# Lorenz Equations and Equilibrium

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Show that for , the equilibrium (x*, y*, z*) = (0, 0, 0) is globally (nonlinearly) stable for the Lorenz system. That is, any (x(t), y(t), z(t)) would eventually approach (0, 0, 0) as .

Consider the "volume"

(a) Show that, using the Lorenz equations:

(b) Show that is strictly negative unless one reaches (x, y, z) = (0, 0, 0). Thus argue that the point (0, 0, 0) is the final destination of all trajectories (x, y, z) for

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Show that for , the equilibrium (x*, y*, z*) = (0, 0, 0) is globally (nonlinearly) stable for the Lorenz system. That is, any (x(t), y(t), z(t)) would eventually approach (0, 0, 0) as .

Consider the ...

#### Solution Summary

Lorenz Equations and Equilibrium are investigated.