Before addressing the unstable situation and incorporating a controller to eliminate chaos in system (1) when, we begin with an example with an R value so that are locally stable. For the example, we choose. For this value of R, we have the rest points and. The eigenvalues of the Jacobian of (1) evaluated at are −20.3408, 9.34082, and −2.266667. Because of the positive eigenvalue, is unstable. At the eigenvalues of the Jacobian are −13.3571, and so are locally stable.

In Figure 1, we illustrate the basins of attraction for. For the given value of, if is in the gray region, the solution satisfying the initial conditions converges to. On the other hand, if is in the black region, the solution satisfying the initial conditions converges to. Comment: All graphics and computations in this paper were done using Mathematica 10.0 [12] . If you would like a copy of the code, send a request to Jim Braselton at jbraselton@georgiasouthern.edu.

Note the “speckled” regions in Figure 1, in these regions we see that system (1) is highly sensitive to initial conditions. For some initial conditions, the solutions converge to but a slight change in the conditions can cause the solution to converge to and vice-versa.

The complexity of how the initial conditions affect the convergence of the solution to or is further illustrated in Figures 2-4. Because the z-coordinate of and are equal, the plot of z as a function of t in Figure 4 is the same regardless of whether the solution converges to or.

Interestingly, it is possible for to be locally stable and yet the Lorenz equations will have a strange attractor.

For the and values typically used when studying the Lorenz system, for, are stable and coexist with a single strange attractor [13] . To illustrate the interesting situation, we set. For this value of R, we have the rest points and. The eigenvalues of the Jacobian of (1) evaluated at are −21.7558, 10.7558, and −2.266667. Because of the positive eigenvalue, is unstable. At the eigenvalues of the Jacobian are −13.6462, and so are locally stable.

In Figure 5 we illustrate the basins of attraction for. For the given value of, if

is in the gray region, the solution satisfying the initial conditions converges to. On the other hand, if is in the black region, the solution satisfying the initial conditions converges to. However, in the speckled region, the solution satisfying the initial conditions converges to the strange attractor which is illustrated in Figure 6.

Solving

results in the equilibrium points

The Jacobian for (8) is

after dividing by, the characteristic polynomial of (11) evaluated at the equilibrium points (10) has the form, where

If, , , and are all positive and

then by the Routh-Hurwitz theorem, the real part of the roots of the characteristic polynomial of (11) evaluated at the equilibrium points (10) will all have negative real part and, consequently, will be locally stable.

The roots of have negative real part under the following conditions.

・ and

- and or

-

or

・ .

If these conditions are not satisfied, then the real part of the roots of the characteristic polynomial of (11) evaluated at the equilibrium points (10) will have a positive real part and, consequently, will be unstable. If initial conditions, , cause system (8) to converge to one of, then

because of the term in system (8), must be greater than 0 to avoid discontinuities in the solution to

the system.

The values and are frequently used when studying the Lorenz equations. Using these values we can rewrite the conditions above as

Figure 7 shows a portion of the region where or For points in

the shaded region, the equilibrium points (10) will be locally stable. Also, if k is sufficiently large, will be locally stable, which is not illustrated in Figure 7. However, from a physical interpretation, smaller k values are preferred as they require less energy than larger k values. Large k-values can be interpreted as a “heavy-handed controller:” if enough power is applied to the system to cause it to stop, it will.

Using, , and, with, we find the rest points and . Evaluated at, the eigenvalues of the Jacobian of (8) when are and so that are unstable. Evaluated at, the Jacobian has eigenvalues, , and so is unstable as well (see Figure 8).

To stabilize the system, we first solve, which shows us that if,

will be locally stable (see Figure 9). We set. In this case,. Note that the addition of the controller changes the location of the rest points. The eigenvalues of the Jacobian of

evaluated at are, and so are locally stable (although their location has moved slightly from when their was no controller).

Numerically, we see that there is a ball centered at each of for which all trajectories starting in the ball converge to. However, the basin of attraction for each point is quite complex. Figure 10 shows the plot of for various values of. First, we solve system (8) using the initial conditions, and. We then plot in gray if the solution converges to and in black if the solution converges to.

The complexity of the basins of attractions are even more apparent when zooming in near one of the equilibria or zooming out as shown in Figure 11.

These figures provide numerical evidence that controlled system has no limit cycles. For those initial

conditions in the black region, the solutions converges to while for those solutions in the gray region, the solutions converge to. Of interest is the “speckled” regions. In these regions we see that system (8) is highly sensitive to initial conditions. For some initial conditions, the solutions converges to but a slight change in the conditions can cause the solution to converge to and vice-versa.

The initial conditions have a major affect on the convergence of a solution to or as is illustrated in Figures 12-14. Because the z-coordinate of and are equal, the plot of z as a function of t in Figure 14 is the same regardless of whether the solution converges to or.

To investigate the existence of a strange attractor as observed in Section 2.2, for each value in Figure 10, we drew a line from slightly before to slightly after. We then computed solutions satisfying, , and for 50 equally spaced points on each line and graphed them. In every case, the observed solution converged to or. For points in the “speckled” region, the solutions varied as to whether they converged to or. Thus, for this R value, we were not able to produce a strange attractor as we did in Section 2.2. Our conjecture is that periodic orbits do not exist and that all solutions converge to or.

Using, , and, with, we find the rest points and . Evaluated at, the eigenvalues of the Jacobian of (8) when are and so that are unstable. Evaluated at, the Jacobian has eigenvalues, , and so is unstable as well. From Figure 7, we see that we will be able to stabilize the system for k-values greater than approximately 0.35. In Section 3.4 that follows, we chose a small k-value to stabilize the system. In this case, choose a large k-value and choose. We choose this k-value because if a trajectory converges to or,

and for R-values in this range (), it is possible for to be stable and yet the system may have a strange attractor as we saw in Section 2.2. Thus, we expected that this controlled system may have a strange attractor as we saw with the corresponding non-controlled system. For this k-value, we obtain. The eigenvalues of the Jacobian evaluated at are,

, and.

Figure 15 illustrates the basin of attraction for various values of. In this case, observe that with large k, the boundary between and is specific and there is not a “speckled” region that was observed for smaller k values.

To investigate the stability of and, we followed the exact same procedure as in Section 2. Once again, we were not able to find a strange attractor leading us to believe that the stability of the limit set is global. The numerical evidence suggest that with a controller of this type, solutions will converge to either or. Even though converges to a number in the range where the uncontrolled system can have two stable equilibria and a strange attractor, for the controlled system we were surprised to see that the controlled system did not produce a strange attractor.

To investigate how implementing our controller affects the stability of solutions of the Lorenz equation, we have examined an extreme range of R and k-values hoping that these simulations indicate the the global behavior that we have numerically observed.

Our last example involves a large R-value and a small k-value. We repeat the example in Section 3.4 except we use. For this k-value, we obtain. The eigenvalues of the Jacobian evaluated at are and.

The situation appears to be similar to the situation discussed previously with small k-values. In Figure 16, observe the “speckled” region as before where solutions may converge to or with slight differences in the initial conditions.

As we did with the previous examples, to investigate the existence of a strange attractor as observed in Section 2.2, for each value in Figure 16, we drew a line from slightly before to slightly after. We then computed solutions satisfying, , and for 50 equally spaced points on each line and graphed them. In every case, the observed solution converged to or. For points in the “speckled” region, the solutions varied as to whether they converged to or. Thus, for this R value, we were not able to produce a strange attractor as we did in Section 2.2. We suspect that the stability is global in that all

solutions converge to or.

Note that we were able to produce “almost strange attractors” in the sense that the trajectories appeared to be chaotic at first-but eventually converged to either or in all our simulations.