Acta Polytechnica


doi:10.14311/AP.2018.58.0128
Acta Polytechnica 58(2):128–143, 2018 © Czech Technical University in Prague, 2018

available online at http://ojs.cvut.cz/ojs/index.php/ap

ŽAMPA’S SYSTEMS THEORY: A COMPREHENSIVE
THEORY OF MEASUREMENT IN DYNAMIC SYSTEMS

Renata Rychtáriková∗, Jan Urban, Dalibor Štys

University of South Bohemia in České Budějovice, Faculty of Fisheries and Protection of Waters, South
Bohemian Research Center of Aquaculture and Biodiversity of Hydrocenoses, Kompetenzzentrum
MechanoBiologie in Regenerativer Medizin, Institute of Complex Systems, Zámek 136, 373 33 Nové Hrady,
Czech Republic

∗ corresponding author: rrychtarikova@frov.jcu.cz

Abstract. The article outlines in memoriam Prof. Pavel Žampa’s concepts of system theory which
enable us to devise a measurement in dynamic systems independently of the particular system behaviour.
From the point of view of Žampa’s theory, terms like system time, system attributes, system link,
system element, input, output, sub-systems, and state variables are defined. In Conclusions, Žampa’s
theory is discussed together with another mathematical approaches of qualitative dynamics known
since the 19th century. In Appendices, we present applications of Žampa’s technical approach to
measurement of complex dynamical (chemical and biological) systems at the Institute of Complex
Systems, University of South Bohemia in České Budějovice.

Keywords: system theory, dynamic system, theory of measurement, complex systems, cybernetics.

1. Introduction
Human thinking has been evolving to be constructive.
It creates models of the Nature perceived by senses, di-
rectly or indirectly, and communicates them [1]. The
overall spread of technical products makes them a
kind of language which is much more general than
any of the national languages. The only competi-
tor of technology in the field of general languages is
mathematics which has the advantage of not being
connected to a particular technical solution.
For an overwhelming part of humans, the mathe-

matical notation is difficult for comprehension which
leads to the preference of representation of mathe-
matical results by the approximation of formulae to
functions of technical tools, in addition, in the current
computer age, in the form of computer visualizations.
This leads to identification of these technical con-
structs with mathematical expressions. Such a way of
thinking has been more widespread in theoretical and
experimental physics than in technology, chemistry
or other sciences which are closer to reality. In latter
sciences, the mathematical formulae are rather useful
constructive tools whose outputs need to be further
verified experimentally.

No mathematical physicists of the late 19th century
understood their equations as a model of any ultimate
truth. They developed mathematical tools as mutu-
ally consistent descriptions. In the beginning of the
19th century, there were many thinkers who criticized
concrete mechanistic approximations and suggested
that theories should be purely mathematically de-
duced [2]. The nicest expression of this approach can
be found in Poincaré’s Science and Method [3] where

is written “. . . it is economy of thought that we should
aim at, and therefore it is not sufficient to give models
to be copied.” In other words, in the Nature, there
seem to be general rules, which are more constrained
than the intellectual freedom of mathematicians, and
mathematics is a way how to describe them.
The observation of qualitatively dynamic rules in

the Nature became known by the advent of Poincaré’s
qualitative dynamics in the late 19th century [4, 5].
It showed that non-linear dynamic systems can be
highly unpredictable so that, upon small changes in
starting conditions, their behaviour can be deviated
wildly. However, in contrast, a phase space of a dy-
namic system is always segregated into zones of at-
traction [4] within which the system travels into a
limit set – a region of space within which it stays
forever. The famous Poincaré-Bendixson Theorem
applied to a two-dimensional space [6, 7] states: “In a
differentiable real dynamic system defined on an open
subset of a (two-dimensional) plane, every non-empty
compact ω-limit set of an orbit, which contains only
finitely many fixed points is either a fixed point, a
periodic orbit, or a connected set composed of a fi-
nite number of fixed points together with homoclinic
and heteroclinic orbits connecting these.” At higher
dimensions, you can have other types of behaviour.
For instance, chaotic behaviour can only arise in con-
tinuous dynamic systems whose phase spaces have
three or more dimensions. In other words, majority
of the dynamic objects do not die but, for infinite
amount of time, occupy a certain volume of space.
Understanding of the Nature can mean a classification
of dynamic systems. For this classification, we must
understand the process of measurement.

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vol. 58 no. 2/2018 Žampa’s Systems Theory

Apart from outlining the work of Prof. Pavel
Žampa1, this article aims to demonstrate that it is
still, or perhaps increasingly, possible to devise a mea-
surement in dynamic system with a full mathematical
rigour. Žampa devised a rigorous concept of measure-
ment which is independent of the particular system
behaviour, including any observer, and explores limits
of experimental cognition.

2. The system
Žampa concluded that the systems theory is not stabi-
lized in full exactness2. Some authors understand it as
a theory of real systems [10] while others understand it
more as a theory of abstract models of real systems [11].
. . . Some of the theories are too narrow and do not
include all occurring examples [11–13], while others
are too broad and in principle inconsistent [14–16] and
must be corrected in an appropriate manner. . . . In
most cases, there are subjectively motivated mathe-
matical requirements which can reflect real needs only
sparsely [17]. This situation requires a urgent solution,
mainly in case of continuous systems, in particular
the stochastic ones.

Žampa constructs his systems theory by definition
of a class of abstract systems which model all real sys-
tems. An abstract system exists for each real system.
Next, Žampa addresses the question how such a

general adequate system should look like. He con-
cludes that a system, demarcated on the basis of the
given paradigm, has no input which could influence
the system in any way. It says that an input and its
properties are part of the system. Instead, as shown
later, the system can be split into sub-systems which
are connected by bonds. The latter approach enables
the system to be represented by measurable quantities
at any time.
Certain substantialities, which are perceived via per-

ceptions and caused by outer environment, are consid-
ered as natural phenomena. They demonstrate certain
qualities of properties of this environment which is
represented by certain objects. Such a quality is, e.g.,
temperature, location or speed of a certain object. We
do not perceive only one perception but their ensembles
which we organize in our minds in a certain way and
between which we find certain relations. It is rather
a certain construct in our mind, which attributes our
concepts about the organization of the world to the
real world, than a perception of real relations between
objects.

1Pavel Žampa (1936–2006), a former head of the Department
of Cybernetics at the Faculty of Applied Science, University of
West Bohemia in Pilsen, spent his life in developing technical
cybernetics. He graduated in 1962, gained his PhD in 1973,
became an assistant professor in 1990 and a full professor in
1997. Since 1990, his publications in English are found to be
only sparse [8]. The most comprehensive source of his ideas is
the revised version of his habilitation which he adopted for his
co-workers and students in late 1990s [9].

2In the text, direct translations of Žampa’s work are high-
lighted by italics.

While the natural phenomena can be, to a certain
extent, perceived immediately, their mutual relations
cannot. Therefore, a recognition of these relations is
much more complicated than the recognition of natural
phenomena themselves. In this way, the definition of
real system depends on our reasoning.

In general, we assume that a set of all events con-
tains primary events, which we immediately consider,
and secondary events, which are all other events by
which the primary events are affected. It depends on
our intuition and, in general, on the correctness of
our concepts about the system which events are in-
cluded into the system. In any case, it is a matter
of compromise between the required precision and the
complexity of the model. In fact, it is always an incom-
plete set. However, the only way how to demarcate the
system is via the approach of trials and errors which
is stopped upon achievement of a certain precision. In
this way, we obtain a set of events which we further
consider as independent of all other events. This does
not mean that events in the system could not affect
events outside the system. Such an affection should be
only unidirectional and should not affect the behaviour
of the system in any way.

In addition, Žampa’s systems theory is an examina-
tion of the role of measurement in the system analysis.
Moreover, it offers a methodological approach to the
mathematical description of cognition. This would
like to be demonstrated on the selection of items from
Zampa’s work.

2.1. Abstract system
2.1.1. System time and attributes
System events occur at certain objects of the real world
and represent themselves by changes of certain quali-
ties which we are able to perceive in a certain way and
organize in our mind. A special quality which enables
us to organize these events is time.

Neither us humans, nor any measuring instrument
is able to record external quantities in infinitely many
time intervals. An adequate model of the perceived
time is a real time t whose definition set

t ∈ T (1)

is a non-empty set T of all time events with K as an
appropriate index set (Figure 1). If we stay in the
realm of experimentally verifiable realities, we assume
only a limited set of experimental times from 0 to F ,
i.e.,

K = {0, 1, 2, . . . , F}. (2)

In the same time, we require that a complete sharp
order

t0 < t1 < t2 < · · · < tF , (3)

which we shall interpret as a precedence of one time
instant before the other, is defined at the set T .
It should be emphasized that it is not necessary

to interpret the set T as the set of time instants.
Sometimes it is more appropriate to assume it as a

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R. Rychtáriková, J. Urban, D. Štys Acta Polytechnica

Figure 1. The scheme of the concept of the sys-
tem time, system attributes, and variables. The
system time T is an ordered set of time instants ti,
i ∈ {0, 1, 2, . . . , F}, in which a sharp order exists. The
system time can be also understood as an order in the
set of system attributes and variables. The system
attribute ai,j , i ∈ {0, 1, 2, . . . , F}, j ∈ {1, 2, . . . , n}, is
a certain property of the system, e.g., the position
in space, velocity, acceleration. The system variable
vi,j , i ∈ {0, 1, 2, . . . , F}, j ∈ {1, 2, . . . , n}, is a value
of scalar or vector character (depicted) by which the
system attribute is characterized, i.e., the value of x,
y, and z position, velocity vector, etc.

set which determines the order of events in a given
system. A special case can be a set which contains
only one element. It is important to realize that the
system is defined only at the set T . Outside this set,
the system does not exist for us.

An abstract system, a good model of a real system,
has to include models of all its attributes. Abstract
attributes will be denominated by symbols

ai, where i = 1, 2, . . . , n, i ∈ I (4)

and denote names or designations of studied attributes
such as coordinates of position, coordinate of speed,
position of a switch, and verity of a statement. A set
of all abstract attributes is designated by symbol A.
An adequate model of such a i-th attribute ai is an
abstract variable of the i-th attribute

vi ∈ Vi, where i ∈ I, (5)

whose definition set is a non-empty set Vi with ele-
ments called values of the i-th attribute.
Žampa’s repeated emphasis on the distinction be-

tween the system, the measurable variables and model

attributes is the key to understanding the contempo-
rarity of Žampa’s thoughts. The fact that Žampa
introduced the mathematical formalism into these
general philosophical terms enabled him to formal-
ize the whole theory and measurement of dynamic
systems.
In Appendix A, Figure 7 shows an example of a

system’s simulation where we know the elementary
time step. The trajectory which is in detail reported
in [18, 19] is by statistical analysis segmented into
regions of oscillations between two clusters which can
be considered as natural sections of the system’s evo-
lution. The time decimation of the system trajectory
influenced the result of statistical analysis. This is
due to a dominating low-frequency oscillation which
affects the value of the variable point divergence gain.
By the decimation of the series, i.e., of the evolving
system, we loose information about this low-frequency
component. The time decimation probably disables to
infer the correct model of the system from the given
dataset.

2.1.2. System trajectory
The obvious property of the dynamic system is its
trajectory. To define it, we shall first introduce the
term system variable v

v = (v1, v2, . . . , vn), (6)

which is an ordered set of n variables of system at-
tributes. Its definition set

v ∈ V, where V = V1 × V2 × · · · × Vn. (7)

An ordered tuple (T,V) is then the basis of the math-
ematization of the definition of the system trajectory.
In order to define the term system trajectory, we
must state that, at each system time instant t ∈ T ,
each attribute ai, i ∈ I of a real system acquires ex-
actly one value vi ∈ Vi. A system trajectory (Figure 2)
is then a mapping

z : T × I →
⋃
i∈I

Vi such that z(t, i) ∈ Vi,

where i ∈ I, (8)

from the set T into the sets A and V .
The mapping z is not generally unique. The set

of all system trajectories will be designated Ω. If
the trajectory realized by the system falls into a given
sub-set B of the set Ω, i.e., B ⊂ Ω, we shall say that
an event B occurred on the system. In all other cases,
the event B did not happen.
The system event is usually a set of trajectories z

which have a certain property V (z). For instance, if
at time tk the trajectory of the attribute ai passes a
point vi = z(tk, i), we can say that the event

B = { z | z(tk, i) = vi } (9)

happened. The set of all sub-sets B of the set Ω is
labelled by the symbol B, where B = {B | B ⊂ Ω },
and is called a set of all system events.

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vol. 58 no. 2/2018 Žampa’s Systems Theory

Figure 2. The scheme of a system trajectory
z as a mapping from the set of time instants ti,
i ∈ {0, 1, 2, . . . , F} to the set of attributes ai,j , i ∈
{0, 1, 2, . . . , F}, j ∈ {1, 2, . . . , n}.

3. General abstract system
3.1. Definition
The general abstract system is an ensemble of all
trajectories which are outcomes of the mapping z. If
a set B is given, all its elements as well as the set Ω of
all mappings z are also given. Then, the definition set
and the set of values of this mapping and thus also
sets T , I, Vi, i ∈ I, and V are also demarcated. On
the contrary, by these sets, i.e., by the sets T and V ,
it is possible to define the set B and thus the whole
abstract system. The abstract system will be further
denoted S . The system is unambiguously defined by
a tuple of T and V , i.e.,

S = (T, V ). (10)

3.2. System behaviour
In the previous chapter, system behaviour which, be-
sides other aspects, describes also qualitative relations
between events was defined as the general abstract sys-
tem S = (T, V ). . . it is useful to complement our
knowledge by causal relations between events to deter-
mine, e.g., for two selected events, how the first one
affects the second one and vice versa or whether their
mutual affections are neutral.

Dependencies of this kind will be called qualitative
dependencies. . .

. . . The qualitative relations are very important for
the definition of the orientation of the cause between

Figure 3. The scheme of a system event as a sub-
set Bi, i ∈ {0, 1, 2, . . . , n} of all trajectories Ω which
satisfy a certain condition. For a selected sub-set of
attributes Ai ∈ A, it can acquire a certain set of values
of variables.

system events. They will lead us to the demarcation
of structural terms in the systems theory such as sys-
tem link, sub-system, system element, input,
output, state variables of the system and sys-
tem structure. Later, it will enable us to explain
certain experimental experience which was so far ex-
plained by a mythical rather than scientific argumen-
tation. . .

It should be noted that the term qualitative dynam-
ics was coined by Poincaré about 130 years ago [4, 5].
It deals with the qualitative differences in the be-
haviour of non-linear systems, in Poincaré’s times
mainly continuous. But it was Poincaré himself who
showed how, using so-called Poincaré sections, the
continuous systems can be turned into discrete deter-
ministic and discrete stochastic systems [4]. Žampa
had obviously limited knowledge of these findings, if
any. He came up with an idea of qualitative relations
in dynamic systems and held it from a quite different
point of view.
Poincaré concludes that we are considerably re-

stricted in the scope of general behaviour of dy-
namic systems. Žampa describes how systems can
be constructed in order to be confronted with a
perceivable system. Both describe principal math-
ematical reasons for such restrictions. The current
research on dynamics of discrete systems indicates
that the scope of available models can be even more
constrained [18, 20] and it is sensible to expect an
observation of only a small set of state space trajecto-
ries.

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R. Rychtáriková, J. Urban, D. Štys Acta Polytechnica

3.3. Deterministic systems
An ordered triple

D = (T, V, z) (11)

is called a deterministic abstract system.

3.4. Stochastic systems
If we define P (B), B ∈ B, as a probability of each
event B in the set B of all sets of events, then the
ordered set

P = (T, V, P ) (12)

is called a stochastic (abstract) system.

3.5. Causal system
We generally assume that the system trajectory is
defined as a mapping z with a definition set

D = T × I (13)

which is by its internal mechanism defined in parts.
We assume that a set

I = {I1, I2, . . . , Im}, (14)

is a decomposition of the set I at which is defined
a convenient order < by the order of determination.
Thus

I =
m⋃
i=1

Ii, where Ii, Ij ∈ I,
Ii 6= Ij =⇒ Ii ∩ Ij = ∅. (15)

This aspect is illustrated in Figure 4.
We saw that the trajectory was determined sequen-

tially on different sub-sets of the set D which forms
a certain ordered decomposition D of the set D. It
is enough to define each segment z | Dk,l of the tra-
jectory z exactly once dependently on the segment
z | Ck,l, where

Ck,l ⊂
⋃

(i,j)<(k,l)

Di,j . (16)

This requirement can be interpreted mainly as a cer-
tain natural property according to which the trajec-
tory segment z | Dk,l is determined by only preceding
and not future segments. This is in agreement with the
causality principle, where each event has its own
cause, which precedes its consequence. Further, we as-
sume that the cause Ck,l determines the consequence
Dk,l completely and is understood as a complete
immediate cause of the consequence Dk,l.

In other words, Žampa’s definition is the first com-
plete mathematical – qualitative – definition of a
causal abstract system. It opens the possibility to
fully and exactly discuss the real causal system for
which the causal system is an adequate constructive
model. In the text below, we shall in full examine the
definition of the term adequate.

Figure 4. The general concept of a causal sys-
tem. The system mapping z determines an order
of evolution of state variables vi,j , i ∈ {0, 1, 2, . . . , F},
j ∈ {1, 2, . . . , n} in time. Ck,l is a complete imme-
diate cause which determines the consequence Dk,l,
k ∈ {2, 3, . . . , F}, l ∈ {1, 2, . . . , F − 1}.

3.6. Phenomenological system
It is never possible to measure values for all system
attributes. Žampa’s approach of phenomenological
system states that we do not search for any other
variables than those which we can measure, otherwise
we consider the system as the “complete” system
presented above.
In our opinion, the phenomenological system is a

broader issue than Žampa considers. We demonstrate
that the measurable – phenomenological – variables
themselves include two system models: an idealized
system model of a measurement and a real system
model of a measuring device, cf. [21].
A simple illustration of the influence of a phe-

nomenological variable point divergence gain on image
processing of two consecutive micrographs obtained
using widefield brightfield optical microscope from the
focal plane of a live cell in order to localize interior
cell objects of the size below the Abbe diffraction
limit [22] is shown in Appendix B.

3.7. General system in a broad context
It should be noted that, by the introduction of the
causal system, Žampa begins to constrain our set
of possible abstract models. The scope of adequate
models begins to be confined to those which are causal.
The discussions on the origin of the arrow of time
remains extensive, but Žampa considers the causality

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vol. 58 no. 2/2018 Žampa’s Systems Theory

to be one of the primitive assumptions.
We would like to notify two important ideas about

the arrow of time. One of them is the textbook prin-
ciple of entropy [23] which states that entropy of the
universe increases. Boltzmann [24] suggested that the
reason for the increase of the entropy was the fact
that, in a system of large number of elements, we
prevalently observe a few most frequently occurring
system states and the others can be neglected. In
other words, instead of examining the dynamics of
each molecule, we can consider a value of state variable
such as temperature or pressure as the phenomeno-
logical variable.

Since the advent of non-linear dynamics, it is known
that, even in very simple systems such as the Rössler
flow [25] or the Lorenz attractor [26], we observe
a trajectory from any given set of states into one
limit set of states, while, from the rest of the set of
states, another limit set is reached. Such limit sets
often exhibit a kind of periodic behaviour leading to
a different type of ergodic state [27] than Boltzmann
expected. This state might be structured in space and
observed phenomena might be distributed in space
unevenly.
The later research in discrete dynamics [20, 28]

showed that discrete systems travel through the state
space via a set of well-defined structures. Thus,
Žampa’s causality principle reports only a natural fact
that all dynamic systems follow a defined trajectory
on their way to the limit set, where they either stop
evolution or obey a probabilistically determinis-
tic trajectory. Causality might be more inherent to
dynamic systems than understood currently. It is a
matter of ongoing research to find out whether each
discrete dynamical system has a corresponding con-
tinuous system whose set of Poincaré sections is our
discrete system. A trajectory of the discrete causal
system is discussed in Appendix A.
In the text below, we shall demonstrate that

Žampa’s causality principle enables us to define con-
ditions which allow to separate the system from the
rest of the world. This enables us to construct the
abstract model of the system, neither complete nor
comprehensive, but always measurable.

4. State theory of systems
As stated above, Žampa’s understanding of the system
means to define a (complete or incomplete) set of
measurable attributes, system time, and the mapping
which describe the system’s evolution. Attributes used
by the state theory are called state attributes ai, i ∈
I. Values, which are acquired by these attributes, are
called state variables of the system vi ∈ Vi, i ∈ I.
The state of the system is then defined as an ordered
set of n values of the system state attributes.

4.1. System state trajectory
In order to discriminate between causes and conse-
quences, we differentiate two groups of state variables:

Figure 5. The concept of causal relations and the
importance of system model [29] depicted a set of
variables measured at a time instant (represented by
circles and ovals) and causal relation in the behaviour
between measuring times. To determine the set of
measured values at time tl we must consider not only
limited a set of measured values at times tj < tl but
also an appropriate causality in intermediate times,
not defined by the set of system time T . A set of
variables at the time of measurement Dk,l is then de-
fined by two indexes k, l, where the first expresses the
causality within the measurement time interval, while
the other is the index of the system time from the set I.
Then, unity Ck,l ⊂

⋃
(i,j)<(k,l) Di,j is the complete

immediate cause, the set of all variables necessary
for prediction of the set Dk,l. From that, among other
conclusions, arises that, at least for technical reasons,
the system cannot be understood without the knowl-
edge of an appropriate system model.

The group of inertial variables whose attributes re-
quire time for the change of their values and the group
of non-inertial or, not fully adequately, static vari-
ables whose attributes (in ideal case) do not need time
for change of their values. If the set I as an ordered
decomposition of the set of attributes ai, i ∈ I into
sub-sets Ii, i = 1, 2, . . . ,m, m ≤ n is in agreement
with the approach in which we determine inertial and
non-inertial variables, each sub-set Ii, i = 1, 2, . . . ,m
determines either inertial or non-inertial variables.
The system state trajectory z is defined by

Equation 8. The state trajectory can be decomposed
into subsets D analogously as previously (Figure 5).
According to the causality principle, for each defini-
tion set Dk,l ∈ D, there exists exactly one definition
set Ck,l for the complete immediate cause z | Ck,l.
There exists mapping K,

K : D → P(D) (17)

which now, according to the causality law, attributes
to each definition set Dk,l ∈ D exactly one definition
set Ck,l = K(Dk,l), for which

Ck,l ⊂
m⋃
j=1

Dk−1,j , l = 1, 2, . . . , r,

k = 0, 1, 2, . . . , F (18)

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R. Rychtáriková, J. Urban, D. Štys Acta Polytechnica

in case of inertial variables and non-empty definition
set Ck,l and

Ck,l ⊂
l−1⋃
j=1

Dk,j , l = r + 1, r + 2, . . . ,m,

k = 0, 1, 2, . . . , F (19)

in case of non-inertial (static) variables.

4.2. Definition of the causal system
The definition of the complete immediate cause is en-
abled by the uniqueness of demarcation of the system
trajectory. The complete immediate cause can be in-
terpreted as a model of the mechanism, according to
which the system trajectory in a real system is formed.
For the abstract system S = (T, V ), there is defined
the mapping K which attributes the definition set of
its complete immediate cause Ck,l to each definition
set of the consequence Dk,l unambiguously. Such a
system is called a causal system K and is identified
with ordered triple (T, V,K), i.e.,

K = (T, V,K). (20)

The ordered tuple (Ck,l, Dk,l), Dk,l ∈ D, is called
a system causal relation (Figure 5). A set of all
causal relations is then represented by mapping K. If
the cause Ck,l is not defined for some Dk,l ∈ D, the
system S = (T, V ) is not considered as the causal
system.

We should note that there can be cases when z | Dk,l

is determined by an empty mapping z | φ, which
is unique and independent of z, and is determined
independently of any other parts of the trajectory. The
segment z | D0,1 which is called an initial condition is
always defined independently.

The demarcation of the set or mapping K of causal
relations in the system is given by the modelled reality.
In certain cases, such a demarcation can be compli-
cated. However, in some cases, the demarcation is not
needed and it is enough to define an ordered decom-
position I into I as a trivial decomposition, where
I = {I}. Such a decomposition demarcates the triv-
ial discrimination level of causal relations which does
not enable us to model oriented relations between
variables inside one time instant. This also means
that, in such a system, it is not possible to define
terms such as input, output, feedback bind or bond
and consequently also all structural terms. Although,
despite their potential incorrectness, such definitions
are artificially constructed.
We show below that, in the system theory, the

non-trivial decomposition of I of the set I enables
us to precisely define structural terms which can be
considered as fundamental terms.

4.3. Complete immediate cause
in a self-organizing system

An interesting aspect of the complete immediate cause
can be illustrated on a model of spatially distributed

system, a stochastic cellular automaton [19, 30] de-
scribed in Appendix A. This system is a typical
probabilistically deterministic system P. The evolu-
tion rule is such that the complete immediate cause
Ck = Ck−1 = C1 for any k. There is no evolution
within the time interval and the state of the system in
the next time step can be completely determined from
the previous time step. This description is completely
true only in the final – ergodic – phase of the sys-
tem’s evolution which occurs after more than 25,000
elementary time steps. Then the actual configuration
is dependent on the number and mutual positions of
ignition points. In other words, the complete immedi-
ate cause Ck covers all time steps of the simulation,
where the stochastic element is added in any step as
well. When the ergodic phase is reached, Ck = Dk,ν ,
where ν is determined by the rule which governs the
system.

5. Structure of the causal system
So far we have considered the system as a certain
enclosed and unchanged entity. In practice, it is use-
ful to discriminate certain parts of the system and
to be able to interfere in the system and change its
properties. Obviously, we shall not allow any other de-
structive interference than those, by which the system
can be decomposed into certain, from the cybernetic
point of view, meaningful parts with the possibility that
certain parts can be replaced by others. Later, such
parts will be called sub-systems. These will be parts
which are either completely isolated from the others
or bound to them by special and easily disconnectable
bonds transmitting only information but not mass and
energy. These bonds are called information or cyber-
netic bonds. Parts of the system, which exchange mass
and energy, are parts which are inseparable from the
cybernetic point of view.

5.1. Bond of the causal system
Let us have a causal system

K = (T, V,K) (21)

(no matter whether in phenomenological or state sense)
and, for all k, k = 0, 1, 2, . . . , F , let us have such a
l, l ∈ {1, 2, . . . ,m} that a cause Ck,l and a conse-
quence Dk,l is represented by exactly one attribute
il ∈ I and jl ∈ I, respectively.

Further, if the value of the attribute il at the time
tk is given by a one-component variable vil(tk) ∈ Vil
and the value of the attribute jl at the time tk by
a one-component variable vjl

(tk), then such an ac-
tion by which the causal relation (Ck,l, Dk,l) can be
disconnected in the causal system K is assumed to
be enabled. It means that, for given k, l and conse-
quence Dk,l, any cause Ck,l will not exist. Then such
a causal relation is called a (information) bond of
the system cl(tk). This information bond is oriented
(Figure 6).

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Figure 6. The scheme of an (oriented) informa-
tion bond c.,b in the system as a causal relation
which is represented by exactly one attribute ai,j
which is symbolized by one component variable vi,j ,
i ∈ {0, 1, 2, . . . , F}, j ∈ {1, 2, . . . , n}.

We also assume that, at each time instant, the bond
is either connected or disconnected. We should also
note that it makes sense to include the attribute ai
into the system only when a respective causal relation
(Ck,l, Dk,l) is disconnectable.

5.2. Sub-systems of the causal system
“In each system, parts which do not communicate mu-
tually or which communicate only by information, not
energetic, bonds can be generally found. Such parts
are called sub-systems of the causal system.” In
the real world, we must only carefully classify to which
extent a particular bond satisfies the requirements for
the information bond. In many interesting systems
from molecules up to society, such requirement can
be fulfilled to a good extent.
Let us assume a causal system K (Equation 21)

together with a set C of all information bonds

cl(tk) ∈ C. (22)

In this system, if we disconnect one of the sub-sets C′
of the set C of all bonds, the system will split into a
set of isolated parts, where each of them can become
a system itself due to an appropriate choice of at-
tributes. Namely, it will be such a sub-set of the set of
all attributes of the original system whose elements lie
in the given isolated part. By disconnection of each in-
formation bond, the definition set of its cause Ck,l and
thus also its cause z | Ck,l, k = 0, 1, 2, . . . , F stop be-
ing defined for a given definition set of a consequence
Dk,l. This is in contradiction with the causality prin-
ciple, according to which a cause for each consequence
exists. By this disconnection, the system stops being
causal. According to the definition above, the new
system can be considered as a sub-system which will
become a causal system via complementing by an
appropriate bond. It is useful to call this system a
causal sub-system.

A causal sub-system is a part of a causal system
which, upon the disconnection of appropriate informa-
tion bonds and upon selection of appropriate primary
attributes, corresponds to the basic definition of gen-
eral abstract system. The causal sub-system will be
marked Kι, where ι is an element of an appropriate
index set. The causal sub-system is often demarcated
by ordered triple

Kι = (T, Vι,Kι), (23)

where Vι is a Cartesian product of an appropriate
sub-system of a set of sub-sets {V1, V2, . . . , Vn} and
Kι is an appropriate sub-set of the set K of all causal
relations of the original causal system K .
It is important to note that the steps described

above serve only for a decomposition of the system into
sub-systems. A system is always causal and, therefore,
also physically possible. A non-causal situation leads
to a result without a cause. This situation occurs
only in one part of the system trajectory – in the
initial state, at the beginning of the state trajectory.
Here, we distinguish two kinds of system’s behaviour
– static and dynamic. To show that the system is
static, it is necessary to have at least two identical
consecutive states. The different situation is in case
of the dynamical systems, where, for the definition of
the initial state, it is necessary to define the initial
conditions with a cause resulting from the initial state.
However, the initial conditions are given – they are
ad hoc results – without any cause, even if there
were causal relations which would lead to the initial
conditions in a given way.

5.3. Inputs, outputs and internal
attributes

An attribute of a causal sub-system, which is a conse-
quence of a certain system bond, is called an input
of this sub-system and a variable, which represents
a value of the input at each time instant tk ∈ T , is
called the input variable of the sub-system u(tk).
An attribute of the causal system, which is a cause of

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R. Rychtáriková, J. Urban, D. Štys Acta Polytechnica

a certain bond, will be called an output of a given
sub-system and a variable, which represents a value
of the output at each time instant tk ∈ T , will be
called an output variable of the sub-system y(tk).
A general attribute is not necessarily an input or out-
put of a causal sub-system. An attribute, which is
neither input nor output of a sub-system, is called an
internal attribute of the sub-system and the corre-
sponding variable is called an internal variable of
the sub-system x(tk).

Since all variables are defined in the causal system,
each input must be connected to one of the outputs.
From that it is clear that the causal system (not a
sub-system) has as many bonds as inputs. From the
definition of outputs also comes out that any (as well
as zero) number of inputs can be connected to each
output.

The precise definition of the system bond and the
information bond is the key concept. In order to
construct an abstract dynamic system, we can easily
make a mistake in examining the system which is in-
separable from other parts of the physical nature. Any
measurement in such a system is obviously impossible.

5.4. Measurable sub-systems
The concept of decomposition of the system into sub-
systems was developed for the purpose of the system
control. However, in the system control, the errors
introduced by omission of the properties of a control
device can be often overlooked. The decomposition
into sub-systems is much more critical in the case of
measurement.
In Appendix C, we show an analysis of the signal

of the HPLC-MS device as a prominent example of
the usage of the device in the chemical analysis.

6. Conclusions
If we intend to use mathematics for the description of
dynamic systems, the best known guidance is qualita-
tive dynamics [4, 5]. Qualitative dynamics teaches us
that a phase space is sectioned into zones of attraction
of different limit sets. It tells us, which types of be-
haviour we can anticipate, if we wait sufficiently long,
until a system arrives at its limit set and stays there.
The quality of the limit set is ergodic behaviour [27],
i.e., that a during sufficiently long time the system
visits each of its states at least once. In layman’s
terms, the system resides in a stable, oscillating, or
other repeatedly changing state.
It seems that most systems are on the trajectory

towards the limit set [21]. Qualitative analysis of limit
sets can be transformed into discrete problem by sec-
tioning of the trajectory by Poincaré sections. They
can be equally well used for sectioning the trajectories
which lead to the limit sets. But these trajectories
are seldom analyzed. Thus, it is not clear whether
the timely structured states and “confluences” of tra-
jectories, which originate from different gardens of
Eden in discrete dynamics [20], do always have their

continuous dynamic counterparts related by Poincaré
sectioning. If it was so, our freedom in choosing a
model would be constrained severely [18]. The only
question which remains is whether the Nature is con-
strained by mathematics.

It is certain that Žampa himself focused his thoughts
on technical systems. He never searched for limit
sets and did not discuss the qualitative dynamics.
Instead, Žampa examined limits for the existence
of the causal system and conditions by which the
system can be separated from the rest of the universe.
Thus, Žampa’s state theory complements our scope
of objective constraints for the choice of models.

Žampa’s state theory is also very close to measure-
ment. The phenomenological system addresses the
problem directly, but it is certain that an adequate
abstract system cannot be constructed only using vari-
ables which we – by chance – measure. By accepting
this, we can construct a machine which performs what
we engineered. But, to our surprise, the machine will
fail so often and the final marketed construction will
include many aspects which the first constructor did
not anticipate. Also, the same is true for many ele-
ments which the constructor did not even notice and
are hidden inside his equations and software which
he used. The chief constructor can remain confident
that his model is still the core of his machine, but
the outside reality, following the rules of qualitative
dynamics outside the limit set, will continue by its
own way.
May we know how to proceed? The knowledge

of qualitative dynamics, as Predrag Cvitanovič ex-
pressed, has “holes large enough to steam a Eurostar
train through them” [4]. Qualitative dynamics con-
cerns with limit sets and seldom studies the trajectory
through the zone of attraction towards them. The
exceptions are Wuensche’s 8-level discrete dynamic
networks and multilevel systems [18]. As proved [18],
the scope of types of qualitatively different trajectories
is limited and much smaller than usually considered
for potential technical constructions.
Žampa gives a part of the recipe. First, we must

be able to dissect the system into the sub-systems
which are mutually bound only by an information
bond. This gives us the structure of the system for
studying. A human in the society or a bird in a flock
can survive independently of the context, at least
during the time of the measurement. Its scope of tra-
jectories can be studied. In societal context, when the
information bond(s) are connected, trajectories which
in disconnected systems are sparse or non-existent can
suddenly prevail.
Second, after the dissection of the system into the

sub-systems, we must look for a system trajectory.
For each set of values of a particular variable, we
must determine the complete immediate cause. In
traditional dynamics, for a system of the unknown
technical composition, this is a neverending process.
Coming from technical environment, Žampa consid-

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ers knowledge of the technical limit of the range of
attribute states. The first advice for examination of
the systems which we do not know is to search for the
complete immediate cause. Any relevant model must
consider an adequate structure of inertial and non-
inertial variables such that each trajectory’s segment
has its own complete immediate cause. Unfortunately,
in case of, e.g., many biological or societal systems,
the complete immediate cause precedes the start of
our experiment or observation [21].

7. Appendices
System description is not only useful for artificial
constructed systems, but also for the data measured
in the experiments, especially, if they are of biologi-
cal or chemical origin. In Appendices, we introduce
three illustrative examples of data description, which
help biophysical understanding of the measured nat-
ural phenomena. The first example is the famous
dynamic Belousov-Zhabotinsky reaction, which is ob-
served in the discrete time and where variables de-
rived from the Rényi entropies are considered as state
variables describing the observed states. The evo-
lution of the system is thus reflected in changes of
information of the underlying physico-chemical phe-
nomena. The next model considers microscopy as
an information channel, where time-spatial evolution
of the point spread function of a live biological spec-
imen is approximated by an image z-stack of the
light distribution in the in-focus region. The last
system is a model of liquid chromatography in tan-
dem with mass spectrometry, where the retention
time of the chromatographic elution represents the
evolution of the compound composition in the de-
pendency on the set-up of the chromatograph. The
additional conditional segmentation of the dataset
into sub-systems follows the system approach and
represents relevant chemical decomposition of the
measurement. In all cases, the adopted system ap-
proach leads to the unscrambling of the system. The
demonstrated system decompositions can provide
meaningful information about the measured exper-
iments.

7.1. Appendix A: The role of the system
time: the analysis of a model series
of the Belousov-Zhabotinsky
reaction

The Belousov-Zhabotinsky reaction is a prominent
experimental example of the self-organization in the
nature [31, 32]. One of its possible models is a noisy
hodgepodge machine [19, 30] which can simulate its
complete trajectory. This type of simulation provides
an image series with the shortest possible time interval
between two consecutive images.

For analysis of the image series, we have developed a
method of the calculation of the point divergence gain
entropy (Iα) and the point divergence gain entropy
density (Pα) which are cumulative variables derived

from the absolute values of the point divergence gain
(ωα,x,y) [33–35]. The parameter-dependent spectra
of these macroscopic variables characterize stepwise
differences in image series and evaluate the evolution
of the system between two consecutive data points.
In Figure 7, the set of parameter α contained val-
ues 0.1, 0.3, 0.5, 0.7, 0.99, 1.3, 1.5, 1.7, 2.0, 2.5,
3.0, 3.5, 4.0. Next, the image series, where each
image is described by the parameter-dependent spec-
trum, is characterized by cluster analysis (k-means
with squared Euclidian distance) which separates the
series into groups of similar images. Then, each
group represents the section of the trajectory (Fig-
ure 4).

The statistical analysis in Figure 7 compares results
for Iα and Pα which are computed from full-image
series and from series, where the divergences were cal-
culated for every 10th image. The analysis of the full
dataset is typical of oscillations between clusters (Fig-
ure 7a,c). The detailed inspection (Figure 7e–f) found
that these oscillations are due to the dominant color
of structures which surround the ignition point. These
changes are slower than the frequencies of oscillations
of dense square waves which dominate the early phase
(Figure 7e) and of the circular waves which dominate
the late phase (Figure 7f). In the decimated trajectory,
this feature is unrecognizable, although the change of
the prevalent structure is determined properly.
The differences between the original and the dec-

imated series calculated from Pα (Figure 7a–b) are
more pronounced than in the case of Iα (Figure 7c–
d). This is a consequence of the fact that, in the Iα
calculation, we sum the values ωα,x,y for all points
(pixels) and the result is dominated by frequent values
whose occurrences do not change significantly over 10
consecutive images. The Pα sums only different levels
and, therefore, is sensitive to subtle changes which
give rise to new unique ωα,x,y values. For higher num-
ber of clusters, the results of the clustering vectors Iα
and Pα are dramatically different.

Despite the complete and the decimated trajectories
are similar, at first sight, the sole usage of the sparse
system time T leads to the completely different struc-
ture of the trajectory. It demonstrates that the choice
of the system time is not voluntary, but, in many cases,
the time have to be measured at elementary system
events. Simple rules such as the Shannon-Nyquist
Sampling Theorem give a good advice for that. Never-
theless, as the given example shows, even if we detect
the high-frequency changes of variables, there can be
low-frequency changes of variables which might not
be fully understood and which significantly change
the result of the analysis.
From many real datasets, where the sampling fre-

quency is technically limited, the model of the system
can be never deduced in the straightforward manner.
This exactly illustrates the idea of the complete im-
mediate cause Dk,l (Figure 4). For the prediction
of the succeeding system’s behaviour, it is necessary

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R. Rychtáriková, J. Urban, D. Štys Acta Polytechnica

Figure 7. k-Means clustering (squared Euclidian distance) of an image series of a stochastic hodgepodge model
of the Belousov-Zhabotinsky reaction according to similar structures using a point divergence gain entropy density
Pα (a,b) and point divergence gain entropy Iα (c–d) for α = {0.1, 0.3, 0.5, 0.7, 0.99, 1.3, 1.5, 1.7, 2.0, 2.5, 3.0, 3.5, 4.0}.
Analysis of a full-image series (a,c) and of every 10th image of the full-image series (b,d). Row 1, a–d: Examples of
time courses of Pα and Iα, respectively, for α = {0.99, 2.0, 0.5, 4.0}. Rows 2–5, a–d: Clustering of images into 3–6
groups, respectively. Figures e–f: Sections of the trajectories shown in Row 5c,d. Number of image corresponds to
the time unit.

to know not only the system’s state at a few preced-
ing time instants, but also a good system’s model
which predicts the system’s behaviour between the
measured time points l, i.e., between the values of the
system time, at the time instants k. In the partic-
ular case of the model of the Belousov-Zhabotinsky
reaction, which anticipates time and spatial discrete-
ness, we know the probability distribution function
of the time element. The question to which extent
the existence of the element of time unit is general
remains to be answered by the fundamental research
of physics.

7.2. Appendix B: Phenomenological
variables unravel superresolved
structures in a series of standard
brightfield micrographs
of a live cell

The only procedure, which we can use for the descrip-
tion of the cell dynamics, is an extensive analysis of
the image information in the brightfield micrographs.
The brightfield optical microscope transmits light,
modifies the wavefront, and the intensity of the elec-
tromagnetic field (i.e., the probability amplitude of

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Figure 8. Superresolved localization of the internal structures of a living MG-63 cell’s nucleolus. (a) The original
image of a living MG-63 cell in the focal region of a brightfield microscope. The image was captured by a colour digital
camera as a 12-bit raw matrix. For visual inspection, the 12-bit image was transformed into an 8-bit image by the
Least Information Lost (LIL) algorithm [36]. The red square highlights the section of nucleolus used for calculation
of ω0.99,x,y values. (b) ω0.99,x,y-Transformation of intensities in the red camera channel of two consecutive images in
the cell’s focal region. Upper row – Original intensities from which the ω0.99,x,y values (lower row) were calculated.
Lower row – Negative (left) and positive (right) values of ω0.99,x,y. The yellow demarcated regions corresponds to the
structures which are attributed to be chromatin by electron microscopy. Scaling of the original 16-bit intensity image
depicting ω0.99,x,y into an 8-bit computer screen by the simple sectioning of the scale (lower left) and by the LIL
transformation with the fully utilized 8-bit scale (lower right). The brightest and darkest intensities correspond to the
ω0.99,x,y values for the time-stable and unstable structures, resp. The size of the image voxel is 64× 64× 130 nm3.

the occurrence of photons at a given position in the
space) is captured by a detector as a near-continuous
signal. The number of resulted charge transfers is
detected and transformed by an analog-digital con-
verter into a matrix of numbers. Therefore, we need to
analyze a phenomenological discrete variable (color of
an image pixel) instead of a near-continuous variable
(number of photons in space).

Aspects of capturing an image by a digital camera,
which is a sub-system on its own, are discussed in [36].
The observed intensities represent a sum of changes of
the intensities which pass a sample and a optical path
of the microscope. An actual shape of the observed
object is determined by diffraction properties and by
the deformation of the electromagnetic field along the
optical paths. According to the Mie theory [37], a
larger object does not always scatter more light and its
scattered image is not always larger. The dependency
of the scattering efficiency on particle’s size has sev-
eral maxima. The Nijboer-Zernike theory [38] tries to
explain the distortion of the wavefront by its traveling
along the optical path. In fact, the description of the
behaviour of a complicated – patterned – wavefront
is not satisfactory. We do not have any theory which

would describe the resulted pattern sufficiently, not
even for a very simple diffracting object. Thus, the
observed object together with the microscope repre-
sent a system which has to be examined from Žampa’s
point of view.

However, since both the diffraction and the passage
of light through the optical system of the microscope
are changes of the electromagnetic field, they can be
hardly disconnected into sub-systems. In addition,
it was found that it is possible to detect objects far
below the diffraction limit of light and the diffraction
limit decreases with increasing light intensity [39].
The latter observation is difficult to explain by any
existent theory.
Articles [34, 35] describe a method how to obtain

superresolved micrographs, i.e., locations of objects,
with the precision below 50 nm, from a series of the
most ordinary brightfield widefield optical microscopic
images. The series was obtained by movement of
the biological specimen along optical axis of the mi-
croscope. As similar to Appendix A, the change of
information (intensity) between two consecutive im-
ages is for each camera pixel analyzed using the point
divergence gain. Wherever the information remains

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unchanged, we assume that we have localized an ob-
ject whose response is larger than a voxel, i.e., than
the area of the camera pixel multiplied by the z-step.
The intensity change between two consecutive z-stack
images of a live cell has basically twofold origin: (1)
the diffraction response of the observed objects is thin-
ner than the z-step or (2) the object moved during the
image capture. Our next assumption for the choice of
a macroscopic phenomenological variable was that the
image structure is multifractal. The chosen method of
analysis [34] is further based on an assumption that
two image points of identical intensity lying at two
levels directly above each other represent (with a high
probability) the same information. The α of different
values used in the calculation result in quite different
histograms of (ωα,x,y): a low α separates rare points,
while a high α separates more frequent points. The
lower row of Figure 8b illustrates values of the point di-
vergence gain at α = 0.99 (ω0.99,x,y) which correspond
to the information difference due to the replacement
of a pixel’s intensity in the first image by the pixel’s
intensity at the same position in the second image for
a couple of images in the focal region differing from
each other by the z-step of 130 nm (the higher the
ω0.99,x,y, the brighter the pixel). The most important
result of this image recalculation is the highlighting
a fine structure inside a nucleolus which is similar to
granular and fibrilar structures reported by canonical
electron microscopy [41]. This fine structure is not
visible in a 8-bit color representation but in a 12-bit
intensity file. This shows that, inside the image of
the organelle, there are different values of ω0.99,x,y
which bring information about different diffracting
objects. It confirms a finding [39] that, in a brightfield
optical micrograph, objects of the diameter of 25 nm
is discriminable [40].

The analysis described above indicates that any in-
formation about a live cell provided by the microscope
is a highly phenomenological quantity which reflects
the content of chemical compounds at the particular
point and at the neighboring points. In other words,
the transformation of the color-coded image series
into ωα,x,y-coded images provides a set of measurable
(phenomenological) variables which maximally yields
the information brought by the biological experiment.
A visual inspection of an uncalibrated brightfield

micrograph and the phenomenological character of
the image information-entropic variables also have a
purely technical reason (see Fig 8a). Vice-bit images
are typically stored and visualized in a 8-bit format.
As described in [36], even in a lossless compression, a
series images are transformed by an algorithm which
differs for each image. The alternative – a simple
sectioning of original (12- or 16-bit) image into 256
levels – can leave most levels unoccupied. It allows
us to conclude that a majority of images analyzed in
the world is transformed in an uncontrolled way and,
therefore, useless for an exact analysis.

7.3. Appendix C: Information bond
in a dataset obtained from liquid
chromatography with mass
spectrometry (HPLC-MS)

In HPLC-MS analysis, we consider an information ob-
tained from a measured signal as a system [42] which
can be separated into sub-systems. The LC-MS mea-
surement [43] is a combination of a physico-chemical
chromatographic separation of individual compounds
on a chromatographic column, a separation of ionized
molecules upon flight through the electromagnetic
field, and a detection of the number of molecules by a
detector. There are numerous technical realizations of
this experiment. The analysis which is presented here
is general and is not dependent on a concrete technical
set-up. The resulted dataset is a two-dimensional set
of values (Figure 9a–b).
The key idea is that the detected signal is a set

of system attributes ai, i ∈ I, where I is a set of
measured time instants. At each time instant t, we
obtain one intensity value y(m, t) for each value of
mass m. Each point (m, t) is assigned at least into
one of these sub-systems:

(1.) random chemical noise r(m, t),
(2.) systematic noise q(m, t),
(3.) signal s(m, t).

The random chemical noise r(m, t) is prevalently a
combination of chemical compounds randomly eluted
from the column with electrical noise on the detector.
This kind of noise can be modelled by a standard
distributions, e.g., by a log-normal distribution as in
the given case (Figure 9). The information about
the random chemical noise is contained in ca. 93%
of data points. It means that the properties of free
random chemical noise can be determined with a high
reliability.

The systematic noise q(m, t) and the signal s(m, t)
are formed by systematic responses of specific mol-
ecules which reach the detector. The systematic
noise q(m, t) represents molecules which are constantly
present in the solvent. This noise occurs as forms of
ridges at the positions of specific m/z (molecular mass
to charge) ratios as well as of chromatographic peaks
in each empty run. The ridges can be clearly assigned
to the systematic noise and form the signal component
q(m, t).
The signal s(m, t) is the sought response, i.e., the

signal originating from molecules (analytes) which
were added to the chromatographic apparatus and
should be detected. The signal of the analyte appears
as a chromatographic peak of a specificm/z value. For
the purpose of the analysis, despite the qualitatively
different origin, the chromatographic peaks of both
the background and the analyte are summed in the
signal s(m, t).

The signal components r(m, t), q(m, t), and s(m, t)
are not independent. Namely, we observe a decrease

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Figure 9. Signal processing in HPLC-MS.

of the q(m, t) and an increase of the r(m, t) at the
position of the non-zero s(m, t) (Figure 9c). Moreover,
the s(m, t) can be further separated into signals of
individual molecules which appear as a few parallel
peaks of different adducts, isotopologues, etc.
Since the r(m, t), the q(m, t), and components of

the s(m, t) represent sub-systems, the signal y(m, t) is
a sum y(m, t) = r(m, t)+q(m, t)+s(m, t). It should be
noted that the mapping y : T×M → I is probabilistic.
Each contributing component, including the responses
of individual compounds, shows a probability density
function which is not generally known.
The first step of the signal analysis is to find the

envelope of the distribution function of the r(m, t)
which can be determined with a high level of confi-
dence. Any signal outside this envelope belongs to
either the q(m, t), the s(m, t) or an isolated electrical
spike. These spikes are confined to one point, they
are not surrounded by other signals differing from the
r(m, t) and are easily eliminated. The noise q(m, t)
appears for almost all t values, while the s(m, t) ap-
pears only timely. For both cases, an algorithm had
to be found in order to analyze the features of the
signal. Figure 9e shows an example of a course of
a probability peak s(m, t) after subtraction of the
r(m, t).
Despite the fact that the r(m, t), the q(m, t), and

the s(m, t) are strongly interconnected and affect each
other, the analysis using the assumption of observation
of the probabilistic system which can be separated
into sub-systems was successful. Figure 9d depicts
that each peak of analyte was separated and many
signals originally hidden in the background noise were
found. Moreover, compounds identified in the control
experiments (i.e., empty run or blank) were identified,
used for calibration, and removed from the final set of
analytes’ signals. The true information bond is never
practically realized in physical systems. Nevertheless,
it can be often assumed in the analysis of a real dataset.
This explains the success of system analyses of systems
with an input and an output in which the properties
of the input and the output are not considered.

List of symbols
ai Abstract attribute, e.g., a coordinate of position,

a coordinate of speed, verity of statement
A Set of all abstract attributes
B Sub-set of the set of all systems trajectories Ω
B Set of all system events
cl(tk) Information bond of the system
C Set of all information bonds
Ck,l Cause
D Definition set of a mapping of a system trajectory z
Dk,l Consequence
D Ordered decomposition of the set D
D Deterministic abstract system
I Set of indexes of attributes
Iα Point divergence gain entropy

141



R. Rychtáriková, J. Urban, D. Štys Acta Polytechnica

I Set defined above Ij , j = 1, 2, . . . ,m with a causal
condition <

K Causal mapping
K Causal system
m Molecular weight
m/z Mass-to-charge ration
P (B) Probability of each event B in the set B
P Stochastic abstract system
q(m, t) Systematic noise
r(m, t) Random chemical noise
s(m, t) Signal
S Abstract system
t Real time
T K-element set of real times with indexes 0, 1, . . . , F
u(tk) Input variable of the sub-system
v System variable
V Definition set of system variables
vi Abstract variable
Vi Definition set of abstract variables
y(m, t) Additive signal
x(tk) Internal variable of the sub-system
y(tk) Output variable of the sub-system
z System state trajectory
α Rényi coefficient
ι Element of an appropriate index set for a causal

sub-system
ν Order of the consequence in the ergodic phase
φ Empty set
ωα,x,y Point divergence gain
Pα Point divergence gain entropy density
Ω Set of all system trajectories

Acknowledgements
This work was supported by the Ministry of Education,
Youth and Sports of the Czech Republic projects CE-
NAKVA (No. CZ.1.05/2.1.00/01.0024), CENAKVA II
(No. LO1205 under the NPU I program), the CENAKVA
Centre Development (No. CZ.1.05/2.1.00/19.0380), by the
European Regional Development Fund in frame of the
project Kompetenzzentrum MechanoBiologie (ATCZ133)
in the Interreg V-A Austria – Czech Republic programme
and by TA ČR Gama TG 03010027 subproject 03-24.

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	Acta Polytechnica 58(2):128–143, 2018
	1 Introduction
	2 The system
	2.1 Abstract system
	2.1.1 System time and attributes
	2.1.2 System trajectory


	3 General abstract system
	3.1 Definition
	3.2 System behaviour
	3.3 Deterministic systems
	3.4 Stochastic systems
	3.5 Causal system
	3.6 Phenomenological system
	3.7 General system in a broad context

	4 State theory of systems
	4.1 System state trajectory
	4.2 Definition of the causal system
	4.3 Complete immediate cause in a self-organizing system

	5 Structure of the causal system
	5.1 Bond of the causal system
	5.2 Sub-systems of the causal system
	5.3 Inputs, outputs and internal attributes
	5.4 Measurable sub-systems

	6 Conclusions
	7 Appendices
	7.1 Appendix A: The role of the system time: the analysis of a model series of the Belousov-Zhabotinsky reaction
	7.2 Appendix B: Phenomenological variables unravel superresolved structures in a series of standard brightfield micrographs of a live cell
	7.3 Appendix C: Information bond in a dataset obtained from liquid chromatography with mass spectrometry (HPLC-MS)

	List of symbols
	Acknowledgements
	References