Center manifold

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Center (red) and unstable (green) invariant manifolds of saddle-node equilibrium point

In mathematics, the center manifold of an equilibrium point of a dynamical system consists of orbits whose behavior around the equilibrium point is not controlled by either the attraction of the stable manifold or the repulsion of the unstable manifold. The first step when studying equilibrium points of dynamical systems is to linearize the system. The eigenvectors corresponding to eigenvalues with negative real part form the stable eigenspace, which gives rise to the stable manifold. Similarly, eigenvalues with positive real part yield the unstable manifold.

This concludes the story if the equilibrium point is hyperbolic (i.e., all eigenvalues of the linearization have nonzero real part). However, if there are eigenvalues whose real part is zero, then these give rise to the center manifold. The behavior on the center manifold is generally not determined by the linearization and thus more difficult to study.

Center manifolds play an important role in bifurcation theory because interesting behavior takes place on the center manifold.

Definition[edit]

Let

\dot{\textbf{x}} = f(\textbf{x})

be a dynamical system with equilibrium point x^*.

The linearization of the system at the equilibrium point is

\dot{\textbf{x}} = A\textbf{x}, \qquad \text{where } A = \frac{df}{dx}(x^*).

The matrix A defines three subspaces:

  • the stable subspace, which is spanned by the generalized eigenvectors corresponding to the eigenvalues λ with Re λ < 0;
  • the unstable subspace, which is spanned by the generalized eigenvectors corresponding to the eigenvalues λ with Re λ > 0;
  • the center subspace, which is spanned by the generalized eigenvectors corresponding to the eigenvalues λ with Re λ = 0.

These spaces are all invariant subspaces of the linearized equation.

Corresponding to the linearized system, the nonlinear system has invariant manifolds, consisting of orbits of the nonlinear system. There is an invariant manifold tangent to the stable subspace and with the same dimension; this manifold is the stable manifold. Similarly, the unstable manifold is tangent to the unstable subspace, and the center manifold is tangent to the center subspace.[1] If, as is common, the eigenvalues of the center subspace are all precisely zero, rather than just real part zero, then a center manifold is often called a slow manifold.

The center manifold theorem[edit]

The center manifold theorem states that if ƒ is Cr (r times continuously differentiable), then at every equilibrium point there is a unique Cr stable manifold, a unique Cr unstable manifold, and a (not necessarily unique) Cr−1 center manifold.[2]

In example applications, a nonlinear coordinate transform to a normal form (mathematics) can clearly separate these three manifolds.[3] A web service [1] currently undertakes the necessary computer algebra for a range of finite-dimensional systems.

There is theory for the existence and relevance of center manifolds in infinite-dimensional dynamical systems.[4] The general theory currently only applies when the center manifold itself is of finite dimension. However, some applications, such as to shear dispersion, can justify and construct an infinite-dimensional center manifold.[5]

Center manifold and the analysis of nonlinear systems[edit]

As the stability of the equilibrium correlates with the "stability" of its manifolds, the existence of a center manifold brings up the question about the dynamics on the center manifold. This is analyzed by the center manifold reduction, which, in combination with some system parameter μ, leads to the concepts of bifurcations.

Correspondingly, two web services currently undertake the necessary computer algebra to construct just the center manifold for a wide range of finite-dimensional systems (provided they are in multinomial form).

  • One web service [2] constructs slow manifolds for systems which are linearly diagonalised, but which may be non-autonomous or stochastic.[6]
  • Another web service [3] constructs center manifolds for systems with general linearisation, but only for autonomous systems.[7]

Examples[edit]

The Wikipedia entry on slow manifolds gives more examples.

A simple example[edit]

Consider the system

 \dot x=x^2,\quad \dot y=y.

The unstable manifold at the origin is the y axis, and the stable manifold is the trivial set {(0, 0)}. Any orbit not on the stable manifold satisfies an equation on the form y=Ae^{-1/x} for some real constant A. It follows that for any real A, we can create a center manifold by piecing together the curve y=Ae^{-1/x} 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[edit]

Another example shows how a center manifold models the Hopf bifurcation that occurs for parameter a\approx 4 in the delay differential equation {dx}/{dt}=-ax(t-1)-2x^2-x^3. Strictly, the delay makes this DE infinite-dimensional.

Fortunately, we may approximate such delays by the following trick that keeps the dimensionality finite. Define u_1(t)=x(t) and approximate the time delayed variable, x(t-1)\approx u_3(t), by using the intermediaries {du_2}/{dt}=2(u_1-u_2) and {du_3}/{dt}=2(u_2-u_3).

For parameter near critical, a=4+\alpha, the delay differential equation is then approximated by the system

 \frac{d\vec u}{dt} =\left[\begin{array}{ccc} 0&0&-4\\
2&-2&0\\ 0&2&-2 \end{array}\right] \vec u +
\left[\begin{array}{c}-\alpha u_3-2u_1^2-u_1^3\\ 0\\
0\end{array}\right].

Copying and pasting the appropriate entries, the web service [4] finds that in terms of a complex amplitude s(t) and its complex conjugate \bar s(t), the center manifold

 \vec 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{i}2e^{i2t}s +\frac{i}2e^{-i2t}\bar s 
\end{array}\right]
+{O}(\alpha+|s|^2)

and the evolution on the center manifold is

  \frac{ds}{dt}= \left[
\frac{1+2i}{10}\alpha s
-\frac{3+16i}{15}|s|^2s 
\right] +{ O}(\alpha^2+|s|^4)

This evolution shows the origin is linearly unstable for \alpha>0\ (a>4), but the cubic nonlinearity then stabilises nearby limit cycles as in classic Hopf bifurcation.

Notes[edit]

  1. ^ Guckenheimer & Holmes (1997), Section 3.2
  2. ^ Guckenheimer & Holmes (1997), Theorem 3.2.1
  3. ^ Murdock, James (2003), Normal forms and unfoldings for local dynamical systems, Springer-Verlag 
  4. ^ Mariana Haragus and Gerard Iooss (2011), Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, Springer 
  5. ^ A.J. Roberts (1988). "The application of centre manifold theory to the evolution of systems which vary slowly in space". J. Austral. Math. Soc. B 29: 480–500. doi:10.1017/S0334270000005968. 
  6. ^ A.J. Roberts (2008). "Normal form transforms separate slow and fast modes in stochastic dynamical systems". Physica A 387: 12–38. arXiv:math/0701623. Bibcode:2008PhyA..387...12R. doi:10.1016/j.physa.2007.08.023. 
  7. ^ A.J. Roberts (1997). "Low-dimensional modelling of dynamics via computer algebra". Computer Phys. Comm. 100: 215–230. Bibcode:1997CoPhC.100..215R. doi:10.1016/S0010-4655(96)00162-2. 

References[edit]

External links[edit]