0)}. Any orbit not on the stable manifold satisfies an equation of the form for some real constant A. It follows that for any real A, we can create a center manifold by piecing together the curve for x > 0 with the negative x axis (including the origin). Moreover, all center manifolds have this potential non-uniqueness, although often the non-uniqueness only occurs in unphysical complex values of the variables. Delay differential equations often have Hopf bifurcations Another example shows how a center manifold models the Hopf bifurcation that occurs for parameter in the delay differential equation . Strictly, the delay makes this DE infinite-dimensional. Fortunately, we may approximate such delays by the following trick that keeps the dimensionality finite. Define and approximate the time-delayed variable, , by using the intermediaries and . For parameter near critical, , the delay differential equation is then approximated by the system Copying and pasting the appropriate entries, the web service finds that in terms of a complex amplitude and its complex conjugate , the center manifold and the evolution on the center manifold is This evolution shows the origin is linearly unstable for , but the cubic nonlinearity then stabilises nearby limit cycles as in classic Hopf bifurcation. See also Invariant manifold Stable manifold Lagrangian coherent structure Normally hyperbolic invariant manifold Notes References . External links Dynamical systems