Separatrix

2020 Mathematics Subject Classification: Primary: 34C45,34M35 Secondary: 37E35 [MSN][ZBL]

A term used in the qualitative theory of differential equations, used for several closely related types of integral curves (solutions).

Separatrices of a saddle

A singular point $x=0$ of a smooth vector field $v(x)=Ax+\cdots$ on the plane, $x\in(\R^2,0)$ is called (nondegenerate) saddle, if the linearization matrix has two real eigenvalues of different sign. Such vector field has two smooth invariant curves through the origin, transversal to each other. These curves are called separatrices of the saddle.

A saddle singularity is topologically equivalent to the standard saddle defined by the vector field $v(x,y)=-x\partial x+y\partial y$ (or the Pfaffian equation $x\rd y+y\rd x=0$) in the coordinates $(x,y)\in(\R^2,0)$. The coordinate axes are separatrices of the standard saddle. The standard saddle has the analytic first integral $f(x,y)=xy$, with the critical level curve $\{f=0\}$ being the union of two separatrices. The $x$-axis consists of the points $a\in(\R^2,0)$ which tend to the singularity, $f^t(a)\to0$, when moved by the flow $\{f^t=\exp tv\}$ of the vector field $v$ as $t\to+\infty$. It is referred to as the stable separatrix. Conversely, points of the $y$-axis are moved away from the singularity by the flow, but $f^t(a)\to0$ as $t\to-\infty$, hence the name unstable separatrix is used for it. The two separatrices "separate" a small punctured neighborhood of the saddle into four hyperbolic sectors, see below.

Separatrices of a self-map

Less frequently the term "separatrix" is used for stable and unstable invariant curves of a hyperbolic self-map $f\in\operatorname{Diff}(\R^2,0)$ with two real eigenvalues $\lambda,\mu\in\R$, one of which is contracting, $|\lambda|<1$, another repelling, $|\mu|>1$.

Separatrices of multidimensional saddles

Sometimes the name "separatrix" by extension of the meaning is used for the stable and unstable invariant manifolds of hyperbolic singularities even if the dimension of these invariant manifolds is higher than 1.

Separatrices of degenerate real singularities

Singular points with degenerate linear part may have a more complicated topological structure which, nevertheless, can be described in terms of sectors of three different type (elliptic, parabolic, hyperbolic) characterized by different asymptotic behavior of integral curves as $t\to\pm\infty$. In this case the trajectories separating sectors of different types are also called separatrices [A1].

Example. A generic vector field on the plane with one zero and one nonzero eigenvalue is topologically equivalent to the saddle-node, the vector field defined by the vector field $x^2\partial_x\pm y\partial_y$. For the standard saddle-node any small neighborhood of the origin consists of one parabolic sector and two hyperbolic sectors, separated by the coordinate (semi)-axes. If the vector field is analytic, then the separatrices are $C^\infty$-smooth, and the separatrix separating the elliptic sector from its two hyperbolic neighbors (the $y$-axis for the standard saddle) is analytic, but in general the separatrix between two hyperbolic sectors (the positive $x$-semiaxis for the standard field $x^2\partial_x-y\partial_y$) is only $C^\infty$-smooth, see Centre manifold.

Separatrices of analytic (real and complex) foliations

In the analytic (real or complex) settings it is more convenient to consider foliations on $(\R^2,0)$, resp., $(\C^2,0)$ defined by the distribution of null spaces of an analytic Pfaffian form $\omega=a(x,y)\rd x+b(x,y)\rd y$ with an isolated singularity (common root of analytic coefficients $a$ and $b$) at the origin.

A separatrix of $\omega$ is "an analytic particular solution" of the Pfaffian equation $\omega=0$, or, in the geometric terms, the germ of an analytic curve $\gamma=\{f=0\}$ defined by a nonconstant irreducible analytic germ $f$ and tangent to the null spaces of the form: $$ \omega\land \rd f=f\Theta,\qquad f\in\mathscr O(\C^2,0),\ \Theta\in\Lambda^2(\C^2,0). $$ Existence of separatrices was proved by Briot and Bouquet for analytic saddles, who formulated the problem for arbitrary isolated analytic singularities.

The problem was solved in 1982 by C. Camacho and P. Sad [CS] who proved that any holomorphic singular foliation on $(\C^2,0)$ always admits an analytic separatrix. The proof of this very deep result is obtained by Bendixson desingularization and delicate combinatorial arguments which were later considerably simplified by J. Cano, see [JC] and [IY, Sect. 14] for a detailed exposition.

Separatrices and dynamics

The role of separatrices in the study of dynamical systems follows from their description as (Hausdorff) limits of trajectories spending a long time near a singular (rest) point but eventually leaving it.

For instance, let $\{\gamma_s\}_{s\in\N}$ be a family of (parameterized) trajectories of a vector field in a small neighborhood $U=(\R^2,0)$ of a singular point, such that:

• all of these trajectories start and end on the boundary of the neighborhood, $\gamma(0),\gamma(T_s)\in\partial U$;
• they spend all the remaining time inside, $\gamma_s(t)\in U$ for all $t\in(0,T_s)$;
• the lifetimes $T_s$ of the trajectories $\gamma_s$ tend to infinity, $T_s\to+\infty$ as $s\to\infty$.

Then the images $\gamma_s([0,T_s])$ must have an accumulation set (in the sense of the Hausdorff metric) which consists of at least two separatrices, one stable and one unstable.

In particular, limit cycles of planar or spherical vector fields can accumulate (again in the Hausdorff sense) only to separatrix polygons formed by cyclically enumerated separatrices which are bi-asymptotic to one or several singular points, see limit set. Separatrix polygons also play very important role in bifurcation of limit cycles [A2].

Characteristic trajectory

A rather closely related notion is that of a characteristic trajectory. An (oriented) integral curve $\gamma(t)$ of planar a vector field $v$, $v(0)=0$, is called characteristic, if it tends (in the positive or negative time) to the singular point tangent to a certain limit direction $V$: $$ \lim_{t\to+\infty}\gamma(t)=0, \qquad \lim_{t\to+\infty}\frac{\gamma(t)}{\|\gamma(t)\|}=V\in\mathbb S^1=\{\|V\|=1\}\subset\R^2. $$ Apart from the conditions on the trajectory and its limit secant, no conditions on the analyticity of the image of $\gamma$ is required.

It should be stressed that while an analytic singularity may exhibit at most finitely many separatrices unless it has a meromorphic first integral, the number of characteristic trajectories may well be uncountable.

Absence of characteristic trajectories guarantees (in the analytic category) that the singularity is monodromic, i.e., that the Poincare return map is well defined on some analytic cross-section passing through the singular point. A monodromic singularity of an analytic vector field is necessarily either center or focus.

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References

[A1]	A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, Qualitative theory of second-order dynamic systems, Wiley/Israel Program Sci. Transl. (1973), MR0350126
[A2]	A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, Theory of bifurcations of dynamic systems on a plane, Wiley/Israel Program Sci. Transl. (1973), MR0344606.
[IY]	Ilyashenko, Yu. and Yakovenko, S. Lectures on analytic differential equations, Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008. MR2363178
[CS]	Camacho, C. and Sad, P. Invariant varieties through singularities of holomorphic vector fields, Ann. of Math. (2) 115 (1982), no. 3, 579--595. MR0657239
[JC]	Cano, J. Construction of invariant curves for singular holomorphic vector fields, Proc. Amer. Math. Soc. 125 (1997), no. 9, 2649--2650. MR1389507