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k1
BA
(d)(c)
(b)(a)
k1k1k1
k3
k2
k2
k3
k1k1
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k3
BA
k1k2
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in real time
system
in advance
parameter
control state
optimality criteria
runtime parameters
generation
trajectory
controller
on-line
information
region and parameter
recipe
control objectives
system model (ODE)
tolerance
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[origin destination] pair
(synthesis)(analysis)
stack of state-space portraits
tolerance
first passonly
dynamicsclassifier
tolerancechecker
MAPPER PATH FINDER
model
region information (R, m)
parameter information (range, step)
control objectives (A, B)
iteration depthparameter infooptimality procedures
model
FIRST
PASS
SETUP
A
B C
trajectory length and step
region corners and grid spacing
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o
second pass
first pass
d
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orig
dest
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A B
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u
theta
--
motor
velocitysense
positionsense
g1 g2
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(c)
Re
Im
0a2
a2 = 0
2
(a)
a2 = (a1/2)
(a1/2)2
a2
(b)
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different from
range?
k out of
previous portrait?
]
section 4.4
section 4.3
section 4.2
k by
increment k
hl
[A, B]; corners, m; ODE model;t, length;
generate
trajectories
dynamicsclassify
trajectories
discretize
stack of
portraits
average k
and old k
Y
N
N
Y
Y
N
Di
D
k; [k k
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(0 0) (3 0)
(3 3)
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3
parameter scale
d
d
d
bifurcation tochaotic attractor
chaotic attractorapproaches objectives
limit cycle
pass
number
1
1
1
1
1
2
2
3
4
4
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[A B]
Y
section 5.3
section 5.4
section 5.2
find best
segment
splice intopath and check
tolerance
compute new
region info
invoke
mapper
pass?
junction info
compute
recipe
tolerance
set of mapsoptim. cost
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a
b
(a)
[h, b][d, g][f, c][a, e]
[d, b][a, c]
[a, b]
(b)
= path finder invocation
a
b
c
d
e
f
g
h
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dest
orig
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(3 3)
(0, 3)
(3 0)
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xx x
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box of [A, B] by
overrange factor
t; trajectory length;
expand bounding
search region corners
B
A
]hl k [k k
;00
mRODE model;
trajectory length
timestep and
determine
[A, B]; tolerance;
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(b)(a)
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x(0)
+ Ix(0)
D
x(0)
x(t)
(b)(a)
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(c)(a) (b)
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T
L
T’
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(b) (c)(a)
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r3 = 10
(8.82, -4.37, 57.44)
(8.8230, -4.3716, 57.2139)
S3(8.10, 29.3, 63.7)
r1 = 40
r0 = 50
S0
S1
(8, 29, 64)
A(-24.5, -20, 68)
B
r2 = 10S2
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frequency
1)(per
A xxxx
x
x
(per = 1)periodic
x
20
3
2
15105
1
drive
drive
periodic chaotic
amplitude
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amplitude magnitude
1
1
T-T time
fre
w 2woo
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range
VCO input voltage
phase
difference
lock range
capture
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7/31/2019 Taming Chaotic Circuits - 1992 Elizabeth Bradley
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7/31/2019 Taming Chaotic Circuits - 1992 Elizabeth Bradley
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7/31/2019 Taming Chaotic Circuits - 1992 Elizabeth Bradley
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7/31/2019 Taming Chaotic Circuits - 1992 Elizabeth Bradley
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