Jan Sanders

Vrije Universiteit Amsterdam, Mathematics, Emeritus

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Papers by Jan Sanders

The Lie algebraic structure of colored networks

arXiv (Cornell University), Sep 23, 2021

In the computation of the normal form of a colored network vector field, following the semigroup(... more In the computation of the normal form of a colored network vector field, following the semigroup(oid) approach in , one would like to be able to say something about the structure of the Lie algebra of linear colored network vector fields. Unlike the purely abstract approach in [10], we describe here a concrete algorithm that gives us the Levi decomposition. If we apply this algorithm to a given subalgebra, it does put the elements in the subalgebra in the block form given by the Levi decomposition, but this need not be the Levi decomposition of the given subalgebra. We show that for N -dimensional vector fields with C colors (different functions describing different types of cells in the network) this Lie algebra netC,N is isomorphic to the semidirect sum of a semisimple part, consisting of two simple components slC and slB, with B = N -C, which we write as a block-matrix and a solvable part, consisting of two elements representing the identity C in c ≃ gl C and B in b ≃ gl B , and an abelian algebra a ≃ Gr(C, N ), the Grassmannian, consisting of the C-dimensional subspaces of R N . The methods in this paper can be immediately applied to study the linear maps of colored networks.

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Journal of Nonlinear Mathematical Physics 2002, Volume 9, Supplement 1, 213--233 Proceedings: Hong Kong A List of 1 + 1 Dimensional Integrable Equations and Their Properties

ABSTRACT We introduce the notion of a ghost characteristic for nonlocal di#erential equations.

Amplified Hopf bifurcations in feed-forward networks

arXiv (Cornell University), Nov 20, 2012

the authors developed a method for computing normal forms of dynamical systems with a coupled cel... more the authors developed a method for computing normal forms of dynamical systems with a coupled cell network structure. We now apply this theory to one-parameter families of homogeneous feed-forward chains with 2-dimensional cells. Our main result is that Hopf bifurcations in such families generically generate branches of periodic solutions with amplitudes growing like ∼ |λ| 1 2 , ∼ |λ| 1 6 , ∼ |λ| 1 18 , etc. Such amplified Hopf branches were previously found in a subclass of feed-forward networks with three cells, first under a normal form assumption and later by explicit computations [8], . We explain here how these bifurcations arise generically in a broader class of feed-forward chains of arbitrary length.

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Coupled cell networks and their hidden symmetries

arXiv (Cornell University), Apr 4, 2013

Dynamical systems with a coupled cell network structure can display synchronous solutions, spectr... more Dynamical systems with a coupled cell network structure can display synchronous solutions, spectral degeneracies and anomalous bifurcation behavior. We explain these phenomena here for homogeneous networks, by showing that every homogeneous network dynamical system admits a semigroup of hidden symmetries. The synchronous solutions lie in the symmetry spaces of this semigroup and the spectral degeneracies of the network are determined by its indecomposable representations. Under a condition on the semigroup representation, we prove that a one-parameter synchrony breaking steady state bifurcation in a coupled cell network must generically occur along an absolutely indecomposable subrepresentation. We conclude with a classification of generic one-parameter bifurcations in monoid networks with two or three cells.

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Versal normal form for nonsemisimple singularities

Journal of Differential Equations, Aug 1, 2019

The theory of versal normal form has been playing a role in normal form since the introduction of... more The theory of versal normal form has been playing a role in normal form since the introduction of the concept by V.I. Arnol'd in . But there has been no systematic use of it that is in line with the semidirect character of the group of formal transformations on formal vector fields, that is, the linear part should be done completely first, before one computes the nonlinear terms. In this paper we address this issue by giving a complete description of a first order calculation in the case of the two-and three-dimensional irreducible nilpotent cases, which is then followed up by an explicit almost symplectic calculation to find the transformation to versal normal form in a particular fluid dynamics problem and in the celestial mechanics L 4 problem.

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Equivariant decomposition of polynomial vector fields

arXiv (Cornell University), Sep 4, 2019

To compute the unique formal normal form of families of vector fields with nilpotent linear part,... more To compute the unique formal normal form of families of vector fields with nilpotent linear part, we choose a basis of the Lie algebra consisting of orbits under the linear nilpotent. This creates a new problem: to find explicit formulas for the structure constants in this new basis. These are well known in the 2D case, and recently expressions were found for the 3D case by ad hoc methods. The goal of the present paper is to formulate a systematic approach to this calculation. We propose to do this using a rational method for the inversion of the Clebsch-Gordan coefficients. We illustrate the method on a family of 3D vector fields and compute the unique formal normal form for the Euler family both in the 2D and 3D case.

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Multilinear Hirola operators, modular forms and the Heisenberg algebra

Hereditary automorphic Lie algebras

Communications in Contemporary Mathematics, Dec 20, 2019

We show that Automorphic Lie Algebras which contain a Cartan subalgebra with a constant spectrum,... more We show that Automorphic Lie Algebras which contain a Cartan subalgebra with a constant spectrum, called hereditary, are completely described by 2-cocycles on a classical root system taking only two different values. This observation suggests a novel approach to their classification. By determining the values of the cocycles on opposite roots, we obtain the Killing form and the abelianisation of the Automorphic Lie Algebra. The results are obtained by studying equivariant vectors on the projective line. As a byproduct, we describe a method to reduce the computation of the infinite dimensional space of said equivariant vectors to a finite dimensional linear computation and the determination of the ring of automorphic functions on the projective line.

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Vector potential normal form classification for completely integrable solenoidal nilpotent singularities

Journal of Differential Equations, Jun 1, 2019

We introduce a sl 2 -invariant family of nonlinear vector fields with a non-semisimple triple zer... more We introduce a sl 2 -invariant family of nonlinear vector fields with a non-semisimple triple zero singularity. In this paper we are concerned with characterization and normal form classification of these vector fields. We show that the family constitutes a Lie algebra structure and each vector field from this family is solenoidal, completely integrable and rotational. All such vector fields share a common quadratic invariant. We provide a Poisson structure for the Lie algebra from which the second invariant for each vector field can be readily derived. We show that each vector field from this family can be uniquely characterized by two alternative representations; one uses a vector potential while the other uses two functionally independent Clebsch potentials. Our normal form results are designed to preserve these structures and representations. The results are implemented in Maple in order to compute vector potential and the Clebsch potential normal forms of a given vector field from this family. Some practical normal form coefficient formulas for degrees of up to four are presented.

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Second quantization and averaging: Fermi resonance

Journal of Chemical Physics, May 15, 1981

We show how to compute from the perturbed classical Hamiltonian the eigenvalues of the quantized ... more We show how to compute from the perturbed classical Hamiltonian the eigenvalues of the quantized problem in the case of resonance. This is done by averaging the classical system up to a given order and then applying second quantization on the reduced, averaged equations. The results show agreement with numerical calculations with an application to a model problem, analyzed in Noid et al. (Ref. 2).

On the number of critical points of the period

Journal of Differential Equations, Aug 1, 1986

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Report on the driven Josephson equation

Springer eBooks, Apr 6, 2008

The (Driven) Josephson equation: An exercise in asymptotics

Lecture Notes in Mathematics, 1983

V : <$> lim i ITs : (t)dt. T ~ ~ ~ 0 That is to say, one wants to know the rotation number ... more V : <$> lim i ITs : (t)dt. T ~ ~ ~ 0 That is to say, one wants to know the rotation number of a given solution ¢. Of course, this quantity depends in general on the initial condition. In fact, this gives an interesting example of what one might call 'topological voltage' since, if there is a ...

On the Fundamental Theorem of Averaging

Siam Journal on Mathematical Analysis, 1983

In this paper we present a simplified proof of the validity of first and second order averaging i... more

Asymptotic Approximations and Extension of Time-Scales

Siam Journal on Mathematical Analysis, Jul 1, 1980

Asymptotic Approximations and Extension of Time-Scales. [SIAM Journal on Mathematical Analysis 11... more

Normal Form in Filtered Lie Algebra Representations

Acta Applicandae Mathematicae, May 1, 2005

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The 1:2:1-resonance, its periodic orbits and integrals

Lecture Notes in Mathematics, 1979

Consider a dynamical system with three degrees of freedom, which, in the linear approximation, ha... more

Projection blocks in homogeneous coupled cell networks

Dynamical Systems-an International Journal, Jan 2, 2017

We introduce a special subset of the graph of a homogeneous coupled cell network, called a projec... more We introduce a special subset of the graph of a homogeneous coupled cell network, called a projection block, and show that the network obtained from identifying this block to a single point can be used to understand the generic bifurcations of the original network. This technique is then used to describe the bifurcations in a generalized feed-forward network, in which the loop can contain more than one cell.

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Unique normal forms: The nilpotent Hamiltonian case

Journal of Differential Equations, Aug 1, 1991

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Normal form theory and spectral sequences

Journal of Differential Equations, Aug 1, 2003

The concept of unique normal form is formulated in terms of a spectral sequence. As an illustrati... more The concept of unique normal form is formulated in terms of a spectral sequence. As an illustration of this technique some results of Baider and Churchill concerning the normal form of the anharmonic oscillator are reproduced. The aim of this paper is to show that spectral sequences give us a natural framework in which to formulate normal form theory

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