Periodic solution

of an ordinary differential equation or system

A solution that periodically depends on the independent variable $ t $. For a periodic solution $ x( t) $( in the case of a system, $ x $ is a vector), there is a number $ T \neq 0 $ such that

$$ x( t+ T) = x( t) \ \textrm{ for } t \in \mathbf R . $$

All possible such $ T $ are called periods of this periodic solution; the continuity of $ x( t) $ implies that either $ x( t) $ is independent of $ t $ or that all possible periods are integral multiples of one of them — the minimal period $ T _ {0} > 0 $. When one speaks of a periodic solution, it is often understood that the second case applies, and $ T _ {0} $ is simply termed the period.

A periodic solution is usually considered for a system of ordinary differential equations where the right-hand sides either are independent of $ t $( an autonomous system):

$$ \tag{1 } \dot{x} = f( x) ,\ \ x \in U, $$

where $ U $ is a region in $ \mathbf R ^ {n} $, or else periodically depend on $ t $:

$$ \tag{2 } \dot{x} = f( t, x),\ \ f( t+ T _ {1} , x) = f( t, x),\ \ x \in U . $$

(In a system with a different type of dependence on $ t $ for the right-hand sides there is usually no periodic solution.) In case (2) the period $ T _ {0} $ of the periodic solution usually coincides with the period $ T _ {1} $ of the right-hand side or is an integer multiple of $ T _ {1} $; other $ T _ {0} $ are possible only in exceptional cases. Periodic solutions with periods $ T _ {0} = kT _ {1} $, $ k > 1 $, describe subharmonic oscillations (see Forced oscillations) and therefore are themselves sometimes called subharmonic periodic solutions (or subharmonics).

System (2) determines the Poincaré return map $ F $( dependent on the choice of the initial moment $ t _ {0} $): If $ x( t, \xi ) $ is the solution to (2) with initial value $ x( t _ {0} , \xi ) = \xi $, then

$$ F( \xi ) = x( t _ {0} + T _ {1} , \xi ). $$

The properties of (2) are closely related to those of $ F $; in particular, the value at $ t = t _ {0} $ for the periodic solution with period $ kT _ {1} $ is a fixed point of $ F $ for $ k= 1 $, while for $ k > 1 $ it is a periodic point with period $ k $, i.e. a fixed point for the iterate $ F ^ { k } $. The research on periodic solutions reduces to a considerable extent to examining the corresponding fixed or periodic points of the Poincaré return map.

The following modification of this construction is used for an autonomous system (1): One takes some local section in the phase space at some point on the trajectory of the periodic solution (which is a closed curve), i.e. one takes a smooth manifold $ \Pi $ of codimension 1 transversal to this trajectory, and considers the mapping that converts a point $ \xi \in \Pi $ to the point of intersection of the trajectory of (1) through $ \xi $ with $ \Pi $ that is first in time.

The behaviour of solutions close to a given periodic solution is described in linear approximation by the corresponding variational system (cf. Variational equations). The coefficients in this linear system in that case periodically depend on $ t $, and therefore one can speak of the corresponding monodromy operator and multipliers. The latter are also termed multipliers for the given periodic solution. The linear approximation determines the properties of the periodic solution (stability, invariant manifolds) to the same extent as for an equilibrium solution (cf. Equilibrium position).

The periodic solutions to (1) have some specific features: one of the multipliers is always one (if the periodic solution does not reduce to a constant), which in particular has to be borne in mind when examining the stability of these periodic solutions (see Andronov–Witt theorem), and the period may change in response to small perturbations, which has to be borne in mind in perturbation theory.

The search for periodic solutions and the examination of their behaviour are of interest not only from the purely mathematical point of view but also because the periodic regimes of real physical systems usually correspond to periodic solutions in the mathematical description of these systems (see Auto-oscillation; Forced oscillations; Oscillations, theory of; Non-linear oscillations; Relaxation oscillation). However, this is a very difficult problem, since there are no general methods for establishing whether periodic solutions exist for a particular system. Various arguments and methods are used in different cases. Many of them relate to perturbation theory (the harmonic balance method, the Krylov–Bogolyubov method of averaging or the method of the small parameter, cf. Small parameter, method of the), and they also touch upon research on bifurcation; others relate to the qualitative theory of differential equations. The latter, in particular, establishes the special role of periodic solutions for (1) with $ n= 2 $: Here the periodic solutions, along with certain other types of solutions, completely determine the behaviour of all the solutions (see also Limit cycle). In this connection, there are some special results on the periodic solutions of such systems (for example, on the periodic solutions of the van der Pol equation and its generalizations or modifications: the Liénard equation and the Rayleigh equation).

Comments

A survey on periodic solutions of differential equations is [a1]. For autonomous systems there is translation invariance: if $ x( t) $ is a periodic solution, then $ x( t+ \tau ) $ is also a periodic solution. This is related to the presence of one multiplier with value 1. For periodic solutions with period $ T _ {0} $ one may replace the time variable $ t $ by a phase variable $ \phi = t / T _ {0} $. The phase variable is determined modulo 1. The notion of phase also applies to a trajectory $ x( t) $ near a limit cycle. In that case phase is identified with the phase of the periodic solution to which the trajectory tends for $ t \rightarrow \infty $. This concept plays an important role in cyclic biological processes, see [a2]. In the case of periodic forcing with period $ T _ {1} $ one may relate the periodic motion of the perturbed system with that of the forcing term having a phase $ \theta = \theta _ {0} + t/T _ {1} $. Let the phase of the perturbed system at time $ t = nT _ {1} $ be $ \phi = \phi ( n, \theta _ {0} ) $. Then the rotation number, defined by

$$ \rho ( \theta _ {0} ) = \lim\limits _ {n \rightarrow \infty } \ \frac{\phi ( n , \theta _ {0} ) }{n} , $$

equals the ratio of the period of the forced system and that of the forcing term. When $ \rho $ is a rational number the system is periodic, while for $ \rho $ irrational the limit solution is quasi-periodic (cf. also Quasi-periodic motion). See also [a3], where a system on a torus is analyzed.

References

[a1]	F. Verhulst, "Nonlinear differential equations and dynamical systems" , Springer (1990) MR1036522 Zbl 0685.34002
[a2]	A.T. Winfree, "The geometry of biological time" , Springer (1980) MR0572965 Zbl 0464.92001
[a3]	J.K. Hale, "Differential equations" , Birkhäuser (1982) MR0667126 Zbl 1180.65102 Zbl 1071.34074 Zbl 1005.93026 Zbl 0988.00064 Zbl 1004.37062 Zbl 0830.34055 Zbl 0817.35119 Zbl 0787.34062 Zbl 0780.00043 Zbl 0787.34002 Zbl 0785.00039 Zbl 0753.34048 Zbl 0785.35050 Zbl 0666.35013 Zbl 0644.65050 Zbl 0612.34066 Zbl 0611.34074 Zbl 0638.00015 Zbl 0582.34058 Zbl 0662.34064