Experimental and numerical observation of dark and bright breathers in the band gap of a diatomic electrical lattice

We observe dark and bright intrinsic localized modes (ILMs), also known as discrete breathers, experimentally and numerically in a diatomic-like electrical lattice. The experimental generation of dark ILMs by driving a dissipative lattice with spatially homogenous amplitude is, to our knowledge, unprecedented. In addition, the experimental manifestation of bright breathers within the band gap is also novel in this system. In experimental measurements the dark modes appear just below the bottom of the top branch in frequency. As the frequency is then lowered further into the band gap, the dark ILMs persist, until the nonlinear localization pattern reverses and bright ILMs appear on top of the finite background. Deep into the band gap, only a single bright structure survives in a lattice of 32 nodes. The vicinity of the bottom band also features bright and dark self-localized excitations. These results pave the way for a more systematic study of dark breathers and their bifurcations in diatomic-like chains.

Figure 1

(a) Schematic circuit diagrams of the electrical lattice, where the blank points represents circuit cells of type $\left(a\right)$ and black points represent circuit cells of type $\left(b\right)$. Black arrows represent driving and its phase. (b) Each cell is connected to a periodic voltage source $V_s\left(t\right)$ via a resistor $R$ and grounded. Each point $A$ of an elemental circuit is connected via inductors $L_1$ to the corresponding points $A$ of neighboring cell. Voltages are monitored at point $A$.

Figure 2

(a) Electrical line linear mode frequencies $f$ as function of the wave vector $q=n\pi /N$, where $n=-N/2...N/2$ and $N=N^{\left(a\right)}+N^{\left(b\right)}$, where $N^{\left(a\right)}$ is the number of cells of type $\left(a\right)$, and likewise for $\left(b\right)$. A solid (dashed) line is used to denote the upper (lower) band. Panels (b) and (c) show the ratios between different linear mode amplitudes in the bands, where $A$ correspond to oscillation amplitudes of cells of type $\left(a\right)$ and $B$ to cells of type $\left(b\right)$.

Figure 3

Sketch of the linear modes corresponding to the top and the bottom of the bands shown in Fig. 2. (a) Uniform linear mode corresponding to the bottom of the lower band. (b) Linear mode corresponding to the top of the lower band, where the $L_2^{\left(b\right)}$ sublattice oscillates out of phase and the other sublattice is at rest. (c) Linear mode corresponding to the bottom of the upper band, where the $L_2^{\left(a\right)}$ sublattice oscillates out of phase, and the other sublattice is at rest. (d) Linear mode corresponding to the top of the upper band, where all neighboring cells oscillate out of phase.

Figure 4

(a) Numerical (dotted line) and experimental (circles) DB profile (maximum and minimum amplitude) corresponding to $V_d=3.5$ V and $f=571$ kHz. Blank points represent circuit cells of type $\left(a\right)$ and black points represent circuit cells of type $\left(b\right)$. (b) Floquet multiplier numerical linearization spectrum confirming that, since all multipliers are inside the unit circle, the solution is stable.

Figure 5

(a) Experimental and (b) numerical voltage oscillations of the DB corresponding to $V_d=3.5$ V and $f=571$ kHz: site 16 (black circles), first-neighbor sites 15 (blue diamond), and 17 (red dots), and second neighbors: sites 14 (green crosses) and 18 (yellow pluses).

Figure 6

(a) Numerical (dotted line) and experimental (circles) three–peak DB profile (maximum and minimum amplitude) corresponding to $V_d=3.5$ V and $f=567$ kHz. Blank points represent circuit cells of type $\left(a\right)$ and black points represent circuit cells of type $\left(b\right)$. (b) Floquet multiplier numerical linearization spectrum confirming the stability of the solution since all multipliers are inside the unit circle.

Figure 7

(a) Numerical (dotted line) and experimental (circles) three-peak bright breather profile (maximum and minimum amplitude) corresponding to $V_d=3.5$ V and $f=532$ kHz. Blank points represent circuit cells of type $\left(a\right)$ and black points represent circuit cells of type $\left(b\right)$. (b) Floquet multiplier numerical linearization spectrum confirming the stability of the solution.

Figure 8

(a) Numerical (dotted line) and experimental (circles) one-peak bright breather profile (maximum and minimum amplitude) corresponding to $V_d=3.5$ V and $f=528$ kHz. Blank points represent circuit cells of type $\left(a\right)$ and black points represent circuit cells of type $\left(b\right)$. (b) Floquet multiplier numerical linearization spectrum confirming, similarly as above, the stability of the solution.

Figure 9

Summary of the progression from dark to bright multipeak ILM structures. Panels (a) show the driver frequency as a function of the number of breathers, both dark (circles, red) and bright (squares, blue). At around 550 kHz, we observe an equal mixture of bright and dark breathers. Panels (b)–(d) show the experimental data corresponding to select frequencies.

Figure 10

(a) Numerical (dotted line) and experimental (circles) DB profile (maximum and minimum amplitude) corresponding to $V_d=3.5$ V and $f=280$ kHz. Blank points represent circuit cells of type $\left(a\right)$ and black points represent circuit cells of type $\left(b\right)$. (b) Floquet multiplier numerical linearization spectrum confirming the stability of the solution.

Figure 11

Experimental FSK modulation that switches abruptly between two distinct values of the driving frequency while keeping the amplitude fixed at 3.5 V. Initially, for $f=271$ kHz, two breathers are seen, but then when the frequency switches to 543 kHz, two distinct ILMs at a different location with the latter frequency (centered on odd sites) develop.