Effects of time delay on the collective behavior of globally coupled harmonic oscillators with fluctuating frequency. (English) Zbl 1533.34057
Summary: In the viscoelastic environment, coupling forces between particles often play an essential role. Due to the time delay in the transmission of coupling forces, this paper focuses on investigating the effects of time delay on the collective behaviors of coupled harmonic oscillators, including synchronization, stability, and stochastic resonance. Firstly, we establish a time delay decoupling formula that can complete the accurate solution of the model under a large delay, thus breaking the limitation of previous small delay approximation methods. Secondly, the statistical synchronization between particles is derived, which means the mean-field behavior of the system can be studied through single-particle behavior. Thirdly, the system’s stability condition is given, demonstrating that the stability region expands with increasing time delay. Finally, the output response amplitude gain of the system is obtained, and then the stochastic resonance behavior of the system is studied. In the small delay region, noise competes with the ordered driving force, resulting in the classical stochastic resonance (CSR) behavior consistent with previous research findings. In the large delay region, periodic matching between the time delay and driving frequency gives rise to parameter-induced stochastic resonance (PSR) behavior, which has not been observed in previous studies.
MSC:
| 34K06 | Linear functional-differential equations |
| 37C60 | Nonautonomous smooth dynamical systems |
| 34K20 | Stability theory of functional-differential equations |
| 34K50 | Stochastic functional-differential equations |
| 70K30 | Nonlinear resonances for nonlinear problems in mechanics |
| 34K24 | Synchronization of functional-differential equations |
References:
| [1] | Benzi, R.; Sutera, A.; Vulpiani, A., The mechanism of stochastic resonance. J Phys A, L453 (1981) |
| [2] | Benzi, R.; Parisi, G.; Sutera, A.; Vulpiani, A., Stochastic resonance in climatic change. Tellus, 10-16 (1982) |
| [3] | Nicolis, C., Stochastic aspects of climatic transitions-response to a periodic forcing. Tellus, 1-9 (1982) |
| [4] | Mcnamara, B.; Wiesenfeld, K., Theory of stochastic resonance. Phys Rev A, 4854-4869 (1989) |
| [5] | Cubero, D., Finite-size fluctuations and stochastic resonance in globally coupled bistable systems. Phys Rev E (2008) |
| [6] | Pikovsky, A.; Zaikin, A.; Ma, D. L.C., System size resonance in coupled noisy systems and in the ising model. Phys Rev Lett (2002) |
| [7] | Oyarzabal, R. S.; Szezech, J. D.; Batista, A. M., Stochastic resonance in dissipative drift motion. Commun Nonlinear Sci Numer Simul, 62 (2017) |
| [8] | Atsumi, Y.; Hata, S.; Nakao, H., Phase ordering in coupled noisy bistable systems on scale-free networks. Phys Rev E (2013) |
| [9] | Gitterman, M., Classical harmonic oscillator with multiplicative noise. Physica A, 309-334 (2005) |
| [10] | Li, J. H., Effect of asymmetry on stochastic resonance and stochastic resonance induced by multiplicative noise and by mean-field coupling. Phys Rev E (2002) |
| [11] | Yu, T.; Zhang, L.; Luo, M. K., Stochastic resonance in the fractional langevin equation driven by multiplicative noise and periodically modulated noise. Phys Scr (2013) · Zbl 1278.26015 |
| [12] | Yang, B.; Zhang, X.; Zhang, L., Collective behavior of globally coupled langevin equations with colored noise in the presence of stochastic resonance. Phys Rev E (2016) |
| [13] | Lin, L. F.; Yu, L.; Wang, H., Parameter-adjusted stochastic resonance system for the aperiodic echo chirp signal in optimal frFT domain. Commun Nonlinear Sci Numer Simul, 171-181 (2017) · Zbl 1465.94023 |
| [14] | Berdichevsky, V.; Gitterman, M., Multiplicative stochastic resonance in linear systems: Analytical solution. Europhys Lett, 161-166 (1996) |
| [15] | Fulinski, A., Changes in transition rates due to barrier fluctuations-the case of dichotomic noise. Phys Lett A, 94-98 (1993) |
| [16] | Robertson, B.; Astumian, R. D., Frequency-dependence of catalyzed-reactions in a weak oscillating field. J Chem Phys, 7414-7419 (1991) |
| [17] | Kubo, R., Stochastic liouville equations. J Math Phys, 174 (1963) · Zbl 0135.45102 |
| [18] | Biswas, D.; Banerjee, T., Time-delayed chaotic dynamical systems (2018), Springer International Publishing · Zbl 1397.34001 |
| [19] | Banerjee, T.; Biswas, D.; Sarkar, B. C., Design and analysis of a first order time-delayed chaotic system. Nonlinear Dyn, 721-734 (2012) |
| [20] | Maza, D.; Mancini, H.; Boccaletti, S.; Genesio, R.; Arecchi, T., Control of amplitude turbulence in delayed dynamical systems. Int J Bifurcation Chaos, 1843-1848 (1998) · Zbl 0982.76523 |
| [21] | You, P. L.; Lin, L. F.; Wang, H. Q., Cooperative mechanism of generalized stochastic resonance in a time-delayed fractional oscillator with random fluctuations on both mass and damping. Chaos (2020) · Zbl 1489.34113 |
| [22] | He, M. J.; Xu, W.; Sun, Z. K., Dynamical complexity and stochastic resonance in a bistable system with time delay. Nonlinear Dyn, 1787-1795 (2017) |
| [23] | Shao, R. H.; Chen, Y., Stochastic resonance in time-delayed bistable systems driven by weak periodic signal. Physica A, 977-983 (2009) |
| [24] | Zhong, S. C.; Zhang, L.; Wang, H. Q.; Ma, H.; Luo, M. K., Nonlinear effect of time delay on the generalized stochastic resonance in a fractional oscillator with multiplicative polynomial noise. Nonlinear Dyn, 1327-1340 (2017) |
| [25] | Atsumi, Y.; Hata, H.; Nakao, H., Phase ordering in coupled noisy bistable systems on scale-free networks. Phys Rev E (2013) |
| [26] | Tang, Y.; Zou, W.; Lu, J.; Kurths, J., Stochastic resonance in an ensemble of bistable systems under stable distribution noises and nonhomogeneous coupling. Phys Rev E (2012) |
| [27] | Cubero, D., Finite-size fluctuations and stochastic resonance in globally coupled bistable systems. Phys Rev E (2008) |
| [28] | Nicolis, C.; Nicolis, G., Coupling-enhanced stochastic resonance. Phys Rev E (2017) |
| [29] | Yu, T.; Zhang, L.; Zhong, S.; Lai, L., The resonance behavior in two coupled harmonic oscillators with fluctuating mass. Nonlinear Dyn, 1735-1745 (2019) · Zbl 1437.70032 |
| [30] | Morillo, M.; Goemez-Ordoeez, J.; Casado, J., System size stochastic resonance in driven finite arrays of coupled bistable elements. Eur Phys J, 2010, 211-215 (2010) |
| [31] | Yu, T.; Zhang, L.; Ji, Y. D.; Lai, L., Stochastic resonance of two coupled fractional harmonic oscillators with fluctuating mass. Commun Nonlinear Sci Numer Simul, 26-38 (2019) · Zbl 1464.34083 |
| [32] | Zhang, L.; Xu, L.; Yu, T.; Lai, L.; Zhong, S. C., Collective behavior of a nearest neighbor coupled system in a dichotomous fluctuating potential. Commun Nonlinear Sci Numer Simul (2020) |
| [33] | Jiang, L.; Lai, L.; Yu, T.; Luo, M. K., Collective behaviors of globally coupled harmonic oscillators driven by different frequency fluctuations. Acta Phys Sin (2021) |
| [34] | Zhan, M.; Wang, X. G.; Gong, X. F.; Wei, G. W.; Lai, C.-H., Complete synchronization and generalized synchronization of one-way coupled time-delay systems. Phys Rev E (2003) |
| [35] | Joerg, D. J.; Morelli, L. G.; Ares, S.; Juelicher, F., Synchronization dynamics in the presence of coupling delays and phase shifts. Phys Rev Lett (2014) |
| [36] | Sun, X. J.; Matjaz̆, . P.; Jürgen, K., Effects of partial time delays on phase synchronization in Watts-Strogatz small-world neuronal networks. Chaos (2017) |
| [37] | Shi, H. J.; Miao, L. Y.; Sun, Y. Z., Synchronization of time-delay systems with discontinuous coupling. Kybernetika, 765-779 (2017) · Zbl 1449.34258 |
| [38] | Ali, M.; Usha, M.; Alsaedi, A.; Ahmad, B., Synchronization of stochastic complex dynamical networks with mixed time-varying coupling delays. Neural Pro Lett, 1233-1250 (2020) |
| [39] | Banerjee, T.; Biswas, D., Synchronization in hyperchaotic time-delayed electronic oscillators coupled indirectly via a common environment. Nonlinear Dyn, 2025-2048 (2013) |
| [40] | Banerjee, T.; Biswas, D.; Sarkar, B. C., Complete and generalized synchronization of chaos and hyperchaos in a coupled first-order time-delayed system. Nonlinear Dyn, 279-290 (2013) |
| [41] | Banerjee, T.; Biswas, D.; Sarkar, B. C., Anticipatory, complete and lag synchronization of chaos and hyperchaos in a nonlinear delay-coupled time-delayed system. Nonlinear Dyn, 321-332 (2013) |
| [42] | Shapiro, V. E.; Loginov, V. M., Formulas of differentiation and their use for solving stochastic equations. Physica A, 563-574 (1978) |
| [43] | Kim, C.; Lee, E. K.; Talkner, P., Numerical method for solving stochastic differential equations with dichotomous noise. Phys Rev E (2006) |
| [44] | Thowsen, A., The Routh-Hurwitz method for stability determination of linear differential-difference systems. Internat J Control, 991-995 (1981) · Zbl 0474.93051 |
| [45] | Sugiyama, S., On existence of periodic solutions of difference-differential equations. Proc Japan Acad, 179 (1961) |
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.
