title: Tikhonov's theorem (dynamical systems)
text: # Tikhonov's theorem (dynamical systems)
  
  In applied mathematics, Tikhonov's theorem on dynamical systems is a result on stability of solutions of systems of differential equations. It has applications to chemical kinetics. The theorem is named after Andrey Nikolayevich Tikhonov.
  
  ## Statement
  
  Consider this system of differential equations:
  
      ${\begin{aligned}{\frac {d\mathbf {x} }{dt}}&=\mathbf {f} (\mathbf {x} ,\mathbf {z} ,t),\\\mu {\frac {d\mathbf {z} }{dt}}&=\mathbf {g} (\mathbf {x} ,\mathbf {z} ,t).\end{aligned}}$
  
  Taking the limit as $\mu \to 0$, this becomes the "degenerate system":
  
      ${\begin{aligned}{\frac {d\mathbf {x} }{dt}}&=\mathbf {f} (\mathbf {x} ,\mathbf {z} ,t),\\\mathbf {z} &=\varphi (\mathbf {x} ,t),\end{aligned}}$
  
  where the second equation is the solution of the algebraic equation
  
      $\mathbf {g} (\mathbf {x} ,\mathbf {z} ,t)=0.$
  
  Note that there may be more than one such function $\varphi$.
  
  Tikhonov's theorem states that as $\mu \to 0,$ the solution of the system of two differential equations above approaches the solution of the degenerate system if $\mathbf {z} =\varphi (\mathbf {x} ,t)$ is a stable root of the "adjoined system"
  
      ${\frac {d\mathbf {z} }{dt}}=\mathbf {g} (\mathbf {x} ,\mathbf {z} ,t).$
  