Normal forms

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James Murdock (2006), Scholarpedia, 1(10):1902. doi:10.4249/scholarpedia.1902 revision #91592 [link to/cite this article]
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Curator: James Murdock

A normal form of a mathematical object, broadly speaking, is a simplified form of the object obtained by applying a transformation (often a change of coordinates) that is considered to preserve the essential features of the object. For instance, a matrix can be brought into Jordan normal form by applying a similarity transformation. This article focuses on normal forms for autonomous systems of differential equations (vector fields or flows) near an equilibrium point. Similar ideas can be used for discrete-time dynamical systems (diffeomorphisms) near a fixed point, or for flows near a periodic orbit.


Basic Definitions

The starting point is a smooth system of differential equations with an equilibrium (rest point) at the origin, expanded as a power series \[ \dot x = Ax + a_1(x) + a_2(x) +\cdots, \] where \(x\in{\mathbb R}^n\) or \({\mathbb C}^n\ ,\) \(A\) is an \(n\times n\) real or complex matrix, and \(a_j(x)\) is a homogeneous polynomial of degree \(j+1\) (for instance, \(a_1(x)\) is quadratic). The expansion is taken to some finite order \(k\) and truncated there, or else is taken to infinity but is treated formally (the convergence or divergence of the series is ignored). The purpose is to obtain an approximation to the (unknown) solution of the original system, that will be valid over an extended range in time. The linear term \(Ax\) is assumed to be already in the desired normal form, usually the Jordan or a real canonical form. A transformation to new variables \(y\) is applied, having the form \[ x=y+u_1(y)+u_2(y)+\cdots, \] where \(u_j\) is homogeneous of degree \(j+1\ .\) This results in a new system \[ \dot y = Ay + b_1(y) + b_2(y) +\cdots, \] having the same general form as the original system. The goal is to make a careful choice of the \(u_j\ ,\) so that the \(b_j\) are "simpler" in some sense than the \(a_j\ .\) "Simpler" may mean only that some terms have been eliminated, but in the best cases one hopes to achieve a system that has additional symmetries that were not present in the original system. (If the normal form possesses a symmetry to all orders, then the original system had a hidden approximate symmetry with transcendentally small error.)

Among many historical references in the development of normal form theory, two significant ones are Birkhoff (1996) and Bruno (1989). As the Birkhoff reference shows, the early stages of the theory were confined to Hamiltonian systems, and the normalizing transformations were canonical (now called symplectic). The Bruno reference treats in detail the convergence and divergence of normalizing transformations.

An Example

A basic example is the nonlinear oscillator with \(n=2\) and \[ A=\left[ \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right]. \] In this case it is possible (no matter what the original \(a_j\) may be) to achieve \(b_j=0\) for \(j\) odd and to eliminate all but two coefficients from each \(b_j\) with \(j\) even. More precisely, writing \(r^2=y_1^2+y_2^2\ ,\) a normal form in this case is \[ \dot y = Ay + \sum_{i=1}^{\infty} \alpha_ir^{2i}y + \beta_ir^{2i}Ay \ .\] In polar coordinates this becomes \[ \dot r = \alpha_1 r^3 + \alpha_2r^5+\cdots \] \[ \dot\theta = 1 + \beta_1r^2 + \beta_2r^4+\cdots \ .\] The first nonzero \(\alpha_i\) determines the stability of the origin, and the \(\beta_i\) control the dependence of frequency on amplitude. Also the normalized system has achieved symmetry (more technically, equivariance) under rotation about the origin. Although the classical (or level-one) approach to normal forms stops with the form obtained above for this example, it is important to note that neither the coefficients \(\alpha_i\) and \(\beta_i\) in the equation, nor the transformation terms \(u_j\) used to achieve the equation, are uniquely determined by the original \(a_j\ .\) In fact, by a more careful choice of the \(u_j\ ,\) it is possible to put the nonlinear oscillator into a hypernormal form (also called a unique, higher-level, or simplest normal form) in which all but finitely many of the coefficients \(\alpha_i\) and \( \beta_i\) are zero. Hypernormal forms are difficult to calculate, and from here on we speak only of classical normal forms.

Asymptotic Consequences of Normal Forms

For some systems, the normal form (truncated at a given degree) is simple enough to become solvable. In this case it is of interest to ask whether this solution gives rise to a good approximation (an asymptotic approximation in some specific sense) to a solution of the original equation (say, with the same initial condition). The answer is "sometimes yes". ("Gives rise to" means that the solution of the truncated normal form usually must be fed back through the transformation to normal form.) Some popular books, such as Nayfeh (1993), present the subject entirely from this point of view, without proving any error estimates or noticing that there are cases in which asymptotic validity cannot hold. Several theorems and open questions in this regard are given in chapter 5 of Murdock (2003). The most basic theorem states that an asymptotic error estimate with respect to a small parameter holds if (a) the parameter is introduced correctly, (b) the matrix of the linear term is semisimple (see below) and has all its eigenvalues on the imaginary axis, and (c) the semisimple normal form style (see below) is used. Although the asymptotic use of normal forms is important when it is true, and has many practical applications, the primary importance of normal forms is as a preparatory step towards the study of qualitative dynamics, unfoldings, and bifurcations.

Geometrical Consequences of the Normal Form

It has already been pointed out that a normal form can decide stability questions and establish hidden symmetries. Computing the normal form up to degree \(k\) also automatically computes (to degree \(k\)) the stable, unstable, and center manifolds, the center manifold reduction, and the fibration of the center-stable and center-unstable manifolds over the center manifold. The common practice of computing the center manifold reduction first, and then computing the normal form only for this reduced system, seems to save work but loses many of these results. See chapter 5 of Murdock (2003).

On occasion, the truncation of a normal form produces a simple system that is topologically equivalent to the original system in a neighborhood of the equilibrium, called topological normal form. For instance, in the example above, truncating after the first nonvanishing \(\alpha_i\) will accomplish this, but if all \(\alpha_i\) are zero, the topological behavior is probably determined by a transcendentally small effect that is not captured by the normal form.

Normal forms are important for determining bifurcations of a system, but this requires the inclusion of unfolding parameters.

The Homological Equation and Normal Form Styles

In the general case, we define the Lie derivative operator \(L_A\) associated with the matrix \(A\) by \((L_A v)(x)=v'(x)Ax-Av(x)\ ,\) where \(v\) is a vector field and \(v'\) is its matrix of partial derivatives. Then \(L_A\) maps the vector space \(\mathcal{V}_j\) of homogeneous vector fields of degree \(j+1\) into itself. The relation between the \(a_j\ ,\) \(b_j\ ,\) and \(u_j\) is determined recursively by the homological equations \[ L_A u_j = K_j - b_j \ ,\] where \(K_1=a_1\) and \(K_j\) equals \(a_j\) plus a correction term computed from \(a_1,\dots,a_{j-1}\) and \(u_1,\dots,u_{j-1}\ .\) Let \(\mathcal{N}_j\) be any choice of a complementary subspace to the image of \(L_A\) in \(\mathcal{V}_j\ ;\) then it is possible to choose the \(u_j\) so that each \(b_j\in \mathcal{N}_j\ .\) (Take \(b_j=P_j K_j\ ,\) where \(P_j:\mathcal{V}_j\rightarrow\mathcal{N}_j\) is the projection map, and note that the homological equation can be solved, nonuniquely, for \(u_j\ .\)) The choice of \(\mathcal{N}_j\) is called a normal form style, and represents the preference of the user as to what is considered "simple". The purpose of this procedure is to ensure that the higher-order correction terms, \(u_j\ ,\) are bounded, so that the approximation to the solution, \(x(t)\ ,\) is valid over an extended range in time.

The Semisimple Case; Resonant Monomials

The theory breaks into two cases according to whether \(A\) is semisimple (diagonalizable) or not. The semisimple case, illustrated by the nonlinear oscillator above, is the easiest, and there is only one useful style (in which \(\mathcal{N}_j\) is the kernel of \(L_A\)), ultimately due to Poincaré. It is easy to describe the semisimple normal form if \(A\) is diagonal with diagonal entries \(\lambda_1,\dots,\lambda_n\) (which usually requires introducing complex variables with reality conditions): The \(r\)th equation (for \(\dot y_r\)) of the normalized system will contain only monomials \(y_1^{m_1}\cdots y_n^{m_n}\) satisfying \[ m_1\lambda_1+\cdots+m_n\lambda_n-\lambda_r=0 \ .\] Such monomials are called resonant because for pure imaginary eigenvalues, this equation becomes a resonance among frequencies in the usual sense. An elementary treatment of normal forms in the semisimple case only is by Kahn and Zarmi (1998).

The Nonsemisimple Case

In the nonsemisimple case there are two important styles, the inner product normal form, originally due to Belitskii but popularized by Elphick et al. (1987), and the sl(2) normal form due to Cushman and Sanders. In the inner product style, \(\mathcal{N}_j\) is the kernel of \(L_{A^*}\ ,\) \(A^*\) being the adjoint or conjugate transpose of \(A\ .\) In the sl(2) style, \(\mathcal{N}_j\) is the kernel of an operator defined from \(A\) using the theory of the Lie algebra sl(2). The inner product style is more popular at this time, but the sl(2) style has a much richer mathematical structure with deep connections to sl(2) representation theory and to the classical invariant theory of Cayley, Sylvester and others. Because of this the sl(2) style has computational algorithms that are not available for the inner product style. There is also a simplified normal form style that is derived from the inner product style by changing the projection.

A modern introduction to normal form theory, containing all the styles mentioned here with references and historical remarks, may be found in the monograph by Murdock (2003). Some more recent developments are contained in the last few chapters of Sanders, Verhulst, and Murdock (2007).


  • Poincaré, H., New Methods of Celestial Mechanics (Am. Inst. of Physics, 1993).
  • Birkhoff, G.D., Dynamical Systems (Am. Math. Society, Providence, 1996).
  • Arnold, V.I., Geometrical Methods in the Theory of Ordinary Differential Equations (Springer-Verlag, New York, 1988).
  • Bruno, A.D., Local Methods in Nonlinear Differential equations (Springer-Verlag, Berlin, 1989).
  • Elphick C., Tirapegui E., Brachet M.E., Coullet P., and Iooss G. A simple global characterization for normal forms of singular vector fields. Physica D, 29:95-127(1987).
  • Nayfeh, A.H., Method of Normal Forms. (Wiley, New York, 1993).
  • Kahn P.B. and Zarmi Y., Nonlinear Dynamics: Exploration through Normal Forms. (Wiley, New York, 1998).
  • Murdock J. Normal Forms and Unfoldings for Local Dynamical Systems. (Springer, New York, 2003).
  • Jan Sanders, Ferdinand Verhulst, and James Murdock, Averaging Methods in Nonlinear Dynamical Systems, Springer, New York, 2007, xxiii+431.

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See Also

Bifurcations, Dynamical Systems, Equilibria, Jordan Normal Form, Lie Algebra, Ordinary Differential Equations, Unfoldings

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