Slide1 : Controlling chemical chaos Vilmos Gáspár Institute of Physical Chemistry
University of Debrecen
Debrecen, Hungary Tutorial lecture at the ESF REACTOR workshop
„Nonlinear phenomena in chemistry'
Budapest, 24-27 January, 2003
Slide2 : 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.
Chaos : Chaos The 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
Slide5 : 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
Slide6 : Outline 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
Henon map : 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 map
Slide8 : Dissipative system - the contraction of volume in the state space The 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 map
Slide9 : http://www.robert-doerner.de/en/Henon_system/henon_system.html
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Slide19 : Two fundamental characteristics of chaotic systems
that makes them unpredictable:
Sensitivity dependence on the initial conditions
This 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 map
Slide20 : http://www.robert-doerner.de/en/Henon_system/henon_system.html
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Slide40 : The 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.
Slide41 : The idea of controlling chaos Henon map - Bifurcation diagram x http://mathpost.la.asu.edu/~daniel/henon_bifurcation.html
Slide42 : For chaos control we apply a small parameter perturbation pn≠ po
if and when the system approaches the fixed point. ABC of Chaos Control x y
Slide43 : Shinbrot, T.; Grebogi, C.; Ott, E.; Yorke, J. A.: Nature, 1993, 363, 411.
Slide44 : 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.
Slide45 : The just described strategy for chaos control implies the followings: 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 exampleHenon map : Numerical example Henon map The linearized equation of motion around the fixed point zF
when pn≠ po parameter perturbation is applied: !
Numerical exampleHenon map : Numerical example Henon map The eigenvalues of the fixed point of the unperturbed system are calculated
by solving the following equation: resulting in
Numerical exampleHenon map : Numerical example Henon map
Numerical exampleHenon map : Numerical example Henon map
Numerical exampleHenon map : Numerical example Henon map
Numerical exampleHenon map : Numerical example Henon map
Numerical exampleHenon map : Numerical example Henon map
Numerical exampleHenon map : Numerical example Henon map
Numerical exampleHenon map : Numerical example Henon map
Numerical exampleHenon map : Numerical example Henon map
Numerical exampleHenon map : Numerical example Henon map
Numerical exampleHenon map : Numerical example Henon map
Slide58 : The linearized equation of motion of the system around the fixed point zF : 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.
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Slide60 : The effect of parameter perturbation:
the map, thus the fixed point is shifted
but we assume the same linear dynamics
Slide61 : To achieve chaos control we demand that the
next iterate falls near the stable direction. This
yields the following condition (see figure). =1 =0
Henon map : Henon map
Slide63 : Constant K can be calculated from
experimental 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.
Slide64 : 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.
Slide65 : Simplification of the OGY formula It also means that instead of targeting the stable manifold, we now directly
target the fixed point itself.
This is the so called SPF (simple proportional feedback) algorithm derived
by 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.
Slide66 : Application of the SPF method:
Reconstruct the chaotic attractor
Generate a one-dimensional map
on a Poincaré section
3. Determine the position of the fixed
point. 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.
Slide67 : 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.
Slide68 : Kiss, I. Z.; Gáspár, V.; Nyikos, L.; Parmananda, P.: J. Phys. Chem. A, 1997, 101, 8668.
Slide69 : Petrov, V.; Gáspár, V.; Masere, J.; Showalter, K.: Nature, 1993, 361, 240. Controlling Chaos in the Belousov–Zhabotinsky Reaction
Slide70 : (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
Slide71 : Other (continuous) methods for chaos control: Artificial neural networks
Slide72 : '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)