Namespaces
Variants
Actions

Difference between revisions of "Centre"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
 
(10 intermediate revisions by 2 users not shown)
Line 1: Line 1:
A kind of pattern of the trajectories of an autonomous system of ordinary second-order differential equations (*) in a neighbourhood of a singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c0212301.png" />, where
+
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c0212302.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
(More frequently spelled now as ''Center''.)
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c0212303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c0212304.png" /> is a domain of uniqueness. This kind of pattern is characterized as follows. There exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c0212305.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c0212306.png" /> such that all trajectories of the system starting in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c0212307.png" /> are closed curves going around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c0212308.png" />. Here the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c0212309.png" /> itself is also called a centre. In the figure the point 0 is the centre. The motion along trajectories with increasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123010.png" /> can proceed clockwise (indicated in the figure by arrows) or counterclockwise. A centre is Lyapunov stable (but not asymptotically stable). Its Poincaré index is 1.
+
{{MSC|34C15|34M35}}
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c021230a.gif" />
+
The topological type of a singular point of planar vector field, all of whose trajectories are closed (periodic).
  
Figure: c021230a
+
===Standard center===
 +
The equation of a mathematical pendulum, reduced to the system of two linear ordinary differential equations
 +
$$
 +
\dot x=y,\quad \dot y=-x, \qquad (x,y)\in(\R^2,0),
 +
$$
 +
is the simplest example of a center, the so called ''linear center''. Its trajectories form the concentric circles $x^2+y^2=r^2$ for all $r>0$.
  
A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123011.png" /> is a centre for a system (*), for example when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123013.png" /> is a constant matrix with pure imaginary eigen values. In contrast to simple rest points of other types that occur for linear second-order systems (a [[Saddle|saddle]], a [[Node|node]] or a [[Focus|focus]]), a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123014.png" /> of centre-type does not, generally speaking, remain a centre under a perturbation of the right-hand side of the linear system, whatever the order of smallness of the perturbations relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123015.png" /> and the order of their smoothness may be. It can then change into a focus (stable or unstable) or into a centre-focus (see [[Centre and focus problem|Centre and focus problem]]). For a non-linear system (*) of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123016.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123017.png" />) a rest point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123018.png" /> can be a centre also in the case when the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123019.png" /> has two zero eigen values.
+
A ''nonlinear center'' (or simply center) is any singularity of a planar vector field $v(x)$ which is [[Local normal forms for dynamical systems#topological equivalence|topologically equivalent]] to the linear center.  
  
See also the references to [[Singular point|Singular point]] of a differential equation.
+
<center><img  src="https://www.encyclopediaofmath.org/legacyimages/common_img/c021230a.gif" /></center>
  
====References====
+
===Stability, integrability, reversibility===
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.V. Amel'kin,   N.A. Lukashevich,   A.P. Sadovskii,   "Non-linear oscillations in second-order systems" , Minsk  (1982) (In Russian)</TD></TR></table>
+
Centers are [[Lyapunov stability|Lyapunov stable]], but not asymptotically stable: all trajectories which start close enough to the singularity, never leave a specified (perhaps, larger) neighborhood of the point, yet do not tend to this point as $t\to\infty$ (the so called ''neutral stability'').
 +
 
 +
If a vector field admits a local first integral which exhibits a strict local extremum<ref>A function $f:(\R^2,0)\to(\R,0)$ has a strict local extremum, if $f(x)\ne 0$ for $x\ne 0$. If $f$ has maximum, then $-f$ has a strict local minimum and vice versa.</ref>, then the singular point is necessarily a center.
 +
 
 +
The inverse assertion, although true even in $C^\infty$-smooth category, has no practical meaning since a center-type singularity may have only ''flat'' $C^\infty$-smooth first integral (with the Taylor series vanishing identically). One important exception is the theorem due to H. Poincaré and A. M. Lyapunov.
 +
 
 +
A real linear operator $\R^2\to\R^2$ is called [[elliptic]], if its eigenvalues form a conjugate pair $\pm i\omega$, $\omega\ne 0$. A singular point of vector field $v(x)=Ax+\cdots$ is ''elliptic'', if its linear part $A$ is an elliptic operator.
 +
 
 +
The Poincaré--Lyapunov theorem asserts that if the vector field (polynomial or analytic) has an elliptic singularity, then it necessarily admits an analytic first integral with the nontrivial quadratic terms. If the linear part is the standard rotation as above, one can find the first integral starting with the terms $\tfrac12(x^2+y^2)+\cdots$. Recursive calculation of the higher coefficients of this integral gives an algorithm for eventual decision in the [[center-focus problem]].
 +
 
 +
The linear center is a ''reversible'' vector field: the linear reflection $S:(x,y)\mapsto (x,-y)$ transforms the field into itself with the opposite sign, $\rd S\cdot v=-v\circ S$, changing thus the direction of the time variable. A singularity that is ''locally reversible'' by an invertible germ $S:(\R^2,0)\to(\R^2,0)$, are necessarily centers. The inverse is in general not true, but for singularities defined by a Pfaffian equation $y\rd y+\cdots=0$ each center is analytically reversible<ref>M. Berthier, R. Moussu, ''Reversibilité et classification des centres nilpotents'', Annales de l'institut Fourier, '''44''' no. 2 (1994), p. 465-494, doi:10.5802/aif.1406, {{MR|1296740}}, {{ZBL|0803.34005}}</ref>.
  
 +
====Complex analytic centers====
 +
In the complex domain there is no difference between the elliptic center (with the eigenvalues of the linear part $\pm i)$ and an integrable resonant saddle (with the eigenvalues $\pm 1$). Here integrability means that the vector field $v$,
 +
$$
 +
\dot x=x+\cdots,\qquad \dot y=-y+\cdots,\qquad (x,y)\in(\C^2,0)
 +
$$
 +
has a holomorphic first integral $\mathscr O(\C^2,0)\owns f(x,y)=xy+\cdots$ (here and everywhere else the dots denote converging series with terms of order greater than all explicitly written). The saddle has a diagonal linear part which is more convenient for calculations, see [[center-focus problem]].
  
 +
----
  
====Comments====
+
====Notes====
For a precise topological definition see [[#References|[a1]]], p. 71.
+
<small>
 +
<references/>
 +
</small>
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.V. Nemytskii,  V.V. Stepanov,  "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.A. Andronov,  E.A. Leontovich,  I.I. Gordon,  A.G. Maier,  "Theory of bifurcations of dynamic systems on a plane" , Israel Program Sci. Transl.  (1971)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.I. Arnol'd,  "Geometrical methods in the theory of ordinary differential equations" , Springer  (1983)  (Translated from Russian)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  V.V. Nemytskii,  V.V. Stepanov,  "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  A.A. Andronov,  E.A. Leontovich,  I.I. Gordon,  A.G. Maier,  "Theory of bifurcations of dynamic systems on a plane" , Israel Program Sci. Transl.  (1971)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  V.I. Arnol'd,  "Geometrical methods in the theory of ordinary differential equations" , Springer  (1983)  (Translated from Russian)</TD></TR>
 +
</table>

Latest revision as of 18:57, 29 July 2024


(More frequently spelled now as Center.)

2020 Mathematics Subject Classification: Primary: 34C15 Secondary: 34M35 [MSN][ZBL]

The topological type of a singular point of planar vector field, all of whose trajectories are closed (periodic).

Standard center

The equation of a mathematical pendulum, reduced to the system of two linear ordinary differential equations $$ \dot x=y,\quad \dot y=-x, \qquad (x,y)\in(\R^2,0), $$ is the simplest example of a center, the so called linear center. Its trajectories form the concentric circles $x^2+y^2=r^2$ for all $r>0$.

A nonlinear center (or simply center) is any singularity of a planar vector field $v(x)$ which is topologically equivalent to the linear center.

Stability, integrability, reversibility

Centers are Lyapunov stable, but not asymptotically stable: all trajectories which start close enough to the singularity, never leave a specified (perhaps, larger) neighborhood of the point, yet do not tend to this point as $t\to\infty$ (the so called neutral stability).

If a vector field admits a local first integral which exhibits a strict local extremum[1], then the singular point is necessarily a center.

The inverse assertion, although true even in $C^\infty$-smooth category, has no practical meaning since a center-type singularity may have only flat $C^\infty$-smooth first integral (with the Taylor series vanishing identically). One important exception is the theorem due to H. Poincaré and A. M. Lyapunov.

A real linear operator $\R^2\to\R^2$ is called elliptic, if its eigenvalues form a conjugate pair $\pm i\omega$, $\omega\ne 0$. A singular point of vector field $v(x)=Ax+\cdots$ is elliptic, if its linear part $A$ is an elliptic operator.

The Poincaré--Lyapunov theorem asserts that if the vector field (polynomial or analytic) has an elliptic singularity, then it necessarily admits an analytic first integral with the nontrivial quadratic terms. If the linear part is the standard rotation as above, one can find the first integral starting with the terms $\tfrac12(x^2+y^2)+\cdots$. Recursive calculation of the higher coefficients of this integral gives an algorithm for eventual decision in the center-focus problem.

The linear center is a reversible vector field: the linear reflection $S:(x,y)\mapsto (x,-y)$ transforms the field into itself with the opposite sign, $\rd S\cdot v=-v\circ S$, changing thus the direction of the time variable. A singularity that is locally reversible by an invertible germ $S:(\R^2,0)\to(\R^2,0)$, are necessarily centers. The inverse is in general not true, but for singularities defined by a Pfaffian equation $y\rd y+\cdots=0$ each center is analytically reversible[2].

Complex analytic centers

In the complex domain there is no difference between the elliptic center (with the eigenvalues of the linear part $\pm i)$ and an integrable resonant saddle (with the eigenvalues $\pm 1$). Here integrability means that the vector field $v$, $$ \dot x=x+\cdots,\qquad \dot y=-y+\cdots,\qquad (x,y)\in(\C^2,0) $$ has a holomorphic first integral $\mathscr O(\C^2,0)\owns f(x,y)=xy+\cdots$ (here and everywhere else the dots denote converging series with terms of order greater than all explicitly written). The saddle has a diagonal linear part which is more convenient for calculations, see center-focus problem.


Notes

  1. A function $f:(\R^2,0)\to(\R,0)$ has a strict local extremum, if $f(x)\ne 0$ for $x\ne 0$. If $f$ has maximum, then $-f$ has a strict local minimum and vice versa.
  2. M. Berthier, R. Moussu, Reversibilité et classification des centres nilpotents, Annales de l'institut Fourier, 44 no. 2 (1994), p. 465-494, doi:10.5802/aif.1406, MR1296740, Zbl 0803.34005

References

[a1] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960)
[a2] A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Theory of bifurcations of dynamic systems on a plane" , Israel Program Sci. Transl. (1971) (Translated from Russian)
[a3] V.I. Arnol'd, "Geometrical methods in the theory of ordinary differential equations" , Springer (1983) (Translated from Russian)
How to Cite This Entry:
Centre. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Centre&oldid=13006
This article was adapted from an original article by A.F. Andreev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article