Integrable Vector Fields Research Articles

QuadCover ‐ Surface Parameterization using Branched Coverings

Abstract We introduce an algorithm for the automatic computation of global parameterizations on arbitrary simplicial 2‐manifolds, whose parameter lines are guided by a given frame field, for example, by principal curvature frames. The parameter lines are globally continuous and allow a remeshing of the surface into quadrilaterals. The algorithm converts a given frame field into a single vector field on a branched covering of the 2‐manifold and generates an integrable vector field by a Hodge decomposition on the covering space. Except for an optional smoothing and alignment of the initial frame field, the algorithm is fully automatic and generates high quality quadrilateral meshes.

Algebraic integrability: a survey

We give a concise introduction to the notion of algebraic integrability. Our exposition is based on examples and phenomena, rather than on detailed proofs of abstract theorems. We mainly focus on algebraic integrability in the sense of Adler-van Moerbeke, where the fibres of the momentum map are affine parts of Abelian varieties; as it turns out, most examples from classical mechanics are of this form. Two criteria are given for such systems (Kowalevski-Painlevé and Lyapunov) and each is illustrated in one example. We show in the case of a relatively simple example how one proves algebraic integrability, starting from the differential equations for the integrable vector field. For Hamiltonian systems that are algebraically integrable in the generalized sense, two examples are given, which illustrate the non-compact analogues of Abelian varieties which typically appear in such systems.

Density property for hypersurfaces $$UV = P({\bar X})$$

We study hypersurfaces of $${\mathbb{C}}^{n+2}_{{\bar x},u,v}$$ given by equations of form $$UV = P({\bar X})$$ where the zero locus of a polynomial p is smooth reduced. The main result says that the Lie algebra generated by algebraic completely integrable vector fields on such a hypersurface coincides with the Lie algebra of all algebraic vector fields. Consequences of this result for some conjectures of affine algebraic geometry and for the Oka-Grauert-Gromov principle are discussed.

Polynomial bases for subspaces of vector fields in the unit ball. Method of ridge functions

This paper deals with the construction of orthogonal polynomial bases for particular subspaces of vector fields defined in the unit ball of ℝ3. Our approach uses vector spherical harmonics to construct orthogonal sets of specific solenoidal and potential vector fields by means of ridge functions. It is shown that the approach leads to bases according to the subspaces induced by the Helmholtz–Hodge decomposition of square integrable vector fields.

Breakdown of Heteroclinic Orbits for Some Analytic Unfoldings of the Hopf-Zero Singularity

In this paper we study the exponentially small splitting of a heteroclinic connection in a one-parameter family of analytic vector fields in ${\Bbb R}^{3}.$ This family arises from the conservative analytic unfoldings of the so-called Hopf zero singularity (central singularity). The family under consideration can be seen as a small perturbation of an integrable vector field having a heteroclinic orbit between two critical points along the z axis. We prove that, generically, when the whole family is considered, this heteroclinic connection is destroyed. Moreover, we give an asymptotic formula of the distance between the stable and unstable manifolds when they meet the plane z = 0. This distance is exponentially small with respect to the unfolding parameter, and the main term is a suitable version of the Melnikov integral given in terms of the Borel transform of some function depending on the higher-order terms of the family. The results are obtained in a perturbative setting that does not cover the generic unfoldings of the Hopf singularity, which can be obtained as a singular limit of the considered family. To deal with this singular case, other techniques are needed. The reason to study the breakdown of the heteroclinic orbit is that it can lead to the birth of some homoclinic connection to one of the critical points in the unfoldings of the Hopf-zero singularity, producing what is known as a Shilnikov bifurcation.

Efficient time-symmetric simulation of torqued rigid bodies using Jacobi elliptic functions

If the three moments of inertia are distinct, the solution to the Euler equations for the free rigid body is given in terms of Jacobi elliptic functions. Using the arithmetic–geometric mean algorithm (Abramowitz and Stegun 1992 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (New York: Dover)), these functions can be calculated efficiently and accurately. Compared to standard numerical ODE and Lie–Poisson solvers, the overall approach yields a faster and more accurate numerical solution to the Euler equations. This approach is designed for mass asymmetric rigid bodies. In the case of symmetric bodies, the exact solution is available in terms of trigonometric functions, see Dullweber et al (1997 J. Chem. Phys. 107 5840–51), Reich (1996 Fields Inst. Commun. 10 181–91) and Benettin et al (2001 SIAM J. Sci. Comp. 23 1189–203) for details. In this paper, we consider the case of asymmetric rigid bodies subject to external forces. We consider a strategy similar to the symplectic splitting method proposed in Reich (1996 Fields Inst. Commun. 10 181–91) and Dullweber et al (1997 J. Chem. Phys. 107 5840–51). The method proposed here is time-symmetric. We decompose the vector field of our problem into a free rigid body (FRB) problem and another completely integrable vector field. The FRB problem consists of the Euler equations and a differential equation for the 3 × 3 orientation matrix. The Euler equations are integrated exactly while the matrix equation is approximated using a truncated Magnus series. In our experiments, we observe that the overall numerical solution benefits greatly from the very accurate solution of the Euler equations. We apply the method to the heavy top and the simulation of artificial satellite attitude dynamics.

A symmetric splitting method for rigid body dynamics

It has been known since the time of Jacobi that the solution to the free rigid body Euler equations of motion is given in terms of a certain type of elliptic functions. Using the Arithmetic-Geometric mean algorithm, Abramowitz & Stegun (1992), these functions can be calculated efficiently and accurately. The overall approach yields a faster and more accurate numerical solution to the Euler equations compared to standard numerical ODE and symplectic solvers. In this paper we investigate the possibility of extending this approach to the case of rigid bodies subject to external forces. By using a splitting strategy similar to the one proposed in Reich (1996), we decompose the vector field of our problem in a free rigid body (FRB) problem and another completely integrable vector field. We apply the method to the simulation of the heavy top.

Linearizability and integrability of vector fields via commutation

In this paper, we consider complex smooth and analytic vector fields X in a neighborhood of a nondegenerate singular point. It is proved the equivalence between linearizability and commutation, i.e., the existence of a commuting vector field Y such that the Lie brackets [ X , Y ] ≡ 0 . For complex smooth and analytic vector fields in the plane and in a neighborhood of a nondegenerate singular point, it is also proved the equivalence between integrability and the existence of a smooth vector field Y , such that Y is a normalizer of X , i.e., [ X , Y ] = μ X .

Homoclinic orbits to invariant tori near a homoclinic orbit to center–center–saddle equilibrium

We consider a perturbation of an integrable Hamiltonian vector field with three degrees of freedom with a center–center–saddle equilibrium having a homoclinic orbit or loop. With the help of a Poincaré map (chosen based on the unperturbed homoclinic loop), we study the homoclinic intersections between the stable and unstable manifolds associated to persistent hyperbolic KAM tori, on the center manifold near the equilibrium. If the perturbation is such that the homoclinic loop is preserved (i.e. the perturbation also has a homoclinic loop inherited from the unperturbed one), we establish that, in general, the manifolds intersect along 8, 12 or 16 transverse homoclinic orbits. On the other hand, in a more generic situation (the loop is not preserved; a condition for this fact is obtained by means of a Melnikov-like method), the manifolds intersect along four transverse homoclinic orbits, though a small neighborhood of the loop has to be excluded. In a first approximation, those homoclinic orbits can be detected as nondegenerate critical points of a Melnikov potential defined on the 2-torus. The number of homoclinic orbits is given by Morse theory applied to the Melnikov potential.

Asymptotic theories of differential fields

We relate the integrability of vector fields, and of the vanishing of $p$-torsion, to model-theoretic questions concerning separably closed fields, endowed canonically with a derivation. While each differential field $(F_p(t)^s,D_p)$ is known to be decidable, we show that the asymptotic theory of these fields as a class is undecidable in a strong sense. This precludes a geometric answer to certain generalizations of the Grothendieck-Katz conjecture.

Splitting methods

I thought that instead of the great number of precepts of which logic is composed, I would have enough with the four following ones, provided that I made a firm and unalterable resolution not to violate them even in a single instance. The first rule was never to accept anything as true unless I recognized it to be certainly and evidently such …. The second was to divide each of the difficulties which I encountered into as many parts as possible, and as might be required for an easier solution. (Descartes)We survey splitting methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods arise when a vector field can be split into a sum of two or more parts that are each simpler to integrate than the original (in a sense to be made precise). One of the main applications of splitting methods is in geometric integration, that is, the integration of vector fields that possess a certain geometric property (e.g., being Hamiltonian, or divergence-free, or possessing a symmetry or first integral) that one wants to preserve. We first survey the classification of geometric properties of dynamical systems, before considering the theory and applications of splitting in each case. Once a splitting is constructed, the pieces are composed to form the integrator; we discuss the theory of such ‘composition methods’ and summarize the best currently known methods. Finally, we survey applications from celestial mechanics, quantum mechanics, accelerator physics, molecular dynamics, and fluid dynamics, and examples from dynamical systems, biology and reaction–diffusion systems.

A strong uniqueness theorem for planar vector fields

This work establishes a strong uniqueness property for a class of planar locally integrable vector fields. A result on pointwise convergence to the boundary value is also proved for bounded solutions.

New symmetry solution techniques for first-order non-linear PDEs

Using recent extensions of work of S. Lie (Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen, Leipzig, Teubner, 1891) and É. Cartan (Leçons sur les invariants intégreaux, Hermann, Paris, 1922; Les systèms differentials extérieures et leurs applications géométrique, Hermann, Paris, 1945) for integrating Frobenius integrable vector field distributions via symmetry, we examine some symmetry techniques for finding local solutions of first-order non-linear partial differential equations (PDEs). In the language of exterior differential systems, we develop a technique for solving first-order quasilinear PDEs that is then applied to general, first-order non-linear partial differential equations. Our results are significant inasmuch as we give algorithms for solving first-order PDEs in the presence of symmetry that are essentially mechanical in nature and done in the original coordinate system.

An application of solvable structures to classical and nonclassical similarity solutions

Using exterior differential systems, we extend work by Harrison and Estabrook for deriving similarity solutions of hyperbolic and parabolic partial differential equations (PDEs). We use exterior calculus results to show that a symmetry (isovector) of the differential ideal corresponding to some hyperbolic or parabolic PDE can be used to generate a Cauchy characteristic vector field of a restricted exterior differential system defined on some four-dimensional regular submanifold of the first jet bundle. We then show that this restricted differential ideal has a Frobenius integrable annihilating space, which can be used to yield a similarity solution of the PDE by applying results from Lie and Cartan on integrating Frobenius integrable vector field distributions via symmetry. We also give an extension to conditional symmetries.

Subharmonic resonance in the non-linear Mathieu equation

In this paper, we present an O(ε) perturbation method that utilizes Lie transform perturbation theory and elliptic functions to investigate subharmonic resonances in the non-linear Mathieu equation x ̈ +(δ+ε cos ωt) x+αx 3=0. It is assumed that the parametric perturbation, ε cos ωt , is small and that the coefficient of the non-linear term, α, is positive but not necessarily small. We derive analytic expressions for features (width and location of equilibria) of resonance bands in a Poincaré section of action-angle space that are associated with 2m : 1 subharmonic periodic solutions. In contrast to previous perturbation treatments of this problem, the unperturbed system is non-linear and the transformation to action-angle variables involves elliptic functions. We are, therefore, not restricted to a neighborhood of the origin in our investigation. The Hamiltonian structure of the unperturbed vector field, an integrable vector field, provides us with a framework for developing an analysis of the perturbed orbit structure. The methodolgy revolves around employing Lie transform perturbation theory for constructing the so-called “resonance Kamiltonian”, K r, whose level curves correspond to invariant curves of a Poincaré map for the non-linear Mathieu equation. Explicit knowledge of K r enables us to derive analytic expressions for the resonance bands in a Poincaré section of action-angle space that are associated with 2m : 1 subharmonic periodic solutions. Predictions of the perturbation method are compared to results obtained by direct numerical integration of the non-linear Mathieu equation. The integrable nature of the unperturbed ( ε=0) non-linear Mathieu equation is preserved under the perturbation method. Consequently, the method is unable to predict the appearance of chaos.

An F. and M. Riesz theorem for planar vector fields

We prove that solutions of the homogeneous equation Lu=0, where L is a locally integrable vector field with smooth coefficients in two variables possess the F. and M. Riesz property. That is, if $\Omega$ is an open subset of the plane with smooth boundary, $u\in C^1(\Omega)$ satisfiesLu=0 on $\Omega$ , has tempered growth at the boundary, and its weak boundary value is a measure $\mu$ , then $\mu$ is absolutely continuous with respect to Lebesgue measure on the noncharacteristic portion of $\partial\Omega$ .

Ergodicity for the Randomly Forced 2D Navier–Stokes Equations

We study space-periodic 2D Navier–Stokes equations perturbed by an unbounded random kick-force. It is assumed that Fourier coefficients of the kicks are independent random variables all of whose moments are bounded and that the distributions of the first N 0 coefficients (where N 0 is a sufficiently large integer) have positive densities against the Lebesgue measure. We treat the equation as a random dynamical system in the space of square integrable divergence-free vector fields. We prove that this dynamical system has a unique stationary measure and study its ergodic properties.

Nash triviality in families of Nash mappings

We study triviality of Nash families of proper Nash submersions or, in a more general setting, the triviality of pairs of proper Nash submersions. We work with Nash manifolds and mappings defined over an arbitrary real closed field R. To substitute the integration of vector fields, we study the fibers of such families on points of the real spectrum (R) over tilde (p) and we construct models of proper Nash submersions over smaller real closed fields. Finally we obtain results on finiteness of topological types in families of Nash mappings, and also results on effectiveness of the above constructions.

A Lie Algebraic Approach to Numerical Integration of Stochastic Differential Equations

In this article, "composition methods (or operator splitting methods)" for autonomous stochastic differential equations (SDEs) are formulated to produce numerical approximation schemes for the equations. In the proposed methods, the exponential map, which is given by the solution of an SDE, is approximated by composition of the stochastic flows derived from simpler and exactly integrable vector field operators having stochastic coefficients. The local and global errors of the numerical schemes derived from the stochastic composition methods are investigated. The new schemes are advantageous to preserve the special character of SDEs numerically and are useful for approximations of the solutions to stochastic nonlinear equations. To examine their superiority, several numerical simulations on the basis of the proposed schemes are carried out for SDEs which arise in mathematical finance and stochastic Hamiltonian dynamical systems.

Critical states and minima for an energy with second–order gradients

Coleman, Marcus & Mizel studied the thermodynamical equilibrium of second–order materials, i.e. materials for which the free–energy density depends not only on the concentration but also on its first and second gradients. They showed that in certain regimes such materials must exhibit equilibrium states that are non–uniform by proving that space–periodic solutions can have lower energy than space–uniform solutions. A dynamical systems approach is used here and it is shown that there is a critical value of the mean concentration. When the mean concentration crosses this critical value, the uniform state can lose its minimizing character and periodic or quasi–periodic states can have lower energy. Mathematically speaking, the presence of that critical value corresponds to a 1:1 resonance. The reversible 1:1 resonance has been studied extensively by Iooss & Peroueme, who showed that a reversible system having a 1:1 resonance singularity can be approximated as closely as desired by an integrable vector field. On this integrable field, all bounded solutions can be easily found and they are described by two integrals. However, in the present application, a zero eigenvalue is present in addition to the double imaginary eigenvalues. Bounded solutions are then described by three integrals. Moreover, since the average concentration must remain constant, a non–local constraint is added, which changes the results significantly. Finally, a selection principle for second–order materials stemming from energy considerations is introduced. The energy of all bounded solutions is compared with the energy of the uniform state. In some cases, non–uniform solutions have a lower energy and lead to non–uniform equilibrium states. More precisely, it is shown that space–periodic or space–quasi–periodic states are minima. Quite surprisingly, it may occur that non–uniform states appear by dilution, i.e. by decreasing the concentration.