controlling chemical chaos vilmos gáspár institute of physical chemistry university of debrecen...
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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)