View source for Invariant set

''of the phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i0522801.png" /> of a dynamical system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i0522802.png" />''

A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i0522803.png" /> which is the union of entire trajectories, that is, a set satisfying the condition

<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/i/i052/i052280/i0522804.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i0522805.png" /> is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i0522806.png" /> under the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i0522807.png" /> corresponding to a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i0522808.png" />.

The invariant set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i0522809.png" /> may possess a definite topological structure as a set of the metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228010.png" />; for example, it can be a topological or smooth manifold, a surface, a closed Jordan curve, or an isolated point. One then says that the invariant set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228011.png" /> is an invariant manifold, an invariant surface, an invariant curve, or an invariant point.

An invariant point is usually called a stationary point of the dynamical system, since for this point the motion reduces to rest: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228012.png" /> for all values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228013.png" />. A closed invariant curve not containing any invariant points of the dynamical system is always formed by the trajectory of a periodic motion, that is, a motion satisfying the condition

<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/i/i052/i052280/i05228014.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228015.png" /> and some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228016.png" />. For this reason it is called a periodic trajectory. Examples which occur as invariant manifolds are a sphere, a torus and a disc; invariant surfaces — a cone, a Möbius strip and a sphere with handles; invariant sets — the set of all stationary points, the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228018.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228019.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228020.png" />-limit points of the motion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228021.png" />, respectively, and also the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228022.png" /> of all wandering points or the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228023.png" /> of all non-wandering points (cf. [[Wandering point|Wandering point]]; [[Non-wandering point|Non-wandering point]]).

An invariant point of the dynamical system in the plane

<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/i/i052/i052280/i05228024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

belongs to one of the following four types, according to the nature of the behaviour of the trajectories in a neighbourhood of it: a [[Node|node]], (Fig.a) a [[Focus|focus]], (Fig.b) a [[Saddle|saddle]], (Fig.c) or a [[Centre|centre]] (Fig.d).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i052280a.gif" />

Figure: i052280a

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i052280b.gif" />

Figure: i052280b

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i052280c.gif" />

Figure: i052280c

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i052280d.gif" />

Figure: i052280d

The node and focus are asymptotically stable or unstable, the saddle is unstable, and the centre is stable (cf. [[Asymptotically-stable solution|Asymptotically-stable solution]]). The Poincaré index of the node, centre and focus is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228025.png" />; that of the saddle is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228026.png" />.

In the case when the Jacobi matrix

<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/i/i052/i052280/i05228027.png" /></td> </tr></table>

of the right-hand side of (1) has at the stationary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228028.png" /> eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228030.png" /> with non-zero real parts, the invariant point is: a node if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228032.png" /> are real and of the same sign; a saddle if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228034.png" /> are real with different signs; or a focus if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228036.png" /> are complex conjugates.

In all these cases, the type of the singular point of the system (1) is the same as for the linear system obtained from (1) by expanding its right-hand side in a Taylor series at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228037.png" />, that is, it has the same type as the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228038.png" /> of the system

<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/i/i052/i052280/i05228039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

whose matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228040.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228041.png" />. There is a deeper relation between the trajectories of (1) in a neighbourhood of a singular point of one of the above types and the trajectories of (2) than that already mentioned. Namely, whenever the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228043.png" /> are analytic in a neighbourhood of the invariant point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228044.png" /> and the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228045.png" /> has eigen values with non-zero real parts, there exists a continuously-differentiable change of variables in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228046.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228047.png" />,

<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/i/i052/i052280/i05228048.png" /></td> </tr></table>

<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/i/i052/i052280/i05228049.png" /></td> </tr></table>

that reduces the system (1) to the system (2).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228051.png" /> are imaginary, then the invariant point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228052.png" /> can either be a focus or a centre. In this case, the question of the classification of the type of the singular point is a separate and difficult problem, namely the [[Centre and focus problem|centre and focus problem]], and requires more refined criteria for distinguishing between the centre and the focus (see [[#References|[1]]], [[#References|[7]]]). Similar difficulties occur in the determination of the type of the singular point in the case when the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228053.png" /> is singular.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) {{MR|0121520}} {{ZBL|0089.29502}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G.D. Birkhoff, "Dynamical systems" , Amer. Math. Soc. (1927) {{MR|1555257}} {{ZBL|53.0733.03}} {{ZBL|53.0732.01}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) {{MR|0087811}} {{ZBL|0070.30603}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) {{MR|0658490}} {{ZBL|0476.34002}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> I.G. Malkin, "Theorie der Stabilität einer Bewegung" , R. Oldenbourg , München (1959) (Translated from Russian) {{MR|0104029}} {{ZBL|0124.30003}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> A.M. Lyapunov, "Collected works" , '''2''' , Moscow-Leningrad (1956) (In Russian) {{MR|0121296}} {{MR|0106812}} {{MR|0086006}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> K.S. Sibirskii, "Algebraic invariance of differential equations and matrices" , Kishinev (1976) (In Russian)</TD></TR></table>



====Comments====
The reduction of the system (1) to the system (2) in a neighbourhood of a singular point is often called linearization of the equation. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228055.png" /> have non-zero real parts (in that case the singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228056.png" /> is said to be of hyperbolic type) such a linearization can always be performed by means of a topological (non-smooth) coordinate transformation (i.e., the transformation and its converse are merely continuous), provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228058.png" /> in (1) are of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228059.png" />. This result holds also for autonomous differential equations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228060.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228061.png" />. If certain "non-resonance" conditions between the eigen values are fulfilled, then one can also obtain smooth (under additional conditions even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052280/i05228062.png" /> or analytic) transformations (work of H. Poincaré, C.L. Siegel, S. Sternberg). See [[#References|[4]]], pp. 256-271 and the references given in [[#References|[4]]] on pp. 271-272. These results are related to the theory of normal forms for differential equations (cf. [[Normal form|Normal form]]) and involve problems on [[Small denominators|small denominators]].

For a description of the Poincaré index see [[Singular point|Singular point]].