Poincaré-Bendixson theory

A part of the qualitative theory of differential equations and theory of dynamical systems involving the limiting (when $t\to\pm\infty$) behaviour of trajectories of autonomous systems of two differential equations of the first order:

$$\dot x_i=f_i(x_1,x_2),\quad i=1,2\label{*}\tag{*}$$

(conditions ensuring the existence and uniqueness of solutions are assumed to be satisfied). In the most important case when the system has only a finite number of equilibrium positions in a bounded part of the plane, the basic result of H. Poincaré (see ) and I. Bendixson (see [2]) is that any bounded semi-trajectory (positive or negative) either tends to an equilibrium position or coils round (like a spiral) to a limit cycle, or in an analogous way coils to a closed separatrix or "separatrix contour" consisting of several separatrices "joining" certain equilibrium positions, or is itself an equilibrium position or a closed trajectory. The corollary used most often is: If the semi-trajectory does not leave a given compact domain not containing an equilibrium position, then there is a closed trajectory in this domain. For cases when there is an infinite number of equilibrium positions or when the semi-trajectories are not bounded, there is also a fairly complete, although more complicated, description (see [4]). Finally one can consider a continuous flow in the plane without assuming that it is given by the differential equations \eqref{*}, because in this case it is still possible to use the basic "technical" premises of the Poincaré–Bendixson theory: the Jordan theorem and the Poincaré return map for local cross-sections which are homeomorphic to a segment (their existence was proved in [7]; see also [8]).

The Poincaré–Bendixson theory borders on: the connection, discovered by Poincaré, between the rotation of a certain field on the boundary of a domain and the indices of the equilibrium positions inside it (see Singular point, index of a); results of Bendixson and L.E.J. Brouwer on the possible types of behaviour of trajectories near equilibrium positions (see [2]–[5]); results making the role of "singular trajectories" (equilibrium positions, limit cycles and separatrices) more precise in the "qualitative picture" arising on the phase plane (see [6]).

Although the general theory gives complete information about the possible types of behaviour of the phase trajectories for the system \eqref{*}, this does not answer the question of which type is realized for a certain actual system. A large number of papers in which, as a rule, the general theory is essentially used but which in no way reduce to applying it automatically has been devoted to the solution of such questions (usually not for an individual system but for a certain class of systems).

References

[1a]	H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 7 (1881) pp. 375–422 Zbl 13.0591.01
[1b]	H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 8 (1882) pp. 251–296 Zbl 14.0666.01
[1c]	H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 1 (1885) pp. 167–244 Zbl 14.0666.01 Zbl 13.0591.01
[1d]	H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 2 (1886) pp. 151–217 Zbl 14.0666.01 Zbl 13.0591.01
[2]	I. Bendixson, "Sur les courbes définiés par des équations différentielles" Acta Math. , 24 (1901) pp. 1–88
[3a]	L.E.J. Brouwer, "On continuous vector distributions" Verhand. K. Ned. Akad. Wet. Afd. Nat. I. Reeks , 11 (1909) pp. 850–858
[3b]	L.E.J. Brouwer, "On continuous vector distributions" Verhand. K. Ned. Akad. Wet. Afs. Nat. I. Reeks , 12 (1910) pp. 716–734
[3c]	L.E.J. Brouwer, "On continuous vector distributions" Verhand. K. Ned. Akad. Wet. Afd. Nat. I. Reeks , 13 (1910) pp. 171–186
[4]	V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) MR0121520 Zbl 0089.29502
[5]	P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) MR0658490 Zbl 0476.34002
[6]	A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Qualitative theory of second-order dynamic systems" , Wiley (1973) (Translated from Russian) MR0350126 Zbl 0282.34022
[7]	H. Whitney, "Regular families of curves" Ann. of Math. , 34 : 2 (1933) pp. 244–270 MR1503106 Zbl 0006.37101 Zbl 59.1256.04
[8]	V.V. Nemytskii, "A structure of one-dimensional limit integral manifolds on the plane and in three-dimensional space" Vestn. Moskov. Univ. : 10 (1948) pp. 49–61 (In Russian)

Comments

A complete treatment of the Poincaré–Bendixson theory for continuous flows in the plane that are not defined by differential equations can be found in [a3], Chapt. VIII; see also [a1], Chapt. 2 (where even the use of local cross-sections is avoided). As to $2$-manifolds other than $\mathbf R^2$, Poincaré–Bendixson theory holds for every orientable $2$-manifold for which the Jordan curve theorem holds (e.g., for $S^2$, $S^1\times\mathbf R^1$; not for the $2$-torus $S^1\times S^1$).

An important consequence of the Poincaré–Bendixson theorem (or rather, a consequence of the techniques used in its proof) is the result by H. Bohr and W. Fenchel (1936) that in a continuous flow in the plane every Poisson-stable trajectory is periodic or a rest point ([a3], VIII.1.21). This result can be proved for certain other $2$-manifolds: the Klein bottle [a5] (immediately implying the Kneser theorem) and the projective plane [a4]. For a $C^2$-flow on an arbitrary compact $2$-manifold it can be proved that the closure of a Poisson-stable trajectory either contains a fixed point or is a periodic trajectory, or equals the whole manifold which, in that case, must be the $2$-torus, [a6]. For a further description of the structure of Poisson-stable trajectories, see [a2].

References

[a1]	A. Beck, "Continuous flows in the plane" , Springer (1974) MR0500869 Zbl 0295.54001
[a2]	C. Gutierrez, "Smoothing continuous flows on two-manifolds and recurrences" Ergod. Th. & Dynam. Sys. , 6 (1986) pp. 17–44 MR0837974 Zbl 0606.58042
[a3]	O. Hajek, "Dynamical systems in the plane" , Acad. Press (1968) MR0240418 Zbl 0169.54401
[a4]	P.-F. Lam, "Inverses of recurrent and periodic points under homomorphisms of dynamical systems" Math. Syst. Theory , 6 (1972) pp. 26–36 MR0301718 Zbl 0229.54035
[a5]	N.G. Markley, "The Poincaré–Bendixson theorem for the Klein bottle" Trans. Amer. Math. Soc. , 135 (1969) pp. 159–165 MR234442 Zbl 0175.50101
[a6]	A.J. Schwartz, "A generalization of a Poincaré–Bendixson theorem to closed two-dimensional manifolds" Amer. J. Math. , 85 (1963) pp. 453–458