数理解析研究所講究録
1014 巻 1997 年 118-132

118

Uses of Envelopes for Global and Asymptotic Analysis; geometrical meaning of renormalization group equation
Teiji Kunihiro$(\mathrm{f}\Gamma \mathrm{A}^{-l}\ovalbox{\tt\small REJECT}^{-)}-$

Faculty of Science and Technology, Ryukoku University,
Seta, Ohtsu, 520-21, Japan
Abstract
We give a comprehensive review of the renormalization group method for global
and asymptotic analysis, putting an emphasis on the relevance to the classical theoryof envelopes and the existence of invariant manifolds of the dynamics under consideration. We clarify that an essential point of the method is to convert the problem
from solving differential equations to obtaining suitable initial (or boundary) conditions. We mention that the notion of envelopes is also useful for constructing global
and asymptotic behavior of wave functions of quantum systems such as the ones
with the quartic potential or double-well otential.
$\dot{\mathrm{p}}$1Introduction

equations have a peculiar power to improve the global
The renormalization groupnature of functions obtained in the perturbation theory in quantum field theory [1]: Theshould not depend on
RG equations represent the fact that a physical quantity
the renormalization point ,$(\mathrm{R}\mathrm{G})$$O(p, \alpha, \mu)$$\mu$$\frac{\partial \mathcal{O}(p,\alpha,\mu)}{\partial\mu}.=0$

(1.1)

.

Such a floating renormalization point was first introduced by Gell-Mann and Low in the. We remark that the renormalization point plays a role of the initialcelebrated
point and the renormalization condition for physical quantities such as coupling constants
at the energy scale may be viewed as setting initial values of these quantities.[2]$\mu$$\mathrm{p}\mathrm{a}\mathrm{p}\mathrm{e}\mathrm{r}[1]$$\mu$Recently, the present author has indicated that the RG equation \‘a la Gell-Mann-Low. it was shown that the notion of envelopes iscan be identified as the envelope
useful for improving the perturbative expansions appearing in quantum field theory. This
identification was realized through an examination of the RG method of Chen, Goldenfeld; they applied the RG equation to
and Oono for global analysis of differential
numerous problems and found that the RG equation gives slow dynamics of the system
in question. The new point of their method is to utilize secular terms which usually$\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}[3];$$\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{S}[6]$119appear when perturbation theory is applied to differential equations; this is in contrast
to all previous methods, which are formulated on the principle to avoid the appearanceof secular terms.
Their method was reformulated on the basis of the classical theory of.
It was demonstrated that owing to the very envelope equation, the functions constructedfrom the solutions in the perturbation theory certainly satisfies the differential equation inquestion uniformly up to the order with which local solutions is constructed. It was also
shown [5] in a most general setting that the RG method gives a reduction of dynamics
and clarified that there is a correspondence between the RG method and the reductive
perturbation: Some interesting examples were also worked out in this method,such as the forced Duffing, the Lotka-Volterra and the Lorenz equations; the first example
showed that the method is applicable to not only automonous but also non-automonous
equations, the second one showed that the method gives phase equations and the last
one showed that center manifolds of the dynamics can be extracted in the method. It isnoteworthy that the notion of functional self-similarity (FSS) extracted as the essence of
the RG in [2] is only applicable to autonomous equations; see [8] for an application of thenotion of FSS for deducing phase equations. The above fact suggests that the notion ofenvelopes better represents the underlying mathematics of the powerfulness of the.
It was recognized also that when the unperturbed equation has neutrally stable solutions,
the RG method works well. It implies that the method is applicable when the dynamicalsystem under consideration has invariant manifolds and useful to extract the manifolds
and the dynamics on the manifolds. It was shown that global and asymptotic behavior
is obtained even for discrete systems by constructing “envelopes” when the system has
neutrally stable solution of the unperturbed.
$\mathrm{e}\mathrm{n}\mathrm{v}\mathrm{e}1_{0}\mathrm{P}^{\mathrm{e}\mathrm{s}}[3,4]$$\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{h}_{0}\mathrm{d}[7]$$\mathrm{R}\mathrm{G}$$\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}_{0}\mathrm{n}\mathrm{s}[9]$Subsequently the classical theory of envelopes was applied for getting asymptotic behaviors of wave functions in quantum $mechanics[10]$ . This is an optimized perturbation
theory in which the perturbation theory is combined with a variational method. The key
ingredient is to construct an envelope of a set of perturbative wave functions. This leads
to a condition similar to that obtained from theof minimal. Applications of the method to quantum anharmonic oscillator and the double well potential
show that uniformly valid wave functions with correct as ymptotic behavior are obtainedin the first-order optimized perturbation even for strong couplings.
$\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{C}\mathrm{i}_{\mathrm{P}}\dot{1}\mathrm{e}$$\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}_{\mathrm{V}\mathrm{i}}\mathrm{t}\mathrm{y}[11]$In this report, we will give only basic ingredients of the RG method based on ourformulation putting an emphasis to the relevance to envelopes; for detailed account of the. In this report, we
method and various applications, please refer toalso clarify that the RG method is most lucidly formulated by noting that the method
converts the problem from solving differential equations to obtaining suitable initial (or. In this report, we shall not discuss on partial
boundary) conditions as does the usual
differential equations due to the lack of space. Please refer to [12] as well as [6, 4, 8] forthis subject.
$\mathrm{R}\mathrm{e}\mathrm{f}.’$\mathrm{R}\mathrm{G}$\mathrm{s}[3,4,5,9,10]$

120

2A short review of the classical theory of envelopes
We here give a brief review of the theory of envelopes. Although the theory can be, we consider here envelope curves, for simplicity.
formulated in higher$\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{S}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{S}[4,5]$

is repLet
be a family of curves parametrized by in the x-y plane; here$F(x,y,\tau)=0$. We suppose thathas the envelope $E$ ,resented by the equation
obtain $G(x, y)$ fromwhich is represented by the equation $G(x, y)=0$ . The.
$F(x, y, \tau)$ .
$\{C_{\mathcal{T}}\}_{\tau}$$C_{\tau}$$\tau$$\{C_{\tau}\}_{\tau}$$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{i}_{\mathrm{S}}\mathrm{t}\mathrm{o}\vee\cdot.$

have the common tangent line at $(x, y)=(x_{0,y_{0}})$ , i.e.,
Now let $E$ and a curve,is the point of tangency. Then and are functions of$E$ , there$G(x_{0},y0)=0$
point
Conversely,
.
for
each
on
a
and of course
exists; thenas a function ofparameter . So we can reduce the problem to get$F(x,y,\tau(x,y))=c(x,y)^{1}.\mathcal{T}_{0}(x_{0},y\mathrm{o})$$G(x, y)$ is obtained as
can be obtained as follows.
$C_{\tau_{0}}$$(x_{0}, y\mathrm{o})$$x_{0}$$\tau_{0};X_{0}=\phi(\tau_{0}),$$y_{0}$$y_{0}=\psi(\tau_{0})$$(x_{0}, y_{0})$$(x_{0}, y_{0})$$\tau_{0}$$\tau_{0}$atSince the tangent line of$F(x, y, \tau)=0$ at the same point, one has$E$$(x_{0}, y\mathrm{o})$is perpendicular to the normal direction of

.

(2.1)

$F_{x}(x_{0}, y_{0}, \mathcal{T}_{0})\phi’(\tau_{0)}+F_{y}(X_{0,y_{0},\tau}0)\psi’(\tau_{0})=0$

On the other hand, differentiating$F(x(\tau_{0}), y(\tau 0),$ $\tau 0)=0$

with respect to$F_{x}(x_{0}, y_{0}, \mathcal{T}_{0})\phi/(\tau_{0)}+F_{y}(X_{0,y0}, \mathcal{T}_{0})\psi’(\tau_{0})+F_{\tau_{0}}(x_{0}, y_{0}, \tau_{0})=0$$\tau_{0}$, one also has

.

(2.2)

Combining the last two equations, we have
$F_{\tau_{0}}(x0, y0, \mathcal{T}_{0})\equiv\frac{\partial F(x_{0,y_{0},)}\tau_{0}}{\partial\tau_{0}}=0$

.

(2.3)

This is the basic equation of the theory of envelopes; we call this type of equation envelope
equation where RG and stand for renormalization group andequations and also.
envelope, respectively, becasue the RG equation Eq(l.l) has the same form asand ,One can thus eliminate the parameter to get a relation between$\mathrm{E}$$\mathrm{R}\mathrm{G}/\mathrm{E}$$\mathrm{E}\mathrm{q}.(2.3)$$x_{0}$$\tau_{0}$$y0$$G(x, y)=F(x, y, \tau_{0}(x, y))=0$ ,with the replacement$(x_{0}, y\mathrm{o})arrow(x, y)$(2.4). $G(x, y)$ is called the discriminant of $F(x, y, t)$ .

Comments are in order here: (i) When the family of curves is given by the function; the envelope is given
is reduced tothe condition. (ii) The equation $G(x, y)=0$ may give not only the envelopeby. This is because the condition that
but also a set of singularities of the curves
is also compatible with Eq. (2.3).

$y=f(x, \tau)$ ,$\partial f/\partial\tau_{0}=0$$\mathrm{E}\mathrm{q}.(2.3)$$E$$y=f(x, \tau_{\mathrm{o}(}x))$$\{C_{\tau}\}_{\tau}$$\partial F/\partial x=\partial F/\partial y=0$

1Since there is a relation$G(x_{0}, y\mathrm{o})=0$

between$x_{0}$

and$y_{0},$$\tau_{0}$is actually a function of$x_{0}$

or$y_{0}$.

121

3The RG method; a simplest example
In this section, using a simplest example we show how the RG method works for obtaining
global and asymptotic behavior of solutions of differential equations. We shall present themethod so that the reader will readily see that the notion of envelopes is intrinsicallyrelated to the method. We shall emphasize that an essential point of the method istuning the initial condition at an arbitrary time perturbatively along with solving the
perturbative equations successively. One will see that the reasoning for various steps inthe procedure and the underlying picture are quite different from the original ones given
in [6]. We believe, however, that the present formulation emphasizing the role of initialconditions and the relevance to envelopes of perturbative local solutions straightens theoriginal argument, and is the most comprehensive one.2$t_{0}$Let us take the following simplest example to show our method:$\frac{d^{2}x}{dt^{2}}+\epsilon\frac{dx}{dt}+x=0$

where$\epsilon$

is supposed to be small. The solution to,$\mathrm{E}\mathrm{q}.(3.1)$

(3.1)

reads

$x(t)= \overline{A}\exp(-\frac{\epsilon}{2}t)\sin(\sqrt{1-\frac{\epsilon^{2}}{4}}t+\overline{\theta})$where$\overline{A}$and$\overline{\theta}$

,

(3.2)

are constants.

Now, let us obtain the solution around t.he initial time $t=t_{0}$ in a perturbative way,
expanding as$x$$x(t,$ $t_{0)}=x_{0}(t, t_{0})+\epsilon x_{1}(t,$ $t_{0)}+\epsilon^{2}x_{2}(t, t_{0})+$

...,

(3.3)

where $x_{n}(t, t_{0})(n=0,1,2\ldots)$ satisfy$\ddot{x}_{0}+x_{0}=0$,$\ddot{x}_{n+1}+x_{n+1}=-\dot{X}_{n}$

.

(3.4)

The initial condition may be specified by$x(t_{0,0}t)=W(t_{0})$

.

(3.5)

We suppose that the initial value $W(t_{0})$ is always on an exact solution of. We also expand the initial value $W(t_{0})$ ;$\mathrm{E}\mathrm{q}.(3.1)$

for any$t_{0}$$W(t_{0})=W0(t\mathrm{o})+\epsilon W_{1}(t_{0})+\epsilon^{2}W_{2}(t_{0})+$

...,

(3.6)

and the terms
will be determined so that the perturbative solutions around different
initial times have an envelope. Hence the initial value $W(t)$ thus constructed will givethe (approximate but) global solution of the equation.
$W_{i}(t_{0})$$t_{0}$

2The following is even a refinment of the argument given in $[3, 5]$ where the fact that the RG method.
is a theory manipulating initial conditions were not fully recognized.

122
Let us perform the above program. The lowest solution may be given by$x_{0}(t, t_{0})=A(t_{0})\sin(t+\theta(t_{0}))$

(3.7)

,

where we have made it explicit that the constants and may depend on the initial time
. The initial value $W(t_{0})$ as a function of is specified as$A$$\theta$$t_{0}$$t_{0}$$W_{0}(t_{0})=x_{0}(t0, t_{0)}=A(t_{0})\sin(t_{0}+\theta(t_{0}))$

.

(3.8)

the
is a neutrally stable solution; with the perturbation
We remark that
constants and may move slowly. We shall see that the envelope equation gives theequations describing the slow motion of and .
$\epsilon\neq 0$$\mathrm{E}\mathrm{q}.(3.7)$$A$$\theta$$A$

The first order equation now reads
solution in the following form,$\theta$$\ddot{x}_{1}+x_{1}$$=-A\cos(t+\theta)$ ,$x_{1}(tt_{0}) \}=-\frac{A}{2}\cdot(t-t_{0})\sin(t+\theta)$

and we choose the

,

(3.9)

which means that the first order initial value $W_{1}(t_{0})=0$ so that the lowest order value
approximates the exact value as closely as possible. Similarly, the second ordersolution may be given by$W_{0}(t_{0})$$x_{2}(t)= \frac{A}{8}\{(t-t0)2\sin(t+\theta)-(t-t_{0})\cos(t+\theta)\}$,(3.10)

thus $W_{2}(t_{0})=0$ again for the present linear equation.
It should be noted that the secular terms have appeared in the higher order terms,
which are absent in the exact solution and invalidates the perturbation theory for far
from . However, with the very existence of the secular terms, we could make$(i=1,2)$ vanish and $W(t_{0})=W_{0}(t_{0})$ up to the third order.
$t$$W_{i}(t_{0})$$t_{0}$

Collecting the terms, we have
$x$

( to)
$t,$

$=$

A $\sin(t+\theta)-\epsilon\frac{A}{2}(t-t_{0})\sin(t+\theta)$$+\epsilon^{2_{\frac{A}{8}\{(tt_{0)(t}}}-2\sin+\theta)-(\mathrm{t}-t_{0})\cos(t+\theta)\}$,(3.11)and more importantly(3.12)$W(t_{0})=W0(\mathrm{t}_{0})=A(t_{0})\sin(t_{0}+\theta(t_{0}))$. We remark that $W(t_{0})$ describing the solution is parametrized by possibly
up toin a definite way.
andslowly moving variable$O(\epsilon^{3})$$A(t_{0})$$\phi(t_{0})\equiv t_{0}+\theta(t_{0})$given by functions
Now we have a family of curves
with . They are } on the exact curve $W(t)$ at $t=t_{0}$ up to$\{C_{t_{0}}\}\iota_{0}$$t_{0}$$1$$\mathrm{a}$parametrized, but only valid locally$\{x(t, t_{0})\}_{t0}$$O(\epsilon^{3})$123which contacts withfor near . So it is conceivable that the envelope $E$ ofeach local solution at $t=t_{0}$ will give a global solution. Thus the envelope function
coincides with $W(t)$ ;$t$$\{C_{t_{0}}\}_{t}0$$t_{0}$$x_{E}(t)$$x_{E}(t)=x(t, t)=W(t)$ .

(3.13)

andas functions of so that the family of
Our task is actually to determine
envelope.
tothe classical theory of envelopes given
Accordingthe local solutions has an
by posing that the envelopein the previous’ the above program can be achieved
equation$A(t_{0})$$\theta(t_{0})$$t_{0}$$\mathrm{i}.\mathrm{m}$$\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}_{0}\mathrm{n}.$$\frac{dx(t,t_{0})}{dt_{0}}=0$(3.14),

gives the solution $t_{0}=t$ . From Eq.’s (3.11) and (3.14), we have$\frac{dA}{dt_{0}}+\epsilon A=0$,

where we have used the fact that $dA/dt$ is
the equations, we have$A(t_{0})=\overline{A}\mathrm{e}^{-\epsilon t\mathrm{o}/2}$

where$\overline{A}$and$\overline{\theta}$,$\frac{d\theta}{dt_{0}}+\frac{\epsilon^{2}}{8}=0$$O(\epsilon)$,(3.15)and neglected the terms of$O(\epsilon^{3})$. Solving$\tau$,(3.16)$\theta(t_{0})=-\frac{\epsilon^{2}}{8}t_{0}+\overline{\theta}$

are constant numbers. Thus we get$x_{E}(t)=x(t, t)=W_{0}(t)= \overline{A}\exp(-\frac{\epsilon}{2}i)\sin((1-\frac{\epsilon^{2}}{8})t+\overline{\theta})$,(3.17). Noting that $\sqrt{1-\epsilon^{2}/4}=1-\epsilon^{2}/8+O(\epsilon^{4})$ , one finds that the resultant
up to
; see
envelope function $x_{E}(t)=W_{0}(t)$ is an approximate but global solution to
Eq. (3.2).
$o(\epsilon^{3})$$\mathrm{E}\mathrm{q}.(3.1)$

4

Nonlinear equations

In this section, we treat a couple of examples of systems of ODE’s with nonlinearity toand the. The examples are theshow how the RG method
equation. We shall derive the time dependence of the solution to the LotkaVolterra equation explicitly; a phase equation will be derived by our method. The Lorenz
equation is an example with three degrees of freedom, which shows a bifurcation. We
shall give the center manifolds of this equation around the first bifurcation point.
$\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{S}[5]$$\mathrm{L}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{Z}[14]$$\mathrm{L}\mathrm{o}\mathrm{t}\mathrm{k}\mathrm{a}- \mathrm{V}\mathrm{o}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{a}[13]$124
Lotka-Volterra equation

4.1

The Lotka-Volterra equation$\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{d}\mathrm{s}[13]$;$\dot{x}=ax-\epsilon xy$,$\dot{y}=-by+\epsilon’xy$

,

(4.1)

and are assumed to be positive. It is. well known that thewhere the constants
the conserved quantity, i.e.,..

.as$a,$ $b,$$\epsilon’$$\epsilon$$\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}_{0}.\mathrm{n}\mathrm{h}$$b\ln|x|+a\ln|y|-(\epsilon’x+\epsilon y)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.$

.

(4.2)

The fixed points are given by $(x=0, y=0)$ and $(x=b/\epsilon’, y=a/\epsilon)$ . Shifting andscaling the variables by$x=(b+\epsilon\xi)/\epsilon’$,$y=a/\epsilon+\eta$

,

(4.3)

we get the reduced equation given by the system$( \frac{d}{dt}-L_{0)\mathrm{u}=-\epsilon\xi}\eta$

,

(4.4)

where
$\mathrm{u}=$,$L_{0}=$ .

(4.5)

The eigen value equation(4.6)$L_{0^{\mathrm{U}=\lambda}0^{\mathrm{U}}}$

has the solution$\lambda_{0}=\pm i\sqrt{ab}\equiv\pm i\omega$,$\mathrm{U}=$ .

(4.7)

Let us try to extract the global behavior of the solution around the fixed point. Our
strategy is the following: We suppose that we are on the exact solution at $t=t_{0}$ where. We also suppose that we can applyis arbitrary; we denote the initial value by
perturbation theory for the solution at least in the small neighbo.rhood of $t=t_{0}$ . We
expand the variable in a Taylor series of ;$\mathrm{W}(t_{0})$$t_{0}$$\epsilon$$\mathrm{u}=\mathrm{u}0+\epsilon \mathrm{u}_{1}+\epsilon^{2}\mathrm{u}_{2}+\cdots$

with$\mathrm{u}_{i}={}^{t}(\xi_{i}, \eta_{i})$

(4.8)

,

. An essential point of our method is to expand the initial value, too;$\mathrm{W}(t_{0})=\mathrm{W}_{0}(t_{0)}+\epsilon \mathrm{W}_{1}.(t_{0)}+\epsilon^{2}\mathrm{W}_{2}(t_{0)}+\cdots$

.

(4.9)

Our central task is to extract the initial value as a function of so that the resulting
local solutions starting from the different initial points at $t=t_{0}$ and , say, $t=t_{0}+\triangle t$$t_{0}$

125
are continued smoothly. This condition is found to be nothing but the one that the local
solutions have an envelope. In actual calculations, it is also important to use the factthat the functional form of the initial values can be reduced from the general solutionof the differential equations in a perturbative way; in this procedure, only independentfunctions modulo to secular terms are retained.
The lowest term satisfies the equation,(4.10)$( \frac{d}{d\mathrm{t}}-L_{0)=}\mathrm{u}_{0}\mathrm{o}$which yields the solution$\mathrm{u}_{0}(t;t0)=A(\mathrm{t}_{0)\mathrm{e}}i\omega t\mathrm{U}+\mathrm{c}.\mathrm{c}.$.

(4.11)

Notice that $A$ is a complex number, so one may parametrize it as$A(t_{0})=A(t_{0})/2i\cdot\exp(i\theta(\mathrm{t}_{0))}$.

(4.12)The solution implies that the initial condition is given by.

(4.13)$\mathrm{u}_{0}(t_{0;t}0)=A(t_{0)}\mathrm{e}^{i\omega t0}\mathrm{U}+\mathrm{c}.\mathrm{c}.$It means that in the lowest approximation the solution is parametrized by a complex
functionor a pair of real functions,and. With a small
perturbation, we expect that and will move slowly.
$A(t_{0})$$A(t_{0})$$A$

Noting that$=\alpha \mathrm{U}+\mathrm{c}.\mathrm{c}.$$\phi(t_{0})\equiv\omega t_{0}+\theta(t_{0})$$\theta$, with $\alpha=(1-ib/\omega)/2$ , one finds that the first orderterm satisfies the equation,$( \frac{d}{dt}-L_{0})\mathrm{u}_{1}=\frac{\omega}{b}[iA22i\omega t\mathrm{e}(\alpha \mathrm{U}+\mathrm{c}.\mathrm{c}.)+\mathrm{c}.\mathrm{c}.]$(4.14)

the solution to which may be given by$\mathrm{u}_{1}=\frac{1}{b}[A^{2}(\alpha \mathrm{U}+\frac{\alpha^{*}}{3}\mathrm{U}^{*})\mathrm{e}+\mathrm{C}.\mathrm{c}.]2i\omega t$

Thus the initial value$\mathrm{W}_{1}(t_{0)}$in this order is given$\mathrm{u}_{1}(t_{0},$$\mathrm{t}_{0)}$.

(4.15).

Similarly, the second order solution may be given by$\mathrm{u}_{2}$$=$$[ \frac{b-i\omega}{3b^{2}}|A|^{2}A\{\alpha(t-\mathrm{t}_{0}+i\frac{\alpha^{*}}{2\omega})\mathrm{U}+\frac{\alpha^{*}}{2i\omega}\mathrm{U}^{*}\}\mathrm{e}^{i}\omega t$.

$+ \frac{b+i\omega}{4b^{2}i\omega}A^{3}(2\alpha \mathrm{U}+\alpha^{*}\mathrm{U}^{*})\mathrm{e}^{3i\omega t}]+\mathrm{c}.\mathrm{c}.$(4.16)

Here we have a secular term proportional to the unperturbed solution. Since we wantto make the lowest initial value as close as the exact one, we demand that as many as

126possible terms in the higher order vanish at $t=t_{0}$ . Thus adding unperturbative solutions,
we make the secular term (of the upper component) vanishes at $t–t_{0^{3}}$. Thus neglecting. We impose, andhigher order terms, we have
give the same value at . By taking a limit
andthat the solutions, we have the envelope equation;$\mathrm{W}(t_{0})=\mathrm{u}(t_{0}, t_{0})$$\mathrm{u}(t, t_{0})=\mathrm{u}_{0}+\epsilon \mathrm{u}_{1}+\epsilon^{2}\mathrm{u}_{2}$$t$$\mathrm{u}(t, t_{0}+\Delta t)$$\mathrm{u}(t, t_{0})$$\Delta tarrow \mathrm{O}$.

$\frac{d\mathrm{u}}{dt_{0}}=0$

with$t_{0}=\mathrm{t}$(4.17),. This gives the equation for $A(t)$ as$\frac{dA}{dt}=-i\epsilon^{2_{\frac{\omega^{2}+b^{2}}{6\omega b^{2}}}}|A|2A$

In terms of $A(t)$ and$\theta(t)$(4.18).

, we have$A(t)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{S}\mathrm{t}.$,$\theta(t)=-\frac{\epsilon^{2}A^{2}}{24}(1+\frac{b^{2}}{\omega^{2}})\omega t+\overline{\theta}_{0}$(4.19),

being a constant. Owing to the prefactor in r.h.s. of Eq. (4.18), the absolutewithvalue of the amplitude becomes independent of , while the phase has a t-dependence.
The envelope function is given by$i$$\overline{\theta}_{0}$$A$$\theta$$t$$\mathrm{u}_{E}(\mathrm{t})==\mathrm{u}(t, \mathrm{t})=\mathrm{W}(t)$.

(4.20)In terms of the components, one has$\xi_{E}$$=$A$\sin\ominus(t)-\epsilon\frac{A^{2}}{6\omega}(\sin 2\ominus(t)+\frac{2\omega}{b}\cos 2\ominus(t))$,$- \frac{\epsilon^{2}A^{3}}{32}\frac{3\omega^{2}-b^{2}}{\omega^{2}b^{2}}(\sin 3\ominus(t)-\frac{4\omega b}{3\omega^{2}-b^{2}}.\cos 3(t))$$\eta_{E}$$=$(4.21)$- \frac{\omega}{b}[(A-\frac{\epsilon^{2}A^{3}}{24}\frac{b^{2}-\omega^{2}}{b^{2}\omega^{2}}\mathrm{I}\cos(t)-\frac{\epsilon^{2}A^{3}}{12b\omega}\sin\ominus(t)$$+ \epsilon\frac{A^{2}}{2b}(\sin 2\ominus(t)-\frac{2b}{3\omega}\cos 2(t))-\frac{\epsilon^{2}A^{3}}{8b\omega}(\sin 3\Theta(t)-\frac{3b^{2}-\omega^{2}}{4b^{2}\omega^{2}}\cos 3\ominus(t))]$

,

where
$\ominus(t)\equiv\tilde{\omega}t+\overline{\theta}_{0}$,$\tilde{\omega}\equiv\{1-\frac{\epsilon^{2}A^{2}}{24}(1+\frac{b^{2}}{\omega^{2}})\}\omega$.

(4.22)One sees that the angular frequency is shifted.
We can identify
by construction. We see thatas an approximate solution to
but uniformly valid solution to theapproximate
is an$\mathrm{u}_{E}(t)=(\xi_{E}(t), \eta_{E}(t))=\mathrm{w}(t)$$\mathrm{u}_{E}(t)$$\mathrm{E}\mathrm{q}.(4.4)$

3Although we can make other terms also vanish at $t=t_{0}[15]$ , the resulting dynamics of andbecome more complicated than the present choice. In the theory of reduction of dynamics, one usually.
prefers simpler$A$$\mathrm{d}\mathrm{y}\mathrm{n}\mathrm{a}\mathrm{m}\mathrm{i}_{\mathrm{C}\mathrm{s}}[16]$$\theta$

127. We remark that the resultant trajectory is closed in conformity
equation up towith the conservation law Eq. (4.2).
$O(\epsilon^{3})$“Explicit solutions” of two-pieces of Lotka-Volterra equation were considered by Frame[17]; however, his main concern was on extracting the period of the solutions in an averagemethod. we are not aware of any other work than ours which gives an explicit form of
the solution as given here.

The Lorenz model

4.2

The Lorenz$\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}[14]$for the thermal convection is given by$=$$\dot{\xi}$$\dot{\eta}=$$\sigma(-\xi+\eta)$$r\xi-\eta-\xi\zeta$$=\xi\eta-b\zeta$$\dot{\zeta}$,,.

(4.23)The steady states are give by(A)$(\xi, \eta, \zeta)=(0,0, \mathrm{o})$,(B)$(\xi, \eta, \zeta)=(\pm\sqrt{b(r-1)}, \pm\sqrt{b(r-1)}, r-1)$.

(4.24)

shows that the origin is stable for $0<r<1$ but
The linear stability
unstable for $r>1$ , while the latter steady states (B) are stable for $1<r<\sigma(\sigma+b+$$3)/(\sigma-b-1)\equiv r_{c}$ but unstable for $r>r_{c}$ . In this report, we examine the non-linear
stability around the origin, for $r\sim 1$ and extract a center manifold for the dynamics inthis region.
$\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{y}\mathrm{s}\mathrm{i}\mathrm{S}[18]$We put$r=1+\mu$and$\mu=\chi\epsilon^{2}$,$\chi=\mathrm{s}\mathrm{g}\mathrm{n}\mu$.

(4.25)We expand the quantities as Taylor series of :$\epsilon$$\mathrm{u}\equiv=\epsilon \mathrm{u}_{1}+\epsilon^{2}\mathrm{u}_{2}+\epsilon^{3}\mathrm{u}_{3}+\cdots$

where$\mathrm{u}_{i}={}^{t}(\xi_{i,\eta i}, (_{i})(i=1,2,3, \ldots)$,(4.26). We also expand the initial value at $t=t_{0;}$$\mathrm{u}(t_{0}, t_{0)}=\mathrm{W}(t_{0})=\epsilon \mathrm{W}_{1}+\epsilon^{2}\mathrm{W}_{2}+\epsilon^{3}\mathrm{W}_{3}+\cdots.$(4.27)

The first order equation reads$( \frac{d}{dt}-L_{0})\mathrm{u}1=0$,(4.28)

128
where$L_{0}=$ ,(4.29)

the eigenvalues of which are found to be$\lambda_{1}=0$,$\lambda_{2}=-\sigma-1$,$\mathrm{U}_{2}=$,$\lambda_{3}=-b$.

(4.30)The respective eigenvectors are$\mathrm{U}_{1}=$We are interested in the asymptotic state as
may take only the neutrally stable solution,$\mathrm{U}_{3}=$$tarrow\infty$$\mathrm{u}_{1}(t;t_{0})=A(t_{0})\mathrm{U}_{1}$.

(4.31). In this asymptotic region, one,(4.32)

because the other terms proportional towill decay out at a sufficiently large time.
Here we have made it explicit that the solution may depend on the initial time .
implies that we have taken the initial condition that$\mathrm{U}_{2,3}$$t_{0}$$\mathrm{E}\mathrm{q}.(4.32)$$\mathrm{W}(t_{0})\simeq \mathrm{W}_{1}(t_{0})=A(t_{0})\mathrm{U}_{1}$.

(4.33)In terms of the components,$\xi_{1}(t)=A(t_{0})$,$\eta_{1}(t)=A(t_{0})$,$\zeta_{1}(t)=0$.

.(4.34)In another word, the motion ofis confined or reduced to the one-dimensional manifold ${}^{t}(A, A, 0)$ , although is a constant in this approximation. One expects that the small
perturbation withwill give rise to a s..lo,w motion of as well as a modification of
the slow manifold.
$\mathrm{u}_{1}(t)$

$A$

$A$

$\epsilon\neq 0$

The second order equation reads$( \frac{d}{dt}-L_{0})\mathrm{u}_{2}==A^{2}\mathrm{U}_{3}$,(4.35)

which may yield$\mathrm{u}_{2}(t)=\frac{A^{2}}{b}$U3,(4.36)or in terms of the components$\xi_{2}=\eta_{2}=0$,$\zeta_{2}=\frac{A^{2}}{b}$.

(4.37)

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Here we have retained only functions independent of the ones appearing in the lowest
approximation. Then the third order equation is given by,$( \frac{d}{dt}-L_{0})\mathrm{u}3==\frac{1}{1+\sigma}(\chi A-\frac{1}{b}A^{3})(\sigma \mathrm{U}_{1}-\mathrm{U}_{2})$(4.38)

which may yield.

(4.39)$\mathrm{u}_{3}=\frac{1}{1+\sigma}(\chi A-\frac{1}{b}A^{3})\{\sigma(t-t0+\frac{1}{1+\sigma})\mathrm{U}_{1}-\frac{1}{1+\sigma}\mathrm{U}_{2}\}$

Here we have again retained functions which have not appeared before except for the termwith which the secular terms vanishes at $t=t_{0}$ . Of course, one may have other choices
for the independent functions, but it is found that the present “minimal” choice gives thesimplest dynamics for the amplitude .
$A$

Thus collecting all the terms, one has$\mathrm{u}(t;t_{0})$$\epsilon A(t_{0})\mathrm{U}_{1}+\frac{\epsilon^{2}}{b}A(t_{0})2\mathrm{U}_{3}$$=$$+ \frac{\epsilon^{3}}{1+\sigma}(\chi A(t_{0})-\frac{1}{b}A(t_{0)^{3}})\{\sigma(t-t0+\frac{1}{1+\sigma})\mathrm{U}_{1}-\frac{1}{1+\sigma}\mathrm{U}_{2}\},$

up to$O(\epsilon^{4})$$(4.40)$. Accordingly, the initial value reads$\mathrm{W}(t_{0})=\mathrm{u}(t_{0}, t_{0)} = \epsilon A(t_{0})\mathrm{U}_{1}+\frac{\epsilon^{2}}{b}A(t_{0})2\mathrm{U}_{3}$.

$+ \frac{\epsilon^{3}}{(1+\sigma)^{2}}.(xA(t\mathrm{o})-\frac{1}{b}A(t_{0})^{3})\{\sigma \mathrm{U}1-\mathrm{U}_{2}\}$(4.41)$-$

Demanding that the solutions at different initial times are continued smoothly, we
equation, which reads
have the$\mathrm{R}\mathrm{G}/\mathrm{E}$

$0$

$=$

$=$

$\frac{d\mathrm{u}}{dt_{0}}|_{\iota=t}0$’$\epsilon\frac{dA}{dt}\mathrm{U}_{1}+2\frac{\epsilon^{2}}{b}A\frac{dA}{dt}\mathrm{U}_{3}-\frac{\sigma}{1+\sigma}\epsilon 3(\chi A-\frac{1}{b}A^{3})\mathrm{U}1$,(4.42). Noting that one may self-consistently assume that $dA/dt=O(\epsilon^{2})$ , we haveup tothe amplitude equation$O(\epsilon^{4})$$\frac{dA}{d\mathrm{t}}=\epsilon^{2}\frac{\sigma}{1+\sigma}(xA(t)-\frac{1}{b}A(t)^{3})$.

(4.43)

With this $A(t)$ , the envelope function is given by the initial value$\mathrm{u}_{E}(t)$,$=$$\mathrm{u}(t;t_{0}=t)$$=$$\mathrm{W}(t)$$=$$\epsilon A(t)\mathrm{U}_{1}+\frac{\epsilon^{2}}{b}A(t)2\mathrm{U}3+\frac{\epsilon^{3}}{(1+\sigma)^{2}}(xA(t)-\frac{1}{b}A(t)3)(\sigma \mathrm{U}1-\mathrm{U}_{2})$,, (4.44)

130
or
$-$

$\xi_{E}(t)=\epsilon A(t)$,$\eta_{E}(t)=\epsilon A(t)+\frac{\epsilon^{3}}{1+\sigma}(xA(\mathrm{t})-\frac{1}{b}A(t)^{3})$,$\zeta_{E}(t)=\frac{\epsilon^{2}}{b}A(t)2$.

(4.45)We see that the initial value is obtained as the envelope of the local solutions and becomesa global solution to the Lorenz model.
A remark is in order here;
identified with a center

shows that the slow manifold which may be
is given by$\mathrm{E}\mathrm{q}.(4.45)$$\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{d}[18]$$\eta=(1+\epsilon^{2}\frac{\chi}{1+\sigma})\xi-\frac{1}{b(1+\sigma)}\xi^{3}$,$\zeta=\frac{1}{b}\xi^{2}$.

(4.46)

Notice that the invariant manifold is modified with the perturbation and also the slow
on the manifold is obtained by our method. One thus sees thatdynamics
the RG method is a powerful tool to extract center manifolds in a concrete form. It
is worth mentioning that since the RG method utilizes neutrally stable solutions as theunperturbed ones, it is rather natural that the RG method can extract center manifolds.
when$\mathrm{E}\mathrm{q}.(4.43)$$\mathrm{e}\mathrm{x}\mathrm{i}_{\mathrm{S}}\mathrm{t}[7]$The basis of the RG method for systems

5

In this section, we give an account of our method in a general setting for ordinary equations
as a summary.

Let
and X satisfy the equation$\mathrm{X}={}^{t}(X_{1}, X_{2}, \cdots, X_{n})$and$\mathrm{F}(\mathrm{X}, t;\epsilon)={}^{t}(F_{1}(\mathrm{X}, t;\epsilon),$$\frac{d\mathrm{X}}{dt}=\mathrm{F}(\mathrm{X}, t;\epsilon)$$F_{2}(\mathrm{X}, t;\epsilon),$$\cdots,$$Fn(\mathrm{X}, t;\epsilon))$

,

(5.1)

,

, whereis arbitrary. We remark that the initialwith the initial condition. We also write the solution X as
may be dependent on , i.e.,value( to) so that the initial-time dependence is explicit.
$\mathrm{X}(t_{0})=\mathrm{W}$$\mathrm{W}$$\mathrm{X}$$t_{0}$$t_{0}$$\mathrm{W}=\mathrm{W}(t_{0})$$t;$Let us try to have the perturbation solution of$\mathrm{E}\mathrm{q}.(5.1)$

around $t=t_{0}$ by expanding$\mathrm{X}(t;t_{0})=\mathrm{X}_{0}(t;t_{0})+\epsilon \mathrm{X}_{1}(t;t_{0})+\epsilon^{2}\mathrm{X}_{2}(t;t_{0})\cdots$

(5.2)

.

We also expand the initial value as$\mathrm{X}(t_{00}, t)=\mathrm{W}(t0)=\mathrm{w}\mathrm{o}(t_{0})+\epsilon \mathrm{w}1(t_{0}))+\epsilon^{2}\mathrm{W}2(t_{0})+\cdots$.

. We suppose that an approximate solution
In fact,
is obtained. It implies that the initial value
the equation up to. We also havecoincides with an exact solution up to$\mathrm{X}_{i}(t_{0}, to)=\mathrm{W}_{i}(t_{0)}$$O(\epsilon^{p})$

(5.3)
to
at $t=t_{0}$$\tilde{\mathrm{X}}=\tilde{\mathrm{X}}(t;t_{0})$$\mathrm{W}(t_{0})$$O(\epsilon^{p})$$, \frac{d\tilde{\mathrm{X}}(\mathrm{t}\cdot t0\tilde{\mathrm{W}}(t_{0)})}{dt},=\mathrm{F}(\tilde{\mathrm{X}}(\mathrm{t};t_{0},\tilde{\mathrm{w}}(\mathrm{t}0)),$$t;\epsilon)+O(\epsilon^{p})$

.

(5.4)

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One may say that now we have a family of the orbits given by the functions
with parameterizing the orbits. We see that the envelope of the family of the orbits
which contacts with each curve at $t=t_{0}$ will give an approximate but global solution of
the equation. Thus the envelope function is nothing but the initial value- as a function of
the initial time$\tilde{\mathrm{X}}(t;t_{0},\tilde{\mathrm{W}}(t_{0)})$$\mathrm{E}$$t_{0}$$\mathrm{X}_{E}(t)=\tilde{\mathrm{X}}(t;t,\dot{\mathrm{W}}(t))--\tilde{\mathrm{W}}(t)$The construction of$\mathrm{E}$is performed as follows: We

.

(5.5)
-

$\mathrm{i}\mathrm{m}\mathrm{p}_{\mathrm{o}\mathrm{S}}\mathrm{e}$that the$\mathrm{R}\mathrm{G}$$/\mathrm{E}$equation(5.6)$\frac{d\tilde{\mathrm{X}}}{dt_{0}}=0$

gives the solution $t_{0}=t$ , from which the dynamics of the initial value
is obtained.
may give equations as many as which are independent of each other. In theapplications to describe asymptotic behavior of solutions, the equation usually reduced
to a low-dimensional equation.
$\mathrm{W}(t)$$n$$\mathrm{E}\mathrm{q}.(5.6)$

i.s. .

In accord with the above relation, one can easily show that‘, one has
. In fact,
the original equation uniformly up to$O(\epsilon^{p})$$\frac{d\mathrm{X}_{E}}{dt}|_{t=t_{0}}$

where

6

$\mathrm{E}\mathrm{q}.(5.6)$$\mathrm{x}_{E}(t)=\tilde{\mathrm{W}}(t)$$=$$. \frac{d\tilde{\mathrm{X}}(t,t_{0},\mathrm{w}(t_{0}))}{dt}|_{t=t_{0}}+’\frac{d\tilde{\mathrm{X}}(t\cdot t_{0},\mathrm{w}(t_{0}))}{dt_{0}}|_{t}=t_{0}$$=$$\frac{d\tilde{\mathrm{X}}(t\cdot t0\mathrm{W}(t_{0}))}{d\mathrm{t}},|_{t=t_{0}}$$=$$\mathrm{F}(\mathrm{X}_{E}(\mathrm{t}_{0}), t_{0};\epsilon)+O(\epsilon)p$’satisfies$\forall t_{0}$

’

,
,

(5.7)

has been used in the last equality.

Concluding remarks

We have described the perturbative RG method for global and asymptotic analysis. We
have emphasized its relevance to the classical theory of envelopes and that the method
concerns with the initial or boundary values of differential equations. It should be remarked that the notion of envelopes is also useful for improvement of perturbation series
and for obtaining asymptotic behavior of wave functions inin quantum filed.
quantum mechanics [10].
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132
As a review article, J. Zinn-Justin, Qu an tum Fie.ld Theory and Critical Phenomena(Clarendon Press, Oxford, 1989).
[2] D.V. Shirkov, Russian Math. Survey 49: 5 (1994) 155;
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$\mathrm{h}\mathrm{e}\mathrm{p}-\mathrm{t}\mathrm{h}/9602024$; Int. J. Mod.

[3] T.Kunihiro, Prog.Theor.Phys. 94, 503(1995); 95, $835(1996)(\mathrm{E})$ .
[4] T.Kunihiro, Jpn. J. Ind. Appl. Math. 14, 51 (1997).

[5] T.Kunihiro, Prog. Theor. Phys. 97, 179 (1997).

[6] N. Goldenfeld, O. Martin and Y. Oono, J. Sci. Comp. 4(1989),4; N. Goldenfeld,. O. Martin, Y. Oono and F. Liu, Phys. Rev. Lett.64(1990) 1361;
N. D. Goldenfeld, Lectures on Phase Transitions and the Renormaliza tion Group (Addison-Wesley,
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[8] S. Sasa, preprint$\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{t}-\mathrm{s}\mathrm{o}\mathrm{l}/9608008$, to be published in Physica D.

[9] T. Kunihiro and J. Matsukidaira, Ryukoku preprint,$\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{t}^{-\mathrm{s}}\mathrm{o}1/9705003$.

[10] T. Hatsuda, T. Kunihiro and T. Tanaka, Phys. Rev. Lett.78 (1997) 3229.
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[12] K. Matsuba and K. Nozaki,
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$\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{t}^{-}\mathrm{s}\mathrm{o}1/9702001,$$\mathrm{P}^{\mathrm{a}\mathrm{t}\mathrm{t}^{-}\mathrm{s}\mathrm{o}}1/9703004$[13] A. J. Lotka, J. Amer. Chem. Soc. $42(1920)1595,$
de lalutte pour lavie (Gauthier-Villars, Paris, 1931).

$\mathrm{v}.‘\dot{\mathrm{v}}\mathrm{o}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{r}\dot{\dagger}\cdot\vee\dot{\backslash }\backslash \mathrm{a},$;a tique$T’\dot{h}eo\mathrm{r}ie\overline{m}\sim athe\mathrm{m}$[14] E. N., Lorenz, J. Atmos. Sci. 20 (1963)130.
[15] H.J. de Vega- and
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J..F.J Salgado,$\mathrm{h}\mathrm{e}\mathrm{p}-_{\mathrm{t}}\mathrm{h}/9701104,$

to. be pu.,b liSh.ed in Phys.. Rev.

[16] Y. Kuramoto, Prog. Theor. Phys. Suppl. 99 (1989) 244; Bussei Kenkyu 49(1987)299.

[17] J. S. Frame, J. theor. Biol. 43 (1974)73.
[18] See for example, J. Guckenheimer and P. Holmes, Nonlin ear Oscillators, Dynamical
Systems, and Bifurca tions..Of Vector Fields (Springer-Verlag, 1983).