Agaoglou, M.; Rothos, V. M.; Frantzeskakis, D. J.; Veldes, G. P.; Susanto, H. Bifurcation results for traveling waves in nonlinear magnetic metamaterials. (English) Zbl 1304.34084 Int. J. Bifurcation Chaos Appl. Sci. Eng. 24, No. 11, Article ID 1450147, 12 p. (2014). Summary: We study a model of a one-dimensional magnetic metamaterial formed by a discrete array of nonlinear resonators. We focus on periodic and localized traveling waves of the model, in the presence of loss and an external drive. Employing a Melnikov analysis we study the existence and persistence of such traveling waves, and study their linear stability. We show that, under certain conditions, the presence of dissipation and/or driving may stabilize or destabilize the solutions. Our analytical results are found to be in good agreement with direct numerical computations. Cited in 2 Documents MSC: 34C60 Qualitative investigation and simulation of ordinary differential equation models 78A48 Composite media; random media in optics and electromagnetic theory 34B40 Boundary value problems on infinite intervals for ordinary differential equations Keywords:resonator; metamaterials; periodic waves; localized waves; Melnikov function PDFBibTeX XMLCite \textit{M. Agaoglou} et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 24, No. 11, Article ID 1450147, 12 p. (2014; Zbl 1304.34084) Full Text: DOI References: [1] DOI: 10.1007/978-3-642-52803-3 · doi:10.1007/978-3-642-52803-3 [2] DOI: 10.1002/mop.23122 · doi:10.1002/mop.23122 [3] DOI: 10.1103/PhysRevE.80.036608 · doi:10.1103/PhysRevE.80.036608 [4] Diblik J., Nonlinear Systems and Complexity 7, in: Localized Excitations in Nonlinear Complex Systems (2014) · Zbl 1327.82096 · doi:10.1007/978-3-319-02057-0_17 [5] DOI: 10.1016/j.physleta.2011.01.042 · doi:10.1016/j.physleta.2011.01.042 [6] DOI: 10.1063/1.1814428 · doi:10.1063/1.1814428 [7] DOI: 10.1063/1.2828176 · doi:10.1063/1.2828176 [8] DOI: 10.1126/science.1177031 · doi:10.1126/science.1177031 [9] DOI: 10.1063/1.3356223 · doi:10.1063/1.3356223 [10] Jackson J. D., Classical Electrodynamics (1999) · Zbl 0920.00012 [11] DOI: 10.1103/PhysRevE.75.067601 · doi:10.1103/PhysRevE.75.067601 [12] DOI: 10.1088/0022-3727/41/17/173001 · doi:10.1088/0022-3727/41/17/173001 [13] DOI: 10.1103/PhysRevE.67.065601 · doi:10.1103/PhysRevE.67.065601 [14] DOI: 10.1103/PhysRevLett.97.157406 · doi:10.1103/PhysRevLett.97.157406 [15] DOI: 10.1142/S0218127411029689 · Zbl 1248.78013 · doi:10.1142/S0218127411029689 [16] Marques R., Metamaterials with Negative Parameters. Theory, Design, and Microwave Applications (2008) [17] DOI: 10.1002/mop.20735 · doi:10.1002/mop.20735 [18] DOI: 10.1587/elex.7.608 · doi:10.1587/elex.7.608 [19] DOI: 10.1364/OE.14.009344 · doi:10.1364/OE.14.009344 [20] DOI: 10.1016/j.photonics.2006.01.005 · doi:10.1016/j.photonics.2006.01.005 [21] DOI: 10.1049/el:20020258 · doi:10.1049/el:20020258 [22] DOI: 10.1063/1.1510945 · doi:10.1063/1.1510945 [23] DOI: 10.1088/0022-3727/37/3/008 · doi:10.1088/0022-3727/37/3/008 [24] DOI: 10.1103/PhysRevE.81.026207 · doi:10.1103/PhysRevE.81.026207 [25] DOI: 10.1088/0022-3727/40/22/004 · doi:10.1088/0022-3727/40/22/004 [26] DOI: 10.1103/PhysRevE.74.067601 · doi:10.1103/PhysRevE.74.067601 [27] DOI: 10.1103/PhysRevB.87.195123 · doi:10.1103/PhysRevB.87.195123 [28] DOI: 10.1103/PhysRevLett.91.037401 · doi:10.1103/PhysRevLett.91.037401 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.