Footnotes:

Definition: Let be differentiable and let = , where is the eigenvalue of the DF matrix. It is the largest change in distance in all directions about , after the first n iterations. Then the Lyapunov number is given by, ~ Back

• Let's illustrate by looking at how does the Lyapunov exponent changes with number of iteration for a = 0.2. (has period one orbit) This shows that the Lyapunov exponent does not change with number of iterations. (Notice that the difference of the determent is so small. MatLab® was unable to resolve the difference) The negative sign indicates the attractor is in the period region. ~ Back

• In partice, the size of the matrix get really big while trying to multiply them all together, and one way to put it in control is to multiply the matrix by a constant, call it the scaling factor, every time we iterated this calculation. (When the scaling factor is too small, the matrix will vanish to zero. If the scaling factor is too large, the entries of the matrix will still go to infinity) From several trial and error, one can find the scaling factor for the specific parameter chosen is 0.60558. ~ Back
• Just as check, we should verify that the determent of the Jacobian matrix does not change with number of iterations, and it should have the value:  The right figure shows that the determent of the DF matrix does not change with number of iterations. (Notice that the difference of the determent is so small. MatLab® was unable to resolve the difference) The positive determent indicates the attractor is in the chaotic region. ~ Back

• The understanding of why adding the feedback will re-enforce the system back to period behavior is still unsolved, and this is the final goal of this research. ~ Back