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Periodic Solutions of Non-autonomous Ordinary Differential Equations

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

Many phenomena in nature can be modeled by systems of ordinary differential equations which depend periodically upon time. For example, a linear or nonlinear oscillator can be forced by a periodic external force, and an important question is to know if the oscillator can exhibit a periodic response under this forcing. This question originated from problems in classical and celestial mechanics, before receiving important applications in radioelectricity and electronics. Nowadays, it also plays a great role in mathematical biology and population dynamics, as well as in mathematical economics, where the considered systems are often subject to seasonal variations. The general theory originated with Henri Poincaré’s work in celestial mechanics, at the end of the nineteenth century and has been constantly developed since.

Introduction

To motivate the problem and its difficulties, let us start with the simple linear oscillator with forcing (or input) h ∈ L...

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Abbreviations

Banach fixed point theorem:

If M is a complete metric space with distance d and f : M → M is contractive, i.e., d(f(u), f(v)) ≤ αd(u, v) for some α ∈ [0, 1) and all u, v ∈ M, then f has a unique fixed point u* and u* = lim k → ∞ f k(u 0) for any u 0∈M

Brouwer degree:

An integer d B [f, Ω] which “algebraically” counts the number of zeros of any continuous mapping \( f:\overline{\Omega}\subset {\mathrm{\mathbb{R}}}^n\to {\mathrm{\mathbb{R}}}^n \) such that 0 ∉ f(∂Ω) and is invariant for sufficiently small perturbations of f. If f is of class C 1 and its zeros are nondegenerate, then \( {d}_B\left[f,\Omega \right]={\displaystyle \sum_{x\in {f}^{-1}(0)}\mathrm{sign}\kern1em \det {f}^{\prime }(x)} \)

Brouwer fixed point theorem:

Any continuous mapping f : B → B, with B homeomorphic to the closed unit ball in ℝn, has at least one fixed point

Leray-Schauder degree:

The extension d LS [I − g, Ω] of the Brouwer degree, where Ω is an open bounded subset of the Banach space X, and \( g:\overline{\Omega}\to X \) is continuous, \( g\left(\overline{\Omega}\right) \) is relatively compact, and 0 ∉ (I − g)(∂Ω) is Leray-Schauder-Schaefer fixed point theorem. If X is a Banach space, g : X → X is a continuous mapping taking bounded subsets into relatively compact ones, and if the set of possible fixed points of εg(ε ∈ [0, 1]) is bounded independently of ε, then g has at least one fixed point

Lusternik -Schnirelmann category:

The Lusternik-Schnirelmann category cat (M) of a metric space M into itself is the smallest integer k such that M can be covered by k sets contractible in M

Lower and upper solutions:

A lower (resp. upper) solution of the periodic problem u″ = f(t, u), u(0) = u(T), u′(0) = u′(T) is a function α (resp. β) of class C 2 such that α″(t) ≥ f(t, α(t)), α(0) = α(T), α′(0) ≥ α′(T)(resp. β″(t) ≤ f(t, β(t)), β(0) = β(T), β′(0) ≤ β′(T))

Palais-Smale condition for a C 1 function φ : X → ℝ:

Any sequence (u k ) k ∈ ℕ such that (φ(u k )) k ∈ ℕ is bounded and lim k → ∞ φ′(u k ) = 0 contains a convergent subsequence

Poincaré operator:

The mapping defined in ℝn by P T : y → p(T; y), where p(t;y) is the unique solution of the Cauchy problem x′ = f(t, x), x(0) = y.

Schauder fixed point theorem:

If C is a closed bounded convex subset of a Banach space X, any continuous mapping g : C → C such that g(C) is relatively compact has at least one fixed point

Sobolev inequality:

For any function u ∈ L 2(0, T) such that u′ ∈ L 2(0, T) and \( {\displaystyle {\int}_0^T} \) u(t)dt = 0, one has \( \underset{t\in \left[0,T\right]}{ \max}\left|u(t)\right|\le \left({T}^{1/2}/2\sqrt{3}\right){\left[{\displaystyle {\int}_0^T{\left|{u}^{\mathit{\prime}}(t)\right|}^2\mathrm{d}t}\right]}^{1/2} \)

Wirtinger inequality:

For any function u ∈ L 2(0, T) such that u′ ∈ L 2(0, T) and \( {\displaystyle {\int}_0^T} \) u(t)dt = 0, one has \( {\displaystyle {\int}_0^T} \)|u(t)|2dt ≤ (T 2/4π 2) \( {\displaystyle {\int}_0^T} \)|u′(t)|2dt

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  1. Département de Mathématique, Université Catholique de Louvain, Louvain-la-Neuve, Belgium

    Jean Mawhin

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    Robert A. Meyers

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Mawhin, J. (2013). Periodic Solutions of Non-autonomous Ordinary Differential Equations. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-3-642-27737-5_392-3

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