Let (x(t), y(t), z(t)) be the solution of the system  x' = y z,   y' = −2xz,   z' = −3xy  with initial condition (x0, y0, z0) and S = {(x, y, z)|x2 + y2 + z2 < 1} be the unit open ball in R3

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Let (x(t), y(t), z(t)) be the solution of the system

 x' = y z,  

y' = −2xz, 

 z' = −3xy 

with initial condition (x₀, y₀, z₀) and S = {(x, y, z)|x² + y² + z² < 1} be the unit open ball in R³ . Show that there exists an open set U ⊂ S, such that if (x₀, y₀, z₀) ∈ U, then x(t), y(t), z(t) ∈ S for all t > 0. 

Hint: construct an appropriate quadratic Lyapunov function V (x, y, z), and define the set as U = {(x, y, z) | V (x, y, z) < r₀} for some constant r₀ > 0.