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Bifurcation Results for Traveling Waves in Nonlinear Magnetic Metamaterials

M. Agaoglou, V. M. Rothos, D. J Fratzeskakis, G. P. Veldes, H. Susanto

TL;DR

It is shown that, under certain conditions, the presence of dissipation and/or driving may stabilize or destabilize the solutions of a one-dimensional magnetic metamaterial formed by a discrete array of nonlinear resonators.

Abstract

In this work, 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.

Bifurcation Results for Traveling Waves in Nonlinear Magnetic Metamaterials

TL;DR

It is shown that, under certain conditions, the presence of dissipation and/or driving may stabilize or destabilize the solutions of a one-dimensional magnetic metamaterial formed by a discrete array of nonlinear resonators.

Abstract

In this work, 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.
Paper Structure (9 sections, 2 theorems, 48 equations, 5 figures)

This paper contains 9 sections, 2 theorems, 48 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The nonlinear magnetic metamaterial model
  3. Bifurcation results for Travelling Waves
  4. Case 1: periodic travelling waves
  5. Case 2: localized travelling waves
  6. Numerical results
  7. Persistence and stability of traveling waves of the unperturbed system
  8. Periodic waves due to drives
  9. Conclusion

Key Result

Theorem 1

Suppose $(H1)$ and $(H2)$. If there is a simple zero $a_{0}$ of a Melnikov function, i.e $M^{u/v}$ =0 and $D_{a} M^{u/v} (a_{0})\neq 0$, then there is a $\delta >0$ such that for any $0\neq \varepsilon \in (-\delta,\delta)$ there is a unique $2\pi u-$periodic solution $U(z)$ of (3) with

Figures (5)

  • Figure 1: Top panel: A representative illustration of the nonlinear magnetic metamaterial structure. Bottom panel: the equivalent circuit model (see text for details).
  • Figure 2: (a) A periodic wave in a traveling frame. (b)-(c) Floquet multipliers of the solution corresponding to panel (a) for parameter values as given in the captions. The solution period in all the plots is $\overline{T}=4\pi$. As a guide we also plot the unit circle in thin solid curve. The stability is calculated using 20 sites.
  • Figure 3: (a) Bifurcation diagram of the solution corresponding to Fig. \ref{['fig1']}(a) with $\Delta=7\times10^{-4}$ and $\lambda=10^{-4}$. The vertical axis is the solution norm $N=\sqrt{\int_0^{\overline{T}}U(z)^2\,dz}$. Solid and dashed curve indicate unstable and stable solutions, respectively. The stability is calculated using $20$ sites. The vertical dashed line is the prediction (\ref{['rm2']}). (b) Typical dynamics of the instability. The initial condition is the solution obtained immediately after the first bifurcation point in (a).
  • Figure 4: (a-b) The existence curve of periodic waves due to the traveling drive for (a) varying $\Delta$ with $\gamma=0$ and $\lambda=0.1$; (b) varying $\gamma$ with $\Delta=0.5$ and $\lambda=0.1$. The vertical axis is the solution norm $N=\sqrt{\int_0^{\overline{T}}U^2(z)\,dz}$. Solid and dashed curve indicate unstable and stable solutions, respectively. The stability is calculated using $20$ sites. (c) The solution profile with $\Delta=0.5,\,\lambda=0.1,$ and $\gamma=0.57$. (d) The Floquet multipliers of the solution in panel (c), showing the instability of the solution.
  • Figure 5: Typical dynamics of the instability of a periodic wave due to the drive. The initial condition corresponds to the solution plotted in Fig. \ref{['fig3']}(c).

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2