If a nonlinear system is time dependent the substitution therefore increases the dimension of the system by one but ensures that the solution curves in the phase space don’t change with changing .
\left{\begin{matrix} \dot{x\_{1}} = x\_{2} \\ \dot{x\_{2}} = \frac{1}{m}(-kx\_{1^-bx\_{2}+F\cos x\_{3}}) \\ \dot{x\_{3}} \end{matrix}\right.- The amount of variables needed to characterize the state of a system is equal to the dimension of it’s phase-space