Controlling chemical chaos Vilmos Gáspár Institute of Physical Chemistry University of Debrecen...

Controlling chemical chaosControlling chemical chaos

Vilmos GáspárInstitute of Physical Chemistry

University of DebrecenDebrecen, Hungary

Tutorial lecture at the ESF REACTOR workshop „Nonlinear phenomena in chemistry”

Budapest, 24-27 January, 2003

This lecture is dedicated to the memory of Professor Endre Kőrös

Chaos*Chaos*

“What’s in a name?”Shakespeare, “Romeo and Juliet”

“Chaos A rough, unordered mass of things”Ovid, “Metamorphoses”

The answer is nothing and everything.

Nothing because “A rose by any other name would smell as sweet.”

And yet, without a name Shakespeare would not have been able to write about that rose or distinguish it from other flowers that smell less pleasant.

So also with chaos.

*Ditto, W.L.; Spano, M. L. Lindner, J. F.: Physica D, 1995, 86, 198.

ChaosChaosThe dynamical phenomenon we call chaos has always existed, but until its naming we had no way to distinguishing it from other aspects of nature such as randomness, noise and order.

From this identification then came the recognition that chaos is pervasive in our word.

Orbiting planets, weather patterns, mechanical systems (pendula), electronic circuits, laser emission, chemical reactions, human heart, brain, etc. all have been shown to exhibit chaos.

Of these diverse systems, we have learned to control all of those that are on the smaller scale. Systems on a more universal scale (weather and planets) remain beyond our control.

chaos Math Stochastic behaviour occurring in a deterministic system. Royal Society, London,1986

http://www.cita.utoronto.ca/~dubinski/movies/mwa2001.mpg

A simulation of the Milky Way/Andromeda Collision showing complex (chaotic) motion of heavenly bodies can be seen on the web page of

John Dubinski Dept. of Astronomy and Astrophysics

University of Toronto, CANADA

OutlineOutline

• Chaotic dynamics of discrete systems the Henon map• The idea of controlling chaos• Fundamental equations for chaos control (ABC)• OGY and SPF methods for chaos control• Application of SPF method to chemical systems• Other methods and perspectives - come to my poster

Michele Henon,astronomer, Nice Observatory, France.

During the 1960's, he studied the dynamics of stars moving within galaxies.

His work was in the spirit of Poincare’s approach to the classisical three-body problem: What important geometric structures govern their behaviour?

The main property of these systems is their unpredictable, chaotic dynamics that are difficult to analyze and visualize.

During the 1970's he discovered a very simple iterated mapping that shows a chaotic attractor, now called Henon's attractor, which allowed him to make a direct connection between deterministic chaos and fractals.

Henon mapHenon map

Dissipative system - - the contraction of volume in the state space

nn

nnn

bxy

yaxx

1

21 1

1

3.00

12)(det

b

bb

xn

z

zf

3.0

4.1

b

a

zz

zfd

)(det1

nA

nnV

The aThe asymptotic motion will occur on sets that have zero volumes A set showing stability against small random perturbations: attractor Chaotic attractor - - locally exponential expansion of nearby points on

the attractor

Henon mapHenon map

CAP1

http://www.robert-doerner.de/en/Henon_system/henon_system.html

CAP2

CAP3

CAP4

CAP5

CAP6

CAP7

CAP8

CAP9

CAP20

Two fundamental characteristics of chaotic systems

that makes them unpredictable:

Sensitivity dependence on the initial conditionsThis causes the systems having the same values of control parameters

but slightly differing in the initial conditions to diverge exponentially (on the

average) during their evolution in time..

Ergodicity

A large set of identical systems which only differ in their initial conditions

will be distributed after a sufficient long time on the attractor exactly the same

way as the series of iterations of one single system (for almost every initial condition of this system).

Henon mapHenon map

CAP1

http://www.robert-doerner.de/en/Henon_system/henon_system.html

CAP2

CAP3

CAP4

CAP5

CAP6

CAP7

CAP8

CAP9

CAP10

CAP11

CAP12

CAP13

CAP14

CAP15

CAP16

CAP17

CAP18

CAP19

CAP20

The idea of controlling chaosThe idea of controlling chaos

“All stable processes, we shall predict. All unstable processes, we shall control.”John von Neumann, circa 1950.

Freeman Dyson: Infinite in All Directions, Chapter „Engineers Dreams”, Harper: N.Y., 1988:

“A chaotic motion is generally neither predictable nor controllable.

It is unpredictable because a small disturbance will produce exponentially growing perturbation of the motion.

It is uncontrollable because small disturbances lead only to other chaotic motions and not to any stable and predictable alternative.

Von Neumann’s mistake was to imagine that every unstable motion could be nudged into a stable motion by small pushes and pulls applied at the right places.”

So it happened.

The idea of controlling chaosThe idea of controlling chaosHenon map - Bifurcation diagram

x

http://mathpost.la.asu.edu/~daniel/henon_bifurcation.html

F

FFnn

nn

n

nn

y

x

p

y

x

zzz

zfz

z

:) ( point fixed1-Period

:dynamics ssystem' The

:statessystem'The

o

1

1 ,

)1(1 nnn p BzAz

)()()( 1 oppnFnFn BzzAzz

The linearized equation of motion of the system around the fixed point zF:

F

FnFnzz

zfAzzAzz

1

For chaos control we apply a small parameter perturbation pn≠ po if and when the system approaches the fixed point.

ABC of Chaos ControlABC of Chaos Control

x

y

)1(1 nnn p BzAz

During chaos control – for simplicity – we apply parameter perturbation that is linearly proportional to the system’s distance from the fixed point,where CT is the control vector.

)2(nnp zC T

From equations (1) and (2) we get the linearized equation of motion around the fixed point when chaos control is attempted:

)3()(1 nn zBCAz T

Shinbrot, T.; Grebogi, C.; Ott, E.; Yorke, J. A.: Nature, 1993, 363, 411.

Chaos control is successful if the new system state zn+1(po+δpn)

lies on the stable manifold of the fixed point zF (po) of the unperturbed system.

)3()(1 nn zBCAz T

The just described strategy for chaos control implies the followings:

)5(0

)(

)4(

'

'

ss

u

s

u

eig

eig

TBCA

A

)3()(1 nn zBCAz T

For a successful chaos control, therefore, one has to know:

• the dynamics of the system around the fixed point • the system’s distance from the fixed point• the right value of control vector CT

• the eigenvalue of the fixed point in the stable direction

Numerical exampleNumerical exampleHenon mapHenon map

nn

nnn

xy

byxpx

1

21

The linearized equation of motion around the fixed point zF

when pn≠ po parameter perturbation is applied:

nnF

n

FnFn

nFnFnFFn

pbx

xxyy

ppyybxxxxx

0

1

01

2

)()(

)()()(2)(

1

1

1

zz

o

)1(1 nnn p BzAz

!

Numerical exampleNumerical exampleHenon mapHenon map

The eigenvalues of the fixed point of the unperturbed system are calculated by solving the following equation:

01

2det

bxFA

0

0

2

2

bxx

bxx

FFs

FFu

resulting in

)5(0

)('

'

ss

ueig TBCA

Let’s find the control vector CT such that

Numerical exampleNumerical exampleHenon mapHenon map

01

2 21 bxFTBCAP

The control vector can be calculated by solving for the new eigenvalues, and by applying the rules of the control strategy.

01

2det

'2

'1

bxFP

01

2 bxFA

0

1B

2

1CSuppose:

Numerical exampleNumerical exampleHenon mapHenon map

02

)(4)2()2(

2

)(4)2()2(

22

11'

222

11'

bxx

bxxbxx

FFu

sFFFF

s

which gives

b

bxx FF2

2

1C

Note that C contains parameters characteristics of the system’s dynamics only.

Numerical exampleNumerical exampleHenon mapHenon map

which gives

3.0

84.12

b

bxx FFC

nn

nnn

xy

yxx

1

21 3.029.1

FF

FFF

xy

yxx

3.029.1 2

8384.0

029.17.02

FF

FF

yx

xx

nn

nnn

xy

byxpx

1

21

3.0

29.1

b

po

Numerical exampleNumerical exampleHenon mapHenon map

3.0

84.1C

)2(nnp zC T

According to our control equation the parameter perturbation for successful chaos control should be the following:

)(

)(3.084.1o

Fn

Fnnn yy

xxppp

)(3.0)(84.1 FnFnn yyxxpp o

Numerical exampleNumerical exampleHenon mapHenon map

-2

-1

0

1

2

900 950 1000 1050 1100

n

x n

Numerical exampleNumerical exampleHenon mapHenon map

-2

-1

0

1

2

900 950 1000 1050 1100

n

x n

Numerical exampleNumerical exampleHenon mapHenon map

-2

-1

0

1

2

900 950 1000 1050 1100

n

x n

Numerical exampleNumerical exampleHenon mapHenon map

-0.5

-0.3

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

1.3

1.5

0 0.5 1 1.5

xn

y n

Numerical exampleNumerical exampleHenon mapHenon map

-0.5

-0.3

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

1.3

1.5

0 0.5 1 1.5

xn

y n

Numerical exampleNumerical exampleHenon mapHenon map

-0.5

-0.3

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

1.3

1.5

0 0.5 1 1.5

xn

y n

The linearized equation of motion of the system around the fixed point zF :

opppp nnFnnFn )),(())(( 1 zzAzz

uss

u

yy

xx

eig

yx

yx

F

1Aff

ff

A

z

s

ususu 0

0eeeeA

sss

uuu

eAe

eAe 1

0

0

su

s

usu eeeeA

Can we do better?Can we determine C experimentally?

Answer: OGY theory*

*Ott, E.; Grebogi, C.; Yorke, J. A.: Phys. Rev. Letters, 1990, 64, 1196.

. and 0 demandingby

vectorsbasic parallel"" old the of terms in

vectorsbasic lar"perpendicu"new Define

TTTT 1 uussussu

su

su

eeee

ee

ffff

ff

1

110

01

su

s

u

su

s

ususu

ee

eeee

T

T

T

TT

f

f

f

fff

T

T

T

T

ss

uusu

s

u

s

ususu

s

usu

f

f

f

feeeeeeeeA

0

0

0

0 1

)(TT 6sssuuu ff eeA

The effect of parameter perturbation:

• the map, thus the fixed point is shifted

• but we assume the same linear dynamics

nn

nFnF

pp

F

pp

pp

p

p

n

1

1 )()(~

)( zzzg )(7)()(~)( 11 gzz nnFnF pppp

)()( gzzAgzz )()(~)()( 111 nnFnnnFn pppppp

gz )()( 1 nnF ppp ))((~))(( 111 nFnnFn pp zzAzz )(TT 6sssuuu ff eeA

)(TT 81 gzeegz nnsssuuunn pp ff~

)gzAgz nnnn pp (~)( 1

To achieve chaos control we demand that thenext iterate falls near the stable direction. Thisyields the following condition (see figure).

011 nunFnu p zzz TT ))(( ff

)(TT 'ff~ 81 gzeegz nnsssuuunn pp

gzeegz nnssusuuuuunnu pp TTTTTT fffff~f 01

=1 =0

gzg nnuuun pp TT ff~0 nuuuunp zg TT f~f 1

nnu

u

u

unp zKz

g

T

T

f

f~

1which is the OGY formula

Ott, E.; Grebogi, C.; Yorke, J. A.: Phys. Rev. Letters., 1990, 64, 1196.

nnu

u

u

unp zKz

g

T

T

f

f~

1 gK

T

T

u

u

u

u

f

f

1

Henon mapHenon map

nn

nnn

xy

byxpx

1

21

3.0

29.1

b

po 83840. FF yx

16300

83981

2

2

.

.

bxx

bxx

FFs

FFu

17050

04851

1

12

2

.

.f

ssuu

suu

u

42070

42070

41

412

2

.

.

o

o

pb

pb

py

px

F

Fg

)(3001.0)(8401.1 FnFnn yyxxpp o

nnu

u

u

unp zKz

g

T

T

f

f~

1 gK

T

T

u

u

u

u

f

f

1

Constant K can be calculated fromexperimental data:

u from the slope of the map about the fixed point at po

• g form the displacement of the fixed point with respect to a change in p

• fuT from the eigenvectors in both stable and unstable directions calculated from the linearized map about the fixed point.

Limits of the OGY method:

• When the fixed point is such that fu and g are nearly orthogonal to each other, the control constant increases to infinity. Such fixed points are uncontrollable.

• The method works only for hyperbolic fixed points with a stable eigenvector.

• Determination of fu requires measurement of two (three) system variables, and also a good numerical approximation to the system’s dynamics around the fixed point. However, collecting data along the stable manifold may be experimentally inaccessible. • In real systems there is often noise present preventing the determination of the system’s state and of the control constant with the required accuracy.

Surprisingly, a simplification of the OGY formula provided the right algorithm for successfully controlling chaos in chemical systems.

gK

T

T

u

u

u

u

f

f

1

gK

T

T

u

u

u

u

f

f

1

Simplification of the OGY formula

))(~1 nFnnn ppp zzK (

If the stable eigenvalue is very small (s 0), the 2D map changes to a 1D map, which leads to a much simpler control formula:

))(~1 nFnnn pxxKpp ( gK

1

It also means that instead of targeting the stable manifold, we now directlytarget the fixed point itself.

This is the so called SPF (simple proportional feedback) algorithm derivedby Peng et al. This method has been used most effectively for controlling chaos in chemical systems.

Peng, B.; Petrov, V.; Showalter, K.: J. Phys. Chem., 1991, 95, 4975.

))(~1 nFnnn pxxKpp ( gK

1

Application of the SPF method:

1. Reconstruct the chaotic attractor

2. Generate a one-dimensional map

on a Poincaré section

3. Determine the position of the fixed

point.

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

I (t)

(m

A)

I (t - 0.5 s) (mA)

0.7

0.8

0.9

1.0

1.10.7 0.8 0.9 1.0 1.1

xn (mA)

x n+1

(m

A)

Copper electrode dissolution in phosphoric acid under potentiostatic conditions.

Kiss, I. Z.; Gáspár, V.; Nyikos, L.; Parmananda, P.: J. Phys. Chem. A, 1997, 101, 8668.

))(~1 nFnnn pxxKpp ( gK

1

Application of the SPF method:

4. Generate the map at a different value of p

5. Determine g from the shift of the map

6. Determine , the slope of the maps

7. Calculate K

8. Determine the system’s position on the map

9. Calculate the parameter perturbation

10. Apply the perturbation for on cycle – go to 8.

0.84 0.86 0.88 0.90 0.92 0.94 0.96

0.84

0.86

0.88

0.90

0.92

0.94

0.96

x n+1

(mA

)

xn (mA)

xf1xf2

0 25 50 75 100 125 150

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4OFFON

I(t)

(m

A)

t (s)

-0.4

-0.2

0.0

0.2

0.4 Vn (m

V)

Kiss, I. Z.; Gáspár, V.; Nyikos, L.; Parmananda, P.: J. Phys. Chem. A, 1997, 101, 8668.

Petrov, V.; Gáspár, V.; Masere, J.; Showalter, K.: Nature, 1993, 361, 240.

Controlling Chaos in the Belousov–Zhabotinsky Reaction

(CO + 1 % H2) : O2 = 7,2 : 5,6

Davies; M. L.; Halford-Mawl, P. A.; Hill, J.; Tinsley, M. R.; Johnson, B. R.; Scott, S. K.; Kiss, I. Z.; Gáspár, V.: J. Phys. Chem. A, 2000, 104, 9944-9952.

Control of Chaos in a combustion reaction

Other (continuous) methods for chaos control:

• Delayed-feedback algorithm: )()()0()( txtxKptp

• Resonant control algorithm: )2sin()0()( tAptp

• Artificial neural networks

Come to see my poster

Kazsu, Z.; Kiss, I. Z.; Gáspár, V.: Experiments on tracking unstable steady states and periodic orbits using delayed feedback

“The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.”

Henri Poincaré (1854-1912)