数理解析研究所講究録
第 1786 巻 2012 年 116-121

116

Reconstruction of restoring forces from periods and amplitudes
Yutaka Kamimura and Takeshi Kaneya
Department of Ocean Sciences,
Tokyo University of Marine Science and Technology
Abstract
We consider an inverse problem to reconstruct a restoring force of an autonomous differential
equation. The restoring force is obtained globally by a Lipschitz continuous function of each amplitude to corresponding amplitude of each periodic solution to the differential equation and uniquely-diffeomorphism of each positive amplitude the corresponding
determined by using a map, anegative amplitude of the solution.
$C^{1}$1

Main Theorem

This article is a concise introduction to a recent work [4] by the authors. The subject
is an inverse problem about nonlinear oscillations on the differential equation$\ddot{u}+g(u)=0$

Here a nonlinearity
condition$g$,$\cdot=\frac{d}{dt}$is a continuous function on$\mathbb{R}$.

(1.1)

and satisfies the so-called signal(1.2).
The condition (1.2) means that the nonlinearity works as a restoring force. If thesolution of (1.1) has a periodic motion then we can denote the positive and negativehalf-amplitudes of the solution by and , respectively; a full amplitude is defined by(see Figure 1). Then the period of the solution is a function of .
We consider an inverse problem to reconstmct of (1.1) from $T(A)$ . Namely, $T(A)$is given and is unknown function to be determined from $T=T(A)$ , and, this problem$ug(u)>0$ ,$u\neq 0$

$g$

$u$

$A$

$b$

$a$

$A$

$T$

$A= \frac{a-b}{2}$

$g$

$g$

involves the following three questions:
Problem 1.1(i) Given a function $T=T(A)$ , does there exist(ii) If such$g$exist, how many such

$g$

$g$

realizing$T$?

do there exist?

(iii) What an additional condition determines$g$uniquely?

This problem to determine from a relation between periods and amplitudes has
been studied by many authors. We pick out some results.
$g$

$T$

Urabe [6, 7] established that, given a positive function with a Lipschitz continuous
locally, namely, forderivative, there exist infinitely many functions realizing
sufficiently small amplitudes.
$T$$g$$T$

Alfawicka [1] proved that the local existence of under a weaker assumption thatitself is Lipschitz continuous.
$g$

$T$

117
Cima,and Villadelprat [2] showed that, to determine uniquely, a map
assigning each negative half-amplitude to each positive half-amplitude is useful.
$Ma\tilde{n}osas$$g$

The global existence of realizing the half-period function (a relation between halfperiods and half-amplitudes) was established by Kamimura [3].
$g$

As is mentioned above, the nonlinearity is not determined uniquely only by theperiod function $T$ . Hence, as an additional data, we employ a function which assigns
the negative half-amplitude to the positive half-amplitude . Note that, is a decreasingfunction. We call a pairing function.
We now prepare three notations. Let
denote the set of continuous functions on
the interval, let $Lip_{+}(I)$ denote the set of Lipschitz continuous, positive functions on, and let Diffl $(I, J)$ denote the set of -diffeomorphisms of onto . We suppose thata period function $T\in Lip_{+}[0, A_{\max}]$ and a pairing function $\varphi\in Diff^{1}([0, a_{\max}], [b_{\min}, 0])$ ,where $A_{\max}= \frac{a_{m}-b_{m}}{2}$ . We obtain the following result, when they are given:$g$$\varphi$

$b$

$a$

$\varphi$$\varphi$$C[\alpha, \beta]$$[\alpha, \beta]$$I$$C^{1}$$I$$J$

Theorem 1.2 A nonlinearity $g\in C[b_{\min}, a_{\max}]$ is uniquely reconstructed by

$T$

and .
$\varphi$

Strictly, Theorem 1.2 implies that: let $b_{\min}<0<a_{\max},$ $A_{\max}= \frac{a_{m}-b}{2}$ , then,given a positive, Lipschitz continuous function
on $[0, A_{\max}]$ and a -diffeomorphism$[0, a_{\max}]$ onto$\varphi(0)=0$
,
with
there
exist a unique $g\in C[b_{\min}, a_{\max}]$ withof$A\in[0,A_{\max}]$(1.2) such that (i) for each, the period of the solution to (1.1) with each
amplitude $A\in[0, A_{\max}]$ coincides with ; (ii) the correspondence assigning each negativehalf-amplitude to each positive half-amplitude is given by .
We here present a rough, physical interpretation of our problem and result based
upon our own experience. When the earthquake shook the east-northern area of Japan
at the 11th of March, 2011, the authors observed that tall buildings around our university
located near the Shinagawa station in Tokyo were swinging very hard. The top of thebuilding was swinging from the right edge to the left edge and then, from the left tothe right (see Figure 1, which is drawn somewhat exaggeratedly). We could get a time
(period) which the building took for this one process corresponding to an amplitude .
At the same time we get a correspondence which assigns to . When the swing became
bigger we get another observed data, a set of a period for another amplitude and anothercorrespondence from to . If the data are obtained continuously for $0<A\leq A_{\max}$(of course, it is possible only theoretically) then one can estimate completely how therestoring force of the building worked.
$T$$\varphi$$C^{1}$$[b_{\min}, 0]$$T$$\varphi$

$b$

$a$

$A$

$b$

$a$

$a$

$b$

Figure 1: Physical interpretation

The outline of the proof of Theorem 1.2 will be given in Section 2. In Section 3, weshall explain how we obtain by giving an example. In Section 4, we shall establish a$g$

118
characterization of , and as a special case, we shall study a problem of isochronicity. InSection 5, we shall pick out some open problems related with Theoreml.2.
$g$2

Outline of the proof
The first important equation is the conservation law$\frac{1}{2}\dot{u}(t)^{2}+G(u(t))=E$

(2.1)

,

is a positive constant expressing the total energy of the system (1.1), and,
where. From this equation (2.1),
throughout the paper, we use the notation: $G(u)$we get$E$$:= \int_{0}^{u}g(\xi)d\xi$$\dot{u}^{2}=2(E-G(u))$.

Clearly, this leads to$\frac{dt}{du}=\pm\frac{1}{\sqrt{2(E-G(u))}}$.

Hence, by an elementary calculation, we obtain$T= \sqrt{2}(\int_{0}^{a}\frac{du}{\sqrt{E-G(u)}}-\int_{0}^{b}\frac{du}{\sqrt{E-G(u)}})$,$G(a)=G(b)=E$ ,where $a= \max u(t),$ $b= \min u(t)$ . We call and a positive and a negative halfamplitude of the solution , respectively. If the velocity vanishes, then (2.1) implies
the second condition. From this relation, by the substitution $s=G(u)$ , which is written, we obtain
as $u=x_{+}(s)(u>0)$ and $u=x_{-}(s)(u<0)$ by using a new function
$a$

$b$

$\dot{u}$$u$$x_{\pm}(s)$$T= \sqrt{2}(\int_{0}^{E}\frac{x_{+}’(s)}{\sqrt{E-s}}ds-\int_{0}^{E}\frac{x_{-}’(s)}{\sqrt{E-s}}ds)$.

Therefore, when we define a function $x(E)$ by$x(E):= \frac{x_{+}(E)-x_{-}(E)}{2}$,we arrive at$T(A)=2 \sqrt{2}\int_{0}^{E}\frac{x’(s)}{\sqrt{E-s}}ds$with $A=x(E)$ .

This function $x(E)$ is a key function in this article. Since $x(E)$ gives the amplitude
this equality is written as$T(x(E))=2 \sqrt{2}\int_{0}^{E}\frac{x’(s)}{\sqrt{E-s}}ds$.

$A$

,

(2.2)

We have to solve (2.2). But, instead of (2.2), it is convenient to use -integration of(2.2), where, -integration means to apply the integral operator , which is defined by
the so-called Riemann-Liouville integral operator$\frac{1}{2}$$I^{\frac{1}{2}}$$\frac{1}{2}$$I^{\frac{1}{2}} \phi(t)=\frac{1}{\sqrt{\pi}}\int_{0}^{t}\frac{\phi(s)}{\sqrt{t-s}}ds$,

119of the order . Hence, we get$\frac{1}{2}$$\frac{1}{2\sqrt{2}\pi}\int_{0}^{E}\frac{T(x(s))}{\sqrt{E-s}}ds=x(E)$,(2.3)

which is a key equation of our task. Note that, firstly, $x(E)$ is an unknown function of(2.3) and that, secondly, the numerator is the composition of and , which is unknown,and therefore (2.3) is a nonlinear integral equation. Fortunately (see [3]), (2.3) can besolved by a method of successive approximations, $x(E)$ is a -fUnction with $x’(E)>0$
for $E>0$ , and $x(E)$ attains
at some point which is denoted by .
Now it is possible to show how we construct in an algorithm (see Figure 2). Firstly,
by a given function , we solve the key equation (2.3), and for $A_{\max}>0$ , determine
such that $x(q)=A_{\max}$ . The number means the maximum value of $E$ . Secondly,
is defined by
$T$

$x$

$C^{1}$$A_{\max}$

$q$

$g$

$T$

$q$

$x_{\pm}(E)$$q$$x(E)= \frac{x_{+}(E)-x_{-}(E)}{2}$,$x_{-}(E)=\varphi(x_{+}(E))$

.

(2.4)

Then $x_{+}(q)=a_{\max},$ $x_{-}(q)=b_{\min}$ . At last, by setting $u=x_{\pm}(E)$ , we obtain the nonlinearity on
by the definition$g$$[b_{\min}, a_{\max}]$(2.5)$g(u)=\{\begin{array}{l}\frac{1}{x_{+}^{l}(x_{+}^{-1}(u))}, 0\leq u\leq a_{\max},\frac{1}{x_{-}’(x_{-}^{-1}(u))}, b_{\min}\leq u\leq 0.\end{array}$

When we determine in this way, one can show that the period of to (1.1) coincides
with the given function , and such is unique. This is the outline of the proof of ourmain theorem.
$u$

$g$

$T$

$x(E)= \frac{x+(E)-x-(E)}{2}$$g$,$x_{-}(E)=\varphi(x_{+}(E))$$\underline{Define}$$\downarrow$

Solve key equation$g(u)= \frac{1}{x_{+}’(x_{+}^{-1}(u))}$$(u\geq 0)$$g(u)= \frac{1}{x_{-}’(x_{-}^{-1}(u))}$$(u\leq 0)$

Figure 2: Algorithm

3

ExampleIn order to see how we get , let us now explore an example. We give $T(A)=$for $A\geq 0$ as a period function and also give $\varphi(a)=2\log(2-e^{\frac{a}{2}})$ for$g$$2 \sqrt{2}\pi\cosh\frac{A}{2}$120$0<a<2\log 2$ as a pairing function. By the first step of the algorithm that we gave inthe previous section, we obtain the solution$x(E)= \log\frac{1+\sqrt{E}}{1-\sqrt{E}}$,$0\leq E<1$.

In fact, this function satisfies the key equation,$\int_{0}^{E}\frac{\cosh(\frac{1}{2}\log\frac{1+\sqrt{s}}{1-\sqrt{s}})}{\sqrt{E-s}}ds=\int_{0}^{E}\frac{ds}{\sqrt{E-s}\sqrt{1-s}}=\log\frac{1+\sqrt{E}}{1-\sqrt{E}}$for $0\leq E<1$ . By the second step, namely solving$x(E)= \frac{x_{+}(E)-2\log(2-e^{\frac{x+(E)}{2}})}{2}$we have$x_{\pm}(E)=2\log(1\pm\sqrt{E})$

Therefore, by applying (2.5) to these functions$g(u)=e^{u}-e^{\frac{u}{2}}$,,$0\leq E<1$ .

,$x_{\pm}(E)$, we finally obtain$b_{\min}\leq u\leq a_{\max}$,

for any $b_{\min}<0<a_{\max}<2\log 2$ .

4Characterization

Now we characterize a functionby (2.4), it leads to$G$. By the key function and the pairing function defined$E=x^{-1}( \frac{x_{+}(E)-x_{-}(E)}{2})=x^{-1}(\frac{x_{+}(E)-\varphi(x_{+}(E))}{2})$.

Hence, by the substitution $E=G(u)$ , which is written as $u=x_{+}(E)$ , we have,$G(u)=x^{-1}( \frac{u-\varphi(u)}{2})$$0\leq u\leq a_{\max}$.

Also, in a similar way, we get$G(u)=x^{-1}( \frac{\varphi^{-1}(u)-u}{2})$,$b_{\min}\leq u\leq 0$.

Thus, we obtain the following:

Theorem 4.1$G(u)=x^{-1}(| \frac{u-\sigma(u)}{2}|)$where the function$\sigma(u)$,$b_{\min}\leq u\leq a_{\max}$is defined by$\sigma(u)=\{\begin{array}{ll}\varphi(u), 0\leq u\leq a_{\max},\varphi^{-1}(u), b_{\min}\leq u\leq 0.\end{array}$,(4.1)

121
The function

has the following properties: (i)Id;Id (the identity map),(ii) $\sigma(0)=0$ ; (iii). By (iii), the function can obtain a cusp
at the origin. By differentiating this $G(u)$ , we get a nonlinearity $g(u)$ . Sinceandare determined from and , respectively, the nonlinearity $g(u)$ can be indexed by theperiod function and the pairing function through (4.1).
An important, special case of this characterization appears in a problem of isochronicity (see [2, 5]). If given a period is a constant , then we have the function $x(E)=$. Therefore, we get the following:
Corollary 4.2 The period of all the periodic motions of (1.1) is a constant if and only, whereis a function satisfying the properties (i),if$\sigma(u)$$\sigma 0\sigma=$$\sigma\neq$$\sigma\in C^{1}[0, a_{\max}]\cap C^{1}[b_{\min}, 0]$$\sigma$$x^{-1}$$T$$\sigma$$\varphi$$T$$\varphi$$T$$\omega$$\frac{\omega}{\sqrt{2}\pi}\sqrt{E}$$\omega$$G(u)= \frac{\pi^{2}}{2\omega^{2}}(u-\sigma(u))^{2},$$u\in \mathbb{R}$$\sigma(u)$(ii), and (iii) ‘ $\sigma\in C^{1}[0, \infty)\cap C^{1}(\infty, 0]$ .

5

Future works

Without a doubt there are many open problems concerning this research that should
be settled. One of important problems is to make a stability result. This asks whether thecorrespondence
is continuous in some appropriate topology (metric). Since, inpractical problems, the data set observed is usually discrete, prescribed pair of function
and becomes an approximation necessarily. This is why a good stability result isdesired. Another is to establish global existence theorems corresponding to Theoreml.2
for other, general equations, for example, $\ddot{u}+g(u,\dot{u})=0$ ( depends on the velocity),$(| \frac{du}{dt}|^{p-2}\frac{du}{dt})+g(u)=0$(-Laplacian), $\triangle u+g(u)=0$ (multi-dimensional case), andso on. We point out that, for the equation $\ddot{u}+g(x)sgn\dot{u}+u=0$ , the readers may refer
to [5]. Also, recently, the global existence result for the half-period function concerning$\frac{d}{dt}((\frac{du}{dt})^{2m-1})+g(u)=0$
has been established by Usami and Yoshimi (see [8]).
$(T, \varphi)\mapsto g$$T$$\varphi$$g$

8References[1] B. Alfawicka, Inverse problems connected with periods of oscillations described by$\mathfrak{X}+g(x)=0$ , Ann. Polon. Math. 44 (1984), 297-308.

[2] A. Cima, F. Mar osas, J. Villadelprat, Isochronicity for several classes of Hamiltoniansystems, J. Differential Equations 157 (1999), 373-413.
[3] Y. Kamimura, Global existence of a restoring force realizing a prescribed half-period,
J. Differential Eqs. 248 (2010), 2562-2584.

[4] Y. Kamimura, T. Kaneya, Global determination of a nonlinearity from a periodic
motion, submitted (2011).
[5] F. Manosas, P.J. Torres, Isochronicity of a class of piecewise continuous oscillators,
Proc. Amer. Math. Soc. 133 (2005), 3027-3035.
[6] M. Urabe, Relation between periods and amplitudes of periodic solutions of$g(x)=0$ , Funkc. Ekvacioj 6 (1964), 63-88.

$\mathfrak{X}+$[7] M. Urabe, Nonlinear Autonomous Oscillations, Academic, New York, 1967.
[8] 宇佐美広介，吉見拓郎，準線型常微分方程式の半周期に関する逆問題，日本数学会 2011年度秋季総合分科会，函数方程式論分科会講演アブストラクト (2011), 24-25.