Synchronization in networks of initially independent dynamical systems. (English) Zbl 1514.34113
Summary: The dynamical system becomes initial-dependent when nonlinear quadratic term is considered, which the attractor can be switched between chaotic and periodical states by resetting the initial values even the parameters are fixed. Standard dynamical analysis and Hamilton energy are calculated to confirm the dynamics dependence on the initial setting. Feedback-based initial setting is applied to find the synchronization dependence on selection of initial values for the memory variable \(z\). The nonlinear quadratic term \(z^2y\) can suppress the oscillation of variable \(y\) via negative feedback, thus periodic oscillation can be triggered to tame another oscillator under bidirectional coupling, then periodic synchronization can be reached. Furthermore, the synchronization approach and pattern selection are considered on the network, and the factor of synchronization is calculated to find the synchronization dependence on coupling intensity. It is found that the network synchronization can be enhanced when noise-like disturbance is applied to reset the memory variable \(z\), the potential mechanism is that the local kinetics is effectively adjusted to trigger periodic stimulus on some nodes thus the collective behaviors are controlled to become consensus.
MSC:
| 34H20 | Bifurcation control of ordinary differential equations |
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 34D06 | Synchronization of solutions to ordinary differential equations |
References:
| [1] | Jin, W.; Lin, Q.; Wang, A., Computer simulation of noise effects of the neighborhood of stimulus threshold for a mathematical model of homeostatic regulation of sleep-wake cycles, Complexity, 2017, Article 4797545 pp. (2017) · Zbl 1377.93031 |
| [2] | Guo, S. L.; Tang, J.; Ma, J., Autaptic modulation of electrical activity in a network of neuron-coupled astrocyte, Complexity, 2017, Article 4631602 pp. (2017) |
| [3] | Tang, J.; Zhang, J.; Ma, J., Astrocyte calcium wave induces seizure-like behavior in neuron network, Sci. China Technol. Sci., 60, 1011-1018 (2017) |
| [4] | Perc, M.; Gosak, M.; Marhl, M., Periodic calcium waves in coupled cells induced by internal noise, Chem. Phys. Lett., 437, 143-147 (2007) |
| [5] | Perc, M.; Marhl, M., Different types of bursting calcium oscillations in non-excitable cells, Chaos Solitons Fractals, 18, 759-773 (2003) · Zbl 1068.92017 |
| [6] | Yassen, M. T., Controlling chaos and synchronization for new chaotic system using linear feedback control, Chaos Solitons Fractals, 26, 3, 913-920 (2005) · Zbl 1093.93539 |
| [7] | Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Phys. Rev. Lett., 64, 821-824 (1990) · Zbl 0938.37019 |
| [8] | Boccaletti, S.; Kurths, J.; Osipov, G., The synchronization of chaotic systems, Phys. Rep., 366, 1-101 (2002) · Zbl 0995.37022 |
| [9] | Yassen, M. T., Chaos control of chaotic dynamical systems using backstepping design, Chaos Solitons Fractals, 27, 2, 537-548 (2006) · Zbl 1102.37306 |
| [10] | Wang, C.; Chu, R.; Ma, J., Controlling a chaotic resonator by means of dynamic track control, Complexity, 21, 370-378 (2015) |
| [11] | Zhang, G.; Ma, J.; Alsaedi, A., Dynamical behavior and application in Josephson Junction coupled by memristor, Appl. Math. Comput., 321, 290-299 (2018) · Zbl 1426.94176 |
| [12] | Guo, M.; Xue, Y.; Gao, Z., Dynamic analysis of a physical SBT memristor-based chaotic circuit, Int. J. Bifurcation Chaos, 27, 13, Article 1730047 pp. (2017) · Zbl 1420.94119 |
| [13] | Ren, G.; Zhou, P.; Ma, J., Dynamical response of electrical activities in digital neuron circuit driven by autapse, Int. J. Bifurcation Chaos, 27, 12, Article 1750187 pp. (2017) |
| [14] | Galias, Z., Numerical study of multiple attractors in the parallel inductor-capacitor-memristor circuit, Int. J. Bifurcation Chaos, 27, 11, Article 1730036 pp. (2017) |
| [15] | Rajamani, V.; Sah, M. P.D.; Mannan, Z. I., Third-order memristive Morris-Lecar model of barnacle muscle fiber, Int. J. Bifurcation Chaos, 27, 4, Article 1730015 pp. (2017) · Zbl 1366.34068 |
| [16] | Ma, J.; Wu, F.; Ren, G., A class of initials-dependent dynamical systems, Appl. Math. Comput., 298, 65-76 (2017) · Zbl 1411.37035 |
| [17] | Wu, F.; Zhou, P.; Alsaedi, A., Synchronization dependence on initial setting of chaotic systems without equilibria, Chaos Solitons Fractals, 110, 124-132 (2018) |
| [18] | Sakthivel, R.; Santra, S.; Anthoni, S. M., Synchronisation and anti-synchronisation of chaotic systems with application to DC-DC boost converter, IET Generation Transm. Distrib., 11, 4, 959-967 (2017) |
| [19] | Mathiyalagan, K.; Park, J. H.; Sakthivel, R., Exponential synchronization for fractional-order chaotic systems with mixed uncertainties, Complexity, 21, 1, 114-125 (2015) |
| [20] | Albert, R.; Barabási, A. L., Statistical mechanics of complex networks, Rev. Modern Phys., 74, 49-98 (2002) |
| [21] | Boccaletti, S.; Latora, V.; Moreno, Y., Complex networks: structure and dynamics, Phys. Rep., 424, 175-308 (2006) · Zbl 1371.82002 |
| [22] | Newman, M., The structure and function of complex networks, SIAM Rev., 45, 2, 167-256 (2003) · Zbl 1029.68010 |
| [23] | Ma, J.; Song, X. L.; Tang, J., Wave emitting and propagation induced by autapse in a forward feedback neuronal network, Neurocomputing, 167, 378-389 (2015) |
| [24] | Kaviarasan, B.; Sakthivel, R.; Lim, Y., Synchronization of complex dynamical networks with uncertain inner coupling and successive delays based on passivity theory, Neurocomputing, 186, 127-138 (2016) |
| [25] | Park, J. H.; Lee, T. H., Synchronization of complex dynamical networks with discontinuous coupling signals, Nonlinear Dynam., 79, 1353-1362 (2015) · Zbl 1345.93022 |
| [26] | Sun, X.; Perc, M.; Kurths, J., Effects of partial time delays on phase synchronization in Watts-Strogatz small-world neuronal networks, Chaos, 27, 5, Article 053113 pp. (2017) |
| [27] | Abdurahman, A.; Jiang, H.; Teng, Z., Lag synchronization for Cohen-Grossberg neural networks with mixed time-delays via periodically intermittent control, Int. J. Comput. Math., 94, 2, 275-295 (2017) · Zbl 1368.82024 |
| [28] | Chen, X.; Qiu, J.; Cao, J., Hybrid synchronization behavior in an array of coupled chaotic systems with ring connection, Neurocomputing, 173, 1299-1309 (2016) |
| [29] | Ma, J.; Xu, Y.; Ren, G., Prediction for breakup of spiral wave in a regular neuronal network, Nonlinear Dynam., 84, 497-509 (2016) |
| [30] | Song, X.; Wang, C.; Ma, J., Collapse of ordered spatial pattern in neuronal network, Physica A, 451, 95-112 (2016) · Zbl 1400.92025 |
| [31] | Wang, C.; Lv, M.; Alsaedi, A., Synchronization stability and pattern selection in a memristive neuronal network, Chaos, 27, Article 113108 pp. (2017) |
| [32] | Yao, Y.; Deng, H.; Yi, M., Impact of bounded noise on the formation and instability of spiral wave in a 2D Lattice of neurons, Sci. Rep., 7, Article 43151 pp. (2017) |
| [33] | Davidenko, J. M.; Pertsov, A. V.; Salomonsz, R., Stationary and drifting spiral waves of excitation in isolated cardiac muscle, Nature, 355, 6358, 349 (1992) |
| [34] | Zhang, G.; Wu, F.; Hayat, T., Selection of spatial pattern on resonant network of coupled memristor and Josephson junction, Commun. Nonlinear Sci. Numer. Simul., 65, 79-90 (2018) · Zbl 1508.94127 |
| [35] | Qin, H.; Wang, C.; Cai, N., Field coupling-induced pattern formation in two-layer neuronal network, Physica A, 501, 141-152 (2018) · Zbl 1514.92012 |
| [36] | Wang, C.; Ma, J., A review and guidance for pattern selection in spatiotemporal system, Internat. J. Modern Phys. B, 32, Article 1830003 pp. (2018) · Zbl 1429.37002 |
| [37] | Majhi, S.; Perc, M.; Ghosh, D., Chimera states in a multilayer network of coupled and uncoupled neurons, Chaos, 27, 7, 073109 (2017) |
| [38] | Bera, B. K.; Majhi, S.; Ghosh, D., Chimera states: Effects of different coupling topologies, Europhys. Lett., 118, 1, 10001 (2017) |
| [39] | Rössler, O. E., An equation for continuous chaos, Phys. Lett. A, 57, 397-398 (1976) · Zbl 1371.37062 |
| [40] | Kobe, D. H., Helmholtz’s theorem revisited, Amer. J. Phys., 54, 6, 552-554 (1986) |
| [41] | Sarasola, C.; Torrealdea, F. J.; d’Anjou, A., Energy balance in feedback synchronization of chaotic systems, Phys. Rev. E, 69, Article 011606 pp. (2004) |
| [42] | Ma, J.; Wu, F. Q.; Jin, W. Y., Calculation of Hamilton energy and control of dynamical systems with different types of attractors, Chaos, 27, Article 053108 pp. (2017) · Zbl 1390.37059 |
| [43] | Wang, C. N.; Wang, Y.; Ma, J., Calculation of Hamilton energy function of dynamical system by using Helmholtz theorem, Acta Phys. Sin., 65, Article 240501 pp. (2016), In Chinese |
| [44] | Song, X. L.; Jin, W. Y.; Ma, J., Energy dependence on the electric activities of a neuron, Chinese Phys. B, 24, 12, Article 128710 pp. (2015) |
| [45] | Wu, F.; Ma, J.; Ren, G., Synchronization stability between initial-dependent oscillators with periodical and chaotic oscillation, J. Zhejiang Univ. Sci. A, 19, 12, 889-903 (2018) |
| [46] | Bolhasani, E.; Azizi, Y.; Valizadeh, A., Synchronization of oscillators through time-shifted common inputs, Phys. Rev. E, 95, Article 032207 pp. (2017) |
| [47] | Wei, X.; Emenheiser, J.; Wu, X., Maximizing synchronizability of duplex networks, Chaos, 28, Article 013110 pp. (2018) |
| [48] | Mei, G.; Wu, X.; Wang, Y., Compressive-sensing-based structure identification for multilayer networks, IEEE T Cybern., 48, 2, 754-764 (2018) |
| [49] | Gosak, M.; Markovic, R.; Dolenöek, J., Network science of biological systems at different scales: a review, Phys. Life Rev., 24, 118-135 (2018) |
| [50] | Majhi, S.; Perc, M.; Ghosh, D., Chimera states in a multilayer network of coupled and uncoupled neurons, Chaos, 27, 7, Article 073109 pp. (2017) |
| [51] | Bera, B. K.; Majhi, S.; Ghosh, D., Chimera states: Effects of different coupling topologies, Europhys. Lett., 118, Article 10001 pp. (2017) |
| [52] | Ma, J.; Wu, F.; Wang, C., Synchronization behaviors of coupled neurons under electromagnetic radiation, Internat. J. Modern Phys. B, 31, Article 1650251 pp. (2017) |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
