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Emergent complexity Chaos and fractals

Emergent complexity

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Emergent complexity. Chaos and fractals. c -plane. Uncertain Dynamical Systems. Julia sets. Overcoming computational complexity. What does this have to do with complex systems? . This classic computational problem illustrates an important idea, but in an easily visualized way. - PowerPoint PPT Presentation

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Page 1: Emergent complexity

Emergent complexity

Chaos and fractals

Page 2: Emergent complexity

Uncertain Dynamical Systems

c-plane

Page 3: Emergent complexity

1 1k k kz cz z

0c

planec

bounded open and connectedkc Mset z

bounded is "Cantor dust"kc Mset z

Page 4: Emergent complexity

1 1k k kz cz z c i

boundedkc Mset z

Page 5: Emergent complexity

planez 1 1k k kz cz z

Julia sets

Page 6: Emergent complexity

1 1k k kz cz z 0c

bounded open and connectedkc Mset z

Undecidable No algorithm to determine has bounded run timec Mset

Page 7: Emergent complexity

Overcoming computational

complexity

Page 8: Emergent 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.

Page 9: Emergent complexity

Main idea

1

"Fragile" means membership changes when

the map is perturbed:1k k kz c z z

e.g. the boundary moves.

Page 10: Emergent complexity

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.

Page 11: Emergent complexity

5

10

15

20

25

30

# iterations

Points not in M.

Page 12: Emergent complexity

5

10

15

20

25

30

# iterations

Color indicates number of

iterations of simulation to show point is not in M.

Page 13: Emergent complexity

But simulation cannot show that points are in M.

1 1 k k kz c z z boundedkc Mset z

Page 14: Emergent complexity

planec

1 1k k kz cz z

But simulation is fundamentally limited• Gridding is not scalable• Finite simulation inconclusive

012

z

Page 15: Emergent complexity

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

Page 16: Emergent complexity

1 1k k kz cz z

planec

2

1

22 1 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

Page 17: Emergent complexity

1 1k k kz cz z

2 decreases

1 1

V z z

c z

21

2 11 1 0 1z

2c

1c Mset

Page 18: Emergent complexity

Trivial to prove that these points are in Mandelbrot set.

1c 2 1c

Page 19: Emergent complexity

Main idea

The longer the proof, the more fragile the remaining regions.

The proof of this region is a bit longer.

Page 20: Emergent complexity

Main idea

And so on…

Proof even longer.

Page 21: Emergent complexity
Page 22: Emergent complexity

Easy to prove these points are in Mset.

Page 23: Emergent complexity

Easy to prove these points are not in Mset.

Page 24: Emergent complexity

Proofs get harder.(But all still “easy.”)

What’s left gets more fragile.

Page 25: Emergent complexity

Complexity» Chaos Fractals

Emergent complexity.

Page 26: Emergent complexity

Complexityimplies fragility

What matters to organized

complexity.

Page 27: Emergent complexity

Emergent complexity

Page 28: 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)

Page 29: Emergent complexity

Proof?

New proof methods that is scalable and systematic (can be

automated).

Page 30: Emergent complexity

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

Page 31: Emergent complexity

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 UnsafeSetB x x InitialSet

B f x x StateSpacex

Initial set

Unsafe set

B(x) = 0

Scalable computation using SOS machinery.

Page 32: Emergent complexity

• 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)

Page 33: Emergent complexity

Feedback control

Variable supply/demand

Physical network

Components

Functionalrequirements

Hardware constraints

“Horizontal” Decompositions

“Ver

tical

” la

yerin

g

Unifying role of dual proofs and decomp-ositions

Page 34: Emergent complexity

Main idea

Think of this as a robustness problem.

Page 35: Emergent complexity

1 1

0k k kz cz z

0c

planec

How robust is stability to perturbations in c?

Globally stable.

Page 36: Emergent complexity

1 1k k kz cz z

0c

planec

Region of convergence.

How robust is stability to perturbations in c?

Page 37: Emergent complexity

1 1k k kz cz z planec planez

Page 38: Emergent complexity

planec 1 1k k kz cz z planez

Simulation is fundamentally limited

Page 39: Emergent complexity

planec 1 1k k kz cz z

planez

Simulation is fundamentally limited

012

z

Page 40: Emergent complexity

5

10

15

20

25

30

# iterations

Page 41: Emergent complexity

10

20

30

40

50

60

iterations

Page 42: Emergent complexity

10

20

30

40

50

60

iterations

-2

Page 43: Emergent complexity

10

20

30

40

50

60

iterations

-2

Page 44: Emergent complexity

10

20

30

40

50

60

iterations

Page 45: Emergent complexity

10

20

30

40

50

60

iterations

Page 46: Emergent complexity

70

120

iterations

Page 47: Emergent complexity

120

iterations

60

1 1k k kz cz z

Page 48: Emergent complexity

1 1k k kx cx x

realc

realx

Page 49: Emergent complexity

1 1k k kx cx x

realc

kx kx

0c 0c

1kx 1kx

Page 50: Emergent complexity

realc

Page 51: Emergent complexity

realc

1 1k k kx cx x

realx

Page 52: Emergent complexity

-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

Page 53: Emergent complexity

-3 -2 -1 0 1 2 3-2

-1

0

1

2

3

4 1x cx x realc

realx

Unstable

Stable

Stable

Fixed points

Page 54: Emergent complexity

-3 -2 -1 0 1 2 3-2

-1

0

1

2

3

4 1x cx x realc

realx

Unstable

Stable

Equilibria

Page 55: Emergent complexity

-3 -2 -1 0 1 2 3-2

-1

0

1

2

3

4

1 1k k kx cx x

realc

realx

2

1

22 1 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

Page 56: Emergent complexity

1 1k k kz cz z

planec

2

1

22 1 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

Page 57: Emergent complexity

1 1k k kz cz z

2 decreases

1 1

V z z

c z

21

2 11 1 0 1z

2c

1c Mset

Page 58: Emergent complexity

Trivial to prove that these points are in Mandelbrot set.

1c 2 1c

Page 59: Emergent complexity

-3 -2 -1 0 1 2 3-2

-1

0

1

2

3

4 1x cx x realc

realx

Unstable

Equilibria

Page 60: Emergent complexity

realc

Page 61: Emergent complexity

Bifurcations for 1 1k k kx cx x

c

“last 200 x”

Page 62: Emergent complexity

“last 200 x”

c

Zoom-in

Page 63: Emergent complexity

“last 200 x”

c

Zoom-in

Page 64: Emergent complexity

-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 22 2

x x c x cc c

Bifurcations to chaos

Page 65: Emergent complexity

-2

-1

0

-3 -2 -1 0 1 2 3-2

-1

0

1

2

1

22 1 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

Page 66: Emergent complexity

-2

-1

0

-3 -2 -1 0 1 2 3-2

-1

0

1

2

1

22 1 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

Page 67: Emergent complexity

-3 -2 -1 0 1 2 3-2

-1

0

1

2

3

4

Page 68: Emergent complexity

-3 -2 -1 0 1 2 3-2

-1

0

1

2

3

4

1 1k k kx cx x

realc

realx

2

1

22 1 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

Page 69: Emergent complexity

2 2 22 2 221 1 1 22 1 1 2 1 2

2 22 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

Page 70: Emergent complexity

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 0c x c

c cx x cc c

Page 71: Emergent complexity

-3 -2 -1 0 1 2 3

-2

-1

0

1 1k k kx cx x

Bifurcations to chaos

Page 72: Emergent complexity

Special case of SOS

Page 73: Emergent complexity

Special case of SOS

Page 74: Emergent complexity

Special case of SOS

Special case of SOS

Contradiction!

Page 75: Emergent complexity
Page 76: Emergent complexity

-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 0c x c

c cx x cc c

What is the shortest proof possible?

Can prove the whole yellow region using SOSTOOLS!

Page 77: Emergent complexity

1c 2 1c

Lyapunov argument

Page 78: Emergent complexity

How to prove membership?2-period lobes

Page 79: Emergent complexity

12

3

3

4

4

4Proof

lengths

Page 80: Emergent complexity

Prove membership of 2-period lobe:• Using a stability argument of the 2-period map.• Using an invariance argument.

Page 81: Emergent complexity

-1

Formulate the invarianceProblem as the emptinessof a semialgebraic set.

Then use SOSTOOLS to construct the certificate.

Page 82: Emergent complexity

-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

Page 83: Emergent complexity

Let 0 be an equilibrium of

( ), .Let be a region containing 0 and let

: be a continuously differentiablefunction such that ( ) 0 in

nx f x xD

V D

V x D

V

( ) ( ) 0 in

Then 0 is asymptotically stable.

Vx f x Dx

Lyapunov’s theorem

Page 84: Emergent complexity

( ) 0 in

( ) ( ) 0 in

V x DVV x f x Dx

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

x1

x 2

Page 85: Emergent complexity

Can we test these conditions algorithmically?

Use the Sum of Squares decomposition!

( ) 0 in

( ) ( ) 0 in

V x DVV x f x Dx

Page 86: Emergent complexity

| ( ) 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 DVV x f x Dx

Then equilibrium is asymptotically stable.

Page 87: Emergent complexity

( , ), ,(0, ) 0

( , ) 0 for ,( , ) 0 for ,

x f x p x D p Pf p

V x p x D p PV 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

Page 88: Emergent complexity

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

Page 89: Emergent complexity

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.

Page 90: Emergent complexity

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.

Page 91: Emergent complexity

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

1a = 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

Page 92: Emergent complexity

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 ab b

,b a

Page 93: Emergent complexity

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 ab b

Search for ( , ) satisfyingthe Lyapunov conditions usingSOSTOOLS.

V x y

Page 94: Emergent complexity

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

Page 95: Emergent complexity

Modeling Analysis

Set of possible system behaviors

Set of bad system behaviors

Proof of robustness

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