Appendix B — Some basics about chaotic systems
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 (Rickles, Hawe, and Shiell 2007). 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 ODEOrdinary Differential Equation. In order to solve any differential equation, the initial condition or starting state must be known. Meaning, that the solution to the ODEOrdinary Differential Equation 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.
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.
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.
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.
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 (Argyris et al. 2017). 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.
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 (Argyris et al. 2017). Nevertheless, by doing literature research, one way to visually distinguish between randomness and chaos was found (Boeing 2016). Yet, in (Boeing 2016) the method was only deployed on a logistic map. Hence, further research is required here.
As explained, a clear definition of chaos does not exist. However, some parts of definitions do occur regularly, e.g., the already mentioned Sensitive Dependence on Initial Conditions (SDIC). Other definition parts are the following: Chaotic motion is and based on a system. An aperiodic system is not repeating any previous 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 4.0.1.
One misconception which is often believed (Taylor 2010) and found, e.g., in Wikipedia (“Wikipedia Entry on Chaos Theory” 2021) is that strange attractors would only appear as a consequence of chaos. Yet, Grebogi et al. (Grebogi et al. 1984) proved otherwise. According to 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 (Taylor 2010). 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.
To refute this, consider the Gophy attractor (Grebogi et al. 1984). 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 (Taylor 2010). 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 (Taylor 2010). 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?
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 (Taylor 2010). 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 (Argyris et al. 2017; Kutz 2022; Strogatz 2019; Taylor 2010).