By Sonja Radosavljevic

Dynamical systems theory attempts to understand the changes over time that occur in physical and abstract systems. The defining feature of DST is that it uses mathematical descriptions in the form of (ordinary or partial) differential or difference equations. Dynamical systems have several key components: state variables, rules of evolution, state space (phase) space and phase portrait.

• A state variable represents some physical or other property of the system, such as number of individuals or levels of nutrients, whose changes in time a dynamical system describes and studies. A system can have any number of state variables. 
• State variables evolve over time, starting from the initial conditions following the rule of evolution (or the dynamics) given by the differential equations. The rules of evolution can be derived from e.g. physics, ecology, economy, or can have an abstract mathematical form.
• The state (phase) space of a dynamical system is the collection of all possible states (configurations) of the system. Each state of the system corresponds to a unique point in the state space. The number of variables used to define the system determines the dimension of the state space. 
• Evolution of a dynamical system corresponds to a trajectory (or an orbit) in the phase space. Different initial states result in different trajectories. The set of all trajectories forms the phase portrait of a dynamical system.

By describing the system in this way, we have made an assumption about the way the system evolves. We have assumed that the current state of the system uniquely determines the next state and that the transition from the current state to the next one is independent of the time at which the transition occurs (i.e. independent of the time of experiment). Such systems are deterministic systems.

We might wonder about this abstraction and its usefulness. Passing from a concrete real life system to its abstract representation entails some loss of information. The abstract theory is not meant to replace detailed analysis (such as in a case study), but good reason to study the abstract model is its generality: any technique or concept we can deduce in the abstract model will immediately apply to any concrete case, giving tools that can be used on the entire spectrum of systems (and many case studies).

Stripping away the details of a system also can focus attention on its more essential properties. This is often a crucial step if we want to compare or draw analogies between systems of different kinds. Similarities are often apparent only when we step back and look at the big picture. For example, some models of the stock market and of the motion of billiard balls are very closely related, but we can not see this until we describe them abstractly and forget any interpretation of the variables. There is also another advantage to the abstract model – the details can actually be irrelevant to the problem, and adopting the abstract point of view can lead to new insights that would otherwise be obscured.

Nonlinearity of dynamical systems

Dynamical systems can be classified in two broad categories depending on the properties of their defining equations. Linear dynamical systems are given by systems of linear differential equations which means that the output is proportional to the input. This property is not fulfilled in the case of nonlinear dynamical systems, where the state variables cannot be written as a sum of independent components (and the output is not proportional to the input). While linear dynamical systems can be solved analytically (and their solution can be written as a combination of algebraic, polynomial, trigonometric or exponential functions), nonlinear systems are hard or impossible to solve analytically. Having an analytical solution provides complete information about the system and its behavior, but even without having one, we are able to learn something about the system’s properties using qualitative methods. Several definitions should be mentioned here:

• Equilibrium point is a state of a dynamical system that does not change with time. There are several types of equilibrium points:
  • Attractor is a state (or set of states) of a dynamical system to which trajectories originating from initial conditions converge over time. For each attractor, a basin of attraction is the set of all initial states leading to long-time behavior that approaches that attractor. Bistable systems have two attractors and multistable systems have three or more. Depending on the initial conditions, future states of the system tend to one of the several possible attractors. Because of this, we investigate sensitivity of a system’s behavior to initial conditions.
  • Saddle point only partially fulfils the above mentioned property, meaning that some trajectories converge to this point and others go away from it.
  • Repeller is an equilibrium point from which all trajectories go away.
• Periodic orbit is a special type of solution for a dynamical system, namely it repeats itself in time. 
• Stability is a property of a system to return to an attractor after perturbations. 
• Qualitative behavior of the system, i.e. the number and type of equilibrium points, can change for different choices of parameters. Sensitivity to parameters is studied using bifurcation analysis.

Qualitative analysis of dynamical systems means that we are identifying the number and type of equilibrium points or periodic orbits and conducting bifurcation and stability analysis. 

What does this tell us about causality?

The dynamical systems theory, as a mathematical theory, does not provide causal explanation. It leads to equilibrium explanations (Sober, 1982), where the state space represents all possible configurations of a system, while attractors and basins of attraction list all possible outcomes of the system’s dynamics (starting from any initial condition and following prescribed trajectory). A causal explanation, on the other hand, focuses on the actual trajectory of the system. In this sense, an equilibrium explanation explains more than a causal explanation – it places the actual trajectory in a more general structure of the phase space without giving the actual cause. In other words, equilibrium explanations can be more explanatory than causal explanations even though they provide less information about what the actual cause was. Equilibrium explanations provide understanding, and understanding can be enhanced without providing more details about what the cause was.

According to Lyon and Colyvan (2008), the explanatory power of models is not that they accurately mirror, map, or represent real systems. They are explanatory because there is a detailed story about why the myriad details that distinguish a class of systems are irrelevant to their large-scale behavior. This story demonstrates, rather than assumes, a kind of stability or robustness of the large-scale behavior we want to explain under drastic changes in the various details of the system. For example, a regime shift in a shallow lake is enabled by the existence of a clear and turbid state attractor. When nutrient concentration increases and finally reaches threshold value, a clear lake becomes turbid i.e. regime shift occurs. Existence of a positive feedback loop in the system is a necessary condition for bistability and this is the key mechanism we have to look after in studying lake dynamics. Specific properties of the lake (such as actual aquatic species) may alter the outcome quantitatively or select one of the two possible outcomes, but it is a positive feedback loop that allows for bistability and regime shift.

References

Kuznetsov, A. Elements of Applied Bifurcation Theory. Third Edition. Springer-Verlag (2004) 

Strogatz, S. Nonlinear Dynamics and Chaos. Perseus, Reading  (1994)

Sober, E. Equilibrium explanations. (1982)

Lyon, A. and Colyvan, M. The Explanatory Power of Phase Spaces. (2008)