Phase plane

<h2 id="example-of-a-linear-system">Example of a linear system</h2>
A two-dimensional <a href="/facts/System_of_differential_equations/4Lum06D8">system</a> of <a href="/facts/Linear_differential_equation/nLEbIIVf">linear differential equations</a> can be written in the form:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

d
                      x
                    
                    
                      d
                      t
                    
                  
                
              
              
                
                =
                A
                x
                +
                B
                y
              
            
            
              
                
                  
                    
                      d
                      y
                    
                    
                      d
                      t
                    
                  
                
              
              
                
                =
                C
                x
                +
                D
                y
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=Ax+By\\{\frac {dy}{dt}}&=Cx+Dy\end{aligned}}}

which can be organized into a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> equation:

d
                    
                      d
                      t
                    
                  
                
                
                  
                    [
                    
                      
                        
                          x
                        
                      
                      
                        
                          y
                        
                      
                    
                    ]
                  
                
                =
                
                  
                    [
                    
                      
                        
                          A
                        
                        
                          B
                        
                      
                      
                        
                          C
                        
                        
                          D
                        
                      
                    
                    ]
                  
                
                
                  
                    [
                    
                      
                        
                          x
                        
                      
                      
                        
                          y
                        
                      
                    
                    ]
                  
                
              
            
            
              
              
                
                  
                    
                      d
                      
                        v
                      
                    
                    
                      d
                      t
                    
                  
                
                =
                
                  A
                
                
                  v
                
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}&{\frac {d}{dt}}{\begin{bmatrix}x\\y\\\end{bmatrix}}={\begin{bmatrix}A&B\\C&D\\\end{bmatrix}}{\begin{bmatrix}x\\y\\\end{bmatrix}}\\&{\frac {d\mathbf {v} }{dt}}=\mathbf {A} \mathbf {v} .\end{aligned}}}

where A is the 2 × 2 <a href="/facts/Coefficient_matrix/S5JmDEW6">coefficient matrix</a> above, and v = (x, y) is a <a href="/facts/Coordinate_vector/TEcLqS40">coordinate vector</a> of two <a href="/facts/Independent_variable/dINfzIF2">independent variables</a>.
Such systems may be solved analytically, for this case by integrating:<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

 
 
 
 
 
 
 d
 y
 
 
 d
 x
 
 
 
 =
 
 
 
 C
 x
 +
 D
 y
 
 
 A
 x
 +
 B
 y
 
 
 
 
 
 {\displaystyle {\frac {dy}{dx}}={\frac {Cx+Dy}{Ax+By}}}

although the solutions are <a href="/facts/Implicit_function/UJmQ2Cyk">implicit functions</a> in x and y, and are difficult to interpret.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<h3>Solving using eigenvalues</h3>
More commonly they are solved with the coefficients of the right hand side written in matrix form using <a href="/facts/Eigenvalues/8TjEoT8u">eigenvalues</a> λ, given by the <a href="/facts/Determinant/eGYqJ2TF">determinant</a>:

det
        
          (
          
            
              
                [
                
                  
                    
                      A
                    
                    
                      B
                    
                  
                  
                    
                      C
                    
                    
                      D
                    
                  
                
                ]
              
            
            −
            λ
            
              I
            
          
          )
        
        =
        0
      
    
    {\displaystyle \det \left({\begin{bmatrix}A&B\\C&D\\\end{bmatrix}}-\lambda \mathbf {I} \right)=0}

and <a href="/facts/Eigenvectors/8TjEoT8u">eigenvectors</a>:

[
            
              
                
                  A
                
                
                  B
                
              
              
                
                  C
                
                
                  D
                
              
            
            ]
          
        
        
          x
        
        =
        λ
        
          x
        
      
    
    {\displaystyle {\begin{bmatrix}A&B\\C&D\\\end{bmatrix}}\mathbf {x} =\lambda \mathbf {x} }

The eigenvalues represent the powers of the exponential components and the eigenvectors are coefficients. If the solutions are written in algebraic form, they express the fundamental multiplicative factor of the exponential term. Due to the nonuniqueness of eigenvectors, every solution arrived at in this way has undetermined constants c1, c2, …, cn.
The general solution is:

x
        =
        
          
            [
            
              
                
                  
                    k
                    
                      1
                    
                  
                
              
              
                
                  
                    k
                    
                      2
                    
                  
                
              
            
            ]
          
        
        
          c
          
            1
          
        
        
          e
          
            
              λ
              
                1
              
            
            t
          
        
        +
        
          
            [
            
              
                
                  
                    k
                    
                      3
                    
                  
                
              
              
                
                  
                    k
                    
                      4
                    
                  
                
              
            
            ]
          
        
        
          c
          
            2
          
        
        
          e
          
            
              λ
              
                2
              
            
            t
          
        
        .
      
    
    {\displaystyle x={\begin{bmatrix}k_{1}\\k_{2}\end{bmatrix}}c_{1}e^{\lambda _{1}t}+{\begin{bmatrix}k_{3}\\k_{4}\end{bmatrix}}c_{2}e^{\lambda _{2}t}.}

where λ1 and λ2 are the eigenvalues, and (k1, k2), (k3, k4) are the basic eigenvectors. The constants c1 and c2 account for the nonuniqueness of eigenvectors and are not solvable unless an initial condition is given for the system.
The above determinant leads to the <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a>:

λ
          
            2
          
        
        −
        (
        A
        +
        D
        )
        λ
        +
        (
        A
        D
        −
        B
        C
        )
        =
        0
      
    
    {\displaystyle \lambda ^{2}-(A+D)\lambda +(AD-BC)=0}

which is just a <a href="/facts/Quadratic_equation/9IMnkJAT">quadratic equation</a> of the form:

λ
          
            2
          
        
        −
        p
        λ
        +
        q
        =
        0
      
    
    {\displaystyle \lambda ^{2}-p\lambda +q=0}

where 
 
 
 
 p
 =
 A
 +
 D
 =
 
 t
 r
 
 (
 
 A
 
 )
 
 ,
 
 
 {\displaystyle p=A+D=\mathrm {tr} (\mathbf {A} )\,,}
 
 ("tr" denotes <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace</a>) and 
 
 
 
 q
 =
 A
 D
 −
 B
 C
 =
 det
 (
 
 A
 
 )
 
 .
 
 
 {\displaystyle q=AD-BC=\det(\mathbf {A} )\,.}

The explicit solution of the eigenvalues are then given by the <a href="/facts/Quadratic_formula/j5ftMowG">quadratic formula</a>:

λ
        =
        
          
            1
            2
          
        
        (
        p
        ±
        
          
            Δ
          
        
        )
        
      
    
    {\displaystyle \lambda ={\frac {1}{2}}(p\pm {\sqrt {\Delta }})\,}

where 
 
 
 
 Δ
 =
 
 p
 
 2
 
 
 −
 4
 q
 
 .
 
 
 {\displaystyle \Delta =p^{2}-4q\,.}

<h3>Eigenvectors and nodes</h3>
The eigenvectors and nodes determine the profile of the phase paths, providing a pictorial interpretation of the solution to the dynamical system, as shown next.

The phase plane is then first set-up by drawing straight lines representing the two eigenvectors (which represent stable situations where the system either converges towards those lines or diverges away from them). Then the phase plane is plotted by using full lines instead of direction field dashes. The signs of the eigenvalues indicate the phase plane's behaviour:

<ul><li>If the signs are opposite, the intersection of the eigenvectors is a <a href="/facts/Saddle_point/v5MT6YbM">saddle point</a>.</li>
<li>If the signs are both positive, the eigenvectors represent stable situations that the system diverges away from, and the intersection is an unstable <a href="/facts/Node_(autonomous_system)/xzSP1Sgm">node</a>.</li>
<li>If the signs are both negative, the eigenvectors represent stable situations that the system converges towards, and the intersection is a stable node.</li></ul>
The above can be visualized by recalling the behaviour of exponential terms in differential equation solutions.

<h3>Repeated eigenvalues</h3>
This example covers only the case for real, separate eigenvalues. Real, repeated eigenvalues require solving the coefficient matrix with an unknown vector and the first eigenvector to generate the second solution of a two-by-two system. However, if the matrix is symmetric, it is possible to use the orthogonal eigenvector to generate the second solution.

<h3>Complex eigenvalues</h3>
Complex eigenvalues and eigenvectors generate solutions in the form of <a href="/facts/Sine/gQ6LXK8z">sines</a> and <a href="/facts/Trigonometric_function/czej7ur0">cosines</a> as well as exponentials. One of the simplicities in this situation is that only one of the eigenvalues and one of the eigenvectors is needed to generate the full solution set for the system.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Phase_line_(mathematics)/p2vsBH02">Phase line</a>, 1-dimensional case</li>
<li><a href="/facts/Phase_space/qPe4Fq57">Phase space</a>, n-dimensional case</li>
<li><a href="/facts/Phase_portrait/H3h1eG8p">Phase portrait</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://tutorial.math.lamar.edu/classes/de/phaseplane.aspx">Lamar University, Online Math Notes - Phase Plane, P. Dawkins</a></li>
<li><a href="http://tutorial.math.lamar.edu/classes/de/systemsde.aspx">Lamar University, Online Math Notes - Systems of Differential Equations, P. Dawkins</a></li>
<li><a href="https://web.archive.org/web/20110927183123/http://virtual.cvut.cz/dynlabmodules/ihtml/dynlabmodules/nonlin/node8.html">Overview of the phase plane method</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">D.W. Jordan; P. Smith (2007). Non-Linear Ordinary Differential Equations: Introduction for Scientists and Engineers (4th ed.). Oxford University Press. ISBN 978-0-19-920825-8. <a href="978-0-19-920825-8" target="_blank">978-0-19-920825-8</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">K.T. Alligood; T.D. Sauer; J.A. Yorke (1996). Chaos: An Introduction to Dynamical Systems. Springer. ISBN 978-0-38794-677-1. <a href="978-0-38794-677-1" target="_blank">978-0-38794-677-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">D.W. Jordan; P. Smith (2007). Non-Linear Ordinary Differential Equations: Introduction for Scientists and Engineers (4th ed.). Oxford University Press. ISBN 978-0-19-920825-8. <a href="978-0-19-920825-8" target="_blank">978-0-19-920825-8</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">W.E. Boyce; R.C. Diprima (1986). Elementary Differential Equations and Boundary Value Problems (4th ed.). John Wiley & Sons. ISBN 0-471-83824-1. <a href="0-471-83824-1" target="_blank">0-471-83824-1</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">D.W. Jordan; P. Smith (2007). Non-Linear Ordinary Differential Equations: Introduction for Scientists and Engineers (4th ed.). Oxford University Press. ISBN 978-0-19-920825-8. <a href="978-0-19-920825-8" target="_blank">978-0-19-920825-8</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
</ol>

Phase plane open-in-new

Phase plane