Numerical Discretization  

In general, differential equations must be integrated numerically. To this end numerical discretizations are applied. If not done properly, the discretised "solution" might differ drastically from the true solution of the differential equation. The difference between true solution and numerical approximation can be a quantitative one, or may even show a qualitatively different picture.

This will be illustrated by means of an example: We choose a specific two-dimensional autonomous differential equation. Its two time-dependent variables are denoted u(t) and v(t). Solutions can be plotted in the two-dimensional (u,v)-phase plane. Our choice is the Van der Pol equation, which can be written as the system

   du/dt = v

   dv/dt= 0.05 * (1 - u*u)*v - u

* means multiplication, and the "d" indicates differentiation with respect to time t. This system has a stable periodic solution (u(t),v(t)), a closed orbit in the phase plane, also called "limit cycle". This true solution will be shown with green curves in the following figures. The horizontal axis depicts u-values, and the vertical axis depicts v-values. Apart from this attracting orbit there is an unstable stationary state at (u,v)=(0,0). This stationary state is a repellor, neighboring true solution will move away. 

Crucial for numerical discretization is the chosen step length. To emphasise possible difficulties we choose simplest integrators with step lengths that are intentionally chosen too large. The results of the chosen numerical discretization schemes are shown with red crosses. All simulations start at the point (u,v)=(-1.244, -1.6045) very close to the limit cycle. The true trajectory starting there should approach the green limit cycle even closer.

For a first test of a discretization, we choose the explicit Euler method with step length 0.2 .

The red crosses of the numerical discretization (connected by short straight lines) show a different and false behaviour: They leave the neighborhood of the true (green) attractor and approach an artificial closed orbit with larger amplitude, which is no true solution of the differential equation! The size of the false attractor varies with the chosen step length.

Another test: Now we start a trajectory inside the true limit cycle (blue crosses):

The calculated trajectory that starts inside (blue) tends to the location (0,0) of the stationary solution. The true instability of the differential equation has become stable for this specific discretization.  

As another test, we now choose an implicit Euler scheme, and smaller step length 0.02.

Now both calculated trajectories approach artificial limit cycles.

Such experiments are exaggerated because of the chosen extremely large step lengths. But there should be a warning: A numerical discretization might have artificial attractors different from the differential equation.