The Lorenz system is a system of ordinary differential equations (the Lorenz equations) first studied by Edward N. Lorenz. For certain parameter values and initial conditions, the system of ODEs has chaotic solutions. The solution is then a so-called strange attractor called the Lorenz attractor, discovered by Lorenz in 1962.

The Lorenz equations were developed as a simplified mathematical model for atmospheric convection. They are a system of three coupled ODEs with three states (degrees of freedom), u, v, and w:

Here, the parameters a, b, and c are generally positive scalar numbers. Not all solutions are chaotic, but Lorenz found that the values 10, 8/3, and 28, respectively, give a Lorenz system with a chaotic behavior.

For the parameter values in Table 1, the solution to this system of equations approaches a geometrical object in the phase space defined by the solution coordinates (u, v, w) called a strange attractor or the Lorenz attractor. It can be shown that this object is not of a standard geometrical topology, like a disk (with dimension 2) or a line (with dimension 1), but another strange topology with a fractal dimension larger than 2. Moreover, almost all perturbations to the solution grow exponentially fast with time. As an illustration of this behavior, the model starts from an initial solution close to the attractor and studies how a very small perturbation to this initial data grows. More precisely, the initial solution used here is (u, v, w) = (−10, −4.45, 35.1), which in a second perturbed case is changed to (u, v, w) = (−10+pert, −4.45+pert, 35.1+pert), where pert = 10−11. To ascertain that the difference between the unperturbed and perturbed problems is larger than the numerical errors, the tolerances used when solving the model are very strict. A default absolute tolerance factor of 0.1 is used, meaning that the absolute tolerance is 0.1 times the relative tolerance TOL used in the model.

First consider the numerical errors. Table 2 shows the results for the solution (u, v, w) for two unperturbed simulations with TOL = 10−15 and TOL = 10−16, respectively, at the times t = 20, 25, 30, and 35. It is clear that for t = 20 all 5 displayed digits of the result agree, a strong indication of a highly accurate solution. In contrast, already at t = 30 only two digits remain correct. Both the numerical errors and the perturbations will grow over time; eventually no digits will be correct with this method and these tolerances.

Figure 1 visualizes how the difference between the unperturbed and perturbed problems grows with time. The plot shows the absolute difference between the two solutions for the most accurate (tight) tolerance value as functions of time together with an estimate for the average growth rate given by eλt, where λ = 0.9, the maximal Lyapunov exponent for this model according to the literature. Note that the plot uses a vertical axis with a logarithmic scale. As the plot shows, the growth of the difference is close to the correct value until it reaches O(1), after which it does not grow much further. The reason is that all solutions stay close to the attractor, so the difference is bounded.

With the chosen set of parameter values, the Lorenz system behaves as a Lorenz attractor. The point trajectories plot in Figure 2 of the phase space shows the “butterfly” or “figure eight” characteristic of this attractor.

From the File menu, choose New.

In the New window, click  Model Wizard.

The parameters a, b, and c are the system parameters for the Lorenz system of ODEs. The default values of 10, 8/3, and 28, respectively, are the values that Lorenz used.

In the Study toolbar, click  Show Default Solver.

Use a Parametric Sweep to study the effect of a small perturbation to the initial data and to changes in a very strict solver tolerance.

Use a Global plot to visualize how the solution changes due to the small perturbation and add an exponential growth corresponding to the largest Lyapunov exponent 0.9 reported in the literature for comparison. Note how the growth stops when the deviation is of O(1); this is caused by the attraction effect. The deviations cannot be larger, since all solutions are close to the attractor.

Use a Point Trajectories plot to visualize the solution to the Lorenz system as point trajectories as shown in Figure 2.

In the Home toolbar, click  Add Plot Group and choose 3D Plot Group.