×

zbMATH — the first resource for mathematics

Codimension-two bifurcation, chaos and control in a discrete-time information diffusion model. (English) Zbl 1356.93053
Summary: In this paper, we present a discrete model to illustrate how two pieces of information interact with online social networks and investigate the dynamics of discrete-time information diffusion model in three types: reverse type, intervention type and mutualistic type. It is found that the model has orbits with period 2, 4, 6, 8, 12, 16, 20, 30, quasiperiodic orbit, and undergoes heteroclinic bifurcation near 1:2 point, a homoclinic structure near 1:3 resonance point and an invariant cycle bifurcated by period 4 orbit near 1:4 resonance point. Moreover, in order to regulate information diffusion process and information security, we give two control strategies, the hybrid control method and the feedback controller of polynomial functions, to control chaos, flip bifurcation, 1:2, 1:3 and 1:4 resonances, respectively, in the two-dimensional discrete system.

MSC:
93C55 Discrete-time control/observation systems
93D15 Stabilization of systems by feedback
93B52 Feedback control
83C10 Equations of motion in general relativity and gravitational theory
Software:
AUTO; AUTO-07P
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agiza, HN; ELabbasy, EM; El-Metwally, H; Elsadany, AA, Chaotic dynamics of a discrete prey-predator model with Holling type II, Nonlinear Anal. Real World Appl., 10, 116-129, (2009) · Zbl 1154.37335
[2] Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos: An Introduction to Dynamical Systems. Springer, New York (1997) · Zbl 0867.58043
[3] Cartwright, JHE, Nonlinear stiffness, Lyapunov exponents, and attractor dimension, Phys. Lett. A, 264, 298-302, (1999) · Zbl 0949.37014
[4] Chen, Z; Yu, P, Controlling and anti-controlling Hopf bifurcations in discrete maps using polynomial functions, Chaos Solitons Fractals, 26, 1231-1248, (2005) · Zbl 1093.37508
[5] Doedel, E.J., Oldeman, B.E.: AUTO-07P: continuation and bifurcation software for ordinary differential equations (2012). http://cmvl.cs.concordia.ca/auto
[6] Eckmann, JP; Ruelle, D, Ergodic theory of chaos and strange attractors, Rev. Mod. Phys., 57, 617-656, (1985) · Zbl 0989.37516
[7] ELabbasy, EM; Agiza, HN; EL-Metwally, H; Elsadany, AA, Bifurcation analysis chaos and control in the Burgers mapping, Int. J. Nonlinear Sci., 4, 171-185, (2007) · Zbl 1394.37123
[8] Frederickson, P; Kaplan, JL; Yorke, ED; Yorke, JA, The Liapunov dimension of strange attractors, J. Differ. Equ., 49, 185-207, (1983) · Zbl 0515.34040
[9] Freeman, M; McVittle, J; Sivak, I; Wu, JH, Viral information propagation in the digg online social network, Phys. A, 415, 87-94, (2015) · Zbl 1402.91622
[10] Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical System and Bifurcation of Vector Fields. Springer, New York (1983) · Zbl 0515.34001
[11] Jiang, J; Wilson, C; Wang, X; Sha, WP; Huang, P; Dai, YF; Zhao, BY, Understanding latent interactions in online social networks, ACM Trans. Web, 7, 1-13, (2013)
[12] Kaplan, JL; Yorke, JA, Chaotic behavior of multidimensional difference equations, Lect. Notes Math., 730, 204-227, (1979) · Zbl 0448.58020
[13] Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, vol. 112, 2nd edn. Springer, New York (1998) · Zbl 0914.58025
[14] Kuznetsov, YA; Meijer, HGE, Numerical normal forms for codim 2 bifurcations of fixed points with at most two critical eigenvalues, SIAM J. Sci. Comput., 26, 1932-1954, (2005) · Zbl 1080.37056
[15] Lei, CX; Lin, ZG; Wang, HY, The free boundary problem describing information diffusion in online social networks, J. Differ. Equ., 254, 1326-1341, (2013) · Zbl 1273.35326
[16] Lerman, K., Ghosh, R., Surachawala, T.: Social contagion: an empirical study of information spread on digg and twitter follower graphs. arXiv:1202.3162 (2012) · Zbl 1273.35326
[17] Li, SP; Zhang, WN, Bifurcations of a discrete prey-predator model with Holling type II functional response, Discrete Contin. Dyn. Syst. Ser. B, 14, 159-176, (2010) · Zbl 1200.37043
[18] Li, B; He, ZM, 1:2 and 1:4 resonances in a two dimensional discrete hindmarsh-rose model, Nonlinear Dyn., 79, 705-720, (2014) · Zbl 1331.92031
[19] Li, B., He, Z.M.: 1:3 resonance and chaos in a discrete Hindmarsh-Rose model. J. Appl. Math. Article ID 896478 (2014b)
[20] Liu, XL; Xiao, DM, Bifurcation in a discrete time Lotka-Volterra predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 6, 559-572, (2006) · Zbl 1100.37054
[21] Liu, XL; Xiao, DM, Complex dynamics behaviors of a discrete-time predator-prey system, Chaos Solitons Fractals, 32, 80-94, (2007) · Zbl 1130.92056
[22] Luo, XS; Chen, GR; Wang, BH; Fang, JQ, Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems, Chaos Solitons Fractals, 18, 775-783, (2003) · Zbl 1073.37512
[23] Ogata, K.: Discrete-Time Control Systems, 2nd edn. Prentice Hall, Upper Saddle River (1995)
[24] Peng, MS, Multiple bifurcations and periodic “bubbling” in a delay population model, Chaos Solitons and Fractals, 25, 1123-1130, (2005) · Zbl 1065.92035
[25] Peng, C., Xu, K., Wang, F., Wang, H.Y.: Predicting information diffusion initiated from multiple sources in online social networks. In: Sixth International Symposium on Computational Intelligence and Design (ISCID), pp. 96-99 (2013) · Zbl 0989.37516
[26] Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics and Chaos, 2nd edn. London, New York, Washington (DC): Boca Raton (1999) · Zbl 0914.58021
[27] Vandermeer, J, Period ‘bubbling’ in simple ecological models: pattern and chaos formation in a quartic model, Ecol. Model., 95, 311-317, (1997)
[28] Wang, F., Wang, H.Y., Xu, K.: Diffusive logistic model towards predicting information diffusion in online social networks. In: 32nd International Conference on Distributed Computing Systems Workshops (ICDCS Workshops), pp. 133-139 (2012) · Zbl 1093.37508
[29] Wang, YF; Vasilakos, AV; Ma, JH; Xiong, NX, On studying the impact of uncertainty on behavior diffusion in social networks, IEEE Trans. Syst. Man Cybern. Part B Cybern., 45, 185-197, (2015)
[30] Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. Springer, New York (2003) · Zbl 1027.37002
[31] Wolf, A; Swift, JB; Swmney, HL; Vastno, JA, Determining Lyapunov exponents from a time series, Phys. D., 16, 285-317, (1985) · Zbl 0585.58037
[32] Yang, J; Leskovec, J, Structure and overlaps of ground-truth communities in networks, ACM Trans. Intell. Syst. Technol., 5, 26-35, (2014)
[33] Ye, SZ; Wu, SF, Measuring message propagation and social influence on twitter.com, Int. J. Commun. Netw. Syst. Sci., 11, 59-76, (2013)
[34] Yuan, LG; Yang, QG, Bifurcation, invariant curve and hybrid control in a discrete-time predator-prey system, Appl. Math. Model., 39, 2345-2362, (2015)
[35] Yu, P; Chen, GR, Hopf bifurcation control using nonlinear feedback with polynomial functions, Int. J. Bifur. Chaos, 14, 1683-1704, (2004) · Zbl 1129.37335
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.