KINETIC
INSTABILITIES
IN
L.
MAN-MADE
M.
AND
NATURAL
REACTORS
Pismen
Department
of Chemical
Engineering,
Technfon,
Israel
Institute
of Technology,
Haifa,Israel
and
Institute
of
Applied
Chemical
Physics,
New York,
N.Y.
10031.
CCNY,
ABSTRACT
Kinetic
systems
theory
instabilities
and self-organization
are reviewed
and discussed
from
of non-equilibrium
processes.
.
the
phenomena
point
of
in chemical
view
of the
and biological
unified
qualitative
KEYWORDS
stability,
control;
self-organization;
nonlinear
kinetics;
Reactors;
bifurcations;
propagating
fronts;
symmetry
breaking;
oscillations;
LIST
OF
autocatalysis,
kinetic
waves,
chaos.
CONTENTS
1. Introduction
2.
3.
4.
5.
6.
Origin
of kinetic
instabilities
Theory
3.1.
Asymptotic
states
3.2.
Bifurcations
3.3.
Propagating
fronts
3.4.
Rapid
fronts.
Excitations
and metastability
3.5.
Slow
fronts.
Stationary
patterns
3.6. Kinetic
waves
Phenomenology
4.1.
Homogeneous
kinetics
4.2.
Combustion
4.3.
Heterogeneous
catalysis
4.4.
Porous
catalysts.
Fixed
beds.
4.5.
Electrochemistry.
Photochemistry
4.6.
Nerve
conduction
4.7.
Crystallization.
Polymerization
4.8.
Morphogenesis
Population
genetics
4.9.
Ecology.
4.10
Other
related
problems
Chaos
Outlook
1.
INTRODUCTION
The title
of this
review
calls
for a definition
of the term
reactor,
which
otherwise
using
this word
in a broader
sense
would
be hardly
needed
in this
audience.
Clearly,
is intended.
We shall
call
a reactor
any open
macroscopic
non-equilibrium
system
Any reactor
the chemical
wherein
transport
and transformation
of species
take
place.
be it an industrial
converter,
a burning
particle,
an electengineer
may deal with,
So does
a living
tissue,
and
satisfies
this
definition
or a catalytic
surface,
rode,
In the above
definition,
and so does
the surface
of the earth
so does
an organism,
In place
of chemical
species
and
the adjective
chemical
is deliberately
omitted.
we can substitute
biological
species
and resources
and Interchemical
interactions,
1950
Kmetlc
action
still
between
hold.
them
birth,
mstabrhtres m man-made
death,
consumption,
1951
and natural reactors
predation.
The
broad
definition
will
Evolution
of a reactor
involves
three
types
of processes'
internal
transformation,
and external
transport,
or exchange
with
the environment.
Both
the
internal
transport,
biologist
and the engineer
are primarily
interested
in asymptotic
states
which
a continuously
operating
reactor
would
reach
after
some
stretch
of time
and wherein
it will
remain
unless
dislodged
by a major
disturbance
or change
in external
conditions.
In an
extreme
situation
when
all communication
with
the environment
is cut off,
the reactor,
At the
now being
a closed
system,
will
relax
to a state
of thermodynamic
equilibrium
when
all internal
transformations
are frozen,
the final
state
will
opposite
extreme,
Both
these
asymptotic
states
are
be one of transport
equilibrium
with
the environment.
Starting
from
either
of them,
and both
are unique.
and gradually
switching
stationary,
on external
transport
or reaction
terms,
we will
obtain
two "basic"
branches
(transport
and kinetic)
of stationary
asymptotic
states.
interesting
case
- that
It may
happen
- in the least
these
branches
meet
midway
in such
a manner
that
the stationary
state
remain
unique
and stable
in the whole
range
of
parameters
and external
conditions
that
somewhere
away
from
It may happen,
however,
the two equilibrium
limits
both
thermodynamic
branches
undergo
some
kind
of bifurcation
leading
to their
destabilization
and to the emergence
of a variety
of other
asymptotic
states,
not all of them
being
stationary,
symmetric,
or even
ordered
We embrace
phenomena
of this
kind
by a loose
term
kinetic
instabilities.
Any
reactor
model
amasses
a good
deal
of omissions,
approximations,
lumping
and "coarsegraining"
as it treads
the narrow
path
between
Failure
and Fallacy.
What
is the
touchstone
of a satisfactory
model?
Quantitative
precision
would
be too stringent
a
test,
especially
when
a really
difficult
system
is under
the scrutiny,
but qualitative
The primary
qualitative
characteristic
of a kinetic
conformity
can never
be foregone.
This
alone
makes
the problem
of
system
is the kind
of instabilities
it may exhibit.
kinetic
instabilities
central
for modelling
of reactors,
both
man-made
ones
of the
chemical
engineer
and natural
ones
of the biologist.
The problem
of kinetic
instabilities
has been
traditionally
high
on the ISCRE
agenda
The
entire
field
was widely
and deeply
probed
by Schmitz
six years
ago
[l]. Since
then,
fast
and far-reaching
developments
have
occurred.
These
were
the years
when
scientists
from
different
fields
of research
were
finally
coming
together
and realizing
their
the years
when
a unified
theory
of instabilities
in noncommon
methods
and aims,
equilibrium
systems
was
crystallizing.
The aspects
of this
theory,
as seen
by the
chemical
engineer,
the physical
chemist,
the physicist,
the fluid
dynamicist,
can be
found
in monographs
by Aris
[2], Nicolis
and Prigogine
[3], Haken
[4], Joseph
[S]. A
number
of recent
interdisciplinary
conferences
[6-91
were
devoted
to this
group
of
problems.
I had
long
hesitated
how
to build
up this
review
in such
a way
that
it might
give
an
overview
of both
methods
and facts
of general
interest
and at the same
time
be neither
too abstract
nor
fragmentary.
Such
problems
never
have
a unique
and stable
solution.
After
some
trial
and error,
the bulk
of the review
came
out cut into
two major
parts,
one of which
represents
the unified
qualitative
theory
of
Theory
and Phenomenology,
kinetic
instabilities,
and another,
experimental
facts
taken
from
different
fields
and
looked
upon
through
the prism
of this
theory.
Two splinter
chapters
broke
away
from
one of them,
discussing
chaos,
this
body,
did so just
because
of the unruly
character
of its subject,
which
is the most
novel
and exciting,
in every
sense
of the word,
kinetic
instability
of all.
In a review
of this
kind,
the choice
of material
cannot
be anything
but
subjective.
I
will
avoid
repetition
of topics
covered
in the reviews
well-known
to chemical
engineers
and concentrate
on phenomena
in spatially
distributed
systems,
which
is the
El,lOl,
field
where
the richest
qualitative
behavior
can be observed
and where
most
progress
has been
achieved
in the recent
years.
A model
2.
ORIGIN
of
a reactor
OF
KINETIC
can
be
INSTABILITIES
written
in
a general
form
(1)
1952
L
M
PISMEN
Here 2
is the state
vector
comprising
concentrations
of interacting
species
and other
relevant
variables,
such
as temperature
or potential,
r(u)
is the vector-valued
-kinetic
function,
and
s
is the transport
operator
describing
internal
transport
processes.
We shall
habitually
refer
to chemical
kinetics,
though
the function
r(u)
can
reflect
transformations
of other
than
chemical
species.
If diffusion/conductiyn-ia
the
only
transport
mechanism,
x=
ODp , where
2
is the matrix
of diffusivities.
Terms
of different
structure
may
describe
such mechanisms
as convection,
migration
or chemotaxis.
External
transport
processes,
or exchange
with
the environment,
should
be
accounted
for by boundary
conditions
to Eq.<l).
In principle,
all three
types
of processes
reflected
by the reactor
model
- internal
transformation,
internal
transport
and exchange
with
the environment
- can exhibit
"antithermodynamic"
features
and become
a source
of complicated
dynamic
behavior
far
from
equilibrium.
In the framework
of this
review
we restrict
attention
to kinetic
instabilities
which
are caused
by the nonlfnear
kinetic
term
r(u)
in Eq (1). We can
distinguish
between
two principal
sources
of "antithermodynamiT'-behavior
in chemical
kinetics
autocatalysis
and delayed
feedback.
Each
of the two violates
a specific
feature
of the process
dynamics
near
the equilibrium
autocatalysis
is incompatible
with
the stability
of chemical
equilibrium,
and delayed
feedback
violates
the principle
of non-oscillatory
approach
to equilibrium
The term
autocatalysis
applies
to any reaction
which
is accelerated
by one of its own
products
or inhibited
by one of the reactants.
Reactants
and products
are distinguished by the sign
of their
stoichiometric
coefficients,
and may include
heat
and electric
charge
as well
as chemical
species.
Accordingly,
exothermic
reactions
and electrochemical
reactions
with
non-monotonic
current-voltage
characteristics
should
be also
dubbed
autocatalytic.
Autocatalysis
is apparently
incompatible
with
the mass
action
law
This
contradiction
can be resolved
in two ways.
First,
one of the reactants
in an irreversible
reaction
can be at the same
time
a reaction
product,
being
generated
in larger
amounts
than
it
is consumed.
This
is the propagation
kinetics
which
is characteristic
to branched
chain
and is also
implied
in
reactions
and to all models
of ecology
and population
genetics,
many
intensively
studied
artificial
reaction
schemes,
first
of all,
in the Brusselator
model
[3],
Another,
even
more
important
source
of autocatalytic
action
is quasistationarity
The
actual
reaction
mechanism
may
involve
some
short-lived
or rapidly
equilibrating
intermediary
species,
like
active
complexes,
adsorbed
forms,
catalyst
modifications,
etc.,
which
do not enter
"macroscopic"
models
like
Eq Cl), but
can be either
lumped
together
or eliminated
with
the help
of quasistationary
or equilfbrium
relationships.
Slowly
interacting
reactants
can influence
quaslstationary
or equilibrium
levels
of intermediary
species
in such
a way
that
the overall
reaction
be inhibited,
the opposite
shift
of these
levels
may be responsible
for autocatalytic
action
of reaction
products.
A bimolecular
reaction
with
the Langmuir-Hinshelwood
(or,
synonymously,MichaelisMenten)
kinetics
inhibited
by a strongly
adsorbing
reactant
gives
the simplest
example
of this basic
mechanism.
Other
illustrations
of the same
idea
can be seen
in Selkov's
[ll] scheme
of the glycolvtic
system
and in various
ecological
models
with
limited
resources
[12]
A somewhat
different
example
is given
by kinetic
schemes
involving
several
equilibrating
forms
of enzyme
or catalyst
sites
differing
by their
activity
Exothermicity,
the source
of instability
most
familiar
to the chemical
engineer,
r131
is based
on the same
principle
of hidden
equilibria,
since
it is the shift
of the
equilibrium
molecular
energy
distribution
in the reacting
mixture,
that
causes
thermal
Similar
changes
of hidden
"microscopic"
distrtbutacceleration
of chemical
reactions
ions,
may be responsible
for mechanisms
of intraspecific
interaction
in ecology
In
electrochemistry,
shifting
equilibria
on or near
the electrode
surface
are a usual
source
of non-monotonic
current-voltage
curves.
The delayed
feedback
1s a more
subtle
variety
of "antithermodynamic"
behavior
which
requires
at least
two non-equilibrium
reaction
steps
for Its realization.
In the
delayed
feedback
is caused
by inhibition
of the first
simplest
and most
usual
case,
step
in a chazn
of chemical
transformations
under
the influence
of one of the species
This
can lead
to oscillatory
behavior
which
is forformed
at the following
stages
Such
a mechanism
bidden
near
the equilibrium
by the Onsager
reciprocity
relations.
works
In all processes
involvzng
distinct
stages
of initiation
(nucleation)
and proThe most
important
of these
are polymerization
and crystallization.
pagation
(growth).
Inhibition
of the nucleation
stage
due to depletion
caused
by the growth
of nuclei
already
formed
is the source
of commonly
observed
oscillations
in continuous
crystall-
Kmetlc mstabdhes
m man-made
1953
and naturalreactors
izers and polymerization
reactors. Backward
inhibition
in a chain
is one of widespread
mechanisms
of biochemical
regulation
[14,15]
of enzyme
reactions
An alternative
mechanism
of delayed feedback may involve an inhibitor
formed in a slow
reversible
parallel
reaction,
like oxidation/reduction
or other process bringing
about
modification
of a catalytic
surface or passivation
of an electrode due to formation
of
deposits or oxide layers. In ecology, delayed feedback may be caused by interaction
of
species on different
trophic levels. In the simplest version,
a predator, which, in
chemical terms, is a product of a reaction involving prey as one of the reactants,
inhibits the prey's growth just by consuming
it.
Basic chemical mechanisms
leading to kinetic instabilities
were discussed by Franck
[16,17].
In his classification,
autocatalytic
mechanisms
involving propagation
and
quasistationarity
are defined, respectively,
as non-systemic
and systemic feedback.
There is a considerable
body of work aimed at deducing the qualitative
dynamics of
isothermal
reactions
(in particular,
possibilities
of destabilization
and oscillatory
behavior)
directly
from the structure
of stoichiometric
equations
118-263
Still, the
kineticist
building
a model which would exhibit required behavior
relies mostly on his
intuition propped by few simple rules. Phase diagrams representing
loci of -r(u)=0 in
the space of state variables
u
can serve as an important
link between the behavior
and the underlying
kinetics
[13,27,28] but this method is not formalized.
An encouraging fact is that very complicated
behavior
can result from very simple models including two, at most three, state variables.
Even most primitive
schemes with "flip-flop"
kinetics may bring variegated
and qualitatively
relevant results [29]
In the conventional
reactor theory, transport processes
play a stabilizing
role
counteracting
autocatalytic
action.
However,
one can also encounter
transport
instabilities caused by any transport mechanism
able to destabilize
the uniform spatial distribution of the transferred
quantity. From the mathematical
point of view, kinetic and
transport
instabilities
differ by the functional
form of the terms responsible
for
non-trivial
dynamic behavior.
Transport
terms contain spatial derivatives,
while
kinetic terms do not. This distinction,
however,
is not absolute
Sometimes
a chemical
source of instability
is masquerading
as a mechanism
of anomalous
transport.
For
instance,
a rough phenomenological
model of active transport in biological
or artificial membranes would operate with negative
diffusivities
or strongly non-diagonal
diffusivity
matrices,
while a more precise model would locate kinetic interactions
responsible
for the transport anomaly.
On the other hand, distinctions
in the functional
form of kinetic and transport
terms
may be lost due to spatial lumping.
Any state variable
in Eq.(l) can be assigned a
certain characteristic
relaxation
time, -ri , and a characteristic
transport
length,
Ri = (Di'i)l/2 . If all
Ri exceed the linear dimension
of the reactor (which may be
due to intensive mixing, like in a CSTR), the internal transport
can be eliminated,and
a lumped (point) model used. Spatial lumping may be possible only along one or two
coordinate
axes, rather than throughout
the reactor
This leads to two-dimensional
(membrane,
catalytic
surface, habitat)
or one-dimensional
reactor models (plug flow
reactor, catalytic wire, nerve).
As a result of spatial lumping, transport
terms
responsible
for the exchange with environment
reduce to algebraic
expressions
added to
kinetic terms describing
chemical sources and sinks
In some cases, instabilities
may be caused by interaction
of kinetic and transport
terms neither of which alone would bring about non-trivial
dynamic behavior.
This
phenomenon
may be, for instance,
connected with changes of transport
coefficients
due
to chemical interactions,
which, in their turn, affect the further course of the
reaction.
Teorell's
[30] oscillations
in porous ion exchange membranes
can be relegated to this category.
Purely transport
instabilities
bring us already into the realm
of hydrodynamics.
Turbulization
and onset of natural convection
are likely occasions
in kinetic experiments,
and can both conceal and imitate instabilities
of kinetic
origin (see, e.g. [31]).
3. THEORY
3.1.
Asymptotic
States
Non-equilibrium
systems of different
physical nature and with different
sources of
instability
can display the same universal
types of dynamic behavior.
Asymptotic
states
of a non-equilibrium
system can be classified
according
to elements of temporal and
1954
L M
PISMEN
spatial order which may arise spontaneously
due to specific dynamic features of a
system in question.
These elements of order can be perceived most clearly by considering a system which is intrinsically
homogeneous
and isotropic,
and is neither affected
by tfme- and space-dependent
external factors, nor influenced by boundary
conditions.
The type of a reactor satisfyfng
these requirements
is called a homogeneous
isotropic
reactor (HIR). The definition
implies that all kinetic and transport parameters
are
uniform throughout
the reactor, and there are no preferred
directions
for transport at
any point. An equiaccessible
catalyst surface, an active membrane,
a nerve, a homogeneous habitat give examples of one- or two-dimensional
HIR (a three-dimensional
reactor
may be approximately
homogeneous
and isotropic
in some special cases). Since the entire
reactor is open to the exchange with environment,
external transport
and kinetic terms
can be combined into an overall kinetic function -f(u,&
depending
on some set of parameters
& . Assuming
that the sole internal transport mechanism
is diffusion with a
constant diffusivity
matrix
g , we rewrite Eq.(l) in the form
.
ax/at = gv2g
+ f(L&Jg
(2)
Equation
(2)has one or more (as a rule, an odd number)
states
homogeneous
stationary
(HSS) which correspond
to zeroes of the kinetic function -f(u) . Other possible
asymptotic
states of Eq.(Z) may be either inhomogeneous
or non-stationary,
or both,and
exhibit temporal or spatial organization
which has not been predetermined
by the timeand space-invariant
model of the reactor.
The types of ordered asymptotic
states are
the limit cycle, or the homogeneous
oscillatory
state (HOS), the stationary
pattern,
or the inhomogeneous
stationary
state (ISS), and the wave pattern, or the inhomogeneous
oscillatory
state (10s).
Still another group of asymptotic
solutions
to Eq.(2)
includes chaotic states (see 55).
At given values of process parameters,
the reactor can have one or more asymptotic
states of different nature; some of these states may be stable and some, not. When the
of asymptotic
states change at
parameters
change, the type, number and stability
from one state to another may involve a jump of the
bifurcation
points. The transition
Othervalues of state variables,
then it is called a hard, or first-order
transition
transition.
wise, the transition
may be continuous;
this is a 80ft, or second-order
Kinetic systems able to exhibit spontaneous
transitions
to spatial and/or temporal
[3,32] self-organizing
systems. Ordered states
order can be called, following Prigogine
are also often called dissipative
structures,
as opposed to HSS lying on the
"thermodynamic"
branch.
This terminology
would be misleading
if understood
as implying
rate will
that higher dissipation
rates are inherent to ordered states. The dissipation
be actually the highest on the transport branch of HSS where the reacting conditions
are close to chemical equilibrium
and the asymptotic
state is unique and flat.
Multiple asymptotic
states remind of different phases in equilibrium
thermodynamics,
states come in much greater variety. Processes
near
but, of course, non-equilibrium
the thermodynamic
equilibrium
have one most important property which greatly narrows
There is a functional
of state variables,
called free
the range of allowed behavior.
energy, which decreases
in all evolution
processes
and reaches a minimum at a stable
oscillatory
states are ruled out, but multiplistationary
state. Due to this property,
city of stationary
states and pattern formation are still possible as collective
phenomena
involving
interaction
of a large number of variables.
Whenever multiple
to
equilibrium
phases exist, one can single out the preferred state which corresponds
the absolute minimum of free energy; all other phases are deemed metastable.
For a non-equilibrium
reaction-diffusion
the role of free energy
state variable,
FCu($:)
system described by a model
is played by the functional
= ,[+ Do
+ V(u)]d=
with
a single
(3)
In any process obeying the scalar equation
(2)
where
V = -jf(u)du is the "potential".
with zero-flux boundary
conditions,
dF/dt<o, and F{u(z):)) 1s extremal when u(%) is a
Useful thermodynamic
analogies
follow from this fact (see 53 3)
stationary
state [333.
lose ground when there are two or more state variables,
These analogies,
however,
cannot be represented
as a gradient
inasmuch as a vector kinetic function,generally,
this is the source of greater qualitative
variety of
of a scalar potential.
Actually,
non-equilibrium
states.
1955
Kmetlc mstablhtlesm man-made and naturaireactors
3.2. Bifurcations
and interrelations
of asympThe most powerful method for studying
genesis, stability
The power of this method stems from the
totic states is the bifurcation
analvsisfundamental
connection
between
changes of stability
and emergence of new asymptotic
states. Bifurcation
analysis takes its origin in classical work by Lyapounov
[341,
to
Schmidt
[35], Poincarg
[36], Hopf [37], see also [38-411. Though first applications
problems of chemical kinetics
go back to the 60's 142-451, the method has attracted
attention
of chemical theorists mainly under the influence of later work on bifurcatSome of the best surveys of modern bifurcation
ions in the Brusselator
model [46-481.
but on this level of abstraction,most
theory 15,403 largely use hydrodynamic
material,
distinctions
between the Navier-Stokes
and reaction-diffusion
equations are already
lost
Consider a HSS of Eq.(2). Its stability
for perturbed
solutions
in the form
s=u
Vectors
2
_
can be
+
checked
z exp(iiz
in a standard
way
by
looking
+ At)
(4)
satisfy
(AI + k2g)Z
= ZlE
(5)
is the matrix of derivatives
of the vectorwhere
I
is the unit matrix and gl
u=a
The eigenvalues
X(k)
depend on the absolute
function -f(u) calculated
at
$
Stability
is determined
by the principal
value of the wavenumber
vector
ReX, = max(Re)l).
eigenvalue
(PEV) i.e. that with the largest real part, X=X,&,);
Reh,<O . Changing a
In the thermodynamic
limit, the HSS is unique and stable, so that
certain parameter
p (e.g. the ratio of characteristic
external transport
and reaction
the eigenvalues
in Eq.(5)
rates) removes the reactor from near-equilibrium
conditions;
change on the way, and, given proper kinetics,
at some point
?J=lJ,the real part of
the PEV reaches zero
This is the bifurcation
point where the thermodynamic
branch
terminates
or loses stability,
and a new branch of asymptotic
states commences.
The nature of solutions belonging
to this new branch depends on whether
the PEV is
zero (x=0> or imaginary
(A = *iw,) at
p=uo , and whether
this PEV corresponds
to a
If
wo=ko=O
, the
homogeneous
perturbation
(k,=O) or to an inhomogeneous
one (k,#O).
and
k,=O , they
if
w,#O
asymptotic
states on the bifurcating
branch are also HSS;
are HOS;
if
w,=O
but
this is
k,#O
a branch of ISS emerges, and at
w,#O , k,#O
a branch of IOS.(Fig.l).
From the physical point of view, appearance
of an unstable
mode with the frequency
w.
and the wavelength
k, , brings about, respectively,
temporal and spatial order, uniformity
in time or space is preserved when the respective frequency
or wavelength
is zero.
spatial
order
quasiperiodic
Fig.
1
and
Creation of spatial and/or temporal order through
The lines are wavy
when k,#O and
bifurcations.
hatched when e#O
.
1956
L
ht
PISMEN
Further information
is provided by the nonlinear
analysis of solutions belonging
to
the new branch. Bifurcation
is supercritical
when the new branch goes into the domain
of parameters where the original HSS is unstable or does not exist; in the opposite
case, when the "basic" branch of HSS and the new branch exist in the same parametric
domain, the bifurcation
is subcritical.
A straightforward
rule for bifurcations
at a
simple eigenvalue
is that supercritically
bifurcating
branches
are stable while subcritically bifurcating
ones are not. In the first case, stability
is "transferred"
from the thermodynamic
branch upon the new branch of asymptotic
states. The transition
from one branch to another is a continuous,
second-order
transition.
When the bifurcation is subcritical,
no stable asymptotic
states slightly differing
from the solution
at the bifurcation
point exist in the supercritical
region, and the only remaining
possibility
is "runaway",
i.e. a jump into a state far removed from the pracritical
branch. This is a first-order
transition which implies that stable asymptotic
states
coexist in some parameter
region.
Due to this overlap, retracing back the way the
process parameters
had been changed would not lead to a repeated in the inverse order
sequence of states of the reactor, but would generate a hysteresis
loop.
Some typical bifurcation
diagrams,
i.e.
the bifurcation
parameter
v are shown
plots of a norm
in Fig-P.
of the solution
11~1 1 versus
Generally,
the space of process parameters
can be separated
into domains where certain
of these domains are bifurcation
stable asymptotic
states exist; the boundaries
surfaces.
Only on these surfaces changes in stability may occur; hence, it is always
sufficient
to check stability
of an asymptotic
state near the bifurcation
point
wherefrom
it emerges.
Following
a branch of stable ordered solutions
(HOS, ISS or IOS), one can see that
away from the bifurcation
point the spatial and/or temporal oscillations
cease to be
harmonic,
their amplitude
grows and the period or wavelength
change, but the spontanStability of an ordered state, like stability
of
eously generated
order is preserved.
a HSS, is determined
by the principal
eigenvalue
of the corresponding
linearized
problem
(more precisely,
in the case of oscillatory
states, by the principal
Floquet
exponent, PFE).
While parameters
change, the real part of the PEV or PFE will vanish
at some point, which again will be a bifurcation
point.
Such bifurcations
can be studied analytically
when
This is a secondary bifurcation.
they occur near a primary bifurcation
point on the branch of "trivial" HSS. In the
limit when the primary and the secondary bifurcation
coincide, we have either a
a compound bifurcation
degenerate
bifurcation
(like in Fig.lb) or, on the contrary,
point where two or more modes become unstable. Of course, such a point can be obtained
in addition to the primary bifurcation
parameter
only by fixing one more parameter,
of two bifurcatflxed at the value
!J=uo . In other words, it lies on an intersection
Analysis
of secondary bifurcations
allows to draw, at least qualitation surfaces
bifurcation
diagrams even for relatively
simple kinetic systems
ively, sophisticated
[49-571.
Secondary bifurcations
can lead to asymptotic
states of still higher level of complexIt may happen that the mode which is destabilized
at the secondary bifurcation
ity
point has a frequency and/or wavelength
other than those characteristic
to the ordered
Then the newly emerging branch will contain asymptotic
states, characterized
state.
by quasiperiodic
or chaotic changes of state variables
in time and/or space. Such a
branch can also emerge directly from a branch of HSS as a result of a bifurcation
at a
multiple
(degenerate)
eigenvalue.
Of special Interest is the case when the degeneracy
of the PEV is not "incidental"
but
is caused by inherent symmetries
of the problem
[58-611. In particular,
synrmetrybreaking bifurcation
(k,#O) always occurs at a highly degenerate
eigenvalue,
provided
Physically,
the reactor 1s more than one-dimensional,
and its size far exceeds
kol
this degeneracy
corresponds
to simultaneous
excitation
of differently
oriented waves
interactions
between
these incipient
inhomogeneities
would lead
or patterns. Nonlinear
by
to "survival"
of a particular
pattern; the form of this pattern would be determined
Both regular and chaotic
chemical kinetics rather than by geometrical
factors [62,631.
solutions
can emerge from such a highly degenerate
bifurcation
point.
3 3. Propagating
If asymptotic
fronts
states
are nouns
in the glossary
of kinetic
instabilities,
transition
Kmetlc mstabhtles
m man-made
1957
and natural reactors
Cb)
(=I
II
UII
‘.
1’
\\ /
I
:/-i-
I
b
PO
)I
(f)
IlUll
t
Fig.
1)
2,
subcrltxcal
blfurcatlon,
hard
Blfurcatlon
diagrams.
a, d, f
supercrltlcal
bLfurcatlon,
soft
(first-order)transltlon,
c, e
The
b
degenerate
blfurcatlon.
(second-order)
transition,
for blfurcatlon
of stationary,
dxagrams
a, c, d are typlcal
SolId
lines
represent
stable,
states.
and e, f, of oscillatory
unstable
branches
of solutions
and dashed,
b,
Fig.
3.
Phase
diagrams
a) oscillatory,
b) excitable,
c) blstable
antlcllnal
d) blstable
syncllnal
system
Dashed
llnfs
represent
phase-plane
Images
of
propagating
fronts
(at E<<l
&<<l,w<<l)
or
alternatively,
traJectorles
of
local
phase
points.
L
19.58
M
PISMEN
fronts are the verbs. Combining elementary
nouns and verbs,
"phrases" describing more sophisticated
behavioral
modes.
we
can construct
many
A transition
front is formed at a border of two HSS which are approached
at two extremes of an extended region. Once being formed, the front propagates without changing its
shape In the direction of the "less stable" state until the latter's territory has been
"conquered"
and the spatial homogeneity
restored. Thus, the front is a solitary wave
which effects transition between two alternative
HSS.
A firmly established
theory, initiated by such major figures as Kolmogorov
and Zeldovich ,
describes properties
of plane fronts in systems with a single state variable.
This
theory has been developed mainly in connection with problems of population
genetics
[64-671
and combustion theory [68-731. Propagating
fronts are very stable 172-751. The
time required to form a transition
front is of the same order of magnitude
as the
characteristic
reaction time, T ; the width of the transitional
region is of the order
Both should be much smaller than the size of an extended reactor and the
!?,
= (DT>~/~.
time the reactor can be swept by the solitary wave propagating with the characteristic
velocity of the order c = %/T = (D/-r>li2.
With a single state variable,
the front is always directed in such a way that the
territory of the state with a lower potential
is enlarged-Again,
this is the behavior
typical for phase transition
in equilibrium
systems. As long as Eq.(3) plays the role
of free energy, thermodynamic
analogies are all but exhaustive.
We can speak of
nucleation
of a new state [76], which starts with the formation of a critical nucleus
extended over the characteristic
diffusional
length R and, as in the equilibrium
case,
can be precipitated
by local inhomogeneities.
We can even recognize the dependence
of
the propagation
velocity on the curvature of the front as a counterpart
of the Kelvin
law, and find an analogy of surface tension [33] in the tendency to minimization
of the
front are
at a given enclosed volume which reduces the total "surface energy",
(D&‘$!(Vu>
'ds , in Eq.(3).
The situation becomes much more complicated when there are
at least two state variables
in Eq (2). If, as it often happens,
the velocity scales,
c = (D/-c)l12, are of different order of magnitude,
the question arises which of the
reactants imposes on the transition
front its characteristic
speed. In addition, no
potential function which would define the propagation
direction unequivocally
exists
criteria of relative stability
any more
On the contrary, as the quasithermodynamic
fronts and
lose ground, we can replace them only by the "pecking order" of transition
consider the advancing state to be "more stable" then the receding one. As it turns
out, even this criterion often falls.
Qualitative
analysis of propagating
fronts and other transient processes
in complex
kinetic systems is facilitated
by asymptotic methods based on wide separation
of time,
length and velocity scales characteristic
to different reactants
[13,27,28,77-821
Diversity of scales is typical both in chemical engineering
and biology,
it would
suffice to mention discrepancies
in thermal and material capacity and in ranges of
diffusion and heat conduction.
This diversity
causes greater qualitative
variety of
for the theorist as it
transition processes;
at the same time, it is advantageous
allows to separate rapid and slow temporal and spatial changes.
Let Eq.(2)
be written
in the dimensionless
form
au/at = v2u + f(u,v)
E h/at
(6)
= &2v2v+
g(u,v)
are ratios of the time and length scales characteristic
to the two variabwhere
&,b
les u, v, and f(u,v), g(u,v) are properly scaled kinetic functions. The most varied
behavior
is observed when there is a short-range
fast ("stiff") autocatalytic
reactant
The other reactant which, by contrast, is long(&=l,
E<<l, ag/a+o in some region).
When the scales are widely separated,
range and slow ("soft") may be non-autocatalytic.
the only kinetic information
needed for constructing
the qualitative
behavior
is the
dispositon
of nullcurves
f(u,v)=O. g(u,v)=O in the phase plane (u.v)
Four basic types of phase diagrams with a sigmoidal nullcurve
g(u,v)=O are shown in
Depending
on the position of the nullcurve
f(u,v)=O relative to the sigmoidal
Fig.3.
to the oscillatory
(Fig.3a), excitable
nullcurve, we obtain diagrams corresponding
(Fig.3c) and bistable synclinal
(Flg.3d) kinetics
(Fig.3b), bistable anticlinal
At s<<l,
6<<1,
all propagating
fronts
have
a two-tier
structure
comprising
two regions,
Kmetlc mstabdltlesm man-made and natural reactors
1959
is following
changes, while
v
u
outer and inner. In the outer region, the variable
so that g(u,v)=O holds in zero order. A steep change of
v
It quasistationarily,
is effected in the inner region at a
g(u,v)=O, v=v*(u*)
between
two stable zeroes of
nearly constant level of the other variable,
u-u z . Given certain kinetics, markedly
dissimilar
behavior
can be realized depending
on whether
the stiff or the soft variable
has a larger characteristic
velocity.
3 4
Rapid
fronts.
Excitations
and metastabllity.
with
When the ratio of characteristic
velocities,
u=~/E , is large, slow propagation
of the order of the characteristic
velocity of the soft reactant is
the speed
c=O(l)
possible
if the Inner "stiff" front can find such stable position, u*=uo , that
v+(u,)
I(uo)
=
I
v- (u,)
ho,
v)dv
= 0
(7)
front is either nonexistent
or unstable,
the front accelWhen such a slow "entrained"
erates until it reaches the ultimate position where the level of the soft variable
either coincides with the stationary
value corresponding
to the receding state or
front is formed, which
a "spearhead"
reaches the extremal level umax or umln . Thus
relaxes in the wake of
w>>l * The soft variable
propagates
with the speed of order
with diffusion playing a negligible
role.
the propagating
dlscontmuity,
The analysis of stability
of entrained
fronts reveals a marked distinction
between two
K=(u+-u-)[I(&types of bistable
systems, differing by the sign of the combination
are stationary
levels of the soft variable
[28]. For synclinal
I(u-)] where
u*
velocity.
It means
system (K<O, Flg.3d) there is always a unique front propagation
that the behavior
is qualitatively
the same as in the case of a single state variable,
On the
and we can always tell which
state 1s "more stable" under given conditions.
new phenomena.
anticlinal
kinetics
(K<O, Fig.fc) brings about qualitatively
contrary,
In this case, whenever
an entrained
front exists, it is unstable,
and there are two
It is no more
stable spearhead
fronts propagating
in opposite directions
[13,28].
possible
to say that one state is more stable than another. both can advance or recede,
system,
depending
on initial conditions.
When
u-<u,<u+
, both HSS of an anticlinal
If the reactor is initially
in one of the homogenu=u+, are metastable,
or excitable.
though finite-amplitude
perturbation
can send the
eous states, a brief short-range,
A nucleus of
local phase point to the alternative
branch of the sigmoidal nullcurve.
the alternative
state formed in this way rapidly expands, while in the wake of the
propagating
front the soft variable
relaxes to the value corresponding
to the alternative HSS
in the same manner, forming
The latter, in its turn, can be destabilized
We can say that the system transan expanding
nucleus of the previous HSS (Fig.4)
forms local disturbances
(which can be of thermal or hydrodynamical
origin, or be
connected with surface inhomogenelties)
into long-wavelength,
low-frequency,
highamplitude wave trains [28,82].
In a system with a single excitable HSS (Fig.3b), the effect of local perturbations
will be very much like in the previous
case, with the only difference
that return to
the left-hand branch of the sigmoidal
nullcurve
in the wake of the propagating
front
is spontaneous
Finally, when the nullcurves
are situated as in Fig. 3a, the kinetic system is oscillatory and no extraneous
excitations
are required
for transitions
between the branches
of the sigmoldal
curve
In a lumped reactor (CSTR) the phase diagram of this kind
generates,
at
E<<l , the so-called
"relaxation
oscillations"
with the phase point
rapidly moving along the horizontal
lines and slowly relaxing along the curved segments
the dashed
contour in Fig.3a.
of
In distributed
reactors,
oscillations
are translated (at w>>l) into "trigger waves" formed by alternating
rapid transition
fronts
between
the alternative
branches
of the sigmoidal
nullcurve.
Each transition
can be
precipitated
by local disturbances,
so that, given some level of background
noise,
behavior
of oscillatory,
excitable
and anticlinal bistable
systems would be hardly
distinguishable.
At the same time, all three show a marked contrast wfth synclinal
systems which behave in a "quasithermodynamic"
fashion and are insensitive
to localized
perturbations.
Often we would encounter
a situation when the reactor is not isotropic,
and there is a
pronounced
gradient of the soft variable
caused, for instance, by sources or sinks at
CES
35/!-j
1960
L
M
PISMEN
t
Fig.
4.
(Cl
_
- _ __
--z.I.~-$_______
(d)
----
Generation
of trLgger
waves
a) the low state
1s descabllb) the neucleus
lzed
locally,
of the upper
state
1s formed
and starts
to expand,
c) the
slow
variable
relaxes
in the
wake
of the propagating
front,
d) the upper
state
is destablllzed
e> the
nucleus
of the
lower
state
expands,
f> the slow
varlable
relaxes
back.
Solrd
and dashed
lines
represent
levels
of the stiff
and
soft
variable,
respectively.
b)
duldt
a)
SO
--f=O
f<O
!
u
1%
fJ<o
---
-
--_
--_
fits0
Fxg.
5
duldt<O
dtIdt<O
The trlstable
system
with
a phase
diagram
of the kind
deplcted
here
can exhlbxt
undulating
fronts
separating
regions
with
low and high
values
of the stiff
varzable,
v.
Changes
of the fraction
of area
covered
by the lower
state
and of the level
of the soft
varzable
follow
the traJectorzes
shown
by dashed
curves
(b)
<a>
1%1
Kmetlc mstablhtlesm man-made and naturalreactors
Then the stiff front can be immobilized
the boundaries.
of the soft variable,
u=uo , satisfies Eq.(7) [83].
at a position
where
the level
I(u) increases
or
Stability
of this stationary
front depends on whether the integral
If the stationary
position
turns out to be
decreases
in the direction of the gradient.
into a steep stationary
front
stable, the gradient of the soft reactant is translated
of the stiff component.
u-u0 , it would run away in either
the stiff front is unstable at
If, on the contrary,
direction
until coming into a close vicinity of the border (within a distance about
the characteristic
length scale of the stiff reactant).
Then, in the case the
advancing
state is incompatible
with the boundary
conditions,
the front stops and,
after the soft variable has relaxed to a new profile, may bounce back, overshoot
the
it may be repelled in a
unstable position and approach another boundary, wherefrom
This is one of possible mechanisms
of oscillations
in a distributed
similar way.
reactor (cf. 14.4)
An effect analogous
to excitations
via propagating
spearhead
fronts can be also realized in compartmental
reactors. This is the regime of "echo waves"
[84], when the
excitation
impulse "echoes" back and forth between two interconnected
cells (CSTR)
Oscillations
of this kind may persist even under conditions when neither of the cells
alone would be oscillatory.
The asymptotic
analysis of waves and fronts can be extended to kinetic systems with a
larger number of stationary
states and/or state variables.
For example, a tristable
system with two autocatalytic
reactants
and the phase diagram of the type shown in
of fronts and paradoxical
reversals
of "pecking
Fig.5, can exhibit synchronization
In systems with one
order" in the mutual chase of the three co-existing
states [283.
more complicated
dynamics of slow motions
can lead to
stiff and two soft variables,
such phenomena
as excitation
of a limit cycle which would manifest
itself by encroachment of wave trains into a homogeneous
region [27]
These phenomena
are still largely
unexplored.
While very diverse behavior
can be obtained even with simplest phase
diagrams of the kind shown in Fig.3, in more complicated
kinetic systems "anything
can
happen",
given a proper phase diagram and suitable length, time and velocity
scales
3.5.
Slow
fronts.
Stationary
patterns
The behavior
of propagating
fronts is quite different when the stiff reactant has a
In this case neither multiple
propagating
lower characteristic
velocity
<w<<l).
phenomenon
can occur insolutions nor metastability
appear, but another interesting
immobilization
of fronts in a perfectly
uniform region with the resulting
stead
polarization
or tessellation
of a homogeneous
reacting medium.
Formation
of immobile fronts in a bistable kinetic system can be most readily understood in the following way.
Consider a reactor with the linear extent (measured on
the characteristic
scale of the soft variable)
6<<L<<l . Then the level of the soft
variable will be practically
constant throughout
the reactor, but the stiff variable
may attain alternative
values
v=&(u)
at the opposite extremes.
The resulting
short-range
front will tend to propagate with the speed
c=O(w), but at w<<l
the soft
variable will have sufficient
time to adjust to the changing position of the front
stabilizing
the inhomogeneous
distribution
of the stiff
and act as a "global regulator"
The stationary
position of the front will be achieved at the level of the
reactant.
soft variable,
u , and with the fraction
17 of the reactor occupied by the "lower"
state
v=v-(u)
determined
by the conditions
C82.85.861
?lf(u,v-<u))
+
(l-ll)f(u,v+~u))
= 0 ,
I(u)
= 0
(8)
the first of which is an approximate
balance equation of the soft reactant,
and the
second, the condition of the front stationarity.
The stability
analysis of solutions
to Eq.(8) shows that the ISS is stable for an anticlinal,
but not for a synclinal
system. We see once again that it is hard to give preference
to any of the alternative
HSS In an anticlinal
kinetic system
both are easily destabilized
at
w>>l, while w<<l
they can "peacefully
coexist".
Stable immobilized
fronts can also appear in excitable
and oscillatory
kinetic systems.
In the latter case,
the only possible
stationary
state is inhomogeneous.
With more
sophisticated
phase diagrams,
like that in Fig 5, oscillatory
destabilization
of an
L Iu PISMEN
1962
immobilized
front is possible. The resulting
limit cycle corresponds
to a "swinging
wave" which indulates
around an unstable stationary
position in unison with oscillating
levels of the soft variable
[28,82].
Since only the overall balance of areas is fixed by the dynamic equations,
the boundaries between polarized
regions are liquid and can fluctuate under the influence
of the
background
noise' at the same time, they tend to acquire a minimum length due to the
"surface tension" effect. The alternative
states are not bound to specific
locations
and may swap their territories
after a long stretch of time.
Fixation of spontaneously
arising inhomogeneities
may be effected by another kinetic
mechanism,
called a "landscape
variegator",
which involves an inertial non-diffusive
variable,
in addition to a diffusive
anticatalytic
reactant.
If the latter is originally polarized,
so that v=vi at two extremes of an extended reactor, the non-diffusive
variable may relax in two hinterland
regions to different
states, u=u* (e.g. two
different modifications
of the catalyst may be formed). In this way, a rigid
"territorial
infrastructure"
repelling
the advance of the propagating
front may be
established.
Mathematically,
such behavior
is assured by the "Fabian condition",
I(u+><O
, I(u?>O
, whereby each state advances on its own and recedes at the rival's
territory
[82].
Inhomogeneous
solutions
to reaction-diffusion
equations
can be constructed
in another
way by considering
symmetry-breaking
bifurcations
at
w,=O, ko#O [3,46-48,87-901.
For
a general two-component
reaction system, such a bifurcation
can occur on a branch of
stable HSS when the following
conditions
hold
(a) the nullcurves
are situated near
the stationary
point like in Fig.3a, i.e., they lie In the same quadrant and
g,=ag/av>o
but
fu=af/au<o
, and (b) the autocatalytic
reactant has a longer characteristic time and a lower diffusivity
(E=lgv/ful>l
, 62/&l).
These conditions
are
compatible with the conditions whereunder
the "global regulator"
mechanism
operates,
but are more restrictive.
This indicates
that often only large-amplitude
inhomogeneities are possible, whereas
small-amplitude
ISS are unstable.
Further nonlinear
analysis of bifurcating
inhomogeneous
solutions
shows that in more
than one dimension
incipient
ISS are unstable
even when the above conditions
are met
and stability
can be achieved only at finite amplitudes
[62]. It can be also proven
that a stable pattern in an extended two-dimensional
region has a hexagonal
(honeycomb)
structure'
the same as in the case of B&ard
cells formed after the onset of natural
convection
in a fluid heated from below. Indeed, the mathematical
background
of
symmetry breaking
in hydrodynamics
and chemical kinetics is virtually
identical.
There are a number of computer experiments
demonstrating
stationary
patterns
in model
methods,
this was the
kinetic systems 191-971. Before the advent of scale-separation
only way to construct
large-amplitude
patterns.
Computer results have to be viewed,
however, with some caution, unless care is taken to prevent numerically
stable
calculation
of physically
unstable patterns.
3.6.
Kinetic
waves
We have already seen in 13.4 how relaxation
oscillations
are translated
into wave
of
patterns in an extended reactor. This is an example of the most common mechanism
wave generation
via desynchronization
of phases of local oscillators.
on a
There is a minor paradox in the theory of kinetic waves. A simple bifurcation
branch of HSS in a kinetic system with two state variables
can lead to HOS or ISS but
not to IOS(wave patterns)
It means that transition
to IOS would require a secondary
could occur only in more complex
bifurcation
on a branch of HOS or ISS, or, otherwise,
Taking this into account, one
kinetic systems and under rather restricting
conditions.
compared with homogeneous
would expect kinetic waves to be rather a rare occasion,
oscillations
or stationary
patterns.
Experiment,
however,
tells the opposite.
There
would be no homogeneous
oscillations
in an extended reactor without mixing, and there
At the same
are but few convincing
demonstrations
of spontaneous
symmetry breaking
time, waves are rather a conrmon phenomenon.
The reason for the persistence
of waves is that any oscillatory
process is only
between local oscillators,
neutrally
stable to phase shifts. A phase difference
introduced
due to initial conditions
or a localized disturbance,
would persist,
provided
the reactor is sufficiently
extended.
once
1963
Kmetlc rnstablbtles
m man-made and naturalreactors
The factor counteracting
the desynchronization
is "phase diffusion"
[98,99] which is
effective
at distances
of the same order of magnitude
as the characteristic
length
For trigger waves (alternating
spearhead
scale of the speed-determining
variable.
fronts) considered
in 93.4, this distance is very short, and a localized
disturbance
oscillations
precipitating
transition
to the
would easily desynchronize
homogeneous
alternative
branch of the sigmoidal nullcurve.
Temporal changes are unfolded in space,
into the
as if on the screen of the oscilloscope,
and the period
'c is translated
wavelength
exceeds the Ifnear extent of the reactor, L ,
1Lrc.r. If this wavelength
the outward picture of homogeneous
oscillations
is still retained,
though the propagation stage 1s incorporated
in the periodic process.
If, on the contrary, L>>!Z , the
and the pattern of propagataverages taken over the entire reactor cease to oscillate,
ing waves takes over.
Clearly,
local aberrations
of kinetic and/or transport parameters
strongly influence
the above picture. If at some location,
due to a shift of local values of kinetic
parameters,
the sigmoidal nullcurve
is modified
in such a way that the phase point
slumps onto the alternative
branch earlier than at neighboring
sites, this point will
serve as a leading center, or "pacemaker",
emanating
the train of trigger waves. Such
points can be found even when average kinetics is of excitable or metastable
and
waves
anticlinal
rather than oscillatory
type, and the resulting picture of consecutive
of excitations
would be practically
indistinguishable
from a "true" wave pattern in an
oscillatory
kinetic system.
w<<l
and phases are synchronOn the contrary,
at
local excitations
do not propagate,
ized within a larger region measured
on the characteristic
scale of the soft reactant.
Under these conditions,
waves can appear only in truly oscillatory
kinetic systems of
the type shin Fig.3a.
There are several formalized methods of constructing
wave patterns.
as phase waves
induced by slight inhomogeneities
in an oscillatory
system [98,99]; as long-wave
extensions
of a homogeneous
limit cycle [99]; as small-amplitude
waves bifurcating
from a HSS [99,100].
Unlike wave solutions
to hyperbolic
equations,
where amplitude
and wavelength
(or frequency)
can be chosen independently,
kinetic waves are determined
by a single parameter'
prescrzbing
the amplitude defines both frequency
and wavelength.
Stability
considerations
may eliminate
even this degree of freedom. For instance,
in
the relaxation-oscillation
system discussed
above only a metastable
HOS and an 10s with
are admissfble
at w>>l.
approximately
the same amplitude
and the wavelength
of order w
Kinetic waves come in many patterns,
depending on the presence of external gradients
and inhomogeneities,
geometry of the reactor and initial conditions.
Experiment
often
i.e. a picture of center waves emanating
gives a "target pattern",
from leading center
Though these "pacemakers"
may be connected with local inhomogeneitles
or inclusions,
stable leading centers can also appear spontaneously
in models with three state varrotating around the oscillating
core have been also
iables [loll. Spiral patterns
obtained in computer experiments
on simple models [102-1061. Both center and spiral
waves can be demonstrated
analytically
for so-called
X-w-systems
[107-1101.
This
"kinetic system" has no chemical meaning, but is easy for analysis because it gives
circular limit-cycle
trajectories
in the phase plane. Analogies
to phenomena
of wave
propagation
can be also seen in numerical modeling of coupled and arrayed oscillators
[Ill-11.5].
A variety of small-amplitude
waveforms
can be obtained by the analysis of bifurcations
of this point, the system can be described by the
at
wofo ,koPO
- In the vicinity
universal
equation for complex amplitude,
a , [116,117]
as/at
- yV2a
+ ffa + BIa12a
(9)
where
B,Y,a
are complex parameters
connected
in a non-trivial
way with the kinetic
and transport parameters
of the system.
Applying stability
criteria allows to single
out certain patterns of propagating
and standing waves which would have a preferred
status in an initially homogeneous
system 1631
Under certain conditions,
the patterns
are turbulised,
and the chaotic state takes over (see 55).
4. PHENOMENOLOCY
4 1
We
Homogeneous
owe
the bulk
kinetics
of experimental
data
on kinetic
waves
and patterns
to studies
of the
‘9 .
L
1964
M
PENEN
Belousov-Zhabotinsky
(BZ) reaction
[118-1211. The chemistry,
let alone kinetics,
of
this reaction is still unclear, and there is disagreement
between major investigators
on the structure
of kinetic equations,
as the model suggested by Noyes and co-workers
[122,123] is disputed by Zhabotinsky
[124]. The detailed mechanism
of this celebrated
reaction is not,however,
as important
as the fact that the reaction
itself can serve as
an instructive
model of still more complex biochemical
systems. Following
the discovery of kinetic waves in shallow diffusional
cells [125] numerous
experimental
studies
of the BZ reaction have demonstrated
a variety of fascinating
phenomena
including rotating spiral and scroll waves [126-1281, stationary
patterns
11291, diffusional
coupling and resonance of two oscillating
cells [130-1321, modulation
of an external
signal [131], and onset of apparent chaos [101,132-1371.
Qualitatively,
all these phenomena
can be explained by the three-variable
Oregonator
model Cl231 with separated scales.This
is hardly surprising,
since the wealth of behavior of three-variable
models has yet to be fathomed, Even most sophisticated
models
cannot,however,
achieve quantitative
agreement with the body of experimental
data.
The discovery
of the first homogeneous
oscillatory
reaction dates back to 1921 [138],
but due to prejudices
based on misinterpretation
of thermodynamic
laws, this phenomenon
had not been given due attention
until investigations
of the BZ reaction made the
evidence overwhelming.
In principle,
very simple reaction schemes are sufficient
to
generate oscillations
and other non-trivial
kinetic phenomena
(see §2) but in reality,
basic mechanisms
of autocatalysis
and/or delayed feedback are Invariably
embodied in
complex chemistry.
Enzyme reactions
are especially
prone to oscillatory
behavior,
in
particular,
there is vast literature
on glycolytic
oscillations
1139-1431.
Though it has been proven that, given proper transport parameters,
homogeneous
oscillations always can be translated
znto spatial order, there are very few observations
of
patterns and waves in systems other than the BZ reaction.
One exception
1s spontaneous
pattern formation observed recently in yeast extracts
11441, this system is, however,
too complex to be considered
as purely chemical.
4 2
Combustion
Thermal feedback and chain branching
are two autocatalytic
mechanisms
involved in combustion,
and each of them alone can bring about complicated
dynamic behavior.
The theory
of thermokinetic
instability
was created in the late 30's by Frank-Eamenetsky
and
Zeldovich
[68,69,145].
In particular,
they have shown how the propagating
ignition
front effects transitlon between
two alternative
HSS of this nonlInear
kinetic system
[68,69].
Further studies of propagating
fronts were largely connected with combustion
problems
[70-731, especially
with combustion
and explosion
of condensed
fuels [146,147]. An
Interesting
"non-classical"
development
in the combustion
theory is the discovery
of
pulsating
ignztion fronts [148] leading to a layered structure of burned samples. Recently, pulsating
fronts were constructed
analytically
as bifurcations
from steadily
propagating
solutions
[149]. Pulsating
fronts are also apparently
responsible
for
oscillations
in combustion
of carbon particles
[150,151]. Another Interesting
phenomenon is formation of hexagonal
patterns on propagating
ignition fronts and their subseqThis instability
is of thennokinetlc
origin and 1s
uent turbullzation
[152,153]
observed when the Lewis number 1s small (Le<l).
Chain branching
acts as another source of instability
in gaseous combustion
An interplay of chain branching
and exothermicity
is responsible
for the behavior
of oscillgaseous combustion
1s often accompanied
by
atory cool flames 11541.
In addition,
instabilities
mingle
hydrodynamzc
turbulence
and shock fronts, so that hydrodynamic
with those of kinetic origin [155].
4.3
Heterogeneous
catalysis
It xs natural to expect that complex and variegated
kinetic mechanisms
of heterogeneous
catalysis would provide an inexhaustable
source of complicated
dynamic behavior.
But,
and experimental
work has been all but exclusively
until recently, both theoretlcal
concentrated
on thermokinetlc
Instabilities
in catalytic reactors
Thus mechanism
of positive
feedback
alone
is responsible
for varied
phenomena
traced
Kmetlc mstahlhes
m man-made
1%5
and naturalreactors
Multistability
down on different
levels of the reactor theory C1,2,10,156]
thermokinetic
oscillations
in CSTR were predicted
as early as in the 40's by
[156-1581 and intensively
studied and reviewed ever
Kamenetsky
and Salnlkov
[1,10,45,159],
but some exotic features of dynamic behavior
of this simplest
still being added [160]
and
Franksince
system
are
The simplest model of equiaccessible
catalytic particle is formally identical
to the
that thermoklnetic
oscillations
in heteroCSTR model. It has been argued, however,
geneous processes
are ruled out due to the large thermal capacity of the catalyst
[I611
phase-plane
analysis shows that an autocatalytic
variable
, in this
Indeed, elementary
case temperature,
should have a shorter characteristic
time scale to ensure oscillatory
behavior.
Likewise,
a longer characteristic
range of heat conduction,
as compared with
surface.
the diffusional
range, makes spatial symmetry breaking unlikely on a continuous
Thermal autoacceleration,
of course, can explain all phenomena which are allowed in the
framework of a single-variable
model. One such phenomenon
is multistability
and
a fortiori
propagation
of transition
fronts. Unlike combustion,
in catalysis
reversibIe motion of ignition and extinction
fronts is possible
Propagation
of such fronts
on platinum wires was recently reported by Barelko and co-workers
[162]
It is interesting that the front remains stationary
within a finite interval of reaction parameters
(Fig.6a) rather than at an isolated point where "potentials"
of the low- and hightemperature
states equalize.
This can be attributed
either to hysteretic
effects due
to pre-existing
surface inhomogeneities,
or to reversible
catalyst modification
in cold
and hot regions which would change activity in such a way that the "Fabian condition"
In the case the latter conjecture be true, the observed front
(93.5) be obeyed
immobilization
could be considered
as an experimental
realization
of the "landscape
variegator"
mechanism
of spontaneous
symmetry breaking
[82].
Asymmetric
stationary
states of a compartmental
reactor (realized as two faces of a
heat-conducting
non-porous
slab or other assemblage
of isolated surface spots llnked
by heat conduction)
are also possible
in the framework of a scalar model [163]. But
asymmetric
states on a continuous
uniform surface [163] are always unstable when a sole
state variable
is involved.
The latter fact, though rather obvious intuitively
[33],
turned out to be not so easy to prove
A strict and elegant proof is known in one,
but not in higher dimensions
[SS].
After the disappointment
with thennokinetic
oscillations,
the center of attention has
shifted to isothermal
oscillatory
reactions.
Most experiments
have been performed with
oxidation
of CO and Hp on metals
[164,165] and surface oxidation-reduction
is considered to be the main mechanism
responsible
for oscillatory
behavior
Recent studies are
revealing
ever more sophisticated
oscillatory
patterns
[166-1701, and mechanistic
explanations
lag behind accumulation
of experimental
material.
Observed
transitions
to
apparently
chaotic behavior
[167-1701 as well as somewhat mysterious
disappearance
of
oscillations
with increased
flow rate [166] are of special interest.
Many experiments
in oscillatory
kinetics
are carried out with catalytic wires or foils,
under conditions
when phase shifts along the catalytic surface, hence, formation of
wave patterns,
are not excluded;
at the same time, only changes in bulk variables
are
measured.
If wave patterns do really appear under experimental
conditions,
they can
influence
the observed picture of bulk oscillations
in several ways.
(1) The period
and the shape of oscillations
may (though not necessarily
will) become size-dependent
(2) Oscillations
will be influenced
by factors affecting
the wave propagation
velocity,
such as surface diffusivity,
heat conductivity,
hydrodynamic
mixing along the surface
etc. (3) Oscillations
will be sensitive
to local Inhomogeneities
acting as leading
centers, or "pace-makers".
All this can complicate
the observed picture considerably.
In particular,
oscillations
may become irregularly
shaped or even apparently
chaotic.
The onset or disappearance
of bulk oscillations
may be caused by changes in propagation
velocity
(e.g., by transition
from slow to rapid propagation
or vice versa in a system
with widely separeted
scales) or by changes in number of leading centers. Of course,
purely kinetic factors can bring about rich phenomenology
of oscillations
even in a
truly lumped reactor.
Phase desynchronization
and onset of waves may both imitate and
conceal the action of more sophisticated
kinetic mechanisms.
Realizing
the role of local states of the catalyst surface has prompted
an intriguing
comeback of thermokinetic
oscillation
mechanisms.
Jensen and Ray [171,172] attributed
the picture of chaotic oscillations
to superposition
of inputs from independently
oscillating
crystallites
retaining
a weak thermal link with the bulk of the catalytic
wire.
This model is in line with recent computational
work on onset of chaos in arrays
L
M
PISMEN
a
b
CREEP WARD
BED LNTRANC
a
=o DC%? &mm
ED ‘ET -70
ft 5 mm
TF = ziD*f?
1
a0
-6
Fig.
200
to0
INTERSTITIAL
6
VELOCITY
300
400
, u (ft/mrn)
(a) Experimental
results
of Barelko
and others
11621
showing
the temperature
dependence
of the front
propagation
velocity
on a catalytic
wire
at different
(b) Calculations
of Amundson
and
co-workers
[180]
reactant
concentrations.
showing
contours
of constant
creeping
front
velocities
in a fixed
bed.
Fig.
7.
The temperature
profiles
v(p,T)
m
a porous
particle
during
the
limit
cycle
at Le=O
1
[178j
Interpreted
as
undulating
fronts.
1,2
the system
is in the
low-temperature
state;
3
the nucleus
of the
high-temperature
state
has been
formed,
4
the
transltlon
front
propagates
towards
the
periphery
becomzng
steeper
as It feeds
on
fresh
reactant,
5- the
front
has
stopped
near
the outer
surface
6-9
the front
retreats
and
fades
away
Kmetlc mstabhtles
m man-made
and naturalreactors
1%7
of chemical oscillators
[111,112] as well as with work by the group of Ross on
instabilities
due to interaction
of bulk reactions
and transport
and localized catalyst sites [173,174]
The onset of oscillations
on rough wires can be also explained
in the framework
of a
continuous
model, as a consequence
of decreasing
thermal conductivity
and heat capacity
and, at the same time, increasing
diffusivity
and material
capacity of porous layers,
which, eventually,
reverses
the scales ratlo and brings about the entire spectrum of
kinetic phenomena
characteristic
to systems with a stiff autocatalytic
reactant.
Local
like crystallltes
with far from average characteristics,
can trigger
inhomogeneities,
with the picture described
in 93.4.
propagating
wave fronts, in accordance
There is little doubt that catalytic
surfaces can exhibit a variety of wave patterns
and symmetry-breaking
phenomena.
The wealth of possibilities
should be impressive,
due
to the existence
of several mechanisms
of communication
along the catalytic surfaceelectronic,
thermal, diffusional.
All of them have varied characteristic
ranges and
can act in conjunction
with processes
going on widely separated
time scales, starting
with times of formation
and decay of active complexes
and Intermediate
adsorbed forms,
and going all the way to the scale of slow catalyst modification.
These wave phenomena
of local measurements.
Such measureremain undetected
up to now, due to difficulties
ments, however,
are becoming
urgent, since they alone can tell us what is the actual
origin of the observed multipeak
and chaotic oscillations,
and bring us nearer to
understanding
the kinetic and transport mechanisms
of catalytic
action.
4.4.
Porous
catalysts.
Fixed
beds
Studies of thermokinetic
instabilities
had
catalysts
and beds. Though activity in this
years, it is interesting
to take a look at
view of the qualitative
theory of reaction
also long dominated
the theory of porous
well-trodden
field has subsided in recent
some well-known
results from the point of
fronts.
in
A porous catalytic
particle
is a distributed
and, in the sense of the definition
it largely reproduces
behavior
of simpler
93.1, inhomogeneous
reactor. Qualitatively,
stationary
profiles
taking place of multiple
stationary
states
systems, with multiple
introducing
another exothermic
reactlon
[175] or an additional
[2]. Quite naturally,
stage of heat transport
[44,176,177]
adds a third stable stationary
state. But
oscillatory
behavior
and pattern formation
require combination
of thermal feedback
either with another autocatalytic
mechanism
or with a non-autocatalytic
reactant having
larger characteristic
time and/or length scales
Computations
of transient behavior
of porous particles
at small Lewis
numbers
[178] are
instructive
in this respect, even though the involved values of process parameters
may
be unrealistic-If
the equations
describing
a single irreversible
exothermic
reaction
are written
in the form Eq.(6), with
u
standing for concentration
and
v , for
, m=Rl/2Le-1
the relevant scale ratios are 6=H-1/2, E=H-lLe
temperature,
H being
Frank-Kamenetsky's
exothennicity
parameter.
At Le<<l , H>l , we have
l&
, E<<l ,
which are conditions
whereunder
a spearhead
front can appear (the condition
6<<1
is
not necessary
for this kind of a solitary wave [28]).
The difference with the idealized system considered
in 93.4 is that there is no finite high-temperature
quasistationary
state at constant concentration,
so that, once the instability
threshold
has been passed, thermal runaway follows, and temperature
rises to levels far above
stationary
values (what is called the "wrong way" phenomenon).
Due to this peculiarity,
the propagating
hot front, once excited at the center of the particle,
does not retain
its shape, but grows steeper as it feeds on fresh reactant while propagating
towards
the periphery
The front is stopped near the cooling boundary,
and, after reactant
has been burned off (the slow concentration
variable relaxed), bounces back into the
interior fading away as it retreats
(Fig.7).
This is, qualitatively,
the picture of
oscillations
at Le<<l , it is very much the same as one described
in the end of 93.4
Of course, nothing like this can happen at Le>>l, H-l
when the thermal wave is slow
and cannot raid reactant-rich
locations
unchecked.
Propagating
and stationary
thermal fronts play an important
role in operation
of fixed
bed reactors
[165,179-1831.
Such fronts separate regions where catalytic
particles
operate in high-temperature
(diffusional)
and low-temperature
(kinetic) regimes, and
slowly propagate
due to low, albeit finite conductivity
of the catalytic bed
Qualitatively, these "creeping
profiles"
can be considered
as solitary waves in the onevariable
reaction-diffusion
model [scalar Eq.(2)] propagating
in a parametric
field 1
L h4 PISMEN
1968
slowly varying in space (flow composition
and temperature)
and/or in time (catalytic
activity).
In a bed of constant activity,
the creeping front will arrive at a stationary position if it finds a cross-section
with such values of flow parameters
that the
"Maxwell construction",
Eq (71, be satisfied.
This is a clear-cut example of front
immobilization
in an anisotropic
reactor discussed
in general terms in 13.4. Unlike
the isotropic
case, immobilization
is possible within a wide range of process parameters, as it is seen in Pig.6b 11791.
It can be expected that changes in catalyst's
activity would keep the front in permanent creeping motion following shifts in the position of the "Maxwellian"
cross-section.
Actual picture, however, may be still more complicated.
Characteristic
times of
catalyst modification
are typically
long; on this scale, the autocatalytic
variable temperature
- becomes fast, and creeping profiles graduate into the category of speartheory, such combination
of scales is
head fronts. As we know from the qualitative
conducive
to "wrong way" and oscillatory
phenomena.
This mechanism
is apparently
responsible
for excessive
hot spots observed by Hlavacek and co-workers
during reversible
deactivation
of catalyst beds 11833
Kinetic systems with instabilities
of other than thermokinetic
origin are acquiring
now more attention
in the reactor literature
[184-1871. We can expect this interest to
grow, especially
due to studies of reactions
on inmnobilixed enzymes where isothermal
kinetics
is rich, and many kinds of instabilities
may be observed
[188]. Steady state
multiplicity
results already from simple Langmuir-Hinshelwood
rate expressions
where
forward inhibition
is caused by competition
of reactants
for catalyst sites. Calculations of multiple effectiveness
factors in such systems date back to the 60's [1891,
but all dynamic possibilities
hidden in this simple kinetic scheme, still remain
unexplored.
When autocatalytic
chemical mechanisms
are combined with heat effects, a
variety of non-trivial
dynamic phenomena
can be expected,
due to natural separation
between characteristic
diffusional
and thermal scales. These effects can be obtained
even in endothermic
systems, since temperature,
being a soft controlling
variable,does
not need to be autocatalytic.
4.5
E1ectrochemistry.Photochemisti-y
which are quite often met in electrode
Non-monotonic
current-voltage
dependencies,
can cause, in conjunction
kinetics,
with diffusional
transport,
a full spectrum of
kinetic instabilities.
There is ample literature
on oscillations
in electrochemical
systems
[17,190].
Patterns and waves of kinetic origin do not seem to be given much
aside from the theory of nerve conduction
(84.6). But
attention by electrochemists,
distinct excitation
there is an old expressive
example of an "iron nerve" transmitting
This phenomenon
is caused by passivation
mechanisms
modifying
electropulses [191].
chemical deposition/dissolution
on the surface of an iron wire in a nitric solution.
attention has been drawn to the fact that most celebrated
oscillatory
reactRecently,
and
ions of homogeneous
kinetics
(14.1) involve ions, rather than neutral molecules,
have to be described
accordingly
[192].
This approach has lead to very interesting
predictions
of influence
of electric field on motion of kinetic waves [193j. Interaction of diffusion
and electric field plays an important
role in membrane processes
[194,1951.
Spontaneous
symmetry breaking
and kinetic waves were observed and described
theoretiSensitivity
of complex dynamic phenomena
of
cally in illuminated
systems
[196,1971.
chemical kinetics to illumination
and electric fields may have in the future many
in the design of novel measuring
and control
surprising
applications,
especially,
devices.
4.6.
Nerve
conduction
A huge body of work on kinetic instabilities
has been accumulated
in the theory of
which for a long time was developing
quite independently
of the
nerve conduction,
related studies in chemical kinetics and reactor theory. There are several good reviews
A recent Russian review of kinetic instabilities
[2031 is
of this field [198-2021
largely based on the studies connected with the problem of propagation
of nervous
impulse.
After
the celebrated
Hodgkin-Huxley
model
of nerve
conduction
[204] had been
created,
Kmetrc mstabllltles
m man-made
and natural reactors
1%9
the arrow of theoretical
development
went towards simplest imaginable models which
would still give the characteristic
dynamic behavior
of nerve fibers [205,206]. Most
of the studies are based on different versions
of Eq.(6) with the diffusional
term in
to introduce
a
the first equation dropped. As it has turned out, it Is sufficient
the slow variable may enter
cubic or piecewise-linear
autocatalytic
nonlinearity;
kinetic functions but linearly
A kinetic model of nervous
impulse has to predict the following qualitative
behavior
excitation
requiring
a localized but finite-amplitude
signal; (2) prop(1) "threshold"
agation with a constant speed; (3) relaxation
period, during which no repeated
Qualitative
analysis along the lines of 53.4 shows that all
excitation
is possible.
these features can be obtained
in the framework of the two-variable
model, Eq.(6) with
a phase diagram of the type shown in Fig.3b and scale ratios ensuring existence
of
neither detailed form of
Given these conditions,
spearhead
fronts (~<<l , w>>l).
nor relative diffusional
range of both variables
influence
qualikinetic functions,
tative behavior.
No wonder that propagation
of excitation
pulses can be reproduced
in
kinetic systems much more primitive
than the real nerve filament, starting
from the
Some more complicated
"iron nerve"
[191] and going to the BZ reaction
(14.1)
Ostwald's
of a train of pulses 12071 can be explained
only in the
phenomena,
like "bursting"
framework
of a three-variable
model.
The same systems can reproduce
the occurrence
of rotating waves in two-dimensional
in view of the hypothesis
excitable
media-This
phenomenon is especially
significant,
of Wiener and Rosenblueth
[208] indicating
such spiral waves as the cause of
fibrillation
in the heart muscle.
4.7.
Crystallization
Polymerization
Oscillatory
behavior
of industrial
crystallizers
is well-known
[209-2111. But observations of spatial symmetry breaking
in crystallization
processes has still longer
history going back to the discovery
of Liesegang
rings 12121. The modern theory of
this phenomenon
[213-2153 expounds periodic precipitation
as one more manifestation
of
kinetic instabilities
leading to spatially ordered non-equilibrium
states
Another
example of the same mechanism
in action has been provided by discovery
of layered
mineral deposits formed by precipitation
from a mixture of perfectly mutually
soluble
components
12161.
it has to be pointed out that the kinetic symmetry breaking
To avoid misunderstanding,
has nothing
to do with the symmetry breaking of thermodynamic
origin which occurs any
time when space-ordered
crystals emerge from homogeneous
liquid. Crystallization
can
be also influenced
by instabilities
of hydrodynamic
origin which would lead to
In particular,
the hexagonal
pattern of snowflakes
emergence
of spatial order as well.
has much more to do with the hexagonal
form of B&nard cells than with the shape of
water molecules
[217]
Polymerization
is also often an oscillatory
process [218,219]. Another kind of kinetic
instability
is gelation in branched
condensation
and chain polymerization
[220,221]
as
Formation
of "secondary
structures"
in polymers, which could have been classified
symmetry breaking,is
commonplace;
it always occurs, however,
in a very complex
rheological
set-up, and I am unaware of its being ever attributed
to kinetic instabilities.
Notice, however,
that secondary
structures
in polymerization
actually stand
midway between the corresponding
phenomena
in crystallization
and morphogenesis,
both
of which have been interpreted
as symmetry breaking of kinetic origin
4.8.
Morphogenesis
One of the basic biological
problems
is how differentiated
cells do emerge from uniform
that chemical instabilities
can trigger the
embryonic
tissue. Turing [222] conjectured
morphogenetic
process.
This work gave an impetus to further studies of spontaneous
symmetry breaking by groups of Striven [223-2251 and Prigogine
[3,91,92,226,227].
Striking patterns were obtained in computer studies of a morphogenetically
relevant
kinetic model by Gierer and Meinhardt
[93-951. It was found that pattern formation
results from the combined action of two "morphogens"
* a slowly diffusing
"activator",
i e. an autocatalytic"reactant,"
and rapidly diffusing
"inhibitor".
This coinsides
with the results of qualitative
theory described
in 53.5. There are indications
that
patterns
arising in an extended homogeneous
reacting medium should be slightly meta-
L
1970
M
PISMEN
stable
in the framework
of a two-variable
model
1331.
Fixation
of incipient
inhomogeneities
may
involve
slower
formation
of more
enduring
structures
resembling
the action
of "landscape
variegator"
(93.5).
Such
secondary
differentiation
is widely
discussed
in morphogenetic
literature
[228].
Of course,
simple
twoor three-variable
models
are
but
caricatures
of real
biological
processes;
but sometimes
a good
caricature
catches
the likeness
better
than
an elaborate
portrait
4.9.
Ecology.
Spreading
230].The
propagates
not bring
Population
genetics
of advantageous
genes
is a classical
problem
of propagating
fronts
[229,64-67,
difference
with
the combustion
theory
is that
the stationary
state
the front
into
(absence
of the advantageous
mutant)
is unstable.
This,
however,
does
qualitatively
significant
changes.
Another
classical
kinetic
problem
of ecology
is oscillatory
behavior
in prey-predator
systems.
The famous
Lotka-Volterra
model
[231,232]
gave
actually
the first
example
of
kinetic
oscillations.
The model
itself
has
one quite
unacceptable
property.
It is a
like
dynamic
systems
of classical
mechanics,
and
Hamiltonian
(conservative)
system,
has
a continuum
of neutrally
stable
orbits.
This
behavior
is unrealistic
and structurally
unstable,
and the original
model
has
to be modified
by additional
nonlinear
terms
to get a properly
stable
unique
limit
cycle
All
ecological
dynamic
models
are easily
translated
into
schemes
of chemical
kinetics
which
would
have
a general
appearance
of branched
chain
reactions
or, in some more
sophisticated
versions,
of Langmuir-Hinshelwood-Michaelis-Menten
reaction
mechanisms.
which
has
prevailed
in ecology
until
recently,
are fully
equivalent
Lumped
models,
to CSTR
models
[12,233,234].
Two basic
ecological
systems
have
qualitatively
different
dynamic
features.
Preypredator
systems
are characterized
by phase
diagrams
of oscillatory,
excitable
or
in accordance
with
the qualitative
theory
of 513.4,
3.5,
bistable
synclinal
type,
and,
can exhibit
oscillations,
waves
and spontaneous
symmetry
breaking.
On the other hand,
systems
describing
two competing
species
or two symbionts
are of synclinal
type,
and
the most
prominent
feature
of their
dynamic
behavior
is bistability
and propagating
solitary
waves marking
advance
of the fittest
species
Survival
of the fittest
competitor
is the basic
kinetic
mechanism
of Darwinian
evolution
[235]
Oscillations
due to prey-predator
interaction
are wide-spread
in nature
In distributed
oscillatory
behavior
is translated
into
"flight
waves".
Spreading
of
systems,
epidemics
is also
described
by kinetic
waves
appearing
in prey-predator
(host-parasite)
models
[237].
Spontaneous
symmetry
breaking
is another
phenomenon
common
for such
models,
with
prey
playing
the role
of the"activator",
and predator,
of the "inhibitor"
[883.
Large-amplitude
inhomogeneities
due
to the "global
regulator"
mechanism
(§3.5)
can be easily
realized
in prey-predator
models
with
intraspecific
interactions
(for
an island
where
"rapidly
mixing"
example,
rabbits
are feeding
on the "slowly
diffusing"
grass
can be spontaneously
separated
into
the "meadow"
and "desert"
zones).
This
is one
of the mechanisms
of heterogenization
of environments
which
permanently
occurs
in
- what
is known
as "patchiness"
or "polyclimax"
[236].
Such
spatial
patterning
nature
has enormous
evolutionary
significance,
since
it greatly
enlarges
the variety
of
species
which
adjust
each
in its own way
to conditions
prevailing
on different
patches
and thus
bring
about
still
deeper
heterogenization
of habitats.
Recently,
ecologists
have
discovered,
to field
workers'diamay
and
theorists'
amusement,
lumped
ecological
models
can
that
some
very
simple,
and quite
realistic
at that,
exhibit
chaotic
behavior
[238-2421
In the field
where
reliable
measurements
are very
difficult
and subject
to a high
level
of noise,
this mathematical
fact
sounds
rather
we chemical
engineers
deal with
easily
definable
and
disquieting.
Comparatively,
accessible
systems
- which,
nevertheless,
bring
us many
surprises
and have
still
more
in stock.
4.10.
Other
related
problems
There
are many
non-equilibrium
systems
described
by nonlinear
equations
other
than
the
exhibit
similar
qualitative
behavior,
including
reaction-diffusion
equations
(1). which
first
of all,belong
hydrodynamic
nonlinear
waves,
spatial
patterning
and chaos.
Here,
instabilities
due to the non-linearities
in the Navier-Stokes
and convective
diffusion
Kmetx
mstabhtles
m man-made
1971
and naturalreactors
studied nonlinear
non-equilibrium
system is the
equations
[51. Another intensively
equations
of nonlinearly
interacting
waves closely resemble
laser 163 The amplitude
Similar equations
are prominent
in the modem
nonequations
of diffusional
kinetics
linear field theory which aspires to grasp both the nature of elementary
particles
and
the structure
of the Universe
as a whole [243], The theory of kinetic instabilities
does indeed touch upon the deepest problems of Being, as we dare to think of the
evolving from the prImeva
bifurcation
point.
ultimate natural reactor,
the Universe,
5. CBAOS
The most tantalizing
kind of kinetic instabrlity
is the chaotic behavior
of deterministic systems. Devoid of Its nearly mystical
appeal, chaos can be viewed as still another
kind of an asymptotic
state differing
from stationary
and ordered states by its
"dimensionality".
A HSS 1s represented
by a point in the phase space; it is zeroNext comes a limiting
dimensional.
A HOS is a one-dimensional
limiting trajectory.
trajectory
densely covering a two-dimensional
surface in the phase space or a hyperA phase point moving along such trajectory
never returns
surface of higher dimension
though it will repeatedly
pass in their infinitesimally
to its past positions
exactly,
close vicmity.
The motion of this kind can be generated just by superimposing
two
This is not yet a chaos; a truly
periodic motions with incommensurate
frequencies.
asymptotic
orbit (called a "strange attractor")
chaotic, as opposed to quasiperiodic,
12443 has to satisfy two roughly equivalent
conditions'(l)
the system be able to
"forget" Initial conditions,
starting at neighborlng
in other words, trajectories
points should widely diverge with time, (2) there be a continuous
frequency
spectrum
The above conditions
can be satzsfied
as a result of non-linear
coupling of two
oscillators
Poincarg was the first to notice this fact [2451. The most extensively
studied simple system able to sustain deterministic
chaos was devised
[2461 by expanding into a Fourier series and severely truncating
equations describing
the Bgnard
of equations
convection
This third-order
system, though having an outward appearance
wealth of
of chermcal kinetics
and looking deceivingly
simple, retains the qualitative
behavior
of the original hydrodynamic
equations,
and is able to simulate the transition
turbulence
at high Rayleigh numbers.
For purely geometrical
reasons,
three is the minimum number of variables
required to
obtain deterministic
chaos in a continuous
lumped system. But in a discrete system,
the equation as simple as
even a single state variable may suffice. Astoundingly,
U
with a function
f(u) humped like
provided
the hump is steep enough
n+l
=
in Fig.8a
[247].
f
(u,)
can display
(10)
very
complicated
behavior,
Equation
(10) was studied by ecologists
as a population
model of single species with
non-overlapping
generations
and crowding effects bringing
the birthrate
down at high
densities
[238-2421. Much wider implications
can be, however,
seen if we return to
continuous
systems and consider Eq.(lO) as a Poincarg map connecting
positions
of a
phase point after subsequent
cycles [248-2501.
The following
example illustrates
this
method.
Consider a system with one rapid (z) and two slow (x,y) variables.
Let the behavioral
surface have a folded shape with a cusp (Fig 8b),and suppose that there are no
attractors
at either fold, so that the phase point is repeatedly brought
to the edge
of the fold and dropped onto the alternative
sheet, which yields a typical picture of
relaxation
oscillations.
Whether these oscillations
are ordered or chaotic, can be
decided by seeing how one of the edges is mapped on itself after a relaxation
cycle. A
map similar to one depicted in Fig.8a (wfth
u measured
from the cusp node on an
arbitrary
scale) would require rotating and folding a segment of the edge. This can be
effected by placing an unstable
focus on the upper fold, like in Fig.8b.
PoincarC
maps corresponding
to either periodic or chaotic behavior
can be obtained by modifying
the flow on either sheet
It 1s obvious that the chaotic behavior
is "robust" or
structurally
stable, since it would not disappear
as a result of slight changes in the
dynamic equations.
Deterministic
chaos is not an exotic phenomenon
at all, and, probably,
is met much more
often than reported.
The catch is that it may be impossible
to tell whether
the observed irregular behavior
is caused by extraneous
noise, hidden variables,
experimental
L
1972
PlSMEN
ht
(a)
Fig
8. A Poincar; map
(a) and the corresponding
flow diagram,
(b) generating
the projection
of a folded behavioral
chaotic behavior.
Fig. 8b represents
surface with the cusp 0. Trajectories
on the upper sheet are shown by solid
lines; there is an unstable
focus on this sheet, marked by a star.
Dashed lines show trajectories
on the lo WE k sheet.
p-
A-
.O
0.40
1.00
/3- 0.80
A-
I
I
I
0,s
1.0
;
1.5
p
,o
.O
0.5
1.0
1.5
X
2.0
Lo
1,s
0.40
Fig.9.
.o
0:s
1.00
2.5
I
Chaotic trajectories
generated by sampled-data
control of the level of one
variable
(Y) around an
unstable stationary
state.
The situation visibly
deteriorates
with growing
control parameter,
6, and
sampling period, A .
Kmetlc mstabhtles
m man-made
and natural reactors
1973
The very
nature
of chaos
is such
that
it looks
dynamics
errors
- or by the intrinsic
chaos
is actually
a very
complicated
structurelike
a random
process.
At the same
time,
a pattern
brought
to such
a baroque
sophistication
that
its orderliness
cannot
be
Studying
the process
dynamics
in a wide
range
of parameters
may
recognized
any more.
deterministic
if confined
to a certain
give
a hunch
that
the observed
chaos
is indeed
This would
not yet qualify
as a proof,
range
of parameters
and disappears
elsewhere.
since
other
spontaneously
emerging
dynamic
features,
like
appearance
of
however,
frequencies
and an outward
picture
may be responsible
for additional
metastable
states,
as well
as for enhanced
sensitivity
to the background
noise.
Paradoxically,
of disorder,
the more
precise
technique
is required
the more
erratically
the dynamic
system
behaves,
for its proper
identification
We can foresee
circumstanControl
systems
may be especially
prone
to chaotic
behavior
ces under
which
an apparently
rationally
designed
control
system
aimed
at sustaining
an
unstable
stationary
state
will
turn
instead
into
a source
of chaos
Chaotic
behavior
arises
in a very
natural
way
in sampled-data
ition
to chaos
is prompted
in this
case
by the higher
dynamic
as opposed
to continuous
systems.
Transcontrol
systems.
complexity
of discrete,
Figure
9 [251]
shows
chaotic
tra3ectories
obtained
by sampled-data
proportional
control
[252]
of one of variables
(y) in a lumped
reactor
which
naturally
would
operate
in an
Instead
of reducing
the amplitude
of the limit
cycle,
the control
oscillatory
state.
action
leads
to destabilization
and randomization
of phase
trajectories.
The chaotic
component
is felt
even
stronger
as the control
parameter
mcreases.
When
the period
of
a peak
in chaotic
destabilization
due to a resonance
with
the
sampling
is changed,
Chaos
of this
kind
may be typical
for regulation
of
innate
frequency
is observed.
Such
a system
(which
may be
complex
systems
where
only
few variables
can be monitored.
anything
from
a polymerization
reactor
to a national
economy)
often
cannot
be left
alone
for a period
needed
to recognize
its
inherent
dynamic
features,
but
chaos
induced
This
is a classical
case
by control
smears
the intrinsic
pattern
beyond
recognition
of an evil
circle
where
ignorance
creates
chaos
and chaos
perpetuates
the ignorance.
the situation
may not be as bad,
With
the continuous
control,
aware
that
regulation
generally
makes
the system
more
complex
so
variables
As the dimensionality
of the phase
space
grows,
orbit
do, and a limit
cycle
may
give way
to a chaotic
state.
but we have
to
by introducing
may
that
of an
be always
additional
asymptotic
It is known
This
is especially
likely
to happen
if the controlling
input
is periodic.
is still
more
important,
that
periodic
operation
may
improve
yield
[lo],
and, what
sometimes
can stabilize
an otherwise
unstable
state
[253,254].
On the other
hand,
however,
periodic
forcing
of an intrinsically
oscillatory
system
may
lead
to chaos
[255,256].
we can expect
that
in distriIf chaos
arises
rather
easily
in simple
lumped
systems,
buted
system
it will
be met
even
more
often.
Poincarg's
coupled
oscillators
can be
realized
as two interconnected
CSTR's
under
conditions
when
both
operate
in an
oscillatory
regime
Both
experimental
[131,134]
and computational
[111,112]
evidence
supports
existence
of this
kind
of chaos.
Deterministic
chaos
in distributed
kinetic
systems
has also
been
obtained
computationally
[258-2601
Bifurcation
analysis
can show
the way
to parametric
regions
where
chaotic
behavior
has
to be expected
[259,63].
This
most
complicated
form
of dynamic
behavior
should
be properly
called
kinetic
turbulence.
It is closely
related
to hydrodynamic
turbulence
and other
chaotic
phenomena
originating
from
deterministic
nonlinear
partial
differential
equations.
Turbulence
may be the only major
problem
of macroscopic
physics
where
even
the basic
questions
remain
unresolved,
and reaction-diffusion
systems
are drawing
close
attention
of theorists
as a testing
ground
for probing
into
the nature
of deterministic
chaos
6.
We have
origins,
prominent
fronts,
diffusion
OUTLOOK
scanned
a variety
of phenomena
which,
in spite
of the diversity
of their
have
a lot of common
with
each
other.
If asked
which
single
feature
is most
in the dynamics
of distributed
reactors,
I would
name
propagating
transition
that
particular
kind
of a solitary
wave
appearing
in nonlinear
reactionequations.
Propagating
fronts
effect
transitions
between
alternative
states
1974
L
M
PISMEN
of a multistable
kinetic
system;
when
coupled
or arrayed,
they
form
excitation
pulses
or wave
patterns;
when
immobilized,
inhomogeneous
stationary
states,
their
colliding
and interacting
may
lead
to the onset
of kinetic
turbulence,
and their
undulating
motion
can be responsible
for oscillatory
behavior
Bringing
together
experimental
and computational
facts
assembled
in different
fields
and reassessing
them
from
the point
of view
of the unified
qualitative
theory,
gives
an overwhelming
evidence
of the wealth
of complex
behavioral
modes
and self-organization
phenomena
induced
by kinetic
instabilities
in reaction-diffusion
systems.
When
encountered
by a particular
kind
of dynamic
behavior,
we are able
now
to decide
which
basic
model
would
be sufficient
to describe
this behavior
qualitatively.
Multistability,
propagating
fronts
and symmetry
breaking
in compartmental
systems
can be
explained
in the framework
of a model
with
a single
autocatalytic
variable.
Two
variables,
with
kinetic
functions
of a certain
general
character
and with
a proper
ratio
of characteristic
scales,
are sufficient
to account
for oscillations,
excitation
pulses,
waves,
undulating
fronts,
synunetry
breaking
in continuous
systems
and even
Three
variables
are required
chaotic
behavior
in compartmental
or distributed
reactors.
The origins
of temporal
and
to obtain
some
more
exotic
and largely
unexplored
effects.
spatial
order
are fairly
clear
by now,
but we still
know
too little
on the nature
of
chaotic
states.
A basic
qualitative
model
is still
a far cry from
a precise
model
which
would
provide
a quantitative
agreement
with
the experiment.
Kinetic
mechanism
cannot
be inferred
from
the observed
picture
of kinetic
instabilities
and self-organizing
structures,
just
Complex
dynamic
behavior
does
because
of the universal
character
of these
phenomena.
A basic
model
relates
to an
not imply
a complex
kinetic
scheme,
and vice
versa.
elaborate
kinetic
mechanism
like
the primitive
Marconi's
model
to a modern
expensive
the principle
of behavior
is the same.
hi-fi;
the difference
is obvious,
but
the heart,
While
the theory
and the academic
experiment
closely
interact
and support
each
other,
there
is an obvious
symmetry
breaking
between
academic
research
and industrial
practice
in the reactor
field.
I think
I can conJecture
a kinetic
mechanism
responsible
for
this behavior.
Self-organizing
processes
exhibiting
a wide
spectrum
of kinetic
instabilities
are strongly
negatively
selected
in applied
and industrial
research.
It
Introducing
additional
is not that
such
processes
are inefficient,
quite
the opposite.
in the case
this
feedback
is
feedback
loops
can
save
energy,
space
and materzal,
however,
copious
control
problems.
positive
or delayed,
it would
bring,
Nobody
can tell
how many
efficient
but unruly
processes
may be found
in the wasteOn the other
hand,
nature
"knows"
how to
basket
of chemical
and engineermg
research.
regulate
processes
with
complex
dynamics
and utilize
their
enormous
potential,
indeed,
she uses
self-organizing
reactions
themselves
as the means
of control
and communication
self-organizing
processes
are strongly
positively
selected
in nature
As a result,
This
gap
but not
complex
improve
variety
of man-made
and natural
reactors
is very
"principles
of design"
between
We can hope
that better
understanding
of the dynamics
of
unsurpassable.
processes
can facilitate
a future
overhaul
of the reactor
control
theory
dramatically
our abilities
to design
and control
chemical
processes
with
of in-built
feedback
loops
and rich
qualitative
behavior.
deep
and
a
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