Emergence of a common generalized synchronization manifold in network motifs of structurally different time-delay systems. (English) Zbl 1372.34087
Summary: We point out the existence of a transition from partial to global generalized synchronization (GS) in symmetrically coupled structurally different time-delay systems of different orders using the auxiliary system approach and the mutual false nearest neighbor method. The present authors have recently reported that there exists a common GS manifold even in an ensemble of structurally nonidentical scalar time-delay systems with different fractal dimensions and shown that GS occurs simultaneously with phase synchronization (PS). In this paper we confirm that the above result is not confined just to scalar one-dimensional time-delay systems alone but there exists a similar type of transition even in the case of time-delay systems with different orders. We calculate the maximal transverse Lyapunov exponent to evaluate the asymptotic stability of the complete synchronization manifold of each of the main and the corresponding auxiliary systems, which in turn ensures the stability of the GS manifold between the main systems. Further we estimate the correlation coefficient and the correlation of probability of recurrence to establish the relation between GS and PS. We also calculate the mutual false nearest neighbor parameter which doubly confirms the occurrence of the global GS manifold.
MSC:
| 34D06 | Synchronization of solutions to ordinary differential equations |
| 34K23 | Complex (chaotic) behavior of solutions to functional-differential equations |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 37M05 | Simulation of dynamical systems |
Keywords:
partial generalized synchronization; global generalized synchronization; structurally different time-delay systems; networks of time-delay systemsSoftware:
K2References:
| [1] | Lakshmanan, M.; Senthilkumar, D. V., Dynamics of nonlinear time-Delay systems (2010), Springer: Springer Berlin · Zbl 1159.37392 |
| [2] | Pikovsky, A. S.; Rosenblum, M. G.; Kurths, J., Synchronization - A unified approach to nonlinear science (2001), Cambridge University Press: Cambridge University Press Cambridge, England · Zbl 0993.37002 |
| [3] | Schäfer, C.; Rosenblum, M. G.; Kurths, J.; Abel, H. H., Heartbeat synchronized with ventilation, Nature, 392, 6673, 239-240 (1998) |
| [4] | Amritkar, R. E.; Rangarajan, G., Spatially synchronous extinction of species under external forcing, Phys Rev Lett, 96, 25, 258102 (2006) |
| [5] | Blasius, B.; Huppert, A.; Stone, L., Complex dynamics and phase synchronization in spatially extended ecological systems, Nature, 399, 6734, 354-359 (1999) |
| [6] | Earn, D. J.D.; Rohani, P.; Grenfell, B. T., Persistence, chaos and synchrony in ecology and epidemiology, Proc R Soc London, Ser-B, 265, 1390, 7-10 (1998) |
| [7] | Grenfell, B. T.; Bjornstad, O. N.; Kappey, J., Travelling waves and spatial hierarchies in measles epidemics, Nature, 414, 6865, 716-723 (2001) |
| [8] | Farmer, S. F., Rhythmicity, synchronization and binding in human and primate motor systems, J Physiol, 509, 1, 3-14 (1998) |
| [9] | Sebe, J. Y.; van Berderode, J. F.; Berger, A. J.; Abel, H. H., Inhibitory synaptic transmission governs inspiratory motoneuron synchronization, J Neurophysiol, 96, 1, 391-403 (2006) |
| [10] | Maraun, D.; Kurths, J., Epochs of phase coherence between el niño/southern oscillation and indian monsoon, Geophys Res Lett, 32, 15, L15709 (2005) |
| [11] | Stein, K.; Timmermann, A.; Schneifer, N., Phase synchronization of the el-niño-southern oscillation with the annual cycle, Phys Rev Lett, 107, 12, 128501 (2011) |
| [12] | Van Leeuwen, P.; Geue, D.; Thiel, M.; Cysarz, D.; Large, S.; Romano, M. C., Influence of paced maternal breathing on fetalmaternal heart rate coordination, Proc Natl Acad Sci, 106, 33, 13661-13666 (2009) |
| [13] | Koronovskii, A. A.; Moskalenko, O. I.; Khramov, A. E.; Shurygina, S. A., Specific features of generalized synchronization in unidirectionally and mutually coupled mappings and flows: method of phase tubes, J Commun Tech El, 59, 12, 1412-1422 (2014) |
| [14] | Hung, Y. C.; Huang, Y. T.; Ho, M. C.; Hu, C. K., Paths to globally generalized synchronization in scale-free networks, Phys Rev E, 77, 1, 016202 (2008) |
| [15] | Moskalenko, O. I.; Koronovskii, A. A.; Hramov, A. E.; Boccaletti, S., Generalized synchronization in mutually coupled oscillators and complex networks, Phys Rev E, 86, 3, 036216 (2012) |
| [16] | Shang, Y.; Chen, M.; Kurths, J., Generalized synchronization of complex networks, Phys Rev E, 80, 2, 027201 (2009) |
| [17] | Uchida, A.; McAllister, R.; Meucci, R.; Roy, R., Generalized synchronization of chaos in identical systems with hidden degrees of freedom, Phys Rev Lett, 91, 17, 174101 (2003) |
| [18] | Rogers, E. A.; Kalra, R.; Schroll, R. D.; Uchida, A.; Lathrop, D. P.; Roy, R., Generalized synchronization of spatiotemporal chaos in a liquid crystal spatial light modulator, Phys Rev Lett, 93, 8, 084101 (2004) |
| [19] | Dmitriev, B. S.; Hramov, A. E.; Krasovskii, A. A.; Starodubov, A. V.; Trubetskov, D. I.; Zharkov, Y. D., First experimental observation of generalized synchronization phenomena in microwave oscillators, Phys Rev Lett, 107, 7, 074101 (2009) |
| [20] | Moskalenko, O. I.; Koronovskii, A. A.; Hramov, A. E., Generalized synchronization of chaos for secure communication: remarkable stability to noise, Phys Lett A, 374, 2925 (2010) · Zbl 1237.94079 |
| [21] | Murali, K.; Lakshmanan, M., Secure communication using a compound signal from generalized synchronizable chaotic systems, Phys Lett A, 241, 6, 303-310 (1998) · Zbl 0933.94023 |
| [22] | Basnarkov, L.; Duane, G. S.; Kocarev, L., Generalized synchronization and coherent structures in spatially extended systems, Chaos Solitons Fractals, 59, 9, 35-41 (2014) · Zbl 1348.93152 |
| [23] | Abarbanel, H. D.I.; Rulkov, N. F.; Sushchik, M. M., Generalized synchronization of chaos: the auxiliary system approach, Phys Rev E, 53, 5, 4528-4535 (1996) |
| [24] | Brown, R., Approximating the mapping between systems exhibiting generalized synchronization, Phys Rev Lett, 81, 22, 4835-4838 (1998) |
| [25] | Kocarev, L.; Parlitz, U., Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems, Phys Rev Lett, 76, 11, 1816-1819 (1996) |
| [26] | Pyragas, K., Weak and strong synchronization of chaos, Phys Rev E, 54, 5 (1996) |
| [27] | Rulkov, N. F.; Sushchik, M. M.; Tsimring, L. S.; Abarbanel, H. D.I., Generalized synchronization of chaos in directionally coupled chaotic systems, Phys Rev E, 51, 2, 980-994 (1995) |
| [28] | Acharyya, S.; Amritkar, R. E., Generalized synchronization of coupled chaotic systems, Eur Phys J Special Topics, 222, 3-4, 939-952 (2013) |
| [29] | Chen, J.; Lu, J. A.; Wu, X.; Zheng, W. X., Generalized synchronization of complex dynamical networks via impulsive control, Chaos, 19, 4, 043119 (2009) · Zbl 1311.34110 |
| [30] | Guan, S.; Wang, X.; Gong, X.; Li, K.; Lai, C. H., The development of generalized synchronization on complex networks, Chaos, 19, 1, 013130 (2009) |
| [31] | Guan, S.; Gong, X.; Li, K.; Liu, Z.; Lai, C. H., Characterizing generalized synchronization in complex networks, New J Phys, 12, 7, 073045 (2010) |
| [32] | Hu, A.; Xu, Z.; Guo, L., The existence of generalized synchronization of chaotic systems in complex networks, Chaos, 20, 1, 013112 (2010) · Zbl 1311.34114 |
| [33] | Shahverdiev, E. M.; Shore, K. A., Generalized synchronization in laser devices with electro-optical feedback, IET Optoelectron, 3, 6, 274-282 (2009) |
| [34] | Zheng, Z.; Wang, X.; Cross, M. C., Transitions from partial to complete generalized synchronizations in bidirectionally coupled chaotic oscillators, Phys Rev E, 65, 5, 056211 (2002) |
| [35] | Ouannas, A.; Odibat, Z., Generalized synchronization of different dimensional chaotic dynamical systems in discrete time, Nonlinear Dyn, 81, 1, 765-771 (2015) · Zbl 1347.34086 |
| [36] | Boccaletti, S.; Valladares, D. L.; Kurths, J.; Maza, D.; Mancini, H., Synchronization of chaotic structurally nonequivalent systems, Phys Rev E, 61, 4, 3712-3715 (2000) |
| [37] | Sorino, M. C.; der Sande, G. V.; Fischer, I.; Mirasso, C. R., Synchronization in simple network motifs with negligible correlation and mutual information measures, Phys Rev Lett, 108, 13, 134101 (2012) |
| [38] | Parlitz, U.; Junge, L.; Lauterborn, W.; Kocarev, L., Experimental observation of phase synchronization, Phys Rev E, 54, 2, 2115-2117 (1996) |
| [39] | Zheng, Z.; Hu, G., Generalized synchronization versus phase synchronization, Phys Rev E, 62, 6, 7882-7885 (2000) |
| [40] | Stankovski, T.; McClintock, P. V.E.; Stefanovska, A., Dynamical inference: where phase synchronization and generalized synchronization meet, Phys Rev E, 89, 6, 062909 (2014) |
| [41] | Senthilkumar, D. V.; Lakshmanan, M.; Kurths, J., Transition from phase to generalized synchronization in time-delay systems, Chaos, 18, 2, 023118 (2007) |
| [42] | Senthilkumar, D. V.; Suresh, R.; Lakshmanan, M.; Kurths, J., Global generalized synchronization in networks of different time-delay systems, Europhys Lett, 103, 5, 50010 (2013) |
| [43] | Mackey, M. C.; Glass, L., Oscillation and chaos in physiological control systems, Science, 197, 4300, 287-289 (1977) · Zbl 1383.92036 |
| [44] | Senthilkumar, D. V.; Lakshmanan, M.; Kurths, J., Phase synchronization in time-delay systems, Phys Rev E, 74, 3, R035205 (2006) |
| [45] | Suresh, R.; Srinivasan, K.; Senthilkumar, D. V.; Raja Mohamed, I.; Murali, K.; Lakshmanan, M., Zero-lag synchronization in coupled time-delayed piecewise linear electronic circuits, Eur Phys J Special Topics, 222, 3-4, 729-744 (2013) |
| [46] | Ikeda, K.; Daido, H.; Akimoto, O., Optical turbulence: chaotic behavior of transmitted light from a ring cavity, Phys Rev Lett, 45, 9, 709-712 (1980) |
| [47] | Appeltant, L.; Soriano, M. C.; Van der Sande, G.; Danckaert, J.; Masser, S.; Dambre, J., Information processing using a single dynamical node as complex system, Nature Commun, 2, 468 (2011) |
| [48] | Qin, H.; Ma, J.; Wang, C.; Wu, Y., Autapse-induced spiral wave in network of neurons under noise, PLoS ONE, 9, 6 (2014) |
| [49] | Qin, H.; Ma, J.; Jin, W.; Wang, C., Dynamics of electric activities in neuron and neurons of network induced by autapses, Sci China Tech Sci, 57, 5, 936-946 (2014) |
| [50] | Song, X.; Wang, C.; Ma, J.; Tang, J., Transition of electric activity of neurons induced by chemical and electric autapses, Sci China Tech Sci, 58, 6, 1007-1014 (2015) |
| [51] | Wang, Q.; Chen, G.; Perc, M., Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling, PLoS ONE, 6, 1 (2011) |
| [52] | Wang, Q.; Chen, G., Delay-induced intermittent transition of synchronization in neuronal networks with hybrid synapses, Chaos, 21, 1, 013123 (2011) |
| [53] | Wang, Q.; Shi, X.; Chen, G., Delay-induced synchronization transition in small-world hodgkin-Huxley neuronal networks with channel blocking, Discrete Continuous Dyn Syst Ser B, 16, 2, 607-621 (2011) · Zbl 1222.76093 |
| [54] | Wang, Q.; Perc, M.; Duan, Z.; Chen, G., Impact of delays and rewiring on the dynamics of small-world neuronal networks with two types of coupling, Physica A, 389, 16, 3299-3306 (2010) |
| [55] | Wang, Q.; Lu, Q. S.; Duan, Z., Adaptive lag synchronization in coupled chaotic systems with unidirectional delay feedback, Int J Non Linear Mech, 45, 6, 640-646 (2010) |
| [56] | Wang, Q.; Perc, M.; Duan, Z.; Chen, G., Synchronization transitions on scale-free neuronal networks due to finite information transmission delays, Phys Rev E, 80, 026206 (2009) |
| [57] | Marwan, N.; Romano, M. C.; Thiel, M.; Kurths, J., Recurrence plots for the analysis of complex systems, Phys Rep, 438, 5-6, 237-329 (2007) |
| [58] | Schumacher, J.; Haslinger, R.; Pipa, G., Statistical modeling approach for detecting generalized synchronization, Phys Rev E, 85, 5, 056215 (2012) |
| [59] | Koronvskii, A. A.; Moskalenko, I. O.; Hramov, A. E., Nearest neighbors, phase tubes, and generalized synchronization, Phys Rev E, 84, 3, 037201 (2011) |
| [60] | Liapunov, A. M., Stability of motion (1966), Academic Press: Academic Press New-York & London · Zbl 0161.06303 |
| [61] | Parlitz, U.; Junge, L.; Kocarev, L., Subharmonic entrainment of unstable periodic orbits and generalized synchronization, Phys Rev Lett, 79, 17, 3158-3161 (1997) |
| [62] | Pecora, L. M.; Carroll, T. L.; Jhonson, G. A.; Mar, D. J., Fundamentals of synchronization in chaotic systems, concepts, and applications, Chaos, 7, 4, 520-543 (1997) · Zbl 0933.37030 |
| [63] | Abarbanel, H. D.I.; Brown, R.; Sidorowich, J. J.; Tsimring, L. S., The analysis of observed chaotic data in physical systems, Rev Mod Phys, 65, 4, 1331-1392 (1993) |
| [64] | Packard, N. H.; Crutchfield, J. P.; Farmer, J. D.; Shaw, R. S., Geometry from a time series, Phys Rev Lett, 45, 9, 712-716 (1980) |
| [65] | Hopfield, J. J., Neural networks and physical systems with emergent collective computational abilities, Proc Natl Acad Sci, 79, 8, 2554-2558 (1982) · Zbl 1369.92007 |
| [66] | Meng, J.; Wang, X., Robust anti-synchronization of a class of delayed chaotic neural networks, Chaos, 17, 2, 023113 (2007) · Zbl 1159.37371 |
| [67] | Abraham, E. R., The generation of plankton patchiness by turbulent stirring, Nature, 391, 6667, 577-580 (1998) |
| [68] | Gakkhar, S.; Singh, A., A delay model for viral infection in toxin producing phytoplankton and zooplankton system, Commun Nonlinear Sci Numer Simulat, 15, 11, 3607-3620 (2010) · Zbl 1222.37098 |
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.
