Difference between revisions of "Centre"
From Encyclopedia of Mathematics
(Importing text file) |
m |
||
| Line 1: | Line 1: | ||
| − | + | The topological type of a singular point of a 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 [[topological equivalence|topologically equivalent]] to the linear center. | |
| − | <img | + | <center><img src="https://www.encyclopediaofmath.org/legacyimages/common_img/c021230a.gif" /></center> |
| − | + | ===Stability, integrability, reversibility=== | |
| + | 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. Poincare and A. M. Lyapunov. | |
| − | + | ||
| + | An 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 is ''elliptic'', if its linear part is an elliptic operator. | ||
| + | The Poincare--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. | ||
| + | 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, $\rm d S\cdot v=-v\circ S$, 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|MR1296740}}, {{Zbl|0803.34005}}</ref>. | ||
| + | ---- | ||
| + | <references/> | ||
| − | |||
| − | |||
====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 | + | <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> |
Revision as of 12:21, 5 May 2012
The topological type of a singular point of a 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. Poincare and A. M. Lyapunov.
An 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 is elliptic, if its linear part is an elliptic operator.
The Poincare--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.
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, $\rm d S\cdot v=-v\circ S$, 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].
- ↑ 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.
- ↑ 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, MRMR1296740, Template:Zbl
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=26046
Centre. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Centre&oldid=26046
This article was adapted from an original article by A.F. Andreev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article