emergent complexity chaos and fractals. uncertain dynamical systems c-plane
TRANSCRIPT
Emergent complexity
Chaos and fractals
Uncertain Dynamical Systems
c-plane
1 1k k kz cz z
0c
planec
bounded open and connectedkc Mset z
bounded is "Cantor dust"kc Mset z
1 1k k kz cz z c i
boundedkc Mset z
planez
1 1k k kz cz z
Julia sets
1 1k k kz cz z 0c
bounded open and connectedkc Mset z
Undecidable No algorithm to
determine has bounded run timec Mset
Overcoming computational
complexity
1 1 ,k k kz cz z z c c
boundedkc Mset z
What does this have to do with complex systems?
• This classic computational problem illustrates an important idea, but in an easily visualized way.
• Most computational problems involve uncertain dynamical systems, from protein folding to complex network analysis. Not easily visualized.
• Natural questions are typically computationally intractable, and conventional methods provide little encouragement that this can be systematically overcome.
Main idea
1
"Fragile" means
membership changes when
the map is perturbed:
1k k kz c z z
e.g. the boundary moves.
Main idea
Points near the boundary are “fragile.”
Merely stating the obvious in this case.
But illustrates general principle that can be exploited by the right
algorithms.
5
10
15
20
25
30
# iterations
Points not in M.
5
10
15
20
25
30
# iterations
Color indicates number of
iterations of simulation to show point is not in M.
But simulation cannot show that points are in M.
1 1 k k kz c z z
boundedkc Mset z
planec
1 1k k kz cz z
But simulation is fundamentally limited• Gridding is not scalable• Finite simulation inconclusive
0
1
2z
It’s easy to prove that this disk is in M.
Other points in M are fragile to the definition of the map.
1 1 k k kz c z z
Merely stating the obvious.
Main idea
1 1k k kz cz z
planec
2
1
221 0
1 1
k k
k k k
k
V z z
V z V z
z cz z
c z
2 decreases
1 1
V z z
c z
Sufficient condition
1 1k k kz cz z
2 decreases
1 1
V z z
c z
21
2 11 1 0 1z
2c
1c Mset
Trivial to prove that these points are in Mandelbrot set.
1c 2 1c
Main idea
The longer the proof, the more fragile the remaining regions.
The proof of this region is a bit longer.
Main idea
And so on…
Proof even longer.
Easy to prove these points are in Mset.
Easy to prove these points are not in Mset.
Proofs get harder.(But all still “easy.”)
What’s left gets more fragile.
Complexity»Chaos
Fractals
Emergent complexity.
Complexityimplies fragility
What matters to organized
complexity.
Emergent complexity
cHow might this help
with organized complexity and
“robust yet fragile”?
• Long proofs indicate a fragility.• Either a true fragility (a useful answer) or an
artifact of the model (which must then be rectified)• Potentially fundamentally changes computational
complexity for organized complexity• Brings back together two research areas that have
been separated for decades:• Numerical analysis and ill-conditioning• Computational complexity (P, NP/coNP,
undecidable)
Proof?
New proof methods that is scalable and systematic (can be
automated).
Breaking hard problems
• SOSTOOLS proof theory and software• Nested family of (dual) proof algorithms• Each family is polynomial time• Recovers most “gold standard” algorithms
as special cases, and immediately improves• No a priori polynomial bound on depth
(otherwise P=NP=coNP)• Conjecture: Complexity implies fragility
Safety Verification and Reachability Analysis
• Safety critical applications.• Exhaustive simulation is not exact.• Set propagation is computationally expensive.
Find a barrier certificate B(x)
( ) 0
( ) 0
( ) 0
B x x UnsafeSet
B x x InitialSet
Bf x x StateSpace
x
Initial set
Unsafe set
B(x) = 0
Scalable computation using SOS machinery.
• Parametric• Memoryless• Dynamic (IQC)
Hybrid, Uncertain, Stochastic
Hybrid systems can be handled easily,even for systems with uncertainty:
• Use supermartingales as certificates.• Get guaranteed bound on reach probability.
Also stochastic hybrid systems:
(Prajna, Jadbabaie – HSCC ’04)
(Prajna, Jadbabaie, Pappas – CDC ’04)
Feedback control
Variable supply/demand
Physical network
Components
Functionalrequirements
Hardware constraints
“Horizontal” Decompositions
“Ver
tica
l”
laye
ring
Unifying role of dual proofs and decomp-ositions
Main idea
Think of this as a robustness problem.
1 1
0k k kz cz z
0c
planec
How robust is stability to perturbations in c?
Globally stable.
1 1k k kz cz z
0c
planec
Region of convergence.
How robust is stability to perturbations in c?
1 1k k kz cz z planec planez
planec 1 1k k kz cz z planez
Simulation is fundamentally limited
planec 1 1k k kz cz z
planez
Simulation is fundamentally limited
0
1
2z
5
10
15
20
25
30
# iterations
10
20
30
40
50
60
iterations
10
20
30
40
50
60
iterations
-2
10
20
30
40
50
60
iterations
-2
10
20
30
40
50
60
iterations
10
20
30
40
50
60
iterations
70
120
iterations
120
iterations
60
1 1k k kz cz z
1 1k k kx cx x
realc
realx
1 1k k kx cx x
realc
kx kx
0c 0c
1kx 1kx
realc
realc
1 1k k kx cx x
realx
-3 -2 -1 0 1 2 3-2
-1
0
1
2
3
4
1 1k k kx cx x
realc
1
10,1
x cx x
xc
realx
Fixed points
StableUnstable
Stable
-3 -2 -1 0 1 2 3-2
-1
0
1
2
3
4
1x cx x realc
realx
Unstable
Stable
Stable
Fixed points
-3 -2 -1 0 1 2 3-2
-1
0
1
2
3
4
1x cx x realc
realx
Unstable
Stable
Equilibria
-3 -2 -1 0 1 2 3-2
-1
0
1
2
3
4
1 1k k kx cx x
realc
realx
2
1
221 0
1 1
k k
k k k
k
V x x
V x V x
x cx x
c x
2 decreases
1 1
V z z
c z
1 1k k kz cz z
planec
2
1
221 0
1 1
k k
k k k
k
V z z
V z V z
z cz z
c z
2 decreases
1 1
V z z
c z
Sufficient condition
1 1k k kz cz z
2 decreases
1 1
V z z
c z
21
2 11 1 0 1z
2c
1c Mset
Trivial to prove that these points are in Mandelbrot set.
1c 2 1c
-3 -2 -1 0 1 2 3-2
-1
0
1
2
3
4
1x cx x realc
realx
Unstable
Equilibria
realc
Bifurcations for 1 1k k kx cx x
c
“last 200 x”
“last 200 x”
c
Zoom-in
“last 200 x”
c
Zoom-in
-3 -2 -1 0 1 2 3
-2
-1
0
1 1k k kx cx x 0 realc
Bounded for -2< 0 realc
stable
2 2 2
2 2 221 1 1 22 1 1 2 1 2
2 2x x c x c
c c
Bifurcations to chaos
-2
-1
0
-3 -2 -1 0 1 2 3-2
-1
0
1
2
1
221 0
1 1
k k
k k k
k
V x x
V x V x
x cx x
c x
2 decreases
1 1
V z z
c z
-2
-1
0
-3 -2 -1 0 1 2 3-2
-1
0
1
2
1
221 0
1 1
k k
k k k
k
V x x
V x V x
x cx x
c x
1 1
1 1 1 ?
k
k k
c x
c cx x
-3 -2 -1 0 1 2 3-2
-1
0
1
2
3
4
-3 -2 -1 0 1 2 3-2
-1
0
1
2
3
4
1 1k k kx cx x
realc
realx
2
1
221 0
1 1
k k
k k k
k
V x x
V x V x
x cx x
c x
2 decreases
1 1
V z z
c z
2 2 22 2 221 1 1 2
2 1 1 2 1 22 2
2 0 2 0
x x c x cc c
c c c
-3 -2 -1 0 1 2 3
-2
-1
0
1 1k k kx cx x
Bounded for - 2 0 real?c
stable
Bifurcations to chaos
2 22 2 1 2
2 0
c x c
c c
1 1k k kx cx x Invariant set?
2 22 2 1 1 2c cx x c
Invariance
2 22
2 222 1 22 1 1 2
2 0
c x cc cx x c
c c
2 22
2 222 1 2, 2 1 1 2 ?
2 0
c x cc cx x c
c c
-3 -2 -1 0 1 2 3
-2
-1
0
1 1k k kx cx x
Bifurcations to chaos
Special case of SOS
Special case of SOS
Special case of SOS
Special case of SOS
Contradiction!
-3 -2 -1 0 1 2 3
-2
-1
0
1 1k k kx cx x
2 22
2 222 1 2, 2 1 1 2 ?
2 0
c x cc cx x c
c c
What is the shortest proof possible?
Can prove the whole yellow region using SOSTOOLS!
1c 2 1c
Lyapunov argument
How to prove membership?2-period lobes
12
3
3
4
4
4Proof
lengths
Prove membership of 2-period lobe:• Using a stability argument of the 2-period map.• Using an invariance argument.
-1
Formulate the invarianceProblem as the emptinessof a semialgebraic set.
Then use SOSTOOLS to construct the certificate.
-6 -4 -2 0 2 4 6-6
-4
-2
0
2
4
6
x1
x 2
1 1
1 1 2
2 32 1 20.1 2 0.1
x x x
x x x x x
Discrete → Continuous
Let 0 be an equilibrium of
( ), .
Let be a region containing 0 and let
: be a continuously differentiable
function such that
( ) 0 in
nx f x x
D
V D
V x D
V
( ) ( ) 0 in
Then 0 is asymptotically stable.
Vx f x D
x
Lyapunov’s theorem
( ) 0 in
( ) ( ) 0 in
V x D
VV x f x D
x
-6 -4 -2 0 2 4 6-6
-4
-2
0
2
4
6
x1
x 2
Can we test these conditions
algorithmically?
Use the Sum of Squares decomposition!
( ) 0 in
( ) ( ) 0 in
V x D
VV x f x D
x
| ( ) 0, 1, ,niD x g x i n
( ), SOS, SOS, ( ) 0, ( ) 0i iV x p q x x
1
( ) ( )N
i ii
V x x p g
1
( ) ( )N
i ii
V x x q g
is SOS
is SOS
Find
such that
SOSTOOLS
( ) 0 in
( ) ( ) 0 in
V x D
VV x f x D
x
Then equilibrium is asymptotically stable.
( , ), ,(0, ) 0
( , ) 0 for ,
( , ) 0 for ,
x f x p x D p Pf p
V x p x D p P
V x p x D p P
Robust Stability?
Describe both and as semilalgebraic sets.D P
Use SOSTOOLS to construct V(x,p).
www.cds.caltech.edu/sostools
Chemical oscillator? ,
2 3constant
X A A BX Y X
Y B
2
2
x a x x y
y b x y
Nondimensional state equations
2
2
x a x x y
y b x y
3
Limit cycle for
b a b a
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
a
b
Can be computed analytically, which is not scalable.
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
a
b
0 0.1 0.2 0.3 0.4 0.5 0.6
1
1.5
2
2.5
3
a = 0.1, b = 0.13
2
2
x a x x y
y b x y
Numerical simulation.
1 1.5 2 2.50
0.2
0.4
0.6
0.8
1
x
y
2.2 2.6 3 3.40
0.2
0.4
0.6
0.8
1
a = 1, b = 2
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
a
b
a = 0.6, b = 1.1
(1.1, 0.6) (2, 1)
2
2
x a x x y
y b x y
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
a
b
a a
b b
,b a
2
2
x a x x y
y b x y
2
2
0
0
a x x y
b x y
equilibrium
2
2
x a x x x x y y
y b x x y y
2 2
2 2
x x
y y
a a
b b
Search for ( , ) satisfying
the Lyapunov conditions using
SOSTOOLS.
V x y
0 0.2 0.4 0.6 0.8 1 1.2 1.40
0.2
0.4
0.6
0.8
1
a
b
,b a
4 order ( , ) th V x y
Modeling Analysis
Set of possible system behaviors
Set of bad system behaviors
Proof of robustness
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