http://www.maths.adelaide.edu.au/anthony.roberts/gencm.php provides a web service to algebraically construct center manifolds of any ODE/DDE that you enter. The centre modes may be slow, as in a pitchfork bifuraction, or oscillatory, as in a Hopf bifurcation, or some more complicated superposition. In the case when the fast modes all decay, the centre manifold supplies you with a faithful large time model of the dynamics. For example, this web page could help you analyse the long time dynamics of the system \[
\frac{d\vec u}{dt}=\vec f(\vec u) =\left[\begin{array}{ccc}
2&1&2\\ 1&-1&1\\ -3&-1&-3 \end{array}\right] \vec u
+\varepsilon \left[\begin{array}{c}u_2u_3\\ -u_1u_3\\
-u_1u_2\end{array}\right]. \] As this system is already entered for you, just enter the magic word, then click on the Submit button to see. Alternatively, you can obtain the equivalent modulation equations corresponding to a given set of

ODE/DDEs that have one or more oscillatory modes (a Hopf bifurcation for example). The analysis provides you with equations for the evolution of the complex amplitudes of the oscillators. This approach is better than averaging/homogenisation.
Keywords | Bifurcation analysis, Control, Identification |

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