Stability of the Lorenz System
The Lorenz system is represented by three first-order differential equations (see [1, Section 9.0 and Exercise 9.3]):[more]
This system can exhibit both chaotic and nonchaotic motion depending on the value of .
When , the system has one fixed point (the origin), and it is stable.
When , the system has three fixed points and the origin becomes unstable (it has at least one positive eigenvalue), while the other two fixed points are stable.
For , all the fixed points become unstable (at least one eigenvalue has a positive real part).
The plotted eigenvalues show these phenomena. A Hopf bifurcation occurs at , when the two eigenvalues are purely imaginary and one is real.
Chaotic motion can be found for or , while periodic motion can be found when or .
This Demonstration shows the non-oscillating, periodic and chaotic dynamics of the Lorenz system by varying the value of and the amount of time plotted (e.g. by plotting the final 20 seconds, transient motion is not pictured). In short, this Demonstration shows several important plots for the Lorenz system: the parametric 3D space with vector plot, the eigenvalues for the fixed points, the phase space in three different planes and time history of all three states. The initial conditions are , and , and the dynamics are plotted for .[less]
 S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 2nd ed., Boulder, CO: Westview Press, 2015.