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A closed trajectory in the phase space of an [[Autonomous system|autonomous system]] of ordinary differential equations that is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l0588501.png" />- or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l0588502.png" />-limit set (cf. [[Limit set of a trajectory|Limit set of a trajectory]]) of at least one other trajectory of this system. A limit cycle is called orbit stable, or stable, if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l0588503.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l0588504.png" /> such that all trajectories starting in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l0588505.png" />-neighbourhood of it for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l0588506.png" /> do not leave its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l0588507.png" />-neighbourhood for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l0588508.png" /> (cf. [[Orbit stability|Orbit stability]]). A limit cycle corresponds to a periodic solution of the system, differing from a constant. In order for a periodic solution to correspond to a stable limit cycle it is sufficient that the moduli of all its multipliers, except one, be less than one (cf. [[Characteristic exponent|Characteristic exponent]]; [[Andronov–Witt theorem|Andronov–Witt theorem]]). From the physical point of view, a limit cycle corresponds to periodic behaviour, or an [[Auto-oscillation|auto-oscillation]], of the system (cf. [[#References|[2]]]).
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{{TEX|done}}
  
Suppose that an autonomous system
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An ''isolated'' closed trajectory in the phase space of an [[Autonomous system|autonomous system]] of ordinary differential equations. A limit cycle corresponds to a periodic non-constant solution 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/l/l058/l058850/l0588509.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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===Dynamics===
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Limit cycles represent the simplest (after the steady states) type of behavior of a continuous time [[dynamical system]]. Theoretically all properties of limit cycles (their [[stability]] and [[bifurcation]]s) can be reduced to investigation of the associated [[Poincaré return map]]<ref>Sometimes also the terms ''monodromy'' or ''holonomy'' are used as synonyms to the "first return map".</ref>. In practice, however, the Taylor coefficients of the Poincare map can be obtained only in the form of integrals over the cycle, which may require some quite detailed knowledge of the shape of the cycle itself.
  
defined in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885011.png" /> is a differentiable manifold, e.g. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885012.png" />, has a closed trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885013.png" />. Draw the hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885014.png" /> intersecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885015.png" /> transversally at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885016.png" />. Then every trajectory of the system starting for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885017.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885018.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885019.png" /> a sufficiently small neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885020.png" />, intersects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885021.png" /> again, at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885022.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885023.png" /> increases. The diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885024.png" /> has fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885025.png" /> and is called the [[Poincaré return map|Poincaré return map]]. Its properties determine the behaviour of trajectories of the system in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885026.png" />. A limit cycle, as distinct from an arbitrary closed trajectory, always determines a Poincaré return map that is not the identity. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885027.png" /> is a saddle point of the diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885028.png" />, then the limit cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885029.png" /> is said to be of saddle type. A system having a limit cycle of saddle type can have homoclinic curves, i.e. trajectories for which the limit cycle is both the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885030.png" />- and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885031.png" />-limit set.
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For instance, in the linear approximation if $\gamma:[0,T]\to\R^n$, $t\mapsto\gamma(t)$, is a limit cycle of period $T>0$ for the vector field $v(x)$ associated with the differential equation $\dot x=v(x)$, $x\in\R^n$, one obtains a linear (non-autonomous) system of differential equations
 +
$$
 +
\dot z=A(t)z,\qquad z\in\R^n, \quad A(t)=\biggl(\frac{\partial v}{\partial x}(\gamma(t)\biggr),\ t\in [0,T].
 +
$$
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The corresponding ''[[Cauchy operator|Cauchy]]--[[Floquet theory|Floquet]]'' linear operator $M:\R^n\to\R^n$ maps a vector $a\in\R_n$ into the vector $Ma=z_a(T)$, where $z_a$ is the solution of the above system with the initial value $z_a(0)=a$. If this operator is [[hyperbolic point|hyperbolic]], i.e., has no modulus one eigenvalues ("[[characteristic exponent]]s"), then the stability pattern of the cycle (dimensions of the corresponding stable and unstable [[invariant manifolds]]) is completely determined (and coincides with that of the iterations $M^k$, $k\in\Z$).
  
In the case of a two-dimensional system (*) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885032.png" /> one takes a straight line for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885033.png" /> and considers the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885035.png" />, which is called the Poincaré return function. The multiplicity of the zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885036.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885037.png" /> is called the multiplicity of the limit cycle. A limit cycle of even multiplicity is called semi-stable. The limit cycles, together with the rest points and the separatrices (cf. [[Separatrix|Separatrix]]), determine the qualitative picture of the behaviour of the other trajectories (cf. [[Poincaré–Bendixson theory|Poincaré–Bendixson theory]], as well as [[#References|[3]]], [[#References|[4]]]). In the case of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885038.png" /> the limit cycles belong to one of the following three types: 1) stable; 2) unstable, i.e. stable if the direction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885039.png" /> is reversed; or 3) semi-stable. E.g., the system
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===Limit cycles of planar vector fields===
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On the two-dimensional sphere (and plane) the topological restrictions which forbid intersection of phase trajectories, make limit cycles the only possible limit motion not directly related to singular points (steady states, also known as stationary solutions). More precisely, if the $\Omega$-limit set of a non-periodic point $a\in \R^2$<ref>A closed invariant subset of the plane, defined as
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$$
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\Omega(a)=\bigcap_{T<+\infty}\overline{\{g^t(a)|t\ge T\}},\qquad g^t(a)=\text{the flow map, }\left.\frac{\rd g^t(a)}{\rd t}\right|_{t=0}=v(a).
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$$</ref>contains no singular point of the field $v$, then it must be a limit cycle ([[Poincare-Bendixson theory|Poincare-Bendixson]], 1886<ref>H. Poincaré, ''Memoire sur les courbes définiés par des équations différentielles'', J. de Math. , '''7''' (1881) pp. 375–422, ibid., '''8''' (1882) pp. 251–296, ibid., '''1''' (1885) pp. 167–244, ibid., '''2''' (1886) pp. 151–217. </ref>, 1901<ref>I. Bendixson, ''Sur les courbes définiés par des équations différentielles'', Acta Math., '''24''' (1901) pp. 1–88. </ref>).
  
<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/l/l058/l058850/l05885040.png" /></td> </tr></table>
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If the presence of singular points cannot be excluded, the situation becomes slightly more complicate. Under the assumption of analyticity one can show that the only possible limit sets for vector fields on the sphere<ref>Polynomial vector fields on the plane also satisfy this assertion.</ref> are singular points, limit cycles and [[separatrix|separatrix polygons]], also known as [[polycycle]]s, which consist of singular points and connecting them arcs of [[separatrix|separatrices]].
  
<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/l/l058/l058850/l05885041.png" /></td> </tr></table>
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For the same reasons [[bifurcation]]s of limit cycles, topological changes of the number of limit cycles, are possible only in annular neighborhoods of existing (multiple) cycles, singular points or polycycles.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885044.png" />, has for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885045.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885046.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885047.png" /> odd a stable (unstable) limit cycle of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885048.png" />, while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885049.png" /> even it has a limit cycle of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885050.png" />. In all cases the limit cycle is the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885051.png" />, i.e. the trajectory of the solution
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===Complex limit cycles===
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A polynomial planar vector field after [[complexification]] defines a holomorphic singular [[foliation]] $\mathscr F$ on the complex [[projective plane]] $\C P^2$. Solutions of the differential equation correspond to leaves of this foliation, yet unlike in the real case, the leaves are topologically two-dimensional and can have much richer topological structure.
  
<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/l/l058/l058850/l05885052.png" /></td> </tr></table>
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A limit cycle after complexification corresponds to a nontrivial loop on a leaf of the foliation $\mathscr F$ with a non-identical [[holonomy]] map. This observation may motivate one of the possible generalizations of the notion of limit cycle for complex ordinary differential equations.  
  
If the system (*) is given on a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885053.png" />, then a limit cycle encircles at least one rest point of the system.
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A ''complex limit cycle'' is a noncontractible closed loop on the leaf of a singular holomorphic foliation on $\C P^2$ with a non-identical holonomy. Note that according to this definition, the same leaf may carry many different limit cycles: for instance, generically the infinite line (with deleted singular points) is a multiply connected leaf of a polynomial foliation, and each small loop around the deleted singularity is a complex limit cycle. However, these limit cycles are [[homology|homologically dependent]]: their sum is zero.
 
 
In order to find limit cycles of second-order systems one uses methods based on the following fact: If a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885054.png" /> is directed inwards (outwards) an annular domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885055.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885056.png" /> does not contain rest points, then there is at least one stable (unstable) limit cycle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885057.png" />. The choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058850/l05885058.png" /> is based on physical considerations or results from analytic or numerical computations.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian) {{MR|0140742}} {{ZBL|0112.05502}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.A. Andronov, A.A. Vitt, A.E. Khaikin, "Theory of oscillators" , Dover, reprint (1987) (Translated from Russian) {{MR|0925417}} {{ZBL|0188.56304}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Qualitative theory of second-order dynamic systems" , Wiley (1973) (Translated from Russian) {{MR|0350126}} {{ZBL|0282.34022}} </TD></TR><TR><TD valign="top">[4]</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) {{MR|0344606}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> W.A. Pliss, "Nonlocal problems of the theory of oscillations" , Acad. Press (1966) (Translated from Russian) {{MR|0196199}} {{ZBL|0151.12104}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> N.N. Moiseev, "Asymptotic methods of non-linear mechanics" , Moscow (1969) (In Russian)</TD></TR></table>
 
  
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===Hilbert 16th problem===
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One of the most challenging problems which remains open for over 120 years, is the Hilbert's question on the number and position of limit cycles of a polynomial vector field on the plane ([[Hilbert_problems#Hilbert.27s_sixteenth_problem.|Problem 16, second part]]). Despite considerable progress in the last 25 years, the only known general result states that each polynomial vector field may have only finitely limit cycles (independently Yu. Ilyashenko and J. Ecalle, 1991). It is not known whether this number is uniformly bounded over all polynomial fields of degree $\le d$, even for $d=2$ (fields of degree $1$ cannot exhibit limit cycles at all).
  
 +
It is worth noting that the Hilbert 16th problem has no nontrivial complex version. A generic polynomial vector field after complexification has countably many homologically independent complex limit cycles, see {{Cite|IY|Sect. 28C}}.
  
====Comments====
+
----
All definitions given above can be formulated for arbitrary dynamical systems, not necessarily defined by an autonomous system of ordinary differential equations. Most of the results are also meaningful in that case. For the Poincaré–Bendixson theory, see also e.g. [[#References|[a1]]], Sect. VIII.1. A good additional general reference is [[#References|[a2]]].
+
====Notes====
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<small>
 +
<references/>
 +
</small>
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> O. Hajek, "Dynamical systems in the plane" , Acad. Press (1968) {{MR|0240418}} {{ZBL|0169.54401}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17 {{MR|0069338}} {{ZBL|0064.33002}} </TD></TR></table>
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{|
 +
|-
 +
|valign="top"|{{Ref|E}}||valign="top"|Écalle, J. ''Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac'', Actualités Mathématiques. Hermann, Paris, 1992. {{MR|1399559}}
 +
|-
 +
|valign="top"|{{Ref|H}}||valign="top"|Hilbert, D. ''Mathematical problems'' Reprinted from Bull. Amer. Math. Soc. '''8''' (1902), 437–479. Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 4, 407--436. {{MR|1779412}}
 +
|-
 +
|valign="top"|{{Ref|I91}}||valign="top"|Ilyashenko, Yu. S. ''Finiteness theorems for limit cycles'', Translations of Mathematical Monographs, '''94'''. American Mathematical Society, Providence, RI, 1991. {{MR|1133882}}
 +
|-
 +
|valign="top"|{{Ref|I02}}||valign="top"|Ilyashenko, Yu. ''Centennial history of Hilbert's 16th problem'' Bull. Amer. Math. Soc. (N.S.) '''39''' (2002), no. 3, 301--354. {{MR|1898209}}
 +
|-
 +
|valign="top"|{{Ref|IY}}||valign="top"|Ilyashenko, Yu. and Yakovenko, S. ''Lectures on analytic differential equations'', Graduate Studies in Mathematics, '''86'''. American Mathematical Society, Providence, RI, 2008. {{MR|2363178}}
 +
|-
 +
|valign="top"|{{Ref|R}}||valign="top"|R. Roussarie, ''Bifurcation of planar vector fields and Hilbert's sixteenth problem'' , Birkhäuser (1998). {{MR|1628014}}.
 +
|-
 +
|}

Latest revision as of 09:14, 1 October 2023


An isolated closed trajectory in the phase space of an autonomous system of ordinary differential equations. A limit cycle corresponds to a periodic non-constant solution of the system.

Dynamics

Limit cycles represent the simplest (after the steady states) type of behavior of a continuous time dynamical system. Theoretically all properties of limit cycles (their stability and bifurcations) can be reduced to investigation of the associated Poincaré return map[1]. In practice, however, the Taylor coefficients of the Poincare map can be obtained only in the form of integrals over the cycle, which may require some quite detailed knowledge of the shape of the cycle itself.

For instance, in the linear approximation if $\gamma:[0,T]\to\R^n$, $t\mapsto\gamma(t)$, is a limit cycle of period $T>0$ for the vector field $v(x)$ associated with the differential equation $\dot x=v(x)$, $x\in\R^n$, one obtains a linear (non-autonomous) system of differential equations $$ \dot z=A(t)z,\qquad z\in\R^n, \quad A(t)=\biggl(\frac{\partial v}{\partial x}(\gamma(t)\biggr),\ t\in [0,T]. $$ The corresponding Cauchy--Floquet linear operator $M:\R^n\to\R^n$ maps a vector $a\in\R_n$ into the vector $Ma=z_a(T)$, where $z_a$ is the solution of the above system with the initial value $z_a(0)=a$. If this operator is hyperbolic, i.e., has no modulus one eigenvalues ("characteristic exponents"), then the stability pattern of the cycle (dimensions of the corresponding stable and unstable invariant manifolds) is completely determined (and coincides with that of the iterations $M^k$, $k\in\Z$).

Limit cycles of planar vector fields

On the two-dimensional sphere (and plane) the topological restrictions which forbid intersection of phase trajectories, make limit cycles the only possible limit motion not directly related to singular points (steady states, also known as stationary solutions). More precisely, if the $\Omega$-limit set of a non-periodic point $a\in \R^2$[2]contains no singular point of the field $v$, then it must be a limit cycle (Poincare-Bendixson, 1886[3], 1901[4]).

If the presence of singular points cannot be excluded, the situation becomes slightly more complicate. Under the assumption of analyticity one can show that the only possible limit sets for vector fields on the sphere[5] are singular points, limit cycles and separatrix polygons, also known as polycycles, which consist of singular points and connecting them arcs of separatrices.

For the same reasons bifurcations of limit cycles, topological changes of the number of limit cycles, are possible only in annular neighborhoods of existing (multiple) cycles, singular points or polycycles.

Complex limit cycles

A polynomial planar vector field after complexification defines a holomorphic singular foliation $\mathscr F$ on the complex projective plane $\C P^2$. Solutions of the differential equation correspond to leaves of this foliation, yet unlike in the real case, the leaves are topologically two-dimensional and can have much richer topological structure.

A limit cycle after complexification corresponds to a nontrivial loop on a leaf of the foliation $\mathscr F$ with a non-identical holonomy map. This observation may motivate one of the possible generalizations of the notion of limit cycle for complex ordinary differential equations.

A complex limit cycle is a noncontractible closed loop on the leaf of a singular holomorphic foliation on $\C P^2$ with a non-identical holonomy. Note that according to this definition, the same leaf may carry many different limit cycles: for instance, generically the infinite line (with deleted singular points) is a multiply connected leaf of a polynomial foliation, and each small loop around the deleted singularity is a complex limit cycle. However, these limit cycles are homologically dependent: their sum is zero.

Hilbert 16th problem

One of the most challenging problems which remains open for over 120 years, is the Hilbert's question on the number and position of limit cycles of a polynomial vector field on the plane (Problem 16, second part). Despite considerable progress in the last 25 years, the only known general result states that each polynomial vector field may have only finitely limit cycles (independently Yu. Ilyashenko and J. Ecalle, 1991). It is not known whether this number is uniformly bounded over all polynomial fields of degree $\le d$, even for $d=2$ (fields of degree $1$ cannot exhibit limit cycles at all).

It is worth noting that the Hilbert 16th problem has no nontrivial complex version. A generic polynomial vector field after complexification has countably many homologically independent complex limit cycles, see [IY, Sect. 28C].


Notes

  1. Sometimes also the terms monodromy or holonomy are used as synonyms to the "first return map".
  2. A closed invariant subset of the plane, defined as $$ \Omega(a)=\bigcap_{T<+\infty}\overline{\{g^t(a)|t\ge T\}},\qquad g^t(a)=\text{the flow map, }\left.\frac{\rd g^t(a)}{\rd t}\right|_{t=0}=v(a). $$
  3. H. Poincaré, Memoire sur les courbes définiés par des équations différentielles, J. de Math. , 7 (1881) pp. 375–422, ibid., 8 (1882) pp. 251–296, ibid., 1 (1885) pp. 167–244, ibid., 2 (1886) pp. 151–217.
  4. I. Bendixson, Sur les courbes définiés par des équations différentielles, Acta Math., 24 (1901) pp. 1–88.
  5. Polynomial vector fields on the plane also satisfy this assertion.

References

[E] Écalle, J. Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualités Mathématiques. Hermann, Paris, 1992. MR1399559
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How to Cite This Entry:
Limit cycle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Limit_cycle&oldid=24494
This article was adapted from an original article by L.A. Cherkas (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article