Unfoldings
| James Murdock (2006), Scholarpedia, 1(12):1904. | doi:10.4249/scholarpedia.1904 | revision #91898 [link to/cite this article] |
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Introduction
A mathematical model of a real-world problem is usually based on idealized assumptions. If the model proves inadequate, it can be improved by adding small terms that were neglected at first. A model obtained by adding small parameters to a given system is called an unfolding of the original system. (One may picture the various behaviors of the expanded system as being hidden or "folded up" when the parameters are set to zero.)
For instance, a pair of coupled oscillators that are first modelled as a conservative system in exact resonance might be improved by adding three small parameters representing damping in each oscillator and detuning of the resonance. Perturbation methods may then be used to obtain approximate solutions expanded in the small parameters, and bifurcation analysis may be used to determine the qualitative changes in the behavior of the system in a neighborhood of the original model.
The mathematical theory of unfolding originated in the theory of singularities of mappings and in catastrophe theory. (For an introduction from this point of view, see Bruce and Giblin 1992.) In dynamical systems, unfolding means the attempt to exhibit all possible behaviors for systems close to a given original system (sometimes called the organizing center of the unfolding) by adding a finite number \(k\) of small parameters \(\mu_1,\dots,\mu_k\ .\) The number \(k\) is called the codimension of the organizing center. In order to begin, it is necessary to specify some space of admissible systems (at least a topological space, usually a smooth manifold) and some equivalence relation on this space expressing the idea that two equivalent systems "have the same behavior". Under these conditions it makes sense to specify an original system (or organizing center) and ask whether there exists a \(k\)-parameter family (for some \(k\)) of systems that intersects each equivalence class in a neighborhood of the organizing center. If so, the goal of the theory can be achieved. If not, the organizing center is said to have "infinite codimension".
Unfoldings of Matrices
Many finite-dimensional linear systems can be represented by a square matrix, whether it be the matrix of a linear transformation or of a linear system of differential equations \(\dot x = Ax\ .\) In either case, a natural equivalence relation is similarity. Suppose that \(A_0\) is a given \(n\times n\) matrix, taken as the organizing center. We wish to construct a family \(A(\mu_1,\dots,\mu_k)\) of matrices that depends continuously (or better, smoothly) on \(\mu_1,\dots,\mu_k\ ,\) reduces to \(A_0\) when \(\mu_1=\cdots=\mu_k=0\ ,\) and intersects each similarity class near \(A_0\ .\) We may assume \(A_0\) is in Jordan normal form, but it cannot always be the case that \(A(\mu_1,\dots,\mu_k)\) will be in Jordan form for all \(\mu_1,\dots,\mu_k\) near zero, because the Jordan form of a matrix does not (always) depend continuously on the matrix. Since the similarity class \(M\) of \(A_0\) is a smooth submanifold of \(\R^{n^2}\) (the space of \(n\times n\) matrices), we require that \(A(\mu_1,\dots,\mu_k)\ ,\) for \(\mu_1,\dots,\mu_k\) near zero, be a smoothly embedded submanifold transverse to \(M\ .\) Such an unfolding of \(A_0\) is called versal (an abbreviation of transversal), and automatically intersects all similarity classes near \(A_0\) (even though these classes have various dimensions). The smallest possible number \(k\) of parameters will equal the codimension (in the usual manifold sense) of \(M\) in \({\mathbb R}^{n^2}\ ;\) this explains the use of "codimension" as defined above. A versal unfolding of this kind is called miniversal.
If \(n=2\) and \(A_0=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\ ,\) the codimension is two and two different miniversal unfoldings are \[\begin{pmatrix} \mu_2&1\\ \mu_1&\mu_2\end{pmatrix}\] and \(\begin{pmatrix} 0&1\\ \mu_1&\mu_2\end{pmatrix}\ .\) The first form is known as a striped matrix. We may observe that this striped matrix commutes with \(A_0^*\) (the adjoint or conjugate transpose of \( A_0 \)), and it is always true that a miniversal unfolding of a matrix can be found by obtaining the most general matrix that commutes with \( A_0^*\ .\) The second form illustrates that the striped matrix is not the simplest choice for the unfolding, if "simplest" is interpreted as "having the most zero entries". (The striped matrix has the advantage that this unfolding is not only transverse to \(M\) but is orthogonal with respect to the inner product \(\langle P,Q\rangle = {\rm tr}\, PQ^*\ .\)) These unfoldings (for matrices of any size) are due to Arnold and are explained in Arnold (1988, section 30), Wiggins (2003, section 20.5), and Murdock (2003, chapter 3).
Relation to Normal Forms
There is a close relationship between unfoldings and normal form ideas. Any smooth one-parameter family \(A(\varepsilon)\) of matrices with \(A(0)=A_0\) can be embedded (up to similarity) into any miniversal unfolding \(A(\mu_1,\dots,\mu_k)\) of \(A_0\ ;\) that is, there is a smooth family of matrices \(T(\varepsilon)\) with \(T(0)=I\) such that \[T(\varepsilon)^{-1}A(\varepsilon)T(\varepsilon) = A(\mu_1(\varepsilon),\dots,\mu_k(\varepsilon))\ ,\] where the functions \(\mu_i(\varepsilon)\) are smooth. The form of the unfolding, as well as the power series expansions of \(\mu_i(\varepsilon)\) can be computed by normal form methods. Writing \(U(\varepsilon)=I+\varepsilon U_1+\cdots\) and \(A(\varepsilon)=A_0+\varepsilon A_1 +\cdots\ ,\) and setting \(T(\varepsilon)^{-1}A(\varepsilon)T(\varepsilon)=B(\varepsilon)=A_0 + \varepsilon B_1 +\cdots\ ,\) one finds that \[L_{A_0} U_1 = A_1-B_1\ ,\] where \(L_P Q = QP-PQ\ .\) Similar homological equations exist at higher orders. Choosing a complement to the image of \(L_{A_0}\) (that is, choosing a normal form style) fixes the form of the unfolding to which \(B(\varepsilon)\) belongs, and the rest of the computation determines the \(\mu_i(\varepsilon)\ .\) The striped matrix unfolding comes from the inner product normal form style \(\ker\Lambda_{A_0^*}\ ,\) and the second type of unfolding illustrated above comes from the simplified normal form style.
Unfoldings of Dynamical Systems
For nonlinear dynamical systems, it is much more difficult to define an appropriate space of systems and equivalence relation with which to begin. Any suitable space of systems will be infinite-dimensional, and under the most natural equivalence relations (either topological equivalence or topological conjugacy), most systems turn out to have infinite codimension, so a versal unfolding is impossible. We must either restrict attention to those few (but important) systems that have finite codimension with respect to topological equivalence, or else adopt a coarser equivalence relation. One often-used equivalence relation is static equivalence, in which attention is limited to the equilibrium solutions.
An unfolding of a dynamical system under static equivalence is one that exhibits all possible bifurcations of the equilibrium (rest) points, up to topological equivalence of the set of equilibria. It is easiest to localize the problem to the bifurcations of a single equilibrium point of the organizing center. Since no bifurcations take place in hyperbolic directions, it is enough to unfold the system on its center manifold. The various cases are classified by the eigenvalues of the Jacobian matrix (i.e., the linearized system at the equilibrium) on the imaginary axis.
A single zero eigenvalue
The simplest case is a system with a single zero eigenvalue at the equilibrium, leading to a center manifold of dimension one. Since the behavior of the system should be dominated by the lowest order term, one considers a (scalar) organizing center of the form \(\dot x = x^k\) (for \(k\) a positive integer). An unfolding under static equivalence is \[\dot x = \mu_1 +\mu_2x+\cdots+\mu_{k-1}x^{k-2} + x^k\ ,\] the interesting point being the absence of \(x^{k-1}\ .\) For instance,
- the unfolding of \(\dot x = x^2\) is \(\dot x = \mu_1 + x^2\ ,\) which exhibits a saddle-node bifurcation as \(\mu_1\) is varied.
- The unfolding of \(\dot x = x^3\) is \(\dot x = \mu_1 + \mu_2x + x^3\ .\) If \(\mu_1=0\) this gives a pitchfork bifurcation as \(\mu_2\) is varied; \(\mu_1\) is an "imperfection parameter" that splits the pitchfork into a saddle-node bifurcation and a continuation curve (i.e., a curve of equilibria that does not bifurcate).
This sort of analysis is very close to the original use of unfoldings in singularity theory. For further information see section 6.3 of Murdock (2003), and for complete details of this approach see Golubitsky and Schaeffer (1985) and Golubitsky et al. (1988). (In the last references, one of the unfolding parameters is treated as the bifurcation parameter and is not counted in the codimension.)
A conjugate pair
The organizing center \[\begin{pmatrix} \dot x \\ \dot y\end{pmatrix} =\begin{pmatrix} 0 &-1\\1 &0\end{pmatrix} \begin{pmatrix} x\\y\end{pmatrix}+ \alpha(x^2+y^2) \begin{pmatrix} x\\y\end{pmatrix} + \beta(x^2+y^2)\begin{pmatrix} -x\\y\end{pmatrix}\ ,\] which is in (semisimple) normal form truncated at the quadratic terms and has a conjugate pair of eigenvalues \(\pm \imath\ ,\) takes the form \[\dot r = \alpha r^3\] \[\dot\theta = 1 + \beta r^2\ .\] If \(\alpha\ne 0\ ,\) an unfolding under local topological equivalence (but not topological conjugacy) is \[\dot r = \mu_1 r + \alpha r^3\] \[\dot\theta = 1 + \beta r^2\ .\] This exhibits an Andronov-Hopf bifurcation as \(\mu_1\) is varied.
A nonsemisimple double eigenvalue
For the case of a double zero eigenvalue with a nonsemisimple linear part, the organizing center is \[\begin{pmatrix} \dot x\\ \dot y\end{pmatrix} = \begin{pmatrix} 0&1\\0&0\end{pmatrix}\begin{pmatrix} x\\y\end{pmatrix} + \begin{pmatrix} 0\\ \alpha x^2+\beta xy\end{pmatrix}\ ,\] with quadratic term in (simplified) normal form. Assuming that \(\alpha\ne 0\ ,\) an unfolding is \[ \begin{pmatrix} \dot x\\ \dot y\end{pmatrix} = \begin{pmatrix} 0\\ \mu_1\end{pmatrix} + \begin{pmatrix} 0&1\\0&\mu_2\end{pmatrix}\begin{pmatrix} x\\ y\end{pmatrix} + \begin{pmatrix} 0\\ \alpha x^2+\beta xy\end{pmatrix}\ .\] It is remarkable that this can be proved to be an unfolding under topological equivalence. (The proof is difficult and uses one of or another of several "blowing-up" techniques.) For further discussion see Bogdanov-Takens bifurcation.
Comparing this unfolding to the matrix unfolding of \(\begin{pmatrix} 0&1\\0&0\end{pmatrix}\) given above, it is seen that the codimension is the same but that one unfolding parameter appears in the constant term rather than in the matrix. This phenomenon is typical, as can be seen using asymptotic unfoldings, sketched below.
Additional Examples
For additional examples of unfoldings presented in an elementary manner, see Kuznetsov (1998), Guckenheimer and Holmes (1986), and Wiggins (2003). For a detailed treatment of some unfoldings with respect to topological equivalence, proved via blowup techniques, see Dumortier et al. (1991).
Asymptotic Unfoldings
As in the case of matrices, unfoldings of dynamical systems can be approached from a normal form viewpoint. Beginning with an organizing center \[\dot x = Ax + a_1(x) + a_2(x)+\cdots\] in normal form (of some chosen style), consider an arbitrary one-parameter perturbation of the following form (where the degree of a term is the subscript plus \(1\)): \[\dot x = Ax + a_1(x) + a_2(x)+\cdots\ :\]
- \[+\varepsilon(p + Bx + b_1(x) + b_2(x) + \cdots) + \cdots\ .\]
The final \(\cdots\) refer to higher powers of \(\varepsilon\ .\) Notice that the \(\varepsilon\) part contains a constant term \(p\ ,\) not present in the unperturbed system. Normal form methods can be applied to simplify \(p\ ,\) \(B\ ,\) \(b_i\ ,\) and so forth. Whatever coefficients cannot be eliminated become unfolding parameters expressed as functions of \(\varepsilon\ .\) Stopping the calculation at a finite degree in \(x\) gives an unfolding with finite codimension, but it is (usually) not a versal unfolding with respect to topological equivalence. Nevertheless, it is often possible to prove that the unfolding correctly exhibits specific features of the behavior. Under generic hypotheses on the quadratic terms \(a_2\ ,\) the number of unfolding parameters in the constant and linear terms (coming from \(p\) and \(B\)) always equals the codimension of the matrix unfolding of \(A\ ,\) explaining the remark in the last section. Asymptotic unfoldings have been used informally without a name for many years, and a number of them are computed by Elphick et al. (1992). A general treatment is given in section 6.4 of Murdock (2003). (The restriction to the simplified normal form style has since been removed, see Murdock and Malonza.) This approach to unfoldings makes the computation of unfoldings quite easy, as illustrated (in the references)by an example of codimension 14. (Deriving useful dynamical conclusions from unfoldings of high codimension is another matter altogether.)
References
- Arnold V. I. (1988) Geometrical Methods in the Theory of Ordinary Differential Equations.Springer, New York, second edition.
- Bruce J. W. and Giblin P. G. (1992) Curves and Singularities. Cambridge University Press, Cambridge, England, second edition.
- Dumortier, F. and Roussarie, R. and Sotomayor, J. (1991) Generic three-parameter families of planar vector elds, unfoldings of saddle, focus, and elliptic singularities with nilpotent linear parts. Lecture Notes in Mathematics, 1480:1-164.
- Elphick C., Tirapegui E., Brachet M. E., Coullet P., and Iooss G. (1987) A simple global characterization for normal forms of singular vector fields. Physica D, 29:95–127.
- Golubitsky M. and Schaeffer D.G. (1985) Singularities and Groups in Bifurcation Theory, volume 1. Springer, New York.
- Golubitsky M., Stewart I., and Schaeffer D.G. (1988) Singularities and Groups in Bifurcation Theory, volume 2. Springer, New York.
- Guckenheimer J. and Holmes P. (1986) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Corrected second printing, Springer, N.Y.
- Kuznetsov Y.A. (1998) Elements of Applied Bifurcation Theory, Second edition, Springer, N.Y.
- Murdock J. (2003) Normal Forms and Unfoldings for Local Dynamical Systems. Springer, New York.
- Murdock J. and Malonza D.M. Improved computation of asymptotic unfoldings. In preparation.
- Wiggins S. (2003) Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York, second edition.
Internal references
- Yuri A. Kuznetsov (2006) Andronov-Hopf bifurcation. Scholarpedia, 1(10):1858.
- Jack Carr (2006) Center manifold. Scholarpedia, 1(12):1826.
- James Murdock (2006) Normal forms. Scholarpedia, 1(10):1902.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- Yuri A. Kuznetsov (2006) Saddle-node bifurcation. Scholarpedia, 1(10):1859.
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See Also
Bifurcation, Dynamical Systems, Equilibria, Jordan Normal Form, Normal Forms, Ordinary Differential Equations
