Numerical continuation

Numerical continuation is a method of computing approximate solutions of a system of parameterized nonlinear equations,

F
 (
 
 u
 
 ,
 λ
 )
 =
 0.
 
 
 {\displaystyle F(\mathbf {u} ,\lambda )=0.}
 
<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
The <a href="/facts/Parameter_(mathematics)/zFvVFMv9">parameter</a> 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
 is usually a real <a href="/facts/Scalar_(mathematics)/MkDbA4Eo">scalar</a> and the solution 
 
 
 
 
 u
 
 
 
 {\displaystyle \mathbf {u} }
 
 is an <a href="/facts/N-tuple/BI1HIxL8">n-vector</a>. For a fixed parameter value 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
, 
 
 
 
 F
 (
 ⋅
 ,
 λ
 )
 
 
 {\textstyle F(\cdot ,\lambda )}
 
 maps <a href="/facts/Euclidean_n-space/R2UbzmzM">Euclidean n-space</a> into itself.
Often the original mapping 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is from a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> into itself, and the <a href="/facts/Euclidean_n-space/R2UbzmzM">Euclidean n-space</a> is a finite-dimensional Banach space.
A <a href="/facts/Steady_state/qgrL4BNs">steady state</a>, or <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed point</a>, of a <a href="/facts/Parameterized_family/YoEWw0I8">parameterized family</a> of <a href="/facts/Flow_(mathematics)/CrmJPfeV">flows</a> or <a href="/facts/Map_(mathematics)/FABF1l2f">maps</a> are of this form, and by <a href="/facts/Discretization/JvjSzwyk">discretizing</a> trajectories of a flow or iterating a map, <a href="/facts/Periodic_orbit/9tRHxcrn">periodic orbits</a> and <a href="/facts/Heteroclinic_orbit/VQhvWArq">heteroclinic orbits</a> can also be posed as a solution of 
 
 
 
 F
 =
 0
 
 
 {\displaystyle F=0}
 
.

Numerical continuation open-in-new

Numerical continuation