Center Manifold
| Jack Carr (2006), Scholarpedia, 1(12):1826. | doi:10.4249/scholarpedia.1826 | revision #126955 [link to/cite this article] |
One of the main methods of simplifying dynamical systems is to reduce the dimension of the system. Centre manifold theory is a rigorous mathematical technique that makes this reduction possible, at least near equilibria.
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An Example
We first look at a simple example. Consider \[\tag{1} x' =ax^3 \,, \qquad y' =-y + y^2 \]
where \(a\) is a constant. Since the equations are uncoupled, we see that the stationary solution \( x=y =0\) of (1) is asymptotically stable if and only if \(a < 0\ .\) Suppose now that \[\tag{2} x' =ax^3 + xy - xy^2\,, \qquad y' =-y + bx^2 +x^2 y \]
Since the equations are coupled we cannot immediately decide if the stationary solution \( x=y =0\) of (2) is asymptotically stable. The key is an abstraction of the idea of uncoupled equations.
A curve \(y =h(x)\ ,\) defined for \(|x|\) small, is said to be an invariant manifold for the system \[\tag{3} x' =f(x,y)\,, \qquad y' = g(x,y) \]
if the solution of (3) with \(x(0) =x_0\ ,\) \(y(0) = h(x_0)\) lies on the curve \(y =h(x)\) as long as \(x(t)\) remains small. For the system (1), \(y=0\) is an invariant manifold. Note that in deciding upon the stability of the stationary solution of (1), the only important equation is \(x' = ax^3\ ,\) that is, we need only study a first order equation on a particular invariant manifold.
Center manifold theory tells us that (2) has an invariant manifold \(y =h(x) = \mbox{O}(x^2)\) for small \(x\ .\) Furthermore, the local behaviour of solutions of the two dimensional system (2) can be determined by studying the scalar equation \[\tag{4} u' = au^3 + uh(u) -uh^2(u) \]
The theory also tells us how to compute approximations to the invariant manifold \(y = h(x)\ .\) For (2) we have that \(h(x) = bx^2 + \mbox{O}(x^4)\) and using this information in (4) gives \[\tag{5} u' =(a+b)u^3 + \mbox{O}(u^5) \]
Hence the stationary solution of (2) is asymptotically stable if \(a+b < 0\) and unstable if \(a+b>0\ .\) If \(a+b = 0\) we need a better approximation to the invariant manifold in order to decide on the stability.
Centre Manifolds
Consider the system \[\tag{6} x' =Ax + f(x,y)\,, \qquad y' = By+g(x,y)\,, \qquad (x,y ) \in \R^n \times \R^m \]
where all the eigenvalues of the matrix \(A\) have zero real parts and all the eigenvalues of the matrix \(B\) have negative real parts. The functions \(f\) and \(g\) are sufficiently smooth and \[ f(0,0) =0\,, \qquad Df(0,0) =0\,, \qquad g(0,0) =0\,, \qquad Dg(0,0) = 0 \] where \(Df\) is the Jacobian matrix of \(f\ .\)
If \(f\) and \(g\) are identically zero then (6) has the two obvious invariant manifolds \(x=0\) and \(y=0\ .\) The invariant manifold \(x=0\) is called the stable manifold, and on the stable manifold all solutions decay to zero exponentially fast. The invariant manifold \(y=0\) is called the centre manifold. In general, an invariant manifold \(y = h(x)\) for (6) defined for small \(|x|\) with \(h(0)=0\) and \(Dh(0)=0\) is called a centre manifold. In more physical terms, the dynamics of y follows the dynamics of x and one may say that x enslaves the variable y. This interpretation has been called slaving principle.
Main Results
The general theory states that there exists a centre manifold \(y =h(x)\) for (6) and that the equation on the centre manifold \[\tag{7} u' =Au + f(x,h(u))\,, \qquad u \in R^n \]
determines the dynamics of (6) near \((x, y) =(0,0)\ .\) In particular, if the stationary solution \(u=0\) of (7) is stable, we can represent small solutions of (6) as \(t \rightarrow \infty\) by \[ x(t) =u(t) + \mbox{O}(e^{-\gamma t} )\,, \qquad y(t) =h(u(t)) + \mbox{O}(e^{-\gamma t}) \] where \(\gamma > 0\) is a constant.
To use the above theory, we need to have enough information about the centre manifold \(y = h(x)\) in order to determine the local dynamics of (7). If we substitute \(y(t) = h(x(t))\) into the second equation in (6) we obtain \[\tag{8} N(h(x)) =h'(x)\left[ Ax +f(x,h(x)) \right] - Bh(x) -g(x,h(x)) = 0 \]
The general theory tells us that the solution \(h\) of (8) can be approximated by a polynomial in \(x\ ,\) that is, if \(N(\phi(x)) = \mbox{O}(|x|^q)\) as \(x \rightarrow 0\) then \(h(x) =\phi (x) + \mbox{O}(|x|^q)\ .\)
There is also an \(m\) dimensional invariant manifold \(W^s\) tangential to the y-axis called the stable manifold. On the stable manifold all solutions decay to zero exponentially fast. Figure 1 illustrates the local dynamics for equation (6). The details of the flow on the centre manifold \(y = h(x)\) depend on the higher order terms in equation (7) and we cannot assign directions to the flow without further information.
We have assumed that all of the eigenvalues of the matrix B in (6) have negative real parts. The theory can be extended to the case in which the matrix B has in addition some eigenvalues with positive real parts. In this case the stationary solution \(x=0, y=0\) of (6) is unstable due to the unstable eigenvalues. There exists a centre manifold for (6) which captures the behaviour of small bounded solutions. In particular, this gives a method of studying all sufficiently small equilibria, periodic orbits and heteroclinic orbits.
Local Bifurcations
Centre manifold reduction is central to the development of bifurcation theory. We illustrate this by means of a simple example. Consider \[\tag{9} x' =\epsilon x -x^3 +xy\,, \qquad y' =-y + y^2 -x^2 \]
where \(\epsilon\) is a small scalar parameter. The goal is to study small solutions of (9). The linearised problem about the zero equilibrium has eigenvalues \(-1\) and \(\epsilon\) so the theory does not directly apply. We can write the equations in the equivalent form \[\tag{10} x' =\epsilon x -x^3 +xy\,, \qquad y' = -y + y^2 -x^2 \,, \qquad \epsilon' = 0 \ .\]
When considered as an equation on \(\R^3\) the \(\epsilon x\) term in (10) is nonlinear and the system has an equilibrium at \((x,y,\epsilon) = (0,0,0)\ .\) The linearisation about this equilibrium has eigenvalues \(-1, 0 ,0\ ,\) that is, it has two zero eigenvalues and one negative eigenvalue. . The theory now applies so that the extended system (10) has a two dimensional centre manifold \(y =h(x,\epsilon)\) that can be approximated by a polynomial in \(x\) and \(\epsilon\ .\) The equation on the centre manifold is two dimensional and may be written in terms of the scalar variables \(u\) and \(\epsilon\) as \[ u' =\epsilon u - 2u^3 + \mbox{higher order terms} \,, \qquad \epsilon' = 0 \] and the local dynamics of (10) can be deduced from this equation.
Notes and Further Reading
The ideas for centre manifolds in finite dimensions have been around for a long time and have been developed by Carr (1981), Guckenheimer and Holmes (1983), Kelly (1967), Vanderbauwhede (1989) and others. For recent developments in the approximation of centre manifolds see Jolly and Rosa (2005). Pages 1-5 of the book by Li and Wiggins (1997) give an extensive list of the applications of centre manifold theory to infinite dimensional problems. Mielke (1996) has developed centre manifold theory for elliptic partial differential equations and has applied the theory to elasticity and hydrodynamical problems. Applications to phase transitions in biological, chemical and physical systems have been investigated by Haken (2004).
In addition, it is interesting to note that there is a stochastic extension of the center manifold theorem, which has been introduced by Boxler (1989). In this case, for instance the center and stable manifolds may fluctuate randomly.
References
J. Carr (1981), Applications of Centre Manifold Theory, Springer-Verlag.
J. Guckenheimer and P. Holmes (1983), Nonlinear Oscillations, Dynamical systems and Bifurcations of Vector Fields. Springer-Verlag.
M. S. Jolly and R. Rosa (2005), Computation of non-smooth local centre manifolds, IMA Journal of Numerical Analysis , 25, no. 4, 698-725.
A. Kelly (1967), The stable, center-stable, center, center-unstable and unstable manifolds. J. Diff. Eqns, 3, 546-570.
Li and S. Wiggins (1997), Invariant manifolds and fibrations for perturbed nonlinear Schrödinger equations. Springer-Verlag.
A. Mielke (1996), Dynamics of nonlinear waves in dissipative systems: reduction, bifurcation and stability. In Pitman Research Notes in Mathematics Series, 352. Longman.
A. Vanderbauwhede (1989). Center Manifolds, Normal Forms and Elementary Bifurcations, In Dynamics Reported, Vol. 2. Wiley.
H. Haken (2004), Synergetics: Introduction and Advanced topics, Springer Berlin
P. Boxler (1989), A stochastic version of center manifold theory, Probability Theory and Related Fields, 83(4), 509-545
Internal references
- John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
External links
See Also
Attractor, Bifurcations, Normal Hyperbolicity, Stability, Synergetics,
