Center manifold

===A simple example===

===A simple example===

Consider the system

Consider the system

: <math> \dot x=x^2,\quad \dot y=y.</math>

<math display="block"> \dot x=x^2,\quad \dot y=y.</math>

The unstable manifold at the origin is the ''y'' axis, and the stable manifold is the trivial set {(0,&nbsp;0)}. Any orbit not on the stable manifold satisfies an equation of the form <math>y=Ae^{-1/x}</math> for some real constant ''A''. It follows that for any real ''A'', we can create a center manifold by piecing together the curve <math>y=Ae^{-1/x}</math> for ''x''&nbsp;>&nbsp;0 with the negative ''x'' axis (including the origin).{{sfn|Chicone|2010|p=344}} Moreover, all center manifolds have this potential non-uniqueness, although often the non-uniqueness only occurs in unphysical complex values of the variables.

The unstable manifold at the origin is the ''y'' axis, and the stable manifold is the trivial set {(0,&nbsp;0)}. Any orbit not on the stable manifold satisfies an equation of the form <math>y=Ae^{-1/x}</math> for some real constant ''A''. It follows that for any real ''A'', we can create a center manifold by piecing together the curve <math>y=Ae^{-1/x}</math> for ''x''&nbsp;>&nbsp;0 with the negative ''x'' axis (including the origin).{{sfn|Chicone|2010|p=344}} Moreover, all center manifolds have this potential non-uniqueness, although often the non-uniqueness only occurs in unphysical complex values of the variables.

For parameter near

For parameter near

critical, <math>a=4+\alpha</math>, the [[delay differential equation]] is then approximated by the system

critical, <math>a=4+\alpha</math>, the [[delay differential equation]] is then approximated by the system

:<math> \frac{d\textbf{u}}{dt} =\left[\begin{array}{ccc} 0&0&-4\\

<math display="block"> \frac{d\textbf{u}}{dt} =\left[\begin{array}{ccc} 0&0&-4\\

2&-2&0\\ 0&2&-2 \end{array}\right] \textbf{u} +

2&-2&0\\ 0&2&-2 \end{array}\right] \textbf{u} +

\left[\begin{array}{c}-\alpha u_3-2u_1^2-u_1^3\\ 0\\

\left[\begin{array}{c}-\alpha u_3-2u_1^2-u_1^3\\ 0\\

0\end{array}\right]. </math>

0\end{array}\right]. </math>

In terms of a [[complex amplitude]] <math>s(t)</math> and its complex conjugate <math>\bar s(t)</math>, the center manifold is

In terms of a [[complex amplitude]] <math>s(t)</math> and its complex conjugate <math>\bar s(t)</math>, the center manifold is

:<math> \textbf{u}=\left[\begin{array}{c} e^{i2t}s+e^{-i2t}\bar s\\

<math display="block"> \textbf{u}=\left[\begin{array}{c} e^{i2t}s+e^{-i2t}\bar s\\

\frac{1-i}2e^{i2t}s +\frac{1+i}2e^{-i2t}\bar s\\

\frac{1-i}2e^{i2t}s +\frac{1+i}2e^{-i2t}\bar s\\

-\frac{i}2e^{i2t}s +\frac{i}2e^{-i2t}\bar s

-\frac{i}2e^{i2t}s +\frac{i}2e^{-i2t}\bar s

+{O}(\alpha+|s|^2) </math>

+{O}(\alpha+|s|^2) </math>

and the evolution on the center manifold is

and the evolution on the center manifold is

:<math> \frac{ds}{dt}= \left[

<math display="block"> \frac{ds}{dt}= \left[

\frac{1+2i}{10}\alpha s

\frac{1+2i}{10}\alpha s

-\frac{3+16i}{15}|s|^2s

-\frac{3+16i}{15}|s|^2s