# Periodic Solutions

## Periodic Solutions

Periodic solutions of equations are solutions that describe regularly repeating processes. In such branches of science as the theory of oscillations and celestial mechanics, periodic solutions of systems of differential equations

are of special interest. A periodic solution *y _{i}* =

*Φ*(

_{i}*t*) of (1) consists of periodic functions of

*t*that have the same period. In other words,

*Φ _{i}* (

*t*+

*τ*) =

*Φ*(

_{i}*t*)

for all *t* and for some *τ* > 0; *τ* is called the period of the solution. If the system (1) is autonomous—that is, the functions *f _{i}* =

*F*(

_{i}*y*

_{1}, …,

*y*),

_{n}*i*= 1, …, n, do not explicitly depend on

*t*—then to the periodic solutions there correspond closed trajectories in the phase space (

*y*

_{1}, …,

*y*). A degenerate form of such trajectories are the equilibrium, or critical, points

_{n}where

The critical points correspond to trivial (constant) periodic solutions. The problem of finding nontrivial periodic solutions has been solved only for special types of differential equations.

Systems of two equations

are of particular importance in the theory of nonlinear oscillations. Their phase space is the *xy*-plane. The critical points of system (2) can be found from the system of equations *P* (*x*, *y*) = 0, *Q* (*x*, *y*) = 0. System (2) does not admit of nontrivial periodic solutions if *P′ _{x}* (

*x*,

*y*) +

*Q′*(

_{y}*x, y*) = 0 (Bendixson’s criterion).

The usual method of finding nontrivial periodic solutions of system (2), if they exist, is to construct a bounded annular region *K* (see Figure 1) such that all trajectories enter it as *t* → + ∞ or *t* → – ∞. If *k* does not contain critical points of system (2), it necessarily contains a closed trajectory that corresponds to a nontrivial periodic solution (the Poincaré-Bendixson theorem). Another approach to finding periodic solutions is provided by the study of the behavior of solutions in neighborhoods of critical points: in the neighborhood of a center, the integral curves of system (2) are closed and correspond to nontrivial periodic solutions.

### REFERENCES

Nemytskii, V. V., and V. V. Stepanov.*Kachestvennaia teoriia differentsial’nykh uravnenii*, 2nd ed. Moscow-Leningrad, 1949.

Andronov, A. A., A. A. Vitt, and S. E. Khaikin.

*Teoriia kolebanii*, 2nd ed. Moscow, 1959.

Stoker, J.

*Nelineinye kolebaniia v mekhanicheskikh i elektricheskikh sistemakh*, 2nd ed. Moscow, 1953. (Translated from English.)