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# Some basics about chaotic systems {.appendix #sec-ch_Ap_Chaotic} | |
Since | |
Chaotic systems are the height | |
of intricacy when considering dynamical systems. | |
The reason why the term intricacy was chosen | |
instead of complexity is that chaotic systems can be, but are not necessarily | |
complex. For the relation between complex and | |
chaotic the reader is referred to [@Rickles2007]. | |
The mentioned intricacy of chaotic systems shall be explained by | |
reviewing two reasons. First, | |
chaotic systems are sensitive to their initial conditions. | |
To understand this, imagine we want to solve an \gls{ode}. In order to solve any | |
differential | |
equation, the initial condition or starting state must be known. Meaning, that the | |
solution to the \gls{ode} at the very first initial step, from where the | |
remaining interval is solved, must be identified beforehand. | |
One might believe, a starting point, which is not guessed unreasonably off, | |
should suffice to infer the system's future dynamics.\newline | |
This is | |
an educated attempt, however, it is not true for systems that exhibit | |
sensitivity to initial conditions. These systems amplify any | |
perturbation or deviation exponentially | |
as time increases. From this it can be concluded | |
that even in case the initial value would be accurate to, e.g., 10 decimal places, | |
still after some time, the outcome can not be trusted anymore. | |
Visually | |
this can be comprehended by thinking of initial conditions | |
as locations in space. Let us picture two points with two initial conditions | |
that are selected to be next to each other. Only by zooming in multiple times, | |
a small spatial deviation should be perceivable. | |
As the time changes, the points will leave the location defined through the initial condition. \newline | |
With | |
chaotic systems in mind, both initially neighboring | |
points will diverge exponentially fast from each other. | |
As a consequence of the initial condition not being | |
known with infinite precision, the initial microscopic | |
errors become macroscopic with increasing time. Microscopic mistakes | |
might be considered to be imperceptible and thus have no impact | |
on the outcome, which would be worth to be mentioned. | |
Macroscopic mistakes on the other hand are visible. Depending on | |
accuracy demands solutions might be or might not be accepted. | |
However, as time continues further, the results eventually | |
will become completely unusable and diverge from the actual output on a macroscopic scale.\newline | |
The second reason, why chaotic systems are very difficult | |
to cope with, is the lack of a clear definition. It can be | |
argued that even visually, it is not always possible to | |
unambiguously identify a chaotic system. The idea | |
is that at some time step, a chaotic system appears to | |
be evolving randomly over time. The question then arises, | |
how is someone supposed to distinguish between something which | |
is indeed evolving randomly and something which only appears | |
to be random. The follow-up question most likely is going to be, | |
what is the difference between chaos and randomness, or | |
even if there is a difference. \newline | |
Maybe randomness itself is only | |
a lack of knowledge, e.g., the movement of gas particles | |
can be considered to be chaotic or random. If the | |
velocity and spatial position of each molecule are | |
trackable, the concept of temperature is made | |
redundant. Gibbs only invented the concept of temperature | |
in order to be able to make some qualitative statements | |
about a system [@Argyris2017]. | |
A system that can not be described microscopically. | |
Here the question arises if the movement of the molecules | |
would be random, how is it possible that every time | |
some amount of heat is introduced into a system, the temperature | |
changes in one direction. If a random microscale system | |
always tends to go in one direction within a macroscale view, | |
a clear definition of randomness is required. \newline | |
Laplace once said if the initial condition | |
(space and velocity) of each atom would be known, | |
the entire future | |
could be calculated. In other words, if a system is | |
build on equations, which is a deterministic way | |
to describe an event, the outcome should just | |
depend on the values of the variables. | |
Thus, the future, for as long as it is desired could be predicted | |
or computed exactly. To briefly summarize this conversion, | |
Albert Einstein once remarked that God would not play dice. Nils | |
Bohr replied that it | |
would be presumptuous of us human beings to prescribe to the Almighty | |
how he is to take his decisions. A more in-depth introduction to | |
this subject is provided by [@Argyris2017]. | |
Nevertheless, by doing literature research, one way to | |
visually distinguish between | |
randomness and chaos was found [@Boeing2016]. | |
Yet, in [@Boeing2016] the method was only | |
deployed on a logistic map. Hence, further research | |
is required here. \newline | |
As explained, a clear definition of chaos does not exist. | |
However, some parts of definitions do occur regularly, e.g., | |
the already mentioned \glsfirst{sdic}. Other definition parts are the following: Chaotic | |
motion is \textbf{aperiodic} and based on a \textbf{deterministic} system. | |
An aperiodic system is not repeating any | |
previous \textbf{trajectory} and a deterministic system is | |
described by governing equations. A trajectory is the evolution | |
of a dynamical system over time. For instance, a dynamical system | |
consisting of 3 variables is denoted as a 3-dimensional dynamical system. | |
Each of the variables has its own representation axis. | |
Assuming these | |
3 variables capture space, motion in the x-,y- and z-direction | |
is possible. For each point in a defined time range, there is one set of x, y and z values, which fully describes the output of the dynamical system or the position at a chosen time point. | |
Simply put, the trajectory is the movement | |
or change of the variables of the differential equation over time. Usually, the | |
trajectory is displayed in the phase space, i.e., the axis represents the state or values of the variables of a dynamical system. An example can be observed in section [-@sec-subsec_1_1_3_first_CNMc]. \newline | |
One misconception which is often believed [@Taylor2010] | |
and found, e.g., in | |
Wikipedia [@Wiki_Chaos] is that | |
strange attractors would only appear as a consequence of | |
chaos. Yet, Grebogi et al. [@Grebogi1984] proved | |
otherwise. According to | |
\cite{Boeing2016,Taylor2010} strange attractors exhibit | |
self-similarity. This can be understood visually by imaging any shape | |
of a trajectory. Now by zooming in or out, the exact same shape | |
is found again. The amount of zooming in or out and consequently | |
changing the view scale, will not change the perceived | |
shape of the trajectory. Self-similarity happens to be | |
one of the fundamental properties of a geometry | |
in order to be called a fractal [@Taylor2010]. | |
In case one believes, | |
strange attractors would always be chaotic and knows that by definition strange attractors phase | |
space is self-similar, then | |
something further misleading is concluded. | |
Namely, if a geometry is turned out not only | |
to be self-similar but also to be a fractal, this | |
would demand interpreting every fractal to be | |
chaotic. \newline | |
To refute this, consider the Gophy | |
attractor [@Grebogi1984]. | |
It exhibits the described self-similarity, | |
moreover, it is a fractal, and it is also a | |
strange attractor. However, the Gophy | |
attractor is not chaotic. The reason is found, when | |
calculating the Lyapunov exponent, which is negative | |
[@Taylor2010]. Latter tells us that two neighboring | |
trajectories are not separating exponentially fast | |
from each other. Thus, it does not obey the | |
sensitive dependence | |
of initial conditions requirement and is | |
regarded to be non-chaotic. The key messages are | |
that a chaotic attractor surely is a strange | |
attractor and a strange attractor is not necessarily | |
chaotic. A strange attractor refers to a fractal | |
geometry in which chaotic behavior may | |
or may not exist [@Taylor2010]. | |
Having acquired the knowledge that strange attractors | |
can occur in chaotic systems and form a fractal, | |
one might infer another question. If a chaotic | |
strange attractor always generates a geometry, which | |
stays constant when scaled, can chaos be | |
regarded to be random?\newline | |
This question will not be discussed in detail here, but for the sake of completeness, the 3 known types of nonstrange attractors | |
shall be mentioned. These are | |
the fixed point attractor, the limit cycle attractor, and the | |
torus attractor [@Taylor2010]. | |
A fixed point attractor is one point in the phase space, which attracts or pulls nearby trajectories to itself. | |
Inside the fix-point attractor, there is no motion, meaning | |
the derivative of the differential equation is zero. | |
In simpler words, | |
once the trajectory runs into a fix-point, the trajectory ends there. | |
This is because no change over time can be found here. | |
A limit cycle can be expressed as an endlessly repeating loop, e.g. in the shape of a circle. | |
The trajectory can start at | |
any given initial condition, still, it can go through a place in the phase space, from where the trajectory is continued as an infinitely | |
repeating loop. | |
For a visualization of the latter and the tours, as well more | |
detail the reader is referred to [@Argyris2017; @Kutz2022; @Strogatz2019; @Taylor2010]. |