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Complex dynamics in simple delayed two-parameterized models. (English) Zbl 1276.37048

Summary: In this paper, some geometrical aspects of root distributions in a special polynomial of the form \(\lambda ^{\tau }(\lambda - (1 - \alpha )) - \beta \) are discussed. Equivariant structures are explored in the corresponding systems. Some sufficient and necessary conditions for a pair of complex conjugate roots of the polynomial with \(\tau =3\) lying on the unit circle. A comparison is made between two simple delayed discrete models, where one can be viewed as the perturbation of the other with a delayed feedback. There exist rich dynamics in the perturbed system, such as chaotic, or even hyperchaotic behavior whereas only regular oscillation modes can be observed in the perturbed system. The introduction of delayed feedback can break or increase the special symmetrical/topological structure of the original system, which leads to complexity. Rich dynamics near equivariant bifurcations under the \(\mathbb Z_{4}/\mathbb Z_{8}\) cyclic group action is explored, including multiple bifurcations, multistability, chaos and hyperchaos etc. As the applications, one can find that there exist higher-codimensional bifurcations with 1:1 strong resonance and 1:2 strong resonance in those models with/without special equivariant systems.

MSC:

37M20 Computational methods for bifurcation problems in dynamical systems
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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[1] Niebur, E.; Schuster, H. G.; Kammen, D. M., Collective frequencies and metastability in networks of limit-cycle oscillators with time delay, Phys. Rev. Lett., 67, 2753-2756 (1991)
[2] Erzgräber, H.; Krauskopf, B.; Lenstra, D., Compound laser modes of mutually delay-coupled Lasers, SIAM J. Appl. Dyn. Syst., 5, 1, 30-65 (2006) · Zbl 1097.78014
[3] Alvarez-Ramirez, J.; Rodriguez, E.; Echeverria, J. C., Delays in the human heartbeat dynamics, Chaos, 19, 028502 (2009)
[4] Baldi, P.; Atiya, A., How delays affect neural dynamics and learning, IEEE Tran. Neural Netw., 5, 612-621 (1994)
[5] Hadeler, K. P., Effective computation of periodic orbits and bifurcation diagrams in delay equations, Numer. Math., 34, 457-467 (1980) · Zbl 0419.34070
[6] Peng, M., Bifurcation and chaotic behavior in the Euler method for a Uçar prototype delay model prototype delay model, Chaos Solitons Fractals, 22, 483-493 (2004) · Zbl 1061.37022
[7] Uçar, A., On the chaotic behavior of a prototype delayed dynamical system, Chaos Solitons Fractals, 16, 187-194 (2003) · Zbl 1033.37020
[8] Uçar, A., A, prototype model for chaos studies, Int. J. Eng. Sci., 40, 251-258 (2002) · Zbl 1211.37041
[9] Masoller, C.; Zanette, D. H., Anticipated synchronization in coupled chaotic maps with delays, Physica A, 300, 3-4, 359-366 (2001) · Zbl 0973.37033
[10] Peng, M., Symmetry breaking, bifurcations, periodicity and chaos in the Euler method for a class of delay differential equations, Chaos Solitons Fractals, 24, 1287-1297 (2005) · Zbl 1079.37011
[11] K. Ikeda, O. Akimoto, Instability leading to periodic and chaotic self-pulsations in a bistable optical cavity.; K. Ikeda, O. Akimoto, Instability leading to periodic and chaotic self-pulsations in a bistable optical cavity.
[12] Ikeda, K.; Matsumoto, K., Study of a high-dimensional chaotic attractor, J. Stat. Phys., 44, 5-6, 955-983 (1986) · Zbl 0659.58036
[13] Kim, S.; Park, S. H.; Ryu, C. S., Multistability in coupled oscillator systems with time delay, Phys. Rev. Lett., 79, 2911-2914 (1997)
[14] Koto, T., Periodic orbits in the Euler methods for a class of delay differential equations, Comput. Math. Appl., 42, 1597-1608 (2001) · Zbl 1161.65367
[15] Guo, S.; Tang, X.; Huang, L., Bifurcation analysis in a discrete-time single-directional network with delays, Neurocomputing, 71, 7-9, 1422-1435 (2008)
[16] Guo, S.; Tang, X.; Huang, L., Stability and bifurcation in a discrete system of two neurons with delays, Nonlinear Anal. Real World Appl., 9, 4, 1323-1335 (2008) · Zbl 1154.37381
[17] Kuruklis, S. A., The asymptotic stability of \(x_{n + 1} - a x_n + b x_{n - k} = 0\), J. Math. Anal. Appl., 188, 719-731 (1994) · Zbl 0842.39004
[18] Peng, M.; Huang, L.; Wang, G., Higher-codimension bifurcations in a discrete unidirectional neural network model with delayed feedback, Chaos, 18, 023105 (2008) · Zbl 1307.92032
[19] Peng, M.; Yang, X., New stability criteria and bifurcation analysis for nonlinear discrete-time coupled loops with multiple delays, Chaos, 20, 013125 (2010) · Zbl 1311.93053
[20] Peng, M.; Yuan, Y., Stability, symmetry-breaking bifurcation and chaos in discrete delayed models, Int. J. Bifurcation Chaos, 18, 5, 1477-1501 (2008) · Zbl 1147.39301
[21] Peng, M.; Yuan, Y., Synchronization and desynchronization in a delayed discrete neural network, Int. J. Bifurcations Chaos Appl. Sci. Eng., 17, 3, 781-803 (2007) · Zbl 1146.39015
[22] Wang, G.; Peng, M., Rich oscillation patterns in a simple delayed neuron network and its linear control, Int. J. Bifurcation Chaos Appl. Sci. Eng., 19, 2993-3004 (2009) · Zbl 1179.39009
[23] Eckmann, J. P.; Ruelle, D., Ergodic theory of chaos and strange attractors, Rev. Mod. Phys., 57, 617-656 (1985) · Zbl 0989.37516
[24] Wolf, A.; Swift, J. B.; Swinney, H. L.; Vastano, J. A., Determining Lyapunov exponents from a time series, Physica D, 16, 285-317 (1985) · Zbl 0585.58037
[25] Alligood, K. T.; Sauer, T. D.; Yorke, J. A., Chaos: An Introduction to Dynamical Systems (1996), Springer-Verlag: Springer-Verlag New York · Zbl 0867.58043
[26] Frederickson, P.; Kaplan, J. L.; Yorke, E. D.; Yorke, J. A., The Liapunov dimension of strange attractors, J. Difference. Equ., 49, 185-207 (1983) · Zbl 0515.34040
[27] Kaplan, J. L.; Yorke, Y. A., A regime observed in a fluid flow model of Lorenz, Commun. Math. Phys., 67, 93-108 (1979) · Zbl 0443.76059
[28] Hampton, A.; Zanette, D. H., Measure synchronization in coupled Hamiltonian systems, Phys. Rev. Lett., 83, 11, 2179-2182 (1999)
[29] Ogata, K., Discrete-Time Control Systems (1995), Prentice Hall: Prentice Hall Englewood Cliffs, NJ
[30] Richter, H., The generalized Henon maps: examples for higher-dimensional Chaos, Int. J. Bifurcations Chaos Appl. Sci. Eng., 12, 6, 1371-1384 (2002) · Zbl 1044.37026
[31] Wen, G.; Xu, D.; Han, X., On creation of Hopf Bifurcations in discrete-time nonlinear systems, Chaos, 12, 2, 350-355 (2002) · Zbl 1080.37567
[32] Puu, T.; Marín, M. R., The dynamics of a triopoly Cournot game when the competitors operate under capacity constraints, Chaos Solitons Fractals, 29, 707-722 (2006) · Zbl 1124.91025
[33] Golubitsky, M.; Stewart, I.; Schaeffer, D. G., (Singularities and Groups in Bifurcation theory, Vol. II. Singularities and Groups in Bifurcation theory, Vol. II, Applied Mathematical Sciences, vol. 69 (1988), Springer-Verlag: Springer-Verlag New York) · Zbl 0691.58003
[34] Kuznetsov, Y. A., Elements of Applied Bifurcation Theory (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0829.58029
[35] Arnold, V. I., Geometrical methods in the theory of ordinary differential equations, (Grundlehren der Mathematischen Wissenschaften, vol. 250 (1988), Springer: Springer New York) · Zbl 0507.34003
[36] Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos (2003), Springer-Verlag: Springer-Verlag Berlin · Zbl 1027.37002
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