Order and chaos

The aim of the theory of dynamical systems theory is to create compact mathe-matical descriptions of the behaviour of complex systems like thela ola. In doing

3.1 The fundamentals of dynamical systems so, dynamical systems focus on how a system changes over time based on the state that the system is currently in. To get a general understanding, it is helpful to study the logistic map, a system formulated to describe the development of populations that was made popular by biologist May (1976) as a discretised ver-sion of the demographic model proposed by Belgian mathematician Verhulst in the mid 19th century. The logistic map is given in Equation 3.1.

𝑥_𝑡+1= 𝑘𝑥_𝑡(1 − 𝑥_𝑡) (3.1)

The formula defines how the state𝑥 of the system at a time point 𝑡 + 1 is calculated. Crucially, this future value of the state variable 𝑥 depends on the current state at𝑡. In addition, the system has a parameter 𝑘that represents the growth rate. For example, if𝑘 = 0.7and𝑥₁ = 0.5at the present time point, the system will predict𝑥₂ = 0.7 ⋅ 0.5 ⋅ (1 − 0.5) = 0.175at the next time point. Figure 3.1 shows how the evolution of the system is continued over 19 additional time steps. The graph shows that the points gradually approach zero. In terms of a population model this means extinction of the population. As there is no member of the population left to reproduce, the system will stay in the state with the value zero forever. This state is called theattractorof the system. Regardless of the state of the system in the beginning, the system will end up in this state.

5 10 15 20

t 0

0.5 1

x t

Figure 3.1: Example for the evolution of the logistic map with𝑘 = 0.7 and𝑥₁= 0.5.

Depending on how the growth rate is chosen, the system can exhibit a variety of patterns. Figure 3.2 gives examples for the logistic map with different values for the growth rate𝑘. In all cases,𝑥₁, the initial state, is0.42. In the case of𝑘 = 1.2 (top left), the system monotonically approaches one value. This type of attractor is calledpoint attractor, it is the same kind of attractor as the one in the illustration of Figure 3.1. In the case of𝑘 = 2.9(top right), the system also approaches one steady state, but while approaching the attractor, it alternates from one side to

the other. When𝑘 = 3.3(bottom left), the attractor of the system is not a steady state. Instead, the system oscillates between two values. This is type of attractor is called alimit cycle. Another interesting case is𝑘 = 4in which the system also oscillates but not in a periodic manner. This behaviour is calledchaos(Kaplan &

Glass 1995). When exhibiting chaotic behaviour, the system will never settle in a steady state or limit cycle but oscillate forever without repeating itself.

5 10 15 20

Figure 3.2: Example for the evolution of the logistic map with different values for the growth parameter (top left:𝑘 = 1.2, top right:𝑘 = 2.9, bottom left:𝑘 = 3.3, bottom right:𝑘 = 4).

Figure 3.3 illustrates the possible patterns of the logistic map as the growth parameter is changed. The plot, also known as a bifurcation plot (Feigenbaum 1978), was created by running the logistic map for 2000 iterations for all growth parameter values between 1 and 4 with a step size of 0.01 (i.e. the simulation was first run with𝑘 = 1, then𝑘 = 1.01, then𝑘 = 1.02, and so on). The initial state is 0.42 in all simulations. The axis for the parameter value is the x-axis.

Of the 2000 iterations for each parameter value, the last 100 values are plotted on the y-axis as single tiny dots. For𝑘 < 3, all dots are plotted on top of each other because after a few iterations the system reached a steady state. As the parameter 𝑘 is increased, the system’s attractor is a limit cycle that oscillates between two values. The cycle frequency is then increased to 4, 8, 16 until the

3.1 The fundamentals of dynamical systems system eventually exhibits aperiodic behaviour as described above. There are, however, bands of growth parameter values in between for which the system moves back to periodic cycles (visible as white stripes in the higher regions on the x-axis) (Kaplan & Glass 1995; Spivey 2007). The behaviour of the system as 𝑘 is scaled beyond 3.5 can be observed in more detail in Figure 3.4. This figure shows the same plot as Figure 3.3 with the x-axis zoomed in to the range of 𝑘 values from 3.5 to 4.

Figure 3.3: Bifurcation plot of the logistic map.

The examples presented around the logistic map demonstrate that an attractor can be a single value or a set of values. For the present work, point attractors will be most important. To change the behaviour of the system, the examples adjusted the growth parameter value𝑘. A parameter like𝑘is called acontrol parameter– a very important concept for the understanding of dynamical systems. Control parameters can be thought of as the parameters that “move the system through its repertoire of patterns and cause them to change” (Kelso 2013: 1538). They often represent environmental conditions, like the growth parameter in the logistic map example, but are not restricted to this role (Kelso 2013).

An essential property of dynamical systems is that they often exhibit quali-tative change as the control parameter is scaled. This behaviour is referred to as bifurcation – a term used above already with regard to Figure 3.3 and 3.4.

As the name of the chart displayed in the two figures, bifurcation plot, already suggested, it shows how the logistic map undergoes bifurcation as the control parameter (the growth parameter) is increased. While the system starts with a

Figure 3.4: Bifurcation plot of the logistic map for values of𝑘 ≥ 3.5.

point attractor, it changes into a pattern with a limit cycle as the control pa-rameter passes a critical threshold. Subsequently, for other control papa-rameter thresholds, the periods of the cycles change before the system turns to aperiodic behaviour. It then changes between aperiodic and periodic patterns for different ranges of control parameter values. All these transitions are qualitative changes, i.e. bifurcations, in the dynamical system.

Code used in this section: logistic_map_evolution.m, logistic_map_bifurcation.m