page_title: Method of averaging
section_title: Example: Restricting the time interval
hierarchical_section_title: Method of averaging / Non-autonomous class of systems: more examples / Example: Restricting the time interval
caption_attribution_description: English: The plot represents the necessity to restricting the time interval of the averaged solution. There is comparison between the original and the averaged one. In the end of the time frame, we note that the both solution blow up.
context_page_description: In mathematics, more specifically in dynamical systems, the method of averaging exploits systems containing time-scales separation: a fast oscillation versus a slow drift. It suggests that we perform an averaging over a given amount of time in order to iron out the fast oscillations and observe the qualitative behavior from the resulting dynamics. The approximated solution holds under finite time inversely proportional to the parameter denoting the slow time scale. It turns out to be a customary problem where there exists the trade off between how good is the approximated solution balanced by how much time it holds to be close to the original solution.
  More precisely, the system has the following form
  of a phase space variable The fast oscillation is given by versus a slow drift of. The averaging method yields an autonomous dynamical system
  which approximates the solution curves of inside a connected and compact region of the phase space and over time of.
  Under the validity of this averaging technique, the asymptotic behavior of the original system is captured by the dynamical equation for.
context_section_description: The average theorem assumes existence of a connected and bounded region which affects the time interval  of the result validity. The following example points it out. Consider the
  where .  The averaged system consists of
  which under this initial condition indicates that the original solution behaves like
  where it holds on a bounded region over .
caption: Figure 4: The plot depicts two fundamental quantities the average technique is based on: the bounded and connected region 
    
      
        
          D
        
      
      {\displaystyle D}
    
   of the phase space and how long (defined by the constant 
    
      
        
          c
        
      
      {\displaystyle c}
    
  )  the averaged solution is valid. For this case, 
    
      
        
          
            
              
                z
                ¨
              
            
          
          +
          z
          =
          8
          ε
          cos
          ⁡
          
            (
            t
            )
          
          
            
              
                
                  z
                  ˙
                
              
            
            
              2
            
          
          ,
           
          z
          (
          0
          )
          =
          0
          ,
           
          
            
              
                z
                ˙
              
            
          
          (
          0
          )
          =
          1
          ;
           
          8
          ε
          =
          
            
              2
              15
            
          
        
      
      {\textstyle {\ddot {z}}+z=8\varepsilon \cos {(t)}{\dot {z}}^{2},~z(0)=0,~{\dot {z}}(0)=1;~8\varepsilon ={\frac {2}{15}}}
    
  . Note that both solutions blow up in finite time.  Hence, 
    
      
        
          D
        
      
      {\displaystyle D}
    
   has been chosen accordingly in order to maintain the boundedness of the solution and the time interval of validity of the approximation is 
    
      
        
          0
          ≤
          ε
          t
          <
          L
          <
          
            
              1
              3
            
          
        
      
      {\displaystyle 0\leq \varepsilon t<L<{\frac {1}{3}}}
    
  .
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