A simple way to visualize the solution of the Lotka–Volterra equation is to draw them in the Rabbits–Wolves plane. Indeed at each time t, the solution is a pair of values (R(t), W (t)). This pair represents a single point in the phase plane R − W.

As t increases, the point (R(t), W (t)) traces out a curve in that plane. Poincaré’s revolutionary idea was to consider the shape of these curves. If a solution is constant, it will be a single fixed point in the phase plane. If it is periodic, it will create a periodic orbit in the same plane, that is a closed curve. A simple general question is then: what are the possible shapes in the phase‐plane for solutions of differential equations? This problem is one of the central problems of the mathematical theory of dynamical systems developed in the 20th century as an alternative to finding exact solutions. A particularly important aspect of the theory is that for a large class of systems of two variables that do not depend explicitly on time (such as the Lotka–Volterra system), the answer is remarkably simple. The only possible solutions that remain finite are periodic orbits, fixed points, or curves that connect fixed points and periodic orbits (fixed point to fixed point, fixed point to periodic orbit, or periodic orbit to periodic orbit).

What about systems of three variables? One of the great surprises of applied mathematics was the discovery that for dynamical systems with three variables, the curves drawn in phase space can be extremely complex. Apart from the fixed points and periodic orbits, there are complex solutions known as chaotic orbits. To illustrate the possibility of such orbits, we return to our Lotka–Volterra system and add another predator, say foxes. Our new system now reads

where α, β, and γ are constant positive parameters and F represents the fox population. If these parameters vanish, we recover our initial Lotka–Volterra system. This system is not meant to be a model of a particular ecological system. Instead, it is used here to understand the nature of solutions to be found in dynamical systems.

Depending on the values of parameters and the initial population size, various behaviours, such as fixed points and periodic orbits, can be discovered. More interestingly, in some parameter regimes, the solutions become chaotic, as shown in Figure 13. For these values of the parameters, the solution does not settle on a periodic orbit, but instead is seen to wrap around a spheroidal structure until it reaches its south pole. Then the orbit is propelled directly from the south to the north pole before it starts winding around the ball again. This orbit is called chaotic because of its complexity. For instance, it becomes impossible to predict after a while whether the orbit will be close to the north pole or to the south pole, or how many times it is going to wind around the ball before reaching the north pole. Whereas the solution is fully determined by the initial values, many of its characteristics are similar to those found in random systems. This also implies that two trajectories that start arbitrarily close to each other will eventually look completely different.

A chaotic attractor for the three‐dimensional Lotka–Volterra equations (α = 2, β = 2.9, γ = 3). Left: for small times, an initial solution spirals around and downward until it reaches the south pole. The orbit follows the central core up to the north pole where it starts spiralling down again. Right: for longer times, the solution repeats this motion. However, the number of turns in the spiral changes in an unpredictable manner.

In the 1960s, the discovery that simple dynamical systems could have complex chaotic solutions marked the beginning of a new era in the study of differential equations. It was also a refutation of Laplace’s deterministic view. Yes, the equations of classical mechanics are simple and deterministic, but their solutions can be as complex as random systems. The accumulation of small and simple interactions over long durations can lead to extraordinarily complex behaviours. Equations may be simple, but their solutions can be devilishly complex.