BIFURCATION ANALYSIS OF PARAMETRIC INSTABILITY OF WAVES IN THE 3D ARRAYS
OF MAGNETIC NANOPARTICLE-FILLED CARBON NANOTUBES AT MICROWAVE FREQUENCY RANGE 

Makeeva Galina Stepanovna, Doctor of physical and mathematical sciences, professor, sub-department of radio engineering and radio electronic systems, Penza State University (40 Krasnaya street, Penza, Russia), radiotech@pnzgu.ru
Golovanov Oleg Aleksandrovich, Doctor of physical and mathematical sciences, professor, sub-department of general professional disciplines, Penza branch of the Military Academy of Maintenance Supplies (Penza-5, Russia), golovanovol@mail.ru 

537.874.6 

Background. Investigation of electromagnetic properties of the anisotropic nanostructured materials based on the 3D arrays of magnetic nanoparticle-filled carbon nanotubes is complicated by the presence of nonlinear interactions of electromagnetic waves with magnetostatic and spin waves. The goal of this work is to develop a method of bifurcation analysis and to model parametric interactions of different modes in the nanocomposites based on the 3D arrays of magnetic nanoparticle- filled carbon nanotubes: electromagnetic waves and magnetostatic waves, spin waves for calculation of the nonlinearity thresholds of these anisotropic nanostructured materials at microwave frequency range.
Materials and methods. The mathematical model of the parametric instability of waves in the 3D arrays of magnetic nanoparticle-filled carbon nanotubes is based on solution of the nonlinear 3D diffraction boundary problems for the Maxwell equations complemented by the Landau-Lifshitz equation with the exchange term using decomposition onto the autonomous blocks with virtual Floquet channels (FABs). The numerical technique involves finding the bifurcation points of the nonlinear Maxwell operator using the authors’ original computational algorithm for the numerical analysis of the bifurcation points modified to include the Landau-Lifshitz equation with the exchange term and improved by combining it with the qualitative method of analysis based on the Lyapunov stability theory.
Results. By computing the bifurcation points the authors determined the results of electrodynamic calculation of the threshold magnitudes of the pumping electromagnetic wave, where the nonlinear processes and the parametric instability excitation of magnetostatic and spin waves happen in 3D arrays of magnetic nanoparticlefilled carbon nanotubes, at microwave frequency range.
Conclusions. The results of mathematical modeling show that the nonlinear thresholds are reduced and, with increase of the value of bias magnetic field, occur at lower inputs at the regime of parametric excitation of magnetostatic and spin waves in the magnetic nanoparticle system. The developed algorithm allows to carry out
computer analysis and optimization of the nonlinear properties of a new class of anisotropic nanomaterials based on the 3D arrays of magnetic nanoparticle-filled carbon nanotubes and 3D nanodevices tunable by the external magnetic field, at microwave frequencies. 

bifurcation points, nonlinear Maxwell operator, pumping electromagnetic wave, 3D periodic array, magnetic nanoparticle, carbon nanotubes 

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