experimental generation of intrinsic localized modes in electrical lattices lars q. english...
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Experimental generation of intrinsic localized modes in electrical lattices
Lars Q. EnglishDepartment of Physics & Astronomy
Dickinson College, Carlisle, PA
(movie: 268 kHz driver frequency)
LENCOS 2009, Seville, Spain
Macroscopic lattices Experiments on electrical transmission line
1 unit cell
Fast 24-channel digitizer(now 32 channels)
Lattice
3
031
2
021
0
0
0
0
1
1ln1
)exp(1
)exp()(
Q
Q
Q
Q
Q
Q
Q
QV
VC
Q
VCVC
Energy Localization in a Nonlinear Electrical Lattice The experimental system
Reverse-biased diode
0
100
200
300
400
500
600
0 2 4 6 8 10
Voltage [V]
Cap
acit
an
ce [
pF
]
np
Charge density
x
21
00 )(
1
V
dd
.1
11
23
22
2
32
2
000
31
31
320
21
21
21
2
0
21
1121
320
202
0
202
02
2
dt
dV
Rdt
dQ
Q
Q
Q
Q
RC
QQQQ
QQQQ
QQQ
Qdt
Qd
dnnn
nnn
nnn
nnn
nnnn
CL20 1 1 11 CL 00 CQ
Duffing Oscillators
Spatially uniformdriving term
Inter-site nonlinearity
NN - coupling
Energy Localization in a Nonlinear Electrical Lattice
Soft-nonlinearity:Quadratic term dominates
cubic term
Uniform mode coupled to uniformMode (k=0)
Linear dispersioncurve; no driver/damping ILM
540 kHz
320 kHz
Energy Localization in a Nonlinear Electrical Lattice Observation of Discrete breathers /ILMs
fd=300 kHzA=1.5V
fd=268 kHz
Energy Localization in a Nonlinear Electrical Lattice Parametric excitation / stabilization of ILMs
Red: Node 13Blue: Node 17Black: Node 19
fd=590 kHzA=4.5V
Energy Localization in a Nonlinear Electrical Lattice Slight change in experimental system
One Unit Cell
Observation of Traveling ILMs
Energy Localization in a Nonlinear Electrical Lattice
Response to uniform driving below the dispersion curve
Energy Localization in a Nonlinear Electrical Lattice
Energy Localization in a Nonlinear Electrical Lattice
Modulational Instability
Incoherent Localized Structures
Coherent structure locked to driver
Patterns of Traveling ILMs
Energy Localization in a Nonlinear Electrical Lattice
3 ILMs 4 ILMs
3 ILMs 6 ILMs
464 kHz 484 kHz
464 kHz 502 kHz
Patterns of Traveling ILMs
Energy Localization in a Nonlinear Electrical Lattice
Driver frequency in between regions of integer-number of ILM(here between 3 and 4 ILMs)
Patterns of Traveling ILMs
(472 kHz) (475 kHz)
Energy Localization in a Nonlinear Electrical Lattice
386 kHz
Switching in a blocking capacitor
cap turned on
Experiments on 2D macroscopic lattice
Preliminary results: We believe we have observed 2D localization
Energy Localization in a Nonlinear Electrical Lattice
Energy Localization in a Nonlinear Electrical Lattice
ILM
Preliminary 2d-lattice data
Node
N
ode
Energy Localization in a Nonlinear Electrical Lattice
Conclusions
• Demonstrated the existence of ILMs/discrete breathers in an electrical lattice
• Fast, multichannel electronic data acquisition makes possible a detailed study of ILM profile and behavior (not possible in other systems)
• Both stationary and slow-moving ILM can be excited in these lattices (depending on the detailed electronic make-up of the unit-cell )
• These ILMs can be generated via modulational instability, parametric instability, or by briefly switching in an impurity.
• Multiple localized features, multi-pulses, can be locked to the driver at higher driver frequencies.