One-parameter Families Of Vector Fields Research Articles

Global Bifurcations in Generic One-parameter Families on $$\mathbb{S}^2$$ S 2

In this paper we prove that generic one-parameter families of vector fields on $$\mathbb{S}^2$$ in the neighborhood of the fields of classes AH, SN, HC, SC (Andronov–Hopf, saddle-node, homoclinic curve, saddle connection) are structurally stable. We provide a classification of bifurcations in these families.

The inverse spectral transform for the Dunajski hierarchy and some of its reductions: I. Cauchy problem and longtime behavior of solutions

In this paper we apply the formal inverse spectral transform for integrable dispersionless partial differential equations (PDEs) arising from the commutation condition of pairs of one-parameter families of vector fields, recently developed by S V Manakov and one of the authors, to one distinguished class of equations, the so-called Dunajski hierarchy. We concentrate, for concreteness, (i) on the system of PDEs characterizing a general anti-self-dual conformal structure in neutral signature, (ii) on its first commuting flow, and (iii) on some of their basic and novel reductions. We formally solve their Cauchy problem and we use it to construct the longtime behavior of solutions, showing, in particular, that unlike the case of soliton PDEs, different dispersionless PDEs belonging to the same hierarchy of commuting flows evolve in time in very different ways, exhibiting either a smooth dynamics or a gradient catastrophe at finite time.

Integrable dispersionless PDEs arising as commutation condition of pairs of vector fields

In this paper we review some results about the theory of integrable dispersionless PDEs arising as commutation condition of pairs of one-parameter families of vector fields, developed by the authors during the last years. We review, in particular, the basic formal aspects of a novel Inverse Spectral Transform including, as inverse problem, a nonlinear Riemann – Hilbert (NRH) problem, allowing one i) to solve the Cauchy problem for the target PDE; ii) to construct classes of RH spectral data for which the NRH problem is exactly solvable, corresponding to distinguished examples of exact implicit solutions of the target PDE; iii) to construct the longtime behavior of the solutions of such PDE; iv) to establish in a simple way if a localized initial datum breaks at finite time and, if so, to study analytically how the multidimensional wave breaks. We also comment on the existence of recursion operators and Backlünd – Darboux transformations for integrable dispersionless PDEs.

On the solutions of the second heavenly and Pavlov equations

We have recently solved the inverse scattering problem for one-parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields, such as the heavenly equation of Plebanski, the dispersionless Kadomtsev–Petviashvili (dKP) equation and the two-dimensional dispersionless Toda (2ddT) equation, as well as with the commutation of one-dimensional vector fields, such as the Pavlov equation. We also showed that the associated Riemann–Hilbert inverse problems are powerful tools to establish if the solutions of the Cauchy problem break at finite time, to construct their long-time behaviour and characterize classes of implicit solutions. In this paper, using the above theory, we concentrate on the heavenly and Pavlov equations, (i) establishing that their localized solutions evolve without breaking, unlike the cases of dKP and 2ddT; (ii) constructing the long-time behaviour of the solutions of their Cauchy problems; (iii) characterizing a distinguished class of implicit solutions of the heavenly equation.

The dispersionless 2D Toda equation: dressing, Cauchy problem, longtime behaviour, implicit solutions and wave breaking

We have recently solved the inverse spectral problem for one-parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions, including the second heavenly equation of Plebanski and the dispersionless Kadomtsev–Petviashvili (dKP) equation, arising as commutation of vector fields. In this paper, we make use of the above theory (i) to construct the nonlinear Riemann–Hilbert dressing for the so-called two-dimensional dispersionless Toda equation , elucidating the spectral mechanism responsible for wave breaking; (ii) we present the formal solution of the Cauchy problem for the wave form of it: (exp(φ))tt = φxx + φyy; (iii) we obtain the longtime behaviour of the solutions of such a Cauchy problem, showing that it is essentially described by the longtime breaking formulae of the dKP solutions, confirming the expected universal character of the dKP equation as prototype model in the description of the gradient catastrophe of two-dimensional waves; (iv) we finally characterize a class of spectral data allowing one to linearize the RH problem, corresponding to a class of implicit solutions of the PDE.

On the solutions of the dKP equation: the nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking

We have recently solved the inverse scattering problem for one-parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions, including the second heavenly equation of Plebanski and the dispersionless Kadomtsev–Petviashvili (dKP) equation. We showed, in particular, that the associated inverse problems can be expressed in terms of nonlinear Riemann–Hilbert problems on the real axis. In this paper, we make use of the nonlinear Riemann–Hilbert problem of dKP (i) to construct the longtime behaviour of the solutions of its Cauchy problem; (ii) to characterize a class of implicit solutions; (iii) to elucidate the spectral mechanism causing the gradient catastrophe of localized solutions of dKP, at finite time as well as in the longtime regime, and the corresponding universal behaviours near breaking.

A hierarchy of integrable partial differential equations in 2+1 dimensions associated with one-parameter families of one-dimensional vector fields

We introduce a hierarchy of integrable partial differential equations in 2+1 dimensions arising from the commutation of one-parameter families of vector fields, and we construct the formal solution of the associated Cauchy problems using the inverse scattering method for one-parameter families of vector fields. Because the space of eigenfunctions is a ring, the inverse problem can be formulated in three distinct ways. In particular, one formulation corresponds to a linear integral equation for a Jost eigenfunction, and another formulation is a scalar nonlinear Riemann problem for suitable analytic eigenfunctions.

Desynchronization of one-parameter families of stable vector fields

Given a one-parameter family of vector fields on , Fλ(x), , such that for each λ, Fλ has a global asymptotically stable equilibrium point xλ, we construct a vector field on of the form G(λ, x) = (g(λ, x), Fλ(x)) which exhibits chaotic behaviour. This result is an incursion in the inverse problem of master–slave synchronization.This paper discusses self-disorganization of parameter dependent stable vector fields. Motivations are found in applications to drug design: one way to lead an unfriendly organism to death is to destabilize its metabolism. In this paper we envisage the mathematical aspect of the question. We show that for a very stable system (one globally attracting equilibrium state) depending on one parameter, there is a way for the system to act on the parameter so as to create chaos.

Stability of hyperbolic vector fields

We give a geometrical proof of the global stability theorem for a large class of hyperbolic vector fields (i. e. those satisfying axiom A and the strong transversality condition). The class of hyperbolic vector fields we consider in this paper is constituted by those hyperbolic vector fields for which the attractor and repellor sets are trivial (i. e. singularities and/or periodic orbits) and their non-trivial basic sets are d-separated. We apply the geometrical proof in order to study elementary bifurcations of one-parameter families of vector fields, such that, say for the vector fields are in the class of hyperbolic vector fields described above, and has an elementary generic bifurcation, i. e. (i) a saddle-node or a Hopf singularity; (ii) a saddle-node or a flip periodic orbit; (iii) a quasi-transversal intersection orbit between an unstable and a stable manifold; all of which are generic bifurcations that unfold genetically at the parameter bifurcation value

Unfolding contracting singular cycles

— We show that if we start with a Morse-Smale vector field and move through a generic oneparameter family of vector fields to a contracting singular cycle and beyond, we reach a region filled up mostly with hyperbolic flows: the Lebesgue measure of parameter values corresponding to non Axiom A flows is zero.

Hopf Bifurcation in the Presence of Spherical Symmetry: Analytical Results

This paper considers a one-parameter family of vector fields equivariant under the orthogonal group $O(3)$, with an invariant fixed point. Hopf bifurcation of this family is studied assuming $O(3)$ acts on the eigenspaces belonging to each purely imaginary eigenvalue as an $l = 2$ representation. The dynamics are then reduced to a ten-dimensional center manifold, and the normal form of the vector field is explicitly given up to fifth order. The five different types of bifurcating periodic solutions, predicted geometrically by Golubitsky and Stewart, are derived analytically: a family of axisymmetric solutions, two types of rotating waves (one rotating at twice the speed of the other), a family of standing waves and a family of tetrahedral waves. The stability conditions are given for all these solutions. These stabilities depend on the three coefficients appearing at cubic order in the normal form and on one combination of three coefficients occurring at fifth order. All solutions can be stable except for the fastest family of rotating waves. The slower rotating waves and the axisymmetric solutions may be simultaneously stable. Finally it is shown that a family of quasiperiodic solutions may bifurcate directly from the invariant fixed point together with the periodic solutions.

Stability and regularity in bifurcations of planar vector fields

SynopsisBifurcations appearing in generic one-parameter families of vector fields on a compact planar region are conceptually characterised in terms of their stability and regularity properties.

Generic one-parameter families of vector fields on two-dimensional manifolds

Introduction. This paper is an announcement of a study on the theory of the topological variation of the phase space of one-parameter families of vector fields (ordinary differential equations, flows). This theory, sometimes called bifurcation theory, has been developed since H. Poincare from several points of view; see, for example, [l]> [z]i [3]» W> [5]. Here, we will be mainly interested in a collection of one-parameter families of vector fields which has the following properties: (a) it is large with respect to all the families, and (b) its elements exhibit a topological variation which is amenable to simple description. Collections with properties (a) and (b) are currently called generic and were introduced in the qualitative global analysis of differential equations by M. Peixoto [o], S. Smale [8], and I. Kupka [ l l ] . See S. Smale [9] for a thorough monography on this field. The geometry of the set 2 of structurally stable vector fields and the study of generic one-parameter families of vector fields are closely related. A vector field is structurally stable if its phase space does not change topologically under small perturbations; a oneparameter family of vector fields exhibits: the simpler a phase-space topological variation, the larger the intersection it has with 2 , or equivalently, the smaller the intersection it has with its complement —the set of nonstructurally stable vector fields. The importance of the set of nonstructurally stable vector fields for the study of the topological variation of the phase space of vector fields was noticed by A. Andronov and E. Leontovich. In [12] they defined the concept of first-order structural stability as a possible guide to pursue such study. In compact two-dimensional manifolds, the only case considered here, the set of first-order structurally stable vector fields is an imbedded Banach manifold of class C and codimension one of the Banach manifold of vector fields; see [13]. This fact, although (as follows from [14]) not sufficient to describe completely the generic

Generic one-parameter families of vector fields on two-dimensional manifolds

Introduction. This paper is an announcement of a study on the theory of the topological variation of the phase space of one-parameter families of vector fields (ordinary differential equations, flows). This theory, sometimes called bifurcation theory, has been developed since H. Poincare from several points of view; see, for example, [l]> [z]i [3]» W> [5]. Here, we will be mainly interested in a collection of one-parameter families of vector fields which has the following properties: (a) it is large with respect to all the families, and (b) its elements exhibit a topological variation which is amenable to simple description. Collections with properties (a) and (b) are currently called generic and were introduced in the qualitative global analysis of differential equations by M. Peixoto [o], S. Smale [8], and I. Kupka [ l l ] . See S. Smale [9] for a thorough monography on this field. The geometry of the set 2 of structurally stable vector fields and the study of generic one-parameter families of vector fields are closely related. A vector field is structurally stable if its phase space does not change topologically under small perturbations; a oneparameter family of vector fields exhibits: the simpler a phase-space topological variation, the larger the intersection it has with 2 , or equivalently, the smaller the intersection it has with its complement —the set of nonstructurally stable vector fields. The importance of the set of nonstructurally stable vector fields for the study of the topological variation of the phase space of vector fields was noticed by A. Andronov and E. Leontovich. In [12] they defined the concept of first-order structural stability as a possible guide to pursue such study. In compact two-dimensional manifolds, the only case considered here, the set of first-order structurally stable vector fields is an imbedded Banach manifold of class C and codimension one of the Banach manifold of vector fields; see [13]. This fact, although (as follows from [14]) not sufficient to describe completely the generic