Centre
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, $\rd S\cdot v=-v\circ S$, 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].
- ↑ 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, 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) |
Centre. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Centre&oldid=26049