013), 201-207
##[3]
J. O. Alzabut, Y. Bolat, T. Abdeljawad, Almost periodic dynamics of a discrete Nicholson's blowflies model involving a linear harvesting term, Adv. Difference Equ., 2012 (2012), 1-13
##[4]
P. Amstera, A. De'bolia , Existence of positive T-periodic solutions of a generalized Nicholson's blowflies model with a nonlinear harvesting term, Appl. Math. Lett., 25 (2012), 1203-1207
##[5]
L. Berezansky, E. Braverman, L. Idels, Nicholson's blowflies differential equations revisited: Main results and open problems , Appl. Math. Model., 34 (2010), 1405-1417
##[6]
W. Chen, Permanence for Nicholson-type systems with patch structure and nonlinear density-dependent mortality terms , Electron. J. Qual. Theory Differ. Equ., 2012 (2012), 1-14
##[7]
W. S. C. Gurney, S. P. Blythe, R. M. Nisbet, Nicholsons blowflies revisited, Nature, 287 (1980), 17-21
##[8]
X. Hou, L. Duan, New results on periodic solutions of delayed Nicholsons blowflies models, Electron. J. Qual. Theory Differ. Equ., 2012 (2012), 1-11
##[9]
X. Hou, L. Duan, Z. Huang, Permanence and periodic solutions for a class of delay Nicholson's blowflies models, Appl. Math. Model., 37 (2013), 1537-1544
##[10]
V. L. Kocic, G. Ladas, Oscillation and global attractivity in the discrete model of Nicholson's blowflies, Appl. Anal., 38 (1990), 21-31
##[11]
M. R. S. Kulenović, G. Ladas, Y. S. Sficas, Global attractivity in population dynamics, Comput. Math. Appl., 18 (1989), 925-928
##[12]
J. Li, C. Du, Existence of positive periodic solutions for a generalized Nicholson's blowfies model, J. Comput. Appl. Math., 221 (2008), 226-233
##[13]
W. T. Li, Y. H. Fan, Existence and global attractivity of positive periodic solutions for the impulsive delay Nicholson's blowflies model, J. Comput. Appl. Math., 201 (2007), 55-68
##[14]
B. Liu, S. Gong, Permanence for Nicholson-type delay systems with nonlinear density-dependent mortality terms , Nonlinear Anal., 12 (2011), 1931-1937
##[15]
X. Liu, J. Meng, The positive almost periodic solution for Nicholsontype delay systems with linear harvesting terms, Appl. Math. Model., 36 (2012), 3289-3298
##[16]
F. Long, Postive almost periodic solution for a class of Nicholson's blowflies model with a linear harvesting term, Nonlinear Anal., 13 (2012), 686-693
##[17]
F. Long, M. Yang, Positive periodic solutions of delayed Nicholsons blowflies model with a linear harvesting term, Electron. J. Qual. Theory Differ. Equ., 2011 (2011), 1-11
##[18]
A. J. Nicholson, An outline of the dynamics of animal populations, Austral. J. Zool., 2 (1954), 9-65
##[19]
S. H. Saker, S. Agarwal, Oscillation and global attractivity in a periodic Nicholson's blowflies model, Math. Comput. Model., 35 (2002), 719-731
##[20]
J. W. H. So, J. S. Yu, On the stability and uniform persistence of a discrete model of Nicholson's blowflies, J. Math. Anal. Appl., 193 (1995), 233-244
##[21]
W. Wang, Positive periodic solutions of delayed Nicholson's blowflies models with a nonlinear density-dependent mortality term, Appl. Math. Model., 36 (2012), 4708-4713
##[22]
J. Wei, M. Y. Li , Hopf bifurcation analysis in a delayed Nicholson blowflies equation , Nonlinear Anal., 60 (2005), 1351-1367
##[23]
Z. Yao, Existence and exponential convergence of almost periodic positive solution for Nicholson's blowflies discrete model with linear harvesting term, Math. Methods Appl. Sci., 37 (2014), 2354-2362
##[24]
Z. Yao, Existence and exponential stability of the unique almost periodic positive solution for discrete Nicholson's blowflies model, Int. J. Nonlinear Sci. Numer. Simul., 16 (2015), 185-190
##[25]
B. G. Zhang, H. X. Xu, A note on the global attractivity of a discrete model of Nicholson's blowflies, Discrete Dyn. Nat. Soc., 3 (1999), 51-55
##[26]
W. Zhao, C. Zhu, H. Zhu, On positive periodic solution for the delay Nicholson's blowflies model with a harvesting term, Appl. Math. Model., 36 (2012), 3335-3340
]
Combination complex synchronization among three incommensurate fractional-order chaotic systems
Combination complex synchronization among three incommensurate fractional-order chaotic systems
en
en
The problem of combination complex synchronization among three incommensurate fractionalorder
chaotic systems is considered. Based on the stability theory of incommensurate fractional-order
systems and the feedback control technique, some robust criteria on combination complex synchronization
are presented. Notably, the proposed combination complex synchronization can establish
a link between the incommensurate fractional-order complex chaos and real chaos. Moreover, three
numerical simulations are provided, which agree well with the theoretical analysis.
308
323
Cuimei
Jiang
Changan
Liu
Shutang
Liu
Fangfang
Zhang
Combination complex synchronization
chaotic complex system
fractional-order system
feedback control technique.
Article.2.pdf
[
[1]
F. T. Arecchi, R. Meucci, A. Di Garbo, E. Allaria, Homoclinic chaos in a laser: synchronization and its implications in biological systems, Opt. Lasers Eng., 39 (2003), 293-304
##[2]
S. Bhalekar, V. Daftardar-Gejji, Synchronization of different fractional order chaotic systems using active control, Commun. Nonlinear Sci. Numer. Simulat., 15 (2010), 3536-3546
##[3]
B. Blasius, A. Huppert, L. Stone, Complex dynamics and phase synchronization in spatially extended ecological systems, Nature, 399 (1999), 354-359
##[4]
L. P. Chen, Y. Chai, R. C. Wu , Lag projective synchronization in fractional-order chaotic (hyperchaotic) systems , Phys. Lett. A, 375 (2011), 2099-2110
##[5]
W. H. Deng, C. P. Li, J. H. Lü, Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dynam., 48 (2007), 409-416
##[6]
K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approch for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3-22
##[7]
H. Dimassi, A. Loria, Adaptive unknown-input observers-based synchronization of chaotic systems for telecommunication, IEEE Trans. Circuits Syst.-I Regul. Pap., 58 (2011), 800-812
##[8]
G. H. Erjaee, S. Momani , Phase synchronization in fractional differential chaotic systems, Phys. Lett. A, 372 (2008), 2350-2354
##[9]
I. Grigorenko, E. Grigorenko , Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett., 91 (2003), 1-4
##[10]
A. S. Hegazi, E. Ahmed, A. E. Matouk, On chaos control and synchronization of the commensurate fractional order Liu system, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 1193-1202
##[11]
H. H. C. Iu, C. K. Tse, A study of synchronization in chaotic autonomous Cuk DC/DC converters, IEEE Trans. Circuits Syst. I: Fundam. Theory Appl., 47 (2000), 913-918
##[12]
C. M. Jiang, S. T. Liu, C. Luo , A new fractional-order chaotic complex system and its antisynchronization, Abstr. Appl. Anal., 2014 (2014), 1-12
##[13]
C. M. Jiang, S. T. Liu, D. Wang, Ceneralized combination complex synchronization for fractional-order chaotic complex systems, Entropy, 17 (2015), 5199-5217
##[14]
C. M. Jiang, S. T. Liu, F. F. Zhang, Complex modified projective synchronization for fractional-order chaotic complex systems, Int. J. Autom. Comput., (2017), 1-13
##[15]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B. V., Amsterdam (2006)
##[16]
C.-M. Kim, S. Rim, W.-H. Kye, J.-W. Ryu, Y.-J. Park, Anti-synchronization of chaotic oscillators, Phys. Lett. A, 320 (2003), 39-46
##[17]
C. G. Li, G. R. Chen, Chaos in the fractional order Chen system and its control, Chaos, Solitons Fractals, 22 (2004), 549-554
##[18]
C. G. Li, G. R. Chen, Chaos and hyperchaos in the fractional-order Rössler equations, Phys. A, 341 (2004), 55-61
##[19]
C. D. Li, X. F. Liao, Lag synchronization of Rössler system and Chua circuit via a scalar signal, Phys. Lett. A, 329 (2004), 301-308
##[20]
J. Liu, Complex modifed hybrid projective synchronization of different dimensional fractional-order complex chaos and real hyper-chaos , Entropy, 16 (2014), 6195-6211
##[21]
X. J. Liu, L. Hong, L. X. Yang, Fractional-order complex T system: bifurcations, chaos control, and synchronization, Nonlinear Dynam., 75 (2014), 589-602
##[22]
P. Liu, H. J. Song, X. Li, Observe-based projective synchronization of chaotic complex modified Van der Pol-Duffing oscillator with application to secure communication, ASME J. Comput. Nonlinear Dynam., 10 (2015), 1-7
##[23]
S. T. Liu, F. F. Zhang, Complex function projective synchronization of complex chaotic system and its applications in secure communications, Nonlinear Dynam. , 76 (2014), 1087-1097
##[24]
J. Q. Lu, J. D. Cao, Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters, Chaos, 15 (2005), 1-10
##[25]
C. Luo, X. Y. Wang, Chaos in the fractional-order complex Lorenz system and its synchronization, Nonlinear Dynam., 71 (2013), 241-257
##[26]
C. Luo, X. Y. Wang, Chaos generated from the fractional-order complex Chen system and its application to digital secure communication, Int. J. Modern Phys. C, 24 (2013), 1-23
##[27]
R. Z. Luo, Y. L. Wang, S. C. Deng, Combination synchronization of three classic chaotic systems using active backstepping design, Chaos, 21 (2011), 1-6
##[28]
G. M. Mahmoud, T. M. Abed-Elhameed, M. E. Ahmed, Generalization of combination-combination synchronization of chaotic n-dimensional fractional-orer dynamical systems, Nonlinear Dynam., 83 (2016), 1885-1893
##[29]
G. M. Mahmoud, E. E. Mahmoud, A. A. Arafa, On projective synchronization of hyperchaotic complex nonlinear systems based on passive theory for secure communictations, Phys. Scr., 87 (2013), 1-10
##[30]
R. Mainieri, J. Rehacek, Projective synchronization in three-dimensional chaotic systems, Phys. Rev. Lett., 82 (1999), 3042-3045
##[31]
D. Matignon, Stability results for fractional differential equations with applications to control processing, In: IMACS, IEEE-SMC, Lille , France (1996)
##[32]
P. A. Mohammad, P. A. Hasan, Robust synchronization of a chaotic mechanical system with nonlinearities in control inputs, Nonlinear Dynam., 73 (2013), 363-376
##[33]
L. M. Pecora, T. L. Caroll , Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824
##[34]
M. G. Rosenblum, A. S. Pikovsky, J. Kurths, Phase synchronization of chaotic oscillators, Phys. Rev. Lett., 76 (1996), 1804-1807
##[35]
M. Srivastava, S. P. Ansari, S. K. Agrawal, S. Das, A. Y. T. Leung, Anti-synchronization between identical and non-identical fractional-order chaotic systems using active control method, Nonlinear Dynam., 76 (2014), 905-914
##[36]
J. W. Sun, G. Z. Cui, Y. F. Wang, Y. Shen, Combination complex synchronization of three chaotic complex systems, Nonlinear Dynam., 79 (2015), 953-965
##[37]
J. W. Sun, Y. Shen, Y. Quan, C. J. Xu, Compound synchronization for four memristor chaotic oscillator systems and secure communication, Chaos, 23 (2014), 1-10
##[38]
J. W. Sun, Y. Shen, G. D. Zhang, C. J. Xu, G. Z. Cui, Combination-combination synchronization among four identical or different chaotic systems, Nonlinear Dynam., 73 (2013), 1211-1222
##[39]
M. S. Tavazoei, M. Haeri, Synchronization of chaotic fractional-order systems via active sliding mode controller, Phys. A, 387 (2008), 57-70
##[40]
L. Wang, B. Yang, Y. Chen, X. Q. Zhang, J. Orchard, Improving Neural-Network classifiers using nearest neighbor partitioning, IEEE Trans. Neural Netw. Learn. Syst., 28 (2016), 2255-2267
]
On conformable delta fractional calculus on time scales
On conformable delta fractional calculus on time scales
en
en
In this paper, we introduce and investigate the concepts of conformable delta fractional derivative
and conformable delta fractional integral on time scales. Basic properties of the theory are proved.
324
335
Dafang
Zhao
Tongxing
Li
Conformable delta fractional derivative
conformable delta fractional integral
time scale.
Article.3.pdf
[
[1]
T. Abdeljawad, On conformable fractional calculus , J. Comput. Appl. Math., 279 (2015), 57-66
##[2]
T. Abdeljawad, M. Al Horani, R. Khalil, Conformable fractional semigroups of operators, J. Semigroup Theory Appl., 2015 (2015 ), 1-9
##[3]
I. Abu Hammad, R. Khalil, Fractional Fourier series with applications, Amer. J. Comput. Appl. Math., 4 (2014), 187-191
##[4]
M. Abu Hammad, R. Khalil, Abel's formula and Wronskian for conformable fractional differential equations , Int. J. Differ. Equ. Appl., 13 (2014), 177-183
##[5]
M. Abu Hammad, R. Khalil , Conformable fractional heat differential equations , Int. J. Pure. Appl. Math., 94 (2014), 215-221
##[6]
M. Abu Hammad, R. Khalil , Legendre fractional differential equation and Legendre fractional polynomials, Int. J. Appl. Math. Res., 3 (2014), 214-219
##[7]
R. P. Agarwal, M. Bohner, T. Li, Oscillatory behavior of second-order half-linear damped dynamic equations, Appl. Math. Comput., 254 (2015), 408-418
##[8]
M. Al Horani, M. Abu Hammad, R. Khalil, Variation of parameters for local fractional nonhomogeneous linear-differential equations, J. Math. Computer Sci., 16 (2016), 147-153
##[9]
H. Batarfi, J. Losada, J. J. Nieto, W. Shammakh, Three-point boundary value problems for conformable fractional differential equations, J. Funct. Spaces, 2015 (2015 ), 1-6
##[10]
B. Bayour, D. F. M. Torres , Existence of solution to a local fractional nonlinear differential equation, J. Comput. Appl. Math., 312 (2017 ), 127-133
##[11]
N. Benkhettou, A. M. C. Brito da Cruz, D. F. M. Torres, A fractional calculus on arbitrary time scales: Fractional differentiation and fractional integration, Signal Process., 107 (2015), 230-237
##[12]
N. Benkhettou, A. M. C. Brito da Cruz, D. F. M. Torres, Nonsymmetric and symmetric fractional calculi on arbitrary nonempty closed sets , Math. Methods Appl. Sci., 39 (2016), 261-279
##[13]
N. Benkhettou, S. Hassani, D. F. M. Torres, A conformable fractional calculus on arbitrary time scales, J. King Saud Univ. Sci., 28 (2016), 93-98
##[14]
M. Bohner, T. Li, Oscillation of second-order p-Laplace dynamic equations with a nonpositive neutral coeficient , Appl. Math. Lett., 37 (2014), 72-76
##[15]
M. J. Bohner, R. R. Mahmoud, S. H. Saker, Discrete, continuous, delta, nabla, and diamond-alpha Opial inequalities, Math. Inequal. Appl., 18 (2015), 923-940
##[16]
M. Bohner, A. Peterson , Dynamic Equations on Time Scales: An Introduction with Application, Birkhäuser, Boston (2001)
##[17]
M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston (2003)
##[18]
M. Bohner, S. H. Saker, Sneak-out principle on time scales , J. Math. Inequal., 10 (2016), 393-403
##[19]
A. Carpinteri, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Vienna (1997)
##[20]
R. Herrmann , Fractional Calculus: An Introduction for Physicists, World Scientific, Singapore (2011)
##[21]
S. Hilger, Ein MaBkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. Thesis, Universtät Würzburg (1988)
##[22]
R. Khalil, M. Al Horani, D. Anderson , Undetermined coeficients for local fractional differential equations, J. Math. Computer Sci., 16, 140-146. (2016)
##[23]
R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70
##[24]
T. Li, J. Diblk, A. Domoshnitsky, Yu. V. Rogovchenko, F. Sadyrbaev, Q.-R. Wang, Qualitative analysis of differential, difference equations, and dynamic equations on time scales , Abstr. Appl. Anal., 2015 (2015 ), 1-3
##[25]
T. Li, S. H. Saker , A note on oscillation criteria for second-order neutral dynamic equations on isolated time scales , Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 4185-4188
##[26]
K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations , John Wiley & Sons, Inc., New York (1993)
##[27]
K. B. Oldham, J. Spanier , The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Dover Publications, New York-London (2006)
##[28]
M. D. Ortigueira, Fractional Calculus for Scientists and Engineers, Springer, Dordrecht (2011)
##[29]
A. Peterson, B. Thompson , Henstock-Kurzweil delta and nabla integrals , J. Math. Anal. Appl., 323 (2006), 162-178
##[30]
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999)
##[31]
J. Sabatier, O. P. Agrawal, J. A. T. Machado , Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht (2007)
##[32]
S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, CRC Press, Switzerland (1993)
]
Asymptotic study of a frictionless contact problem between two elastic bodies
Asymptotic study of a frictionless contact problem between two elastic bodies
en
en
We consider a mathematical model which describes the bilateral, frictionless contact between two
elastic bodies. We will establish a variational formulation for the problem and prove the existence
and uniqueness of the weak solution. We then study the asymptotic behavior when one dimension of
the domain tends to zero. In which case, the uniqueness result of the solution for the limit problem
are also proved.
336
350
Y.
Letoufa
H.
Benseridi
M.
Dilmi
A priori inequalities
free boundary problems
nonlinear equation
transmission conditions
Tresca law
variational problem.
Article.4.pdf
[
[1]
A. Atangana, A novel model for the Lassa Hemorrhagic Fever: Deathly Disease for Pregnant Women , Neural Comput. Appl., 26 (2015), 1895-1903
##[2]
A. Atangana, E. F. Doungmo Goufo, A model of the groundwater flowing within a leaky aquifer using the concept of local variable order derivative , J. Nonlinear Sci. Appl., 8 (2015), 763-775
##[3]
H. M. Baskonus, H. Bulut , New hyperbolic function solutions for some nonlinear partial differential equation arising in mathematical physics, Entropy, 17 (2015), 4255-4270
##[4]
G. Bayada, M. Boukrouche , On a free boundary problem for the Reynolds equation derived from the Stokes systems with Tresca boundary conditions, J. Math. Anal. Appl., 282 (2003), 212-231
##[5]
G. Bayada, K. Lhalouani, Asymptotic and numerical analysis for unilateral contact problem with Coulomb's friction between an elastic body and a thin elastic soft layer , Asymptot. Anal., 25 (2001), 329-362
##[6]
H. Benseridi, M. Dilmi, Some inequalities and asymptotic behavior of a dynamic problem of linear elasticity , Georgian Math. J., 20 (2013), 25-41
##[7]
M. Boukrouche, R. El mir, On a non-isothermal, non-Newtonian lubrication problem with Tresca law: Existence and the behavior of weak solutions, Nonlinear Anal., 9 (2008), 674-692
##[8]
M. Boukrouche, G. Lukaszewicz, On a lubrication problem with Fourier and Tresca boundary conditions , Math. Models Methods Appl. Sci., 14 (2004), 913-941
##[9]
M. Boukrouche, F. Saidi, Non-isothermal lubrication problem with Tresca fluid-solid interface law. , Part I, Nonlinear Anal., 7 (2006), 1145-1166
##[10]
M. Dilmi, H. Benseridi, A. Saadallah , Asymptotic analysis of a Bingham fluid in a thin domain with Fourier and Tresca boundary conditions, Adv. Appl. Math. Mech., 6 (2014), 797-810
##[11]
G. Duvaut, J. L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris (1972)
##[12]
N. Hemici, A. Matei , A frictionless contact problem with adhesion between two elastic bodies, An. Univ. Craiova Ser. Mat. Inform., 30 (2003), 90-99
##[13]
J. Koko, Uzawa block relaxation domain decomposition method for the two-body contact problem with Tresca friction, Comput. Methods Appl. Mech. Engrg., 198 (2008), 420-431
##[14]
X. Li, Symmetric Coupling of the Meshless Galerkin boundary node and finite element methods for Elasticity , CMES Comput. Model. Eng. Sci., 97 (2014), 483-507
##[15]
X. Li, H. Chen, Y. Wang, Error analysis in Sobolev spaces for the improved moving least-square approximation and the improved element-free Galerkin method, Appl. Math. Comput., 262 (2015), 56-78
##[16]
A. Saadallah, H. Benseridi, M. Dilmi, S. Drabla, Estimates for the asymptotic convergence of a non- isothermal linear elasticity with friction , Georgian Math. J., 23 (2016), 435-446
]
Multiple solutions for a class of perturbed damped vibration problems
Multiple solutions for a class of perturbed damped vibration problems
en
en
The existence of three distinct weak solutions for a class of perturbed damped vibration problems
with nonlinear terms depending on two real parameters is investigated. Our approach is based on
variational methods.
351
363
Mohamad Reza Heidari
Tavani
Ghasem A.
Afrouzi
Shapour
Heidarkhani
Multiple solutions
perturbed damped vibration problem
critical point theory
variational methods.
Article.5.pdf
[
[1]
G. A. Afrouzi, S. Heidarkhani, S. Moradi , Perturbed elastic beam problems with nonlinear boundary conditions , Annal. Al. I. Cuza Univ. Math., (to appear), -
##[2]
F. Antonacci, P. Magrone, Second order nonautonomous systems with symmetric potential changing sign, Rend. Mat. Appl., 18 (1998), 367-379
##[3]
G. Bonanno, P. Candito , Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differential Equations, 244 (2008), 3031-3059
##[4]
G. Bonanno, G. DAgui, Multiplicity results for a perturbed elliptic Neumann problem , Abstr. Appl. Anal., 2010 (2010 ), 1-10
##[5]
G. Bonanno, R. Livrea, Periodic solutions for a class of second-order Hamiltonian systems, Electron. J. Differential Equations, 2005 (2005), 1-13
##[6]
G. Bonanno, R. Livrea, Multiple periodic solutions for Hamiltonian systems with not coercive potential , J. Math. Anal. Appl., 363 (2010), 627-638
##[7]
G. Bonanno, R. Livrea , Existence and multiplicity of periodic solutions for second order Hamiltonian systems depending on a parameter , J. Convex Anal., 20 (2013), 1075-1094
##[8]
G. Bonanno, S. A. Marano , On the structure of the critical set of non-differentiable functions with a weak compactness condition , Appl. Anal., 89 (2010), 1-10
##[9]
G. Chen, Nonperiodic damped vibration systems with asymptotically quadratic terms at infinity: infinitely many homoclinic orbits , Abstr. Appl. Anal., 2013 (2013 ), 1-7
##[10]
G. Chen, Non-periodic damped vibration systems with sublinear terms at infinity: infinitely many homoclinic orbits , Nonlinear Anal., 92 (2013), 168-176
##[11]
H. Chen, Z. He , New results for perturbed Hamiltonian systems with impulses , Appl. Math. Comput., 218 (2012), 9489-9497
##[12]
G.-W. Chen, J. Wang, Ground state homoclinic orbits of damped vibration problems, Bound. Value Probl., 2014 (2014 ), 1-15
##[13]
G. Cordaro , Three periodic solutions to an eigenvalue problem for a class of second order Hamiltonian systems , Abstr. Appl. Anal., 18 (2003), 1037-1045
##[14]
G. Cordaro, G. Rao, Three periodic solutions for perturbed second order Hamiltonian systems , J. Math. Anal. Appl., 359 (2009), 780-785
##[15]
G. DAgui, S. Heidarkhani, G. Molica Bisci, Multiple solutions for a perturbed mixed boundary value problem involving the one-dimensional p-Laplacian , Electron. J. Qual. Theory Diff. Eqns., 2013 (2013 ), 1-14
##[16]
F. Faraci, Multiple periodic solutions for second order systems with changing sign potential, J. Math. Anal. Appl., 319 (2006), 567-578
##[17]
F. Faraci, R. Livrea, Infinitely many periodic solutions for a second-order nonautonomous system , Nonlinear Anal., 54 (2003), 417-429
##[18]
J. R. Graef, S. Heidarkhani, L. Kong, Infinitely many solutions for a class of perturbed second-order impulsive Hamiltonian systems, Acta Appl. Math., 139 (2015), 81-94
##[19]
J. R. Graef, S. Heidarkhani, L. Kong , Nontrivial periodic solutions to second-order impulsive Hamiltonian systems, Electron. J. Differential Equations, 2015 (2015), 1-17
##[20]
S. Heidarkhani, G. A. Afrouzi, M. Ferrara, G. Caristi, S. Moradi, Existence results for impulsive damped vibration systems , Bull. Malays. Math. Sci. Soc., 2016 (2016 ), 1-20
##[21]
J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York (1989)
##[22]
P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems , Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38
##[23]
P. H. Rabinowitz, Variational methods for Hamiltonian systems, Handbook of Dynamical Systems vol. 1, Part A, 2002 (2002), 1091-1127
##[24]
J. Sun, H. Chen, J. J. Nieto, M. Otero-Novoa, The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects , Nonlinear Anal., 72 (2010), 4575-4586
##[25]
C.-L. Tang, Periodic solutions of non-autonomous second order systems with \(\gamma\)-quasisubadditive potential, J. Math. Anal. Appl., 189 (1995), 671-675
##[26]
C.-L. Tang, Periodic solutions for nonautonomous second order systems with sublinear nonlinearity , Proc. Amer. Math. Soc., 126 (1998), 3263-3270
##[27]
C.-L. Tang, X.-P. Wu, Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems, J. Math. Anal. Appl., 275 (2002), 870-882
##[28]
X. Wu, J. Chen, Existence theorems of periodic solutions for a class of damped vibration problems, Appl. Math. Comput., 207 (2009), 230-235
##[29]
X. Wu, S. Chen, K. Teng, On variational methods for a class of damped vibration problems , Nonlinear Anal., 68 (2008), 1432-1441
##[30]
X. Wu, W. Zhang, Existence and multiplicity of homoclinic solutions for a class of damped vibration problems, Nonlinear Anal., 74 (2011), 4392-4398
##[31]
J. Xiao, J. J. Nieto, Variational approach to some damped Dirichlet nonlinear impulsive differential equations, J. Franklin Inst., 348 (2011), 369-377
##[32]
E. Zeidler, Nonlinear functional analysis and its applications, Vol. II: Linear monotone operators, Springer-Verlag, New York (1985)
##[33]
W. Zou , Variant fountain theorems and their applications, Manuscripta Math., 104 (2001), 343-358
]
A commensal symbiosis model with Holling type functional response
A commensal symbiosis model with Holling type functional response
en
en
A two species commensal symbiosis model with Holling type functional response takes the form
\[\frac{dx}{dt}=x\left(a_1-b_1x+\frac{c_1y^p}{1+y^p}\right),\]
\[\frac{dy}{dt}=y\left(a_2-b_2y\right)\]
is investigated, where \(a_i, b_i, i = 1, 2, p\) and \(c_1\) are all positive constants, \(p \geq 1\). Local and global
stability property of the equilibria is investigated. We also show that depending on the ratio of \(\frac{a_2}{b_2}\),
the first component of the positive equilibrium \(x^*(p)\) may be the increasing or decreasing function
of \(p\) or independent of \(p\). Our study indicates that the unique positive equilibrium is globally stable
and the system always permanent.
364
371
Runxin
Wu
Lin
Li
Xiaoyan
Zhou
Commensal symbiosis model
stability.
Article.6.pdf
[
[1]
F. Chen , Permanence of periodic Holling type predator-prey system with stage structure for prey, Appl. Math. Comput., 182 (2006), 1849-1860
##[2]
F. Chen , Permanence for the discrete mutualism model with time delays, Math. Comput. Modelling, 47 (2008), 431-435
##[3]
L. Chen, F. Chen, L. Chen , Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey refuge, Nonlinear Anal. Real World Appl., 11 (2010), 246-252
##[4]
L. Chen, L. Chen, Z. Li , Permanence of a delayed discrete mutualism model with feedback controls, Math. Comput. Modelling, 50 (2009), 1083-1089
##[5]
F. Chen, W. Chen, Y. Wu, Z. Ma, Permanence of a stage-structured predator-prey system, Appl. Math. Comput., 219 (2013), 8856-8862
##[6]
F. Chen, L. Pu, L. Yang , Positive periodic solution of a discrete obligate Lotka-Volterra model, Comm. Math. Biol. Neurosci., 2015 (2015), 1-9
##[7]
F. Chen, J. Shi, On a delayed nonautonomous ratio-dependent predator-prey model with Holling type functional response and diffusion, Appl. Math. Comput., 192 (2007), 358-369
##[8]
L. Chen, X. Song, Z. Lu, Mathematical models and methods in ecology, Shichuan Science and Technology Press, Beijing (2002)
##[9]
F. D. Chen, H. Wang, Y. Lin, W. Chen, Global stability of a stage-structured predator-prey system, Appl. Math. Comput., 223 (2013), 45-53
##[10]
L. Chen, X. Xie , Permanence of an N-species cooperation system with continuous time delays and feedback controls, Nonlinear Anal. Real World Appl., 12 (2011), 34-38
##[11]
F. Chen, X. Xie, Study on the dynamic behaviors of cooperation population modeling, (in Chinese), Science Press, Beijing (2014)
##[12]
L. Chen, X. Xie, L. Chen , Feedback control variables have no influence on the permanence of a discrete N-species cooperation system, Discrete Dyn. Nat. Soc., 2009 (2009 ), 1-10
##[13]
F. Chen, X. Xie, X. Chen, Dynamic behaviors of a stage-structured cooperation model, Comm. Math. Biol. Neurosci., 2015 (2015), 1-19
##[14]
F. Chen, J. Yang, L. Chen, X. Xie, On a mutualism model with feedback controls, Appl. Math. Comput., 214 (2009), 581-587
##[15]
F. Chen, M. You , Permanence for an integrodifferential model of mutualism, Appl. Math. Comput., 186 (2007), 30-34
##[16]
R. Han, F. Chen , Global stability of a commensal symbiosis model with feedback controls, Comm. Math. Biol. Neurosci., 2015 (2015), 1-15
##[17]
X. Li, W. Yang , Permanence of a discrete model of mutualism with infinite deviating arguments, Discrete Dyn. Nat. Soc., 2010 (2010), 1-7
##[18]
Y. Li, T. Zhang, Permanence of a discrete N-species cooperation system with time-varying delays and feedback controls, Math. Comput. Modelling , 53 (2011), 1320-1330
##[19]
Z. Liu, J. Wu, R. Tan, Y. Chen, Modeling and analysis of a periodic delayed two-species model of facultative mutualism , Appl. Math. Comput., 217 (2010), 893-903
##[20]
Z. Miao, X. Xie, L. Pu, Dynamic behaviors of a periodic Lotka-Volterra commensal symbiosis model with impulsive , Comm. Math. Biol. Neurosci., 2015 (2015 ), 1-15
##[21]
G. Sun, W. Wei , The qualitative analysis of commensal symbiosis model of two populations, Math. Theory Appl., 23 (2003), 64-68
##[22]
Y. Wu, F. Chen, W. Chen, Y. Lin , Dynamic behaviors of a nonautonomous discrete predator-prey system incorporating a prey refuge and Holling type II functional response, Discrete Dyn. Nat. Soc., 2012 (2012 ), 1-14
##[23]
X. Xie, F. Chen, Y. Xue, Note on the stability property of a cooperative system incorporating harvesting, Discrete Dyn. Nat. Soc., 2014 (2014 ), 1-5
##[24]
X. Xie, F. Chen, K. Yang, Y. Xue, Global attractivity of an integrodifferential model of mutualism, Abstr. Appl. Anal. , 2014 (2014 ), 1-6
##[25]
X. Xie, Z. Miao, Y. Xue, Positive periodic solution of a discrete Lotka-Volterra commensal symbiosis model, Comm. Math. Biol. Neurosci., 2015 (2015 ), 1-10
##[26]
Y. Xue, X. Xie, F. Chen, R. Han, Almost periodic solution of a discrete commensalism system, Discrete Dyn. Nat. Soc., 2015 (2015 ), 1-11
##[27]
W. Yang, X. Li, Permanence of a discrete nonlinear N-species cooperation system with time delays and feedback controls, Appl. Math. Comput., 218 (2011), 3581-3586
##[28]
K. Yang, Z. Miao, F. Chen, X. Xie, Influence of single feedback control variable on an autonomous Holling-II type cooperative system, J. Math. Anal. Appl., 435 (2016), 874-888
##[29]
K. Yang, X. Xie, F. Chen , Global stability of a discrete mutualism model, Abstr. Appl. Anal., 2014 (2014 ), 1-7
##[30]
Y. C. Zhou, Z. Jin, J. L. Qin, Ordinary differential equation and its application, Science Press, Beijing (2003)
]
Computational coupled fixed points for \(\mathbf{F}\)-contractive mappings in metric spaces endowed with a graph
Computational coupled fixed points for \(\mathbf{F}\)-contractive mappings in metric spaces endowed with a graph
en
en
The purpose of this work is to present some existence theorems for coupled fixed points of F-type
contractive operator in metric spaces endowed with a directed graph. Our results generalize the main
result obtained by Chifu and Petrusel [C. Chifu, G. Petrusel, Fixed Point Theory Appl., 2014 (2014),
13 pages]. We also present applications to some nonlinear integral system equations to support the
results.
372
385
Phumin
Sumalai
Poom
Kumam
Dhananjay
Gopal
Coupled fixed point
directed graph
nonlinear integral equations.
Article.7.pdf
[
[1]
T. G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65 (2006), 1379-1393
##[2]
C. Chifu, G. Petrusel, New results on coupled fixed point theory in metric spaces endowed with a directed graph , Fixed Point Theory Appl., 2014 (2014 ), 1-13
##[3]
J. Jachymski , The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc., 136 (2008), 1359-1373
##[4]
S. Suantai, P. Charoensawan, T. A. Lampert, Common coupled fixed point theorems for (\theta-\psi\)-contraction mappings endowed with a directed graph , Fixed Point Theory Appl., 2015 (2015 ), 1-11
##[5]
D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012 (2012), 1-6
]
Contact pseudo-slant submanifolds of a Kenmotsu manifold
Contact pseudo-slant submanifolds of a Kenmotsu manifold
en
en
We study the geometry of the contact pseudo-slant submanifolds of a Kenmotsu manifold. Necessary
and sufficient conditions are given for a submanifold to be a pseudo-slant submanifold, contact
pseudo-slant product, mixed geodesic and totally geodesic in Kenmotsu manifolds.
386
394
Süleyman
Dirik
Mehmet
Atceken
Ümit
Yildirim
Kenmotsu manifold
slant submanifold
contact pseudo-slant submanifold.
Article.8.pdf
[
[1]
M. Atçeken, S. Dirik, On the geometry of pseudo-slant submanifolds of a Kenmotsu manifold , Gulf j. math., 2 (2014), 51-66
##[2]
M. Atçeken, S. Dirik, Ü. Yıldırım , Pseudo-slant submanifolds of a locally decomposable Riemannian manifold , J. Adv. Math., 11 (2015), 5587-5599
##[3]
M. Atçeken, S. K. Hui, Slant and pseudo-slant submanifolds in \((LCS)_n\)-manifolds , Czechoslovak Math. J., 63 (2013), 177-190
##[4]
J. L. Cabrerizo, A. Carriazo, L. M. Fernandez, M. Fernandez, Slant submanifolds in Sasakian manifolds , Glasg. Math. J., 42 (2000), 125-138
##[5]
B.-Y. Chen , Geometry of slant submanifolds, Katholieke Universiteit Leuven, Leuven (1990)
##[6]
B.-Y. Chen, Slant immersions, Bull. Austral. Math. Soc., 41 (1990), 135-147
##[7]
U. C. De, A. Sarkar, On pseudo-slant submanifolds of trans sasakian manifolds, Proc. Est. Acad. Sci., 60 (2011), 1-11
##[8]
K. Kenmotsu , A class of almost contact Riemannian manifolds , Tohoku Math. J., 24 (1972), 93-103
##[9]
V. A. Khan, M. A. Khan , Pseudo-slant submanifolds of a Sasakian manifold, Indian J. Prue Appl. Math., 38 (2007), 31-42
##[10]
A. Lotta , Slant submanifolds in contact geometry , Bulletin mathmatique de la Socit des Sciences Mathmatiques de Roumanie, 39 (1996), 183-198
##[11]
N. Papaghuic , Semi-slant submanifolds of a Kaehlarian manifold, An. St. Univ. Al. I. Cuza. Univ. Iasi., 40 (1994), 55-61
]
Dynamic behaviors of two species amensalism model with a cover for the first species
Dynamic behaviors of two species amensalism model with a cover for the first species
en
en
In this paper, a two species amensalism model with a cover for the first species takes the form
\[\frac{dx}{dt}=a_1x(t)-b_1x^2(t)-c_1(1-k)x(t)y(t),\]
\[\frac{dy}{dt}=a_2y(t)-b_2y^2(t),\]
is investigated, where \(a_i, b_i, i = 1, 2\) and \(c_1\) are all positive constants, \(k\) is a cover provided for the
species \(x\), and \(0 < k < 1\). Our study shows that if \(0 \leq k < 1-\frac{a_1b_2}{a_2c_1}\),
then \(E_2(0, \frac{a_2}{b_2})\) is globally stable,
and if \(1>k>1-\frac{a_1b_2}{a_2c_1}\), then \(E_3(x^*, y^*)\) is the unique globally stable positive equilibrium. More
precisely, the conditions which ensure the local stability of \(E_2(0, \frac{a_2}{b_2})\)
is enough to ensure its global
stability, and once the positive equilibrium exists, it is globally stable. Some numerical simulations
are carried out to illustrate the feasibility of our findings.
395
401
Xiangdong
Xie
Fengde
Chen
Mengxin
He
Amensalism model
Lyapunov function
stability.
Article.9.pdf
[
[1]
K. V. L. N. Acharyulu, N. C. Pattabhi Ramacharyulu, On the stability of an ammensal- enemy harvested species pair with limited resources , Int. J. Open Problems Compt. Math., 3 (2010), 241-266
##[2]
K. V. L. N. Acharyulu, N. C. Pattabhi Ramacharyulu , An immigrated ecological ammensalism with limited resources, Int. J. Sci. Adv. Technol., 27 (2011), 78-84
##[3]
K. V. L. N. Acharyulu, N. C. Pattabhi Ramacharyulu , On the carrying capacity of enemy species, inhibition coeficient of ammensal species and dominance reversal time in an ecological ammensalisma special case study with numerical approach, Intern. J. Adv. Sci. Tech., 43 (2012), 49-58
##[4]
K. V. L. N. Acharyulu, P. Rama Mohan, N. C. Pattabhi Ramacharyulu, , An Ecological Mathematical Model of an Immigrated Ammensal and a Migrated Enemy Model with Mortality Rate for Ammensal Species-An Analytical Investigation, Int. J. Pure Appl. Sci. Technol., 4 (2011), 71-84
##[5]
L. J. Chen, F. D. Chen, L. J. Chen , Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey refuge , Nonlinear Anal. Real World Appl., 11 (2010), 246-252
##[6]
L. J. Chen, F. D. Chen, Y. Q. Wang , In uence of predator mutual interference and prey refuge on Lotka-Volterra predator-prey dynamics, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 3174-3180
##[7]
F. D. Chen, L. J. Chen, X. D. Xie, On a Leslie-Gower predator-prey model incorporating a prey refuge , Nonlinear Anal. Real World Appl., 10 (2009), 2905-2908
##[8]
F. D. Chen, W. X. He, R. Y. Han , On discrete amensalism model of Lotka-Volterra, J. Beihua Univ. (Natural Science), 16 (2015), 141-144
##[9]
F. Chen, Z. Ma, H. Zhang, Global asymptotical stability of the positive equilibrium of the Lotka-Volterra prey-predator model incorporating a constant number of prey refuges , Nonlinear Anal. Real World Appl., 13 (2012), 2790-2793
##[10]
F. Chen, Y. Wu, Z. Ma , Stability property for the predator-free equilibrium point of predator-prey systems with a class of functional response and prey refuges, Discrete Dyn. Nat. Soc., 2012 (2012), 1-5
##[11]
F. D. Chen, X. D. Xie , Study on the dynamic behaviors of cooperation population modeling , Science Press, Beijing (2014)
##[12]
F. D. Chen, M. S. Zhang, R. Y. Han , Existence of positive periodic solution of a discrete Lotka-Volterra amensalism model, J. Shengyang Univ. (Natural Science), 27 (2015), 251-254
##[13]
E. Gonzez-Olivares, R. Ramos-Jiliberto , Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecol. Model., 166 (2003), 135-146
##[14]
R. Y. Han, Y. L. Xue, L. Y. Yang, F. Chen , On the existence of positive periodic solution of a LotkaVolterra amensalism model, J. Longyang Univ., 33 (2015), 22-26
##[15]
Y. J. Huang, F. D. Chen, L. Zhong , Stability analysis of a prey-predator model with Holling type III response function incorporating a prey refuge, Appl. Math. Comput., 182 (2006), 672-683
##[16]
L. L. Ji, C. Q. Wu , Qualitative analysis of a predator-prey model with constant-rate prey harvesting incorporating a constant prey refuge, Nonlinear Anal. Real World Appl., 11 (2010), 2285-2295
##[17]
T. K. Kar, Stability analysis of a prey-predator model incorporating a prey refuge , Commun. Nonlinear Sci. Numer. Simul., 10 (2005), 681-691
##[18]
W. Ko, K. Ryu , Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge, J. Differential Equations, 231 (2006), 534-550
##[19]
V. Krivan , On the Gause predator-prey model with a refuge: A fresh look at the history , J. Theoret. Biol., 274 (2011), 67-73
##[20]
Z. Z. Ma, F. D. Chen, C. Q. Wu, W. Chen , Dynamic behaviors of a Lotka-Volterra predator-prey model incorporating a prey refuge and predator mutual interference, Appl. Math. Comput., 219 (2013), 7945-7953
##[21]
Z. H. Ma, W. D. Li, S. F. Wang , The effect of prey refuge in a simple predator-prey model, Ecol. Model., 222 (2011), 3453-3454
##[22]
B. Sita Rambabu, K. L. Narayan, S. Bathul , A mathematical study of two species amensalism model with a cover for the first species by homotopy analysis method, Adv. Appl. Sci. Res., 3 (2012), 1821-1826
##[23]
G. C. Sun , Qualitative analysis on two populations ammenslism model, J. Jiamusi Univ. (Nature Science Edition), 21 (2003), 283-286
##[24]
Y. D. Tao, X. Wang, X. Y. Song , Effect of prey refuge on a harvested predator-prey model with generalized functional response, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1052-1059
##[25]
Y. M. Wu, F. D. Chen, W. L. Chen, Y. H. Lin, Dynamic behaviors of a nonautonomous discrete predator-prey system incorporating a prey refuge and Holling type ii functional response , Discrete Dyn. Nat. Soc., 2012 (2012), 1-14
##[26]
Y. M. Wu, F. D. Chen, Z. Z. Ma, Y. H. Lin , Permanence and extinction of a nonautonomous predator-prey system incorporating prey refuge and Rosenzweig functional response, J. Biomath., 29 (2014), 727-731
##[27]
R. Z. Yang, C. R. Zhang, Dynamics in a diffusive predator-prey system with a constant prey refuge and delay, Nonlinear Anal. Real World Appl., 31 (2016), 1-22
##[28]
Q. Yue , Dynamics of a modified Leslie-Gower predator-prey model with Holling-type II schemes and a prey refuge, SpringerPlus, 5 (2016), 1-12
##[29]
Z. P. Zhang , Stability and bifurcation analysis for a amensalism system with delays, Math. Numer. Sin., 30 (2008), 213-224
##[30]
Z. F. Zhu, Q. L. Chen , Mathematical analysis on commensalism Lotka-Volterra model of populations, J. Jixi University, 8 (2008), 100-101
]
Nonlinear predictive high order sliding mode control for permanent magnet synchronous motor drive system
Nonlinear predictive high order sliding mode control for permanent magnet synchronous motor drive system
en
en
The speed control problem for permanent magnet synchronous motor (PMSM) drive system is
studied in this paper. The predictive control has been proved as an effective solution for the fast transient
response, but the performance will be deteriorated in the presence of model uncertainties and
load disturbance. Thus a composite control method combining the nonlinear generalized predictive
control (GPC) with the sliding mode control is proposed. In view of the chattering problems caused
by the traditional sliding mode control, a high order terminal sliding mode controller is designed,
which can reduce the chattering effectively. In the end, the stability of the system is proved. The
simulation and experimental results show that, compared with PI controllers, the designed controller
has the faster speed response and the stronger robustness, and the chattering is decreased obviously
than the first order sliding mode controller.
402
411
Xudong
Liu
Chenghui
Zhang
Ke
Li
Qi
Zhang
Permanent magnet synchronous motor
generalized predictive control
high order terminal sliding mode control.
Article.10.pdf
[
[1]
I. C. Baik, K. H. Kim, M. J. Youn, Robust nonlinear speed control of PM synchronous motor using boundary layer integral sliding mode control technique, IEEE Trans. Control Syst. Technol., 8 (2000), 47-54
##[2]
S. Bolognani, S. Bolognani, L. Peretti, M. Zigliotto , Design and implementation of model predictive control for electrical motor drives, IEEE Trans. Ind. Electron., 56 (2009), 1925-1936
##[3]
S. Chai, L.Wang, E. Rogers , A cascade MPC control structure for PMSM with speed ripple minimization, IEEE Trans. Ind. Electron., 60 (2013), 2978-2987
##[4]
W. H. Chen, D. J. Ballance, P. J. Gawthrop , Optimal control of nonlinear systems: a predictive control approach, Automatica J. IFAC, 39 (2003), 633-641
##[5]
M. Comanescu, L. Xu, T. D. Batzel , Decoupled current control of sensorless induction-motor drives by integral sliding mode, IEEE Trans. Ind. Electron., 55 (2008), 3836-3845
##[6]
M. Das, C. Mahanta , Optimal second order sliding mode control for linear uncertain systems, ISA Trans., 53 (2014), 1807-1815
##[7]
R. Errouissi, M. Ouhrouche, W. H. Chen, A. M. Trzynadlowski, Robust cascaded nonlinear predictive control of a permanent magnet synchronous motor with antiwindup compensator, IEEE Trans. Ind. Electron., 59 (2012), 3078-3088
##[8]
Y. Huangfu, S. Laghrouche, W. Liu, R. Q. Ma, A. Miraui, Chattering avoidance high order sliding mode control for permanent magnet synchronous motor, Electric Mach. Control, 16 (2012), 7-11
##[9]
R. Krishnan , Electric motor drives: modeling, analysis, and control, Prentice Hall, New York (2001)
##[10]
S. Laghrouche, F. Plestan, A. Glumineau , Higher order sliding mode control based on integral sliding mode, Automatica J. IFAC, 43 (2007), 531-537
##[11]
S. Li, M. Zhou, X. Yu , Design and implementation of terminal sliding mode control method for PMSM speed regulation system, IEEE Trans. Ind. Electron., 9 (2013), 1879-1891
##[12]
H. Liu, S. Li, Speed control for PMSM servo system using predictive functional control and extended state observer, IEEE Trans. Ind. Electron., 59 (2012), 1171-1183
##[13]
H. Sira-Ramírez, J. Linares-Flores, C. García-Rodríguez, M. A. Contreras-Ordaz , On the control of the permanent magnet synchronous motor: an active disturbance rejection control approach, IEEE Trans. Control Syst. Technol., 22 (2014), 2056-2063
##[14]
Y. M. Wang, Y. Feng, Q. L. Lu , Design of free-chattering sliding mode control systems for permanent magnet synchronous motor, Electric Mach. Control, 12 (2008), 514-519
##[15]
Q. Wang, B. Xu, L. You, X. Wang, BPMSM control using a sliding mode controller with fractional order of suspension forces based on differential geometry, Internat. J. Control Autom., 8 (2015), 187-198
##[16]
H. Wang, B. Zhou, S. Fang , A PMSM sliding mode control system based on exponential reaching law , Trans. China Electrotechnical Soc., 24 (2009), 71-77
##[17]
J. Yong-Ho, S. W. Lee , The implementation of speed control on IPMSM using simple nonlinear adaptive back-stepping, Internat. J. Control Autom., 7 (2014), 343-352
##[18]
X. Zhang, L. Sun, K. Zhao, L. Sun, Nonlinear speed control for PMSM system using sliding-mode control and disturbance compensation techniques, IEEE Trans. Ind. Electron., 28 (2013), 1358-1365
]
The reverse order law for EP modular operators
The reverse order law for EP modular operators
en
en
In this paper, we present new conditions that reverse order law holds for EP modular operators.
412
418
Javad
Farokhi-ostad
Mehdi Mohammadzadeh
Karizaki
EP operator
reverse order law
Moore-Penrose inverse
Hilbert \(C^*\)-module
closed range.
Article.11.pdf
[
[1]
T. Aghasizadeh, S. Hejazian , Maps preserving semi-Fredholm operators on Hilbert \(C^*\)-modules, J. Math. Anal. Appl., 354 (2009), 625-629
##[2]
R. Bouldin , The product of operators with closed range , Tôhoku Math. J., 25 (1973), 359-363
##[3]
R. Bouldin , Closed range and relative regularity for products , J. Math. Anal. Appl., 61 (1977), 397-403
##[4]
D. S. Djordjević , Products of EP operators on Hilbert spaces , Proc. Amer. Math. Soc., 129 (2001), 1727-1731
##[5]
D. S. Djordjević, Further results on the reverse order law for generalized inverses, SIAM J. Matrix Anal. Appl., 29 (2007), 1242-1246
##[6]
M. Frank , Self-duality and \(C^*\)-reflexivity of Hilbert \(C^*\)-moduli , Z. Anal. Anwendungen, 9 (1990), 165-176
##[7]
M. Frank , Geometrical aspects of Hilbert \(C^*\)-modules, Positivity, 3 (1999), 215-243
##[8]
T. N. E. Greville , Note on the generalized inverse of a matrix product, SIAM Rev., 8 (1966), 518-521
##[9]
S. Izumino , The product of operators with closed range and an extension of the reverse order law, Tôhoku Math. J., 34 (1982), 43-52
##[10]
E. C. Lance, Hilbert \(C^*\)-modules, A toolkit for operator algebraists, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge (1995)
##[11]
M. Mohammadzadeh Karizaki, M. Hassani, M. Amyari , Moore-Penrose inverse of product operators in Hilbert \(C^*\)-modules, Filomat, 30 (2016), 3397-3402
##[12]
M. Mohammadzadeh Karizaki, M. Hassani, M. Amyari, M. Khosravi, Operator matrix of Moore-Penrose inverse operators on Hilbert \(C^*\)-modules , Colloq. Math., 140 (2015), 171-182
##[13]
K. Sharifi , The product of operators with closed range in Hilbert \(C^*\)-modules , Linear Algebra Appl., 435 (2011), 1122-1130
##[14]
K. Sharifi , EP modular operators and their products, J. Math. Anal. Appl., 419 (2014), 870-877
##[15]
K. Sharifi, B. A. Bonakdar, The reverse order law for Moore-Penrose inverses of operators on Hilbert \(C^*\)-modules, Bull. Iranian Math. Soc., 42 (2016), 53-60
##[16]
Q. Xu, L. Sheng, Positive semi-definite matrices of adjointable operators on Hilbert \(C^*\)-modules, Linear Algebra Appl., 428 (2008), 992-1000
]
Linear interpolation for spatial data grid by newton polynomials
Linear interpolation for spatial data grid by newton polynomials
en
en
This article concerns with an improved interpolation method based on Newton scheme with high
dimensional data. It obtains the interpolation function when the nodes are arranged in a spatial
grid. The spatial interpolation method is systematic and very useful in engineering field. As an
example of application, we use the method to simulate the 3-D temperature field nearby a motor by
27 nodal temperature values. The results show that the interpolation function is quickly obtained
within a limited cube space which realized data visualization. This demonstrates the effectiveness of
the proposed method.
419
434
Yiming
Jiang
Xiaodong
Hu
Sen
Wu
Ancai
Zhang
Fei
Yuan
Lu
Liu
Ridong
Zha
Newton polynomials
spatial interpolation
numerical estimation
data visualization
simulation
temperature field.
Article.12.pdf
[
[1]
B. Bojanov, Y. Xu , On polynomial interpolation of two variables, J. Approx. Theory, 120 (2003), 267-282
##[2]
C. de Boor, A. Ron , On multivariate polynomial interpolation, Constr. Approx., 6 (1990), 287-302
##[3]
L. Bos, M. Caliari, S. De Marchi, Y. Xu , Bivariate Lagrange interpolation at the Padua points: the generating curve approach , J. Approx. Theory, 143 (2006), 15-25
##[4]
M. Caliari, S. De Marchi, M. Vianello , Bivariate polynomial interpolation on the square at new nodal sets, Appl. Math. Comput., 165 (2005), 261-274
##[5]
M. Crainic, N. Crainic , Birkhoff interpolation with rectangular sets of nodes and with few derivatives, East J. Approx., 14 (2008), 423-437
##[6]
L. Du, Comparison of interpolation methods used in thermal recovery of heavy oil and the visualization of temperature field, Dalian University of Technology, (2008)
##[7]
R. Franke, Scattered data interpolation: tests of some methods , Math. Comput., 38 (1982), 181-200
##[8]
M. Gasca, T. Sauer , Polynomial interpolation in several variables, Adv. Comput. Math., 12 (2000), 377-410
##[9]
R. A. Lorentz , Multivariate Birkhoff interpolation , Lecture Notes in Mathematics, Springer-Verlag, Berlin (1992)
##[10]
T. Sauer, Y. Xu, On multivariate Lagrange interpolation , Math. Comput., 64 (1995), 1147-1170
##[11]
J. Su , Research on the modeling theory, method and realization technology of the visualization-oriented temperature field FEA, Zhejiang University, (2002)
##[12]
Z. Tang , 3-D data field visualization, Tsinghua University Press, Beijing (1999)
##[13]
D. N. Varsamis, N. P. Karampetakis , On the Newton bivariate polynomial interpolation with applications, Multidimens. Syst. Signal Process., 25 (2014), 179-209
##[14]
X. Wang, S. Zhang, T. Dong , Newton basis for multivariate Birkhoff interpolation , J. Comput. Appl. Math., 228 (2009), 466-479
##[15]
R. Wu, T. Wu, H. Li, A family of multivariate multiquadric quasi-interpolation operators with higher degree polynomial reproduction, J. Comput. Appl. Math., 274 (2015), 88-108
##[16]
H. Xiong, S. Zeng, Y. Mao , Applied mathematics foundation , Tianjin University Press, Tianjin (1993)
]
New results in quasi cone metric spaces
New results in quasi cone metric spaces
en
en
In this paper, we prove some interesting results using forward and backward convergence in quasi
cone metric spaces. We study forward and backward sequential compactness, sequential countably
compactness, and sequential continuity property in quasi cone metric spaces and give some interesting
results.
435
444
Taja
Yaying
Bipan
Hazarika
Huseyin
Cakalli
Quasi cone metric space
forward convergence
backward convergence
ff-continuous
forward complete
forward sequential countably compactness.
Article.13.pdf
[
[1]
M. Abbas, G. Jungck, Common fixed point results for noncommuting mappings without continuity in cone metric spaces , J. Math. Anal. Appl., 341 (2008), 416-420
##[2]
M. Abbas, B. E. Rhoades, Fixed and periodic point results in cone metric spaces , Appl. Math. Lett., 22 (2009), 511-515
##[3]
T. Abdeljawad, E. Karapinar , Quasicone metric spaces and generalizations of Caristi Kirk's theorem , Fixed Point Theory Appl., 2009 (2009 ), 1-9
##[4]
G. E. Albert , A note on quasi-metric spaces, Bull. Amer. Math. Soc., 47 (1941), 479-482
##[5]
H. Çakalli, Sequential definitions of compactness , Appl. Math. Lett., 21 (2008), 594-598
##[6]
H. Çakalli, On G-continuity, Comput. Math. Appl., 61 (2011), 313-318
##[7]
H. Çakalli, Half quasi Cauchy sequences, arXiv preprint, (2012), -
##[8]
H. Çakalli , Upward and downward statistical continuities , Filomat, 29 (2015), 2265-2273
##[9]
H. Çakalli , On variations of quasi-Cauchy sequences in cone metric spaces, Filomat, 30 (2016), 603-610
##[10]
H. Çakalli, A. Sönmez , Slowly oscillating continuity in abstract metric spaces, Filomat, 27 (2013), 925-930
##[11]
K. P. Chi, T. Van An, Dugundji's theorem for cone metric spaces , Appl. Math. Lett., 24 (2011), 387-390
##[12]
J. Collins, J. Zimmer , An asymmetric Arzelà-Ascoli theorem, Topology Appl., 154 (2007), 2312-2322
##[13]
R. Engelking , General topology, Translated from the Polish by the author, Second edition, Sigma Series in Pure Mathematics, Heldermann Verlag, Berlin (1989)
##[14]
L. G. Huang, X. Zhang , Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), 1468-1476
##[15]
M. Khani, M. Pourmahdian , On the metrizability of cone metric spaces , Topology Appl., 158 (2011), 190-193
##[16]
H.-P. A. Künzi , A note on sequentially compact quasipseudometric spaces, Monatsh. Math., 95 (1983), 219-220
]
On the solution of linear and nonlinear partial differential equations: applications of local fractional Sumudu variational method
On the solution of linear and nonlinear partial differential equations: applications of local fractional Sumudu variational method
en
en
In this paper, the local fractional Sumudu variational iteration method is being used to investigate
the solutions of partial differential equations containing the local fractional derivatives. The present
technique is the combination of the local fractional Sumudu transform and fractional variational
iteration method. Three illustrative examples are given to demonstrate the efficiency of the method.
445
451
Badr S.
Alkahtani
Obaid J.
Algahtani
Pranay
Goswami
Modified Riemann-Liouville derivative
local fractional derivative
fractional variational iteration method
fractional differential equations
local fractional Sumudu transform.
Article.14.pdf
[
[1]
K. Al-Khaled, S. Momani , An approximate solution for a fractional diffusion-wave equation using the decomposition method , Appl. Math. Comput., 165 (2005), 473-483
##[2]
D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus, Models and numerical methods, Series on Complexity, Nonlinearity and Chaos , World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2012)
##[3]
J. Chen, F. Liu, V. Anh, Analytical solution for the time-fractional telegraph equation by the method of separating variables, J. Math. Anal. Appl., 338 (2008), 1364-1377
##[4]
L. Debnath, D. D. Bhatta , Solutions to few linear fractional inhomogeneous partial differential equations in fluid mechanics , Fract. Calc. Appl. Anal., 7 (2004), 21-36
##[5]
M. Dehghan, F. Shakeri , Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astron., 13 (2008), 53-59
##[6]
D. D. Ganji, A. Sadighi, Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations , J. Comput. Appl. Math., 207 (2007), 24-34
##[7]
A. Hanyga , Multidimensional solutions of time-fractional diffusion-wave equations, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458 (2002), 933-957
##[8]
J. H. He, Variational iteration method|a kind of non-linear analytical technique: some examples , Int. J. Nonlinear Mech., 34 (1999), 699-708
##[9]
J. H. He , Variational iteration method-some recent results and new interpretations, J. Comput. Appl. Math., 207 (2007), 3-17
##[10]
H. Jafari, H. K. Jassim, Numerical solutions of telegraph and Laplace equations on Cantor sets using local fractional Laplace decomposition method , Int. J. Adv. Appl. Math. Mech., 2 (2015), 144-151
##[11]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo , Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006)
##[12]
F. Mainardi, Y. Luchko, G. Pagnini , The fundamental solution of the space-time fractional diffusion equation , Fract. Calc. Appl. Anal., 4 (2001), 153-192
##[13]
K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York (1993)
##[14]
S. Momani , An explicit and numerical solutions of the fractional KdV equation, Math. Comput. Simulation, 70 (2005), 110-118
##[15]
Z. Odibat, S. Momani , Approximate solutions for boundary value problems of time-fractional wave equation, Appl. Math. Comput., 181 (2006), 767-774
##[16]
Z. Odibat, S. Momani , The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics , Comput. Math. Appl., 58 (2009), 2199-2208
##[17]
K. B. Oldham, J. Spanier , The fractional calculus, Theory and applications of differentiation and integration to arbitrary order, With an annotated chronological bibliography by Bertram Ross, Mathematics in Science and Engineering, Academic Press, New York-London (1974)
##[18]
H. M. Srivastava, A. K. Golmankhaneh, D. Baleanu, X. J. Yang , Local fractional Sumudu transform with application to IVPs on Cantor sets , Abstr. Appl. Anal., 2014 (2014), 1-7
##[19]
N. H. Sweilam, M. M. Khader, Variational iteration method for one dimensional nonlinear thermoelasticity, Chaos Solitons Fractals , 32 (2007), 145-149
##[20]
V. Turut, N. Güzel, On solving partial differential equations of frational order by using the variational iteration method and multivariate cPadé approximations , Eur. J. Pure Appl. Math., 6 (2013), 147-171
##[21]
X. J. Yang , Local fractional functional analysis and its applications, Asian Academic publisher Limited, Hong Kong (2011)
##[22]
X. J. Yang , Local fractional integral transforms, Prog. Nonlinear Sci., 4 (2011), 12-25
##[23]
X. J. Yang , Advanced local fractional calculus and its applications, World Sci., New York (2012)
##[24]
A. M. Yang, J. Li, H. M. Srivastava, G. N. Xie, X. J. Yang , Local fractional Laplace variational iteration method for solving linear partial differential equations with local fractional derivative , Discrete Dyn. Nat. Soc., 2015 (2015), 1-9
##[25]
A. M. Wazwaz, The variational iteration method: a powerful scheme for handling linear and nonlinear diffusion equations , Comput. Math. Appl., 54 (2007), 933-939
##[26]
A. M. Wazwaz , The variational iteration method for rational solutions for KdV, K(2; 2), Burgers, and cubic Boussinesq equations , J. Comput. Appl. Math., 207 (2007), 18-23
]
Nonlinear stabilization control of Furuta pendulum only using angle position measurements
Nonlinear stabilization control of Furuta pendulum only using angle position measurements
en
en
In this paper, we discuss the stabilization control problem for a nonlinear mechanical system
called Furuta pendulum. A new stabilizing control method that only uses the measurements of angle
position is developed. This method has three successive steps. First, we present the dynamic equation
of Furuta pendulum and change it into an affine nonlinear system by appropriately choosing state
variables. Second, we linearize the nonlinear system around the origin and consider the nonlinear
higher order term to be system's fictitious disturbance. After that, an idea of equivalent input
disturbance is used to design the stabilizing controller for the nonlinear system. The effectiveness of
our proposed control strategy is illustrated via a numerical example.
452
460
Lin
Zhao
Shuli
Gong
Ancai
Zhang
Lanmei
Cong
Nonlinear analysis and control
Furuta pendulum
underactuated mechanical system
equivalent input disturbance.
Article.15.pdf
[
[1]
C. Aguilar-Ibáñez, M. S. Suarez-Castanon , Stabilization of the inverted pendulum via a constructive Lyapunov function , Acta Appl. Math., 111 (2010), 15-26
##[2]
T. Albahkali, R. Mukherjee, T. Das , Swing-up control of the pendubot: An impulse-momentum approach, IEEE Trans. Robot., 25 (2009), 975-982
##[3]
J. Aracil, J. A. Acosta, F. Gordillo , A nonlinear hybrid controller for swinging-up and stabilizing the Furuta pendulum, Control Eng. Pract., 21 (2013), 989-993
##[4]
F. Gordillo, J. A. Acosta, J. Aracil, A new swing-up law for the Furuta pendulum, Internat. J. Control, 76 (2003), 836-844
##[5]
P. X. Hera, L. B. Freidovich, A. S. Shiriaev, U. Mettin , New approach for swinging up the furuta pendulum: theory and experiments, Mechatronics, 19 (2009), 1240-1250
##[6]
C. A. Ibáñez, J. H. S. Azuela , Stabilization of the Furuta pendulum based on a Lyapunov function , Nonlinear Dynam., 49 (2007), 1-8
##[7]
M. Izutsu, Y. Pan, K. Furutac , A swing-up/stabilization control by using nonlinear sliding mode for Furuta pendulum , Proc. 16th IEEE Conf. Control Appl., Singapore, (2007), 1191-1196
##[8]
B. Jakubczyk, W. Respondek , On linearization of control systems , Bull. Acad. Polon. Sci. Sér. Sci. Math., 28 (1980), 517-522
##[9]
X. Z. Lai, J. H. She, S. X. Yang, M. Wu , Comprehensive unified control strategy for underactuated two-link manipulators, IEEE Trans. Syst. Man Cybern. B Cybern., 39 (2009), 389-398
##[10]
S. Nair, N. E. Leonard , A normal form for energy shaping: Application to the Furuta pendulum, Proc. 41th IEEE Conf. Decision Control, Las Vegas, NV, USA, (2002), 516-521
##[11]
R. Olfati-Saber, Fixed point controllers and stabilization of the cart-pole system and the rotating pendulum , Proc. 38th IEEE Conf. Decision Control, Phoenix, AZ, USA, (1999), 1174-1181
##[12]
G. Oriolo, Y. Nakamura , Control of mechanical systems with second order nonholonomic constraints: Underactuated manipulators , Proc 30th IEEE Conf Decision Control, Brighton, UK, (1991), 2398-2403
##[13]
M. S. Park, D. Chwa , Swing-up and stabilization control of inverted-pendulum systems via coupled sliding-mode control method , IEEE Trans. Ind. Electron., 56 (2009), 3541-3555
##[14]
M. Reyhanoglu, A. Van der Schaft, N. H. McClamroch, I. Kolmanovsky , Dynamics and control of underactuated mechanical systems, IEEE Trans. Automat. Contr., 44 (1999), 1663-1671
##[15]
J. H. She, M. Fang, Y. Ohyama, H. Hashimoto, M. Wu , Improving disturbance-rejection performance based on an equivalent-input-disturbance approach , IEEE Trans. Ind. Electron., 55 (2008), 380-389
##[16]
N. Sun, Y. C. Fang, X. B. Zhang, An increased coupling-based control method for underactuated crane systems: theoretical design and experimental implementation, Nonlinear Dynam., 70 (2012), 1135-1146
##[17]
M. Wiklund, A. Kristenson, K. J. Astrom , A new strategy for swinging up an inverted pendulum , Proc. IFAC Symposium, Sydney, Australia, (1993), 151-154
##[18]
X. Xin, Y. N. Liu , Reduced-order stable controllers for two-link underactuated planar robots , Automatica J. IFAC, 49 (2013), 2176-2183
##[19]
A. Zhang, X. Z. Lai, M. Wu, J. H. She , Stabilization of underactuated two-link gymnast robot by using trajectory tracking strategy, Appl. Math. Comput., 253 (2015), 193-204
##[20]
A. Zhang, J. H. She, X. Z. Lai, M. Wu , Motion planning and tracking control for an acrobot based on a rewinding approach, Automatica J. IFAC, 49 (2013), 278-284
]
Fractal dimension of the controlled Julia sets of the output duopoly competing evolution model
Fractal dimension of the controlled Julia sets of the output duopoly competing evolution model
en
en
The output duopoly competing evolution model has an integral role in the study of the economic
phenomenon. In this paper, the basic methods of Julia sets is applied to this model. At first, Julia
set of this model is introduced. Then, two different control methods are taken to control Julia set:
one is the step hysteresis control method and the other is the optimal function control. Meanwhile
box-counting dimensions of the controlled Julia set under these methods are computed to depict the
complexity of Julia sets and the system. The simulation results show the efficacy of these methods.
461
471
Zhaoqing
Li
Yongping
Zhang
Jian
Liu
The output duopoly competing evolution model
Julia set
step hysteresis control method
optimal function control fractal dimension.
Article.16.pdf
[
[1]
H. N. Agiza, A. S. Hegazi, A. A. Elsadany , The dynamics of Bowley's model with bounded rationality , Chaos Solitons Fractals, 12 (2001), 1705-1717
##[2]
L. Budinski-Petković, I. Lončarević, Z. M. Jakšić, S. B. Vrhovac, Fractal properties of financial markets , Phys. A, 410 (2014), 43-53
##[3]
J. G. Du, T. Huang, Z. Sheng , Analysis of decision-making in economic chaos control, Nonlinear Anal. Real World Appl., 10 (2009), 2493-2501
##[4]
K. Falconer, Fractal geometry , Mathematical foundations and applications, John Wiley & Sons, Ltd., Chichester (1990)
##[5]
Y. Lin , Fractal and its application in the securities market, Econ. Probl., 8 (2001), 44-46
##[6]
P. Liu , Spatial fractal control and chaotic synchronization in complex dynamical system, Shandong Univ., (2012), 22-25
##[7]
F. Lu, Fractal theory and its application in economic management , Financ. Manag. Inst., 3 (2011), 91-93
##[8]
Z. Peng, C. Li, H. Li , The research of enterprise and fractal management model in Era of knowledge economy , Business Studies, 255 (2002), 33-35
##[9]
D. Preiss, Nondiffierentiable functions, (Czech) Pokroky Mat. Fyz. Astronom., 28 (1983), 148-154
##[10]
J. Shu, D. Tan, J. Wu , The exploration of fractal structure in China's stock market , J. Southwest Jiaotong Univ., 38 (2004), 212-215
##[11]
A. S. Soliman , Fractals in nonlinear economic dynamic systems , Chaos Solitons Fractals, 7 (1996), 247-256
##[12]
M. de Sousa Vieira, A. J. Lichtenberg , Controlling chaos using nonlinear feedback with delay , Phys. Rev. E, 54 (1996), 1200-1207
##[13]
C. Wang, R. Chu, J. Ma , Controlling a chaotic resonator by means of dynamic track control, Complexity, 21 (2015), 370-378
##[14]
F. Zhang, C. Mu, G. Zhang, D. Lin, Dynamics of two classes of Lorenz-type chaotic systems, Complexity, 21 (2015), 363-369
##[15]
M. Zhang, Z. Wei , The Application of the fractal theory in describing the regional transportation economy coordination, China soft sci., 1 (1996), 112-123
##[16]
C. Zhang, F. Wu, L. Zhao , The application of the fractal theory in economics, Statistics and Decision, 1 (2009), 158-159
##[17]
X. H. Zhu, N. Patel, W. Zuo, X. C. Yang, Fractal analysis applied to spatial structure of chinas vegetation , Chinese Geographical Sci., 16 (2006), 48-55
]