Within science and mathematics, dynamics is the study of how things change with respect to time. As opposed to describing things simply in terms of their static properties, the patterns we observe all around us in how the state of things changes over time is an alternative way through which we can describe the phenomena we see in our world. A state space – also called a phase space – is a model used within dynamical systems to capture this change in a system’s state over time. A state space of a dynamical system is a two or possibly three-dimensional graph in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the state space.
Continuous & Discrete Models
It is possible to model the change in a system’s state in two ways, as continuous or discrete. Firstly, with continuous models, the time interval between our measurements is negligibly small making it appear as one long continuum, and this is done through the language of calculus. Calculus and differential equations have formed a key part of the language of modern science since the days of Newton and Leibniz. Differential equations are great for few elements. They give us lots of information, but they also become very complicated very quickly. On the other hand, we can measure time as discrete meaning there is a discernible time interval between each measurement, and we use what are called iterative maps to do this. Iterative maps give us less information but are much simpler and better suited to dealing with very many entities where feedback is important. Whereas differential equations are central to modern science, iterative maps are central to the study of nonlinear systems and their dynamics as they allow us to take the output of the previous state of the system and feed it back into the next iteration, thus making them well designed to capture the feedback characteristic of nonlinear systems.
The first type of motion we might encounter is simple transient motion, that is to say, some system that gravitates towards a stable equilibrium and then stays there. Such as putting a ball in a bowl, it will roll around for a short period before it settles at the point of least potential gravity, its so-called equilibrium, and then will just stay there until perturbed by some external force.
Another common type of motion we encounter is periodic motion. For example, the motion of the planets around the Sun is periodic. This type of periodic motion is of course very predictable. We can predict far out into the future and way back into the past when eclipses happen. In these systems, small disturbances are often rectified and do not increase to alter the system’s trajectory very much in the long run. The rising and receding motion of the tides or the change in traffic lights are also examples of periodic motion. Whereas in our first type of motion the system simply moves towards its equilibrium point, in this second periodic motion it is more like it is cycling around some equilibrium.
All dynamic systems require some input of energy to drive them. In physics, they are referred to as dissipative systems as they are constantly dissipating the energy being inputted to the system in the form of motion or change. A system in this periodic motion is bound to its source of energy, and its trajectory follows some periodic motion around it, or towards and away from it. For example, the human body requires the input of food on a periodic basis, we consume food then dissipate it through some activity and then consume more and dissipate it again in a somewhat periodic fashion. Like other biological systems, we are bound to cycle through this set of states. The same is true for our car or a business that are constrained by the inputs of fuel or finance. The dissipation and the driving force tend to balance, setting the system into its typical behavior. This typical set of states the system follows around its point of equilibrium is called an attractor.
In the field of dynamical systems, an attractor is a set of values or states toward which a system tends to evolve for a wide variety of starting conditions to the system. System values that get close enough to the attractor remain close even if slightly disturbed. There are many examples of attractors such as the use of addictive substances. While being subject to the addiction our body cycles in and out of its physiological influence but continuously comes back to it in a somewhat periodic and predictable fashion, that is, until it is able to break free from it. A so-called basin of attraction then describes all the points within a state space that will move the system towards a particular attractor. For example, a planet’s gravitational field is a basin of attraction. If we place some matter that is large enough into the gravitational field, it will be drawn into its orbit irrespective of its starting condition. An attractor can be thought of as a subspace of a state space that closes in on itself.
Transience and periodic motion are characteristic of linear systems, relatively simple systems with a single point equilibrium. But the dynamics of nonlinear systems involve the interaction of more than just one attractor, and they may have multiple equilibria, with their long-term trajectory being sensitive to initial conditions making it virtually impossible to predict with any accuracy. This is what is called chaos; sensitive dependence upon initial conditions that allows the behavior of a deterministic nonlinear system to be non-predictable.