Normal form (dynamical systems)

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the normal form of a <a href="/facts/Dynamical_system/782gREl5">dynamical system</a> is a simplified form that can be useful in determining the system's behavior.
Normal forms are often used for determining local <a href="/facts/Bifurcation_theory/sLsN54Rn">bifurcations</a> in a system. All systems exhibiting a certain type of bifurcation are locally (around the equilibrium) <a href="/facts/Topological_conjugacy/vQvYmcIA">topologically equivalent</a> to the normal form of the bifurcation. For example, the normal form of a <a href="/facts/Saddle-node_bifurcation/3fTGpIyt">saddle-node bifurcation</a> is

d
              
              x
            
            
              
                d
              
              t
            
          
        
        =
        μ
        +
        
          x
          
            2
          
        
      
    
    {\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=\mu +x^{2}}

where 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 is the bifurcation parameter. The transcritical bifurcation

d
              
              x
            
            
              
                d
              
              t
            
          
        
        =
        r
        ln
        ⁡
        x
        +
        x
        −
        1
      
    
    {\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=r\ln x+x-1}

near 
 
 
 
 x
 =
 1
 
 
 {\displaystyle x=1}
 
 can be converted to the normal form

d
              
              u
            
            
              
                d
              
              t
            
          
        
        =
        μ
        u
        −
        
          u
          
            2
          
        
        +
        O
        (
        
          u
          
            3
          
        
        )
      
    
    {\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=\mu u-u^{2}+O(u^{3})}

with the transformation 
 
 
 
 u
 =
 x
 −
 1
 ,
 μ
 =
 r
 +
 1
 
 
 {\displaystyle u=x-1,\mu =r+1}
 
.
See also <a href="/facts/Canonical_form/K2jhB44k">canonical form</a> for use of the terms canonical form, normal form, or standard form more generally in mathematics.

Normal form (dynamical systems) open-in-new

Normal form (dynamical systems)