Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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The Trick

We now show that, given a largely arbitrary

dynamical system, it is possible to introduce a

deformed version of it featuring a real constant !,

that has the following properties: for ! = 0, it

coincides with the original, undeformed system; for

!>0, it possesses an open region having full

dimensionality in its phase space such that all

solutions evolving from an initial datum in it are

completely periodic with a period

T which is a finite

integer multiple, or perhaps a simple fraction, of the

basic period

T ¼

2

½1

Let us indeed, consider a quite general dynamical

system which we write as follows:



¼ F ; ðÞ ½2

Here   () is the dependent variable, which

might be a scalar, a vector, a tensor, a matrix, you

name it. The independent variable is , and the main

limitation on the dynamical system [2] is that it be

permissible to treat this variable as complex; this

requires that the derivative with respect to this

complex variable  that appears in the left-hand side

of the evolution equation [2] make sense, namely

that this dynamical system be analytic, entailing that

the dependent variable  be an analytic function of

the complex variable  (but this does not require

() to be a holomorphic nor a meromorphic

function of ; () might feature all sorts of

singularities, including branch points, in the com-

plex -plane, indeed this will generally happen since

we generally assume the evolution equation (??) to

be nonlinear). The quantity F in the right-hand side

of [2] – which has of course the same scalar, vector,

matrix... character as  – might depend (arbitrarily

but analytically) on  as well as on . (Let us also

emphasize that this approach is as well applicable to

more general dynamical systems that also feature

other, ‘‘spacelike’’, independent variables, for

instance are a system of PDEs rather than ODEs;

the interested reader is referred to the literature cited

below).

In spite of the generality of this dynamical system,

[2], there generally holds a result (‘‘Theorem of

existence, uniqueness and analyticity’’) that charac-

terizes the solution () of its initial-value problem

determined by the assignment

ð0Þ¼

Here, for notational simplicity, we assign the initial

datum 

at  = 0; and we assume of course that the

right-hand side of [2] is not singular for  = 0 and

 = 

. The relevant result guarantees, not only for

the initial datum 

, but for a (sufficiently small but

open) set of initial data in its neighborhood, the

existence of a circular disk in the complex -plane,

centered at  = 0 (where the initial data are assigned)

and having a nonvanishing radius , such that the

solutions () corresponding to these initial data are

holomorphic in it, namely for 

<(and note that

if () is a multicomponent object, the property to

be holomorphic is featured by each and everyone of

its components).

Let us now introduce the following changes of

dependent and independent variables:

zðtÞ¼exp i!tðÞðÞ ½3a

   tðÞ¼

exp i!tðÞ1

½3b

This transformation is called ‘‘the trick’’. The

essential part of it is the change of independent

variable [3b]: and let us re-emphasize that, here and

hereafter, the new independent variable t is con-

sidered as the real, ‘‘physical time’’ variable. Note

that [3b] entails

 0ðÞ¼0; _ 0ðÞ¼1

and, most importantly, that (t) is a periodic

function of t with period T, see [1]. More specifi-

cally, as the time t increases from zero onwards, the

complex variable  travels counterclockwise round

and round on the circle C the diameter of which, of

length 2/!, lies on the imaginary axis in the complex

-plane, with one extreme at the origin,  = 0, and

the other at the point  = 2i/!, making a full circle in

the time interval T. As for the prefactor exp(i!t)

that multiplies () in the right-hand side of [3a], its

purpose is to allow, via an appropriate choice of the

parameter , the deformed system, see below, to

have a neater look; however this choice is hereafter

restricted by the condition that  be real and

rational, say



with p and q two coprime integers and q > 0. This

restriction is essential to guarantee, via [3], that if

()isholomorphic in  in the (closed) disk

encircled by the circle C, then z(t)iscompletely

periodic (namely, each and everyone of its compo-

nents is periodic) with the period

T ¼ qT ½4

Isochronous Systems 167

The deformed dynamical system is the one that

obtains from [2] via the trick [3]. It clearly reads as

follows:

z ¼i!z þ exp i  þ 1ðÞ!t½

 F exp i!tðÞz;

exp i!tðÞ1



½5

And it is plain, on the basis of the arguments we just

gave, that this system is isochronous, a sufficient

condition for the complete periodicity with period

T, see [4], of its solutions being provided by the

inequality

<

which can clearly be satisfied by initial data situated

inside an open domain of such data, at least

provided ! is sufficiently large (actually, in all the

examples reported below no restriction on the value

of ! is required, namely such an open domain exists

for any arbitrary value of !>0).

Examples

In this subsection we report tersely several examples

of isochronous dynamical systems; in each case we

also provide the reference where more information

can be found. Except when explicitly otherwise

mentioned, these dynamical systems are to be

considered in the complex context.

The first example we report is a Hamiltonian

N-body problem which is a generalization of a well-

known integrable (indeed, superintegrable)system

(see Integrable Systems: Overview). It is characterized

by the (normal) Hamiltonian

Hð

z; pÞ¼

n¼1

þ !



m;n¼1;m6¼n

k¼1

ðkÞ

 z

ðÞ

½6a

and correspondingly by the Newtonian equations of

motion

€

þ !

m¼1;m6¼n

k¼1

ðkÞ

 z

ðÞ

1þ2k

½6b

Here the

N(N  1)K ‘‘coupling constants’’ f

(k)

are

arbitrary, except for the symmetry restriction

(k)

= f

(k)

(see [6a]).

The next example we report is a real N-body

problem in the horizontal plane, characterized by

the Newtonian equations of motions



¼!

k ^



þ 2

m¼1;m6¼n



þ 



;















hi

½7

Here

 (x

, y

,0) is a real two-vector in the

horizontal plane,

k  (0, 0, 1) is the unit vector

orthogonal to the horizontal plane, the symbol ^

denotes the (three-dimensional) vector product so

that

k ^

= (y

, x

, 0), and we use the short-hand

notation



entailing r

= r

þ r

 2



. Note that these equations are translation-and

rotation-invariant;andtheyareHamiltonian,

although the corresponding Hamiltonian function

is not of normal type (kinetic plus potential

energy).

The N(N 1) ‘‘coupling constants’’ 

and 

are of course real, but they are otherwise arbitrary

except for the symmetry restrictions 

= 



= 

whicharerequiredinorderthatthis

system be Hamiltonian.Ifall these coupling

constants vanish, this dynamical system has a

clear physical interpretation: it describes the

motion of N equal, electrically charged, point

particles, moving in the horizontal plane under

the effect of a magnetic field orthogonal to that

plane (in the approximation in which the electro-

static interparticle interaction is neglected). In that

case each particle moves on a circle, the center and

radius of which depend on the initial data, while

the time taken to go round it is, in all cases, T,see

[1]. If the

N(N  1) coupling constants 

vanish, 

= 0, and the

N(N  1) coupling con-

stants 

all equal unity, 

= 1, the system is a

well-known integrable (indeed solvable)system;

andthisisaswellthecaseifthe

N(N  1)

coupling constants 

vanish, 

= 0, and the

N(N  1) coupling constants 

equal minus one

half, and only act among ‘‘nearest neighbors’’,



= 

(

m, nþ1

þ 

m, n1

)(seetheentryIntegrable

systems in this Encyclopedia).

Because of its many interesting features as well as

the neatness of its equations of motion (especially in

their complex version, see below) the honorary title

of ‘‘goldfish’’ has been attributed to this model,

characterized by the Newtonian equations of motion

in the plane [7]. A more detailed discussion of it – in

particular of its behavior for initial data outside of

the region yielding isochronous motions – is made in

the next section.

Several interesting classes of isochronous dyna-

mical systems are reported in Calogero F. (2004b).

168 Isochronous Systems

We only mention here a remarkably general

example, characterized by the Newtonian equa-

tions of motion

€

z þ i!

z ¼

k¼1

ðkÞ

z þ i!

zðÞ

where

z  (z

, ..., z

) is the N-vector whose com-

plex components z

 z

(t) are the dependent vari-

ables, while the ‘‘forces’’

(k)

(z,

z) are required to be

analytic in all their arguments and to satisfy the

scaling properties

ðkÞ

ðz;

zÞ¼

k

ðkÞ

ðz;

zÞ

which however entail no restriction on the velocity-

dependence of these forces, namely on the depen-

dence of

(k)

(z,

z) on the (components of the)

second,

z, of its two N-vector arguments.

The next example we report is characterized by

the Newtonian equations of motion



þ i!



þ 2!

m¼1;m6¼n

where we assume the N dependent variables



(t) to be three-vectors (although the property of

isochronicity of this deformed system would hold no

less if these were S-vectors, with S an arbitrary

positive integer) and we use the short-hand notation



. This system is (perhaps) remarkable

inasmuch as it represents a (complex) deformation

of the classical N-body gravitational problem, to

which it clearly reduces for ! = 0.

The next example we report is characterized by

the following (first-order) equations of motion of

oscillator type:

 ip

¼ f

ðx; yÞ; n ¼ 1; ...; N

þ iq

¼ g

ðx; yÞ; m ¼ 1; ...; M

½8

Here the N-vector

x, respectively the M-vector y,

have as components the N þMcomplexdependent

variables x

 x

(t), y

 y

(t); the N þM para-

meters p

, q

are all nonnegative integers (or they

could be nonnegative rational numbers); and the

N þ M complex functions f

, g

are restricted by

the following conditions (which are sufficient to

guarantee the isochronicity of this dynamical

system):

(1) f

(x, y) and g

(x, y) are holomorphic at

x = 0, y = 0;

(2) lim

" ! 0

1

f ("x, "y)] = 0, lim

" ! 0

1

g("x, "y)]

= 0;

(3)

f (x, y) and g(x, y) are polynomial in the y

;

(4a) lim

" !0

1p

x, "

q

y)] = nondivergent, n =

1, ..., N;

(4b) lim

"!0

1þq

x,"

q

y)]=nondivergent,m=

1,...,M.

In the conditions (4a) and (4b) the notation "

x indicates

of course the N-vector of components "

,and

likewise "

q

y is the M-vector of components "

q

Note that this dynamical system, [8], includes the

Hamiltonian case characterized by the restrictions

N ¼ M; p

¼ q

; f

ðx; yÞ¼

@Vðx; yÞ

; g

ðx; yÞ

¼

@Vðx; yÞ

which imply that the equations of motion [8] are

just the Hamiltonian equations entailed by the

Hamiltonian function

Hð

x; yÞ¼i!

n¼1

þ Vðx; yÞ

isochronicity being now guaranteed by the following

conditions on the function V(

x, y):

(1) V(

x, y)isholomorphic at x = 0, y = 0;

(2) lim

" !0

2

V("x, "y)] = 0;

(3) V(

x, y)ispolynomial in the y

;

(4) lim

" !0

1

V("

x, "

p

y)] = nondivergent.

The last two examples we report can be char-

acterized as assemblies of non-linear harmonic

oscillators, inasmuch as these two dynamical sys-

tems (which are actually special cases of more

general systems) have the remarkable property that

their generic solutions (namely, all their solutions,

except for a lower-dimensional set of singular

solutions in which one or more of the ‘‘moving

particles’’ shoot off to infinity at a finite time) are

completely periodic with the fixed period T, see [1].

Their Newtonian equations of motion read



 3i!



 2!

¼ c

¼1

¼1

n





m





 3i!



 2!

¼ c

¼1

¼1







These are two (different) systems of NM Newtonian

equations of motion satisfied by the NM complex

S-vectors

(with S an arbitrary positive integer);

hence here the index n runs from 1 to N, and the

index m runs from 1 to M, with N and M two

arbitrary positive integers, while c is of course an

arbitrary complex constant (which might actually be

rescaled away). The dot sandwiched between two

Isochronous Systems 169

S-vectors denotes the standard (Euclidian) scalar

product, entailing the rotation-invariant character,

in S-dimensional space, of these equations of

motion. Since these systems only feature linear and

cubic forces, these models are remarkably close to

physics; and they become even more applicable if

they are written in their real versions, that obtain in

an obvious manner by setting

þ i

; c ¼ a þ ib

In contrast to what we did for the previous examples,

let us outline here the derivation of these results.

Actually the two systems of Newtonian equations

written above are merely special subcases, corres-

ponding to appropriate parametrizations of a square

matrix M (of appropriate rank) in terms of S-vectors, of

the following nonlinear matrix evolution equation:

€

M  3i!

M  2!

M ¼ cM

½9

Hence the findings reported above are merely special

cases of the more general result according to which

the generic solution of this nonlinear matrix evolu-

tion equation – with M  M(t) a square matrix of

arbitrary rank – is periodic with period T, see [1]:

Mðt þ TÞ¼MðtÞ

And this result is an immediate consequence, via the

following matrix version of the trick

MðtÞ¼expði!tÞðÞ; ¼

expði!tÞ1

½10

of a previous result due to V. I. Inozemtsev,

according to which the matrix evolution equation



¼ c

which clearly corresponds to [9] via [10], is

integrable and all its solutions () are mero-

morphic functions of the independent variable .

The Transition to Deterministic Chaos

In this section we illustrate, using the real N-body

problem in the plane characterized by the New-

tonian equations of motion [7], the behavior of an

isochronous system of the kind described above

when the initial data fall outside of the region

yielding isochronous motions.

To do this it is convenient to use the complex

version of the equations of motion [7], that obtain

from [7] by setting

¼ x

þ iy

;

¼ x

; y

; 0ðÞ;

k ¼ð0; 0; 1Þ; a

¼ 

þ i

½11

and read as follows:

€

¼ i!

þ 2

m¼1;m6¼n

 z

½12

The main tool of our analysis is the (particularly

simple) version of the trick appropriate to this

model,

ðtÞ¼

ðÞ; ¼

expði!tÞ1

½13a

entailing

ð0 ¼ 

ð0Þ;

ð0Þ¼

ð0Þ½13b

that relates our equations of motion [12] to the

equations of motion



¼ 2

m¼1;m6¼n



 

½14

These equations of motion, together with the initial

data 

(0), 

(0) (see [13b]) define the solutions 



() in the complex -plane. The ‘‘physical’’

evolution of the points z

 z

(t) as functions of

the real time variable t is then given by the evolution

of the corresponding coordinates 

(), see [13a], as

the complex variable  travels round and round on

the circle C in the complex -plane, the diameter of

which of length 2/!, has one extreme at the origin

 = 0 and the other on the positive imaginary axis at

 = 2i/!. It is therefore clear that the behavior of

(t) as a function of the real, ‘‘physical time’’

variable t depends on the analytic structure of 

()

as function of the complex variable , in particular

of the singularities, if any, of this function 

() that

fall in the disk D encircled by the circle C in the

complex -plane.

Let us tersely review the relevant analysis. We

recall first of all that (it can be proven that) there

exists in phase space an open region of initial data

(0),

(0), characterized by large values of the

moduli jz

(0)  z

(0)j of the initial interparticle

distances and by small values of the moduli of the

initial particle velocities j

(0)j (see [14] and [13b]),

that guarantees (all components 

() of) the

corresponding solution

()of[14]tobeholo-

morphic in (a disk of radius  centered at the origin

 = 0 of the complex -plane that includes) the circle

C, hence the corresponding solution

z(t)tobe

completely periodic with period T, see [13a] and

[1]. This result guarantees the isochronous character

of this model, [12], for any arbitrarily given assign-

ment of the coupling constants a

170 Isochronous Systems

Next, let us restrict, for simplicity, our considera-

tion to models [12] in which the coupling constants

are real and nonnegative,

 0 ½15

Then the singularities of the generic solution

()of

[14] – which occur at values 

of  where two

coordinates 

() coincide, say 



(

) = 



(

) = b

(see the right-hand side of [14]) – are branch points

characterized by the exponent, say,

 ¼ 



1 þ a



½16

so that in their neighborhood, namely for   



ðÞ¼b  c   

ðÞ



þv   

ðÞ

k¼1

‘;m¼0‘þm1

’

ðsÞ

k‘m

  

ðÞ

kþ‘þmð1Þ

s ¼ ;  ½17a



ðÞ¼b

þ v

  

ðÞ

k¼1

‘¼

m¼0

’

ðnÞ

k‘m

  

ðÞ

kþ‘þmð1Þ

n 6¼ ;  ½17b

The  sign in front of c in the right- hand side of the

first, [17a], of these formulas indicates that one sign

must be chosen for s = , the opposite for s = .

Note that here the 4 þ 2(N  2) = 2N constants



, b, c, v, b

, v

are a priori arbitrary – except for

the obvious restrictions b

6¼ b, b

6¼ b

– while the

coefficients ’

(s)

k‘m

, ’

(n)

k‘m

can be computed from these

constants, recursively, by inserting this ansatz, [17],

in the equations of motion [14]. The fact that the

number, 2N,ofa priori undetermined coupling

constants equals the number of arbitrary initial data

for this system of ODEs, [14], indicates that this

kind of branch points, characterized by the expo-

nents 

, see [16], is the typical singularity featured

by the generic solution

()of[14]. (Branch points

with different exponents may appear, but only in

nongeneric solutions

() which, at some value 

, feature the coincidence of more than two

components, say 



(

) = 



(

) = 



(

)).

We conclude therefore that the generic solution

()

of [14] features a, generally infinite, number of branch

points, that generally affect each of its components



(), and which are characterized, for the class of

models to which we are restricting attention here, see

[15]) by (real) exponents 

,see[16], which are then

clearly characterized by the inequalities

0 <

 1

What does this tell us about the generic solution

z(t)

of the equations of motions of primary interest to

us, [12], in particular about its evolution as function

of the real ‘‘time’’ variable t?

To the solution

() is associated a Riemann

surface the structure of which is determined by the

character and distribution of the branch points of

() in the complex -plane (each of which is

generally featured by each component 

()of(),

although generally not in the same way: see [17]),

and we know from [13a] that the values taken by

z(t)ast evolves from t = 0 towards t = 1 coincide

with the values taken by

() as the independent

variable  travels, on that Riemann surface asso-

ciated with

(), counterclockwise round and round

on the circle C defined above (the diameter of which

lies on the imaginary axis in the complex -plane,

with one end at  = 0 and the other at  = 2i/!),

employing a period T, see [1], to make each full

round. Hence the behavior of the solution

z(t)of

[12] depends on the structure of the Riemann

surface associated with the corresponding solution

()of[14], and specifically on the number of

different sheets of that surface that are visited as one

travels on it before returning, if ever, to the main

sheet from which the travel started at t =  = 0.

If no other sheet is visited besides the main one,

the corresponding solution

z(t) is of course periodic

with period T, see [1]and[13a],

ztþ TðÞ¼ztðÞ ½18

This happens provided no branch point is featured

() on its main sheet inside the circle C; and, as

already indicated above, it has been proven (even in

the more general case with arbitrary coupling

constants a

) that there is an open region having

full dimensionality in the phase space of initial data,

see [13b], that yields such an outcome, implying the

isochronicity of the model characterized by the

Newtonian equations of motion [12]. This region

R of initial data has a boundary – a lower-

dimensional domain in the phase space of initial

data – out of which emerge motions leading, at a

time t

smaller than T, to a ‘‘particle collision’’, say



) = z



The character of the solution

z(t) yielded by initial

data outside of the region R depends on the

structure of the Riemann surface associated with

the corresponding solution

(). This is mainly

determined by the values of the branch point

exponents 

, which are themselves determined by

the values of the coupling constants a

, see [17]

and [16]. Let us focus on the (more interesting) case

in which these constants a

are rational numbers,

entailing that the coefficients 

determining the

Isochronous Systems 171

character of the branch points are as well rational,

see [16], so that each of the cuts associated with

them opens the way, in the Riemann surface, to a

finite number of sheets. There are then two

possibilities, each generally characterized by open

regions of initial data having full dimensionality in

phase space, the boundaries of which always are

(lower-dimensional) domains out of which emerge

motions leading, in a time t

smaller than T,toa

‘‘particle collision’’.

One possibility is that the number B of sheets

visited before returning to the main sheet be finite,

B < 1; the corresponding solutions

z(t) are then

completely periodic with period

T = (B þ 1)T,

z(t þ

T) = z(t).

Another possibility is that the number of new

sheets visited be unlimited, namely that the structure

of the Riemann surface be such that, by traveling

round and round on it along the circle C one never

returns back to the main sheet. This can happen,

even if the exponents 

are all rational so that via

the cuts associated to each of them access is gained

to only a finite number of new sheets, because of the

possibility that an infinity of branch points be

located inside the circle C on the infinite sheets

associated to these branch points, via a never ending

mechanism of branch points nesting. Whenever this

happens the corresponding solution

z(t)isaperiodic;

and it is moreover likely that it then be chaotic,in

the sense of displaying a sensitive dependence on its

initial data. Indeed this will happen whenever some

ones out of this infinity of branch points fall

arbitrarily close to the contour C, because then a

minute change in the initial data, to which there will

correspond a minute change in the pattern of these

branch points of

() in the complex -plane, will

cause some relevant branch point to cross over from

outside the circle C to inside it, or viceversa, and this

will eventually affect quite significantly the time

evolution of

z(t), by causing a change in the

sequence of sheets that get visited by traveling

along the circle C on the Riemann surface associated

to the corresponding

().

This phenomenology has a clear ‘‘physical inter-

pretation’’, which can be qualitatively understood

as follows. The N-body problem characterized by

the Newtonian equations of motion [12] generally

yields confined motions, the trajectory of each

particle tending to wind round and round – it

would indeed reduce to a circle were it not for the

interaction with the other particles. A possibility, as

we know, is that this N-body motion be completely

periodic, with the same period T that characterizes

the circular motion of each particle when the two-

body interparticle interaction is altogether missing

= 0). Another possibility, in the case discussed

above with rational coupling constants, is that there

exist other motions which are as well completely

periodic, but with periods which are integer multi-

ples of T. A third possibility, which cannot a priori

be excluded, is that there also exist motions which

are aperiodic but in some way overall ordered,

perhaps featuring trajectories that eventually wind

up around limit cycles. And still another possibility

is that the motions described by the solution

z(t)be

aperiodic and disordered. In this case the physical

mechanism causing a sensitive dependence on the

initial data can be understood as follows. Such

disordered motions necessarily feature near misses,

in which, typically, two particles pass quite close to

each other (while the probability that an actual

collision occur among point particles moving in a

plane is of course a priori nil). Such a near miss in

the motion described by

z(t) corresponds – see the

discussion above – to a branch point of the

corresponding solution

() occurring quite close

to the circle C in the complex -plane (which is the

one-dimensional region of the two-dimensional

complex -plane in which the values of

()

correspond to the values z(t) describing the motion

of physical particles moving as functions of the

time t); and in the generic case of a two-body near

miss, there is a correspondence between the fact

that such a branch point occur just inside,orjust

outside, the circle C, and the way the particles pass,

on one or the other side, by each other. Likewise,

the tiny change in the initial data that causes, in the

context of the solutions

() – see the discussion

above – a branch point of

() to pass from inside

to outside the circle C, or viceversa, corresponds, in

the context of the ‘‘physical’’ solutions

z(t), to a

change occurring in the corresponding near miss,

from the case in which the two particles involved in

it slide by each other on one side to the case in

which they instead slide by each other on the other

side – entailing a significant change in the sub-

sequent motion (indeed, the closer a near miss, the

more it affects the motion, due to the singularity

of the two-body interaction at zero separation,

see [12]).

The phenomenology outlined here does indeed

occur in this goldfish model. It also occurs – rather

similarly if more simply, since in this case only

square-root branch points occur, irrespective of the

values of the coupling constants – in the model [6]

with K = 1. Indeed, it is clear that this phenomen-

ology provides a paradigm of rather general applic-

ability for the transition from isochronicity to

deterministic chaos, indeed perhaps for the generic

onset of deterministic chaos.

172 Isochronous Systems

See also: Bifurcations of Periodic Orbits;

Calogero–Moser–Sutherland Systems of Nonrelativistic

and Relativistic Type; Integrable Systems: Overview;

Quantum Calogero–Moser Systems; Synchronization of

Chaos.

Further Reading

Calogero F (2001) Classical Many-Body Problems Amenable to

Exact Treatments. Heidelberg: Springer.

Calogero F (2004a) Partially superintegrable (indeed isochronous)

systems are not rare. In: Shabat AB, Gonzalez-Lopez A,

Man˜ as M, Martinez Alonso L, and Rodriguez MA (eds.) New

Trends in Integrability and Partial Solvability, NATO Science

Series, II. Mathematics, Physics and Chemistry, Proceedings of

the NATO Advanced Research Workshop held in Cadiz,

Spain, 2–16 June 2002, vol. 132, pp. 49–77. Kluwer.

Calogero F (2004b) Isochronous dynamical systems. Applicable

Anal. (in press).

Calogero F (2004c) Isochronous systems. Proceedings of the VI

International Conference on Geometry, Integrability and

Quantization, Varna, June 2004. Edited by Mladenov IM

and Hirshfeld AC, Sofia, Bulgaria, 2005, pp. 11–61 (ISBN

954-84952-9-5).

Calogero F and Franc˛oise J-P (2002) Periodic motions galore:

how to modify nonlinear evolution equations so that they

feature a lot of periodic solutions. Nonlinear Math. Phys 9:

99–125.

Calogero F and Franc˛oise J-P (2003) Nonlinear evolution ODEs

featuring many periodic solutions. Theor. Mat. Fis 137:

1663–1675.

Calogero F and Franc˛oise J-P (2002) Isochronous motions galore:

nonlinearly coupled oscillators with lots of isochronous

solutions. In: Proceedings of the Workshop on Superintegrability

in Classical and Quantum Systems, Centre de Recherches

Mathe´matiques (CRM), Universite´deMontre´al, September

2002, CRM Proceedings and Lecture Notes, American Mathe-

matical Society, 2004, vol. 37, pp. 15–27.

Chavarriga J and Sabatini M (1999) A survey of isochronous

systems. Qualitative Theory Dyn. Syst 1: 1–70.

Mariani M and Calogero F (2005) Isochronous PDEs. Yader-

naya Fizika (Russian Journal of Nuclear Physics)68:

958–968.

Isomonodromic Deformations

V P Kostov, Universite

de Nice – Sophia Antipolis,

Nice, France

Introduction

In this article we consider families of linear

differential equations whose monodromy data do

not depend on the parameters. Such families are

called isomonodromic deformations of any of the

equations of the family (for the definitions of a

regular and Fuchsian linear system and of

their monodromy groups, see Riemann–Hilbert

Problem).

Schlesinger’s Equation

The best-studied example of an isomonodromic

deformation is the Fuchsian system on Riemann’s

sphere CP

= C [1considered by L Schlesinger:

pþ1

j¼1

t  a

X ½1

Here the poles a

2 C are free parameters and the

matrices-residua A

depend analytically on

a := (a

, ..., a

pþ1

); therefore, system [1] is in fact a

family of linear systems which is an analytic

deformation of the system obtained for a

= a

One can think of system [1] as defined by the

Pfaffian system

dX ¼ !

X;!

pþ1

j¼1

t  a

dðt  a

Þ½2

Suppose first that the poles a

vary within

small nonintersecting disks of the points a

,so

small that the standard system of generators of

the monodromy group could be defined by one

and the same contours for all values of the

parameters a

(see Fig ure 1 from Riemann–Hilbert

Problem). Suppose also that one chooses 1 as

base point and that one has

t¼1

¼ I ½3

(where I is the identity matrix) for all values of the

parameters a

. Finally, suppose that all matrices A

are nonresonant, that is, without two eigenvalues

differing by a nonzero integer. Then the following

conditions are necessary and sufficient for system [1]

to be isomonodromic:

ðaÞ¼

pþ1

j¼1;j6¼i

½A

ðaÞ; A

ðaÞ

 a

dða

 a

i ¼ 1; ...; p þ 1 ½4

This system (called Schlesinger’s equations) results

from the Frobenius integrability condition

= !

^ !

of system [2].

Isomonodromic Deformations 173

Remarks 1

(i) To find the matrices-residua A

as functions of a

and given their values A

a=a

is a Cauchy

problem. It is solvable for a close to a

and

the matrices A

are analytic in a.

(ii) The differential of A

being a commutator

, .], the matrix A

remains within its con-

jugacy class throughout the deformation.

(iii) Schlesinger’s equations are the necessary and

sufficient conditions for isomonodromy also in

the case when system [1] has a logarithmic

pole at 1 whose matrix-residuum does not

change throughout the deformation. In this

case the solution to system [1] in its Levelt’s

decomposition at 1 (see Riemann–Hilbert

Problem) equals U

(1=t)t

D

E

G, where

is a diagonal matrix with integer entries,

is an upper-triangular constant matrix, and

is holomorphic at 1 and such that

(0) = I.

Definition 2 The deformation satisfying condition

[4] with initial condition [3] for the solution to

system [1] is called the normalized Schlesinger

deformation.

Remark 3 When the matrices-residua A

are

nonresonant, then every isomonodromic deforma-

tion of system [1] with a

= a

is either the normal-

ized Schlesinger deformation or is a nonnormalized

Schlesinger deformation, that is, obtained from

the normalized one by a change of variables

X 7!C(a)X, C(a) 2 GL(n, C ). In this way, one has

t=1

= C(a) instead of [3] and the deformation is

described by a Pfaffian system with a form of the

kind !

= !

pþ1

j=1



(a)da

Example 4 The following one-parameter Fuchsian

family is an isomonodromic Schlesinger deformation:

pþ1

j¼1

t  ba

Here the matrices A

are constant and the parameter

b takes nonzero values. Indeed, one either checks

directly that there holds condition [4] or one makes

the change of time (which does not change

monodromy) t 7!bt after which the parameter

b disappears.

A A Bolibrukh has shown that in the resonant

case every isomonodromic deformation of a Fuch-

sian system is described by an integrable Pfaffian

system with 1-form ! = !

þ !

, where the mero-

morphic 1-form !

vanishes at 1 and has poles of

orders r

along the hyperplanes {x  a

= 0}; here r

is the largest nonzero integer difference between two

eigenvalues of the matrix A

Consider now Schlesinger’s equation in the global

situation, that is, when the poles a

belong to the

universal covering Z of the space C

n, where  is

the ‘‘diagonal,’’ that is, the union of all sets

= a

}, i 6¼ j. Suppose that the matrices A

are

nonresonant. There are values of a (their set is

denoted by  ) for which some entries of some of the

matrices-residua A

tend to 1. Typically, at such

points the matrices A

have poles of second order;

this is a result due to Bolibrukh. Indeed, set

= Q

1

, where J

is the Jordan normal form

of A

; hence, this is a constant matrix; we assume

that Q

2 SL(n, C). Typically, at points of  the

matrices Q

and Q

1

have simple poles, which

makes a pole of second order for A

B Malgrange and, independently, T Miwa have

proved that system [4] is completely integrable and

that it has the Painleve´ property: ‘‘The only

movable singularities of its solutions are poles.’’

(The fixed singularities of the solutions are, by

definition, along the points of Z which are over .

The positions of the movable singularities depend

on the initial condition, that is, on the values of the

matrices A

for a = a

.) In other words, the

solutions to Schlesinger’s equation are matrices

meromorphic on Z.

Theorem 5 The set  of movable singular points

of the Schlesinger equation is the set of zeros of a

function  (the Miwa -function) holomorphic on Z

and such that

i;j;i6¼j

trðA

ðaÞA

ðaÞÞdða

 a

¼ d logððaÞÞ

Some improvements of this result are due to

Malgrange and Bolibrukh.

Isomonodromy and Confluence

The idea to consider a linear system of ordinary

differential equations with a pole of order higher than

1 as embedded into a family of Fuchsian systems with

confluence of the poles has been proposed by V I

Arnol’d in 1984 and independently by J-P Ramis in

1988. The idea has been used by A Duval, B Khesin,

A A Glutsyuk, and other authors. In particular, it is

interesting to relate the Stokes multipliers (defined in

the next section) of the system obtained as a result of

a confluence to the monodromy groups of the

174 Isomonodromic Deformations

Fuchsian systems obtained for values of the para-

meters before the confluence occurs.

Example 6 Consider the one-parameter family of

linear systems:

ðt

 ÞdX=dt ¼ðAðÞt þ BðÞÞX ½5

Here the matrices A, B,andX are n  n.

Suppose that t 2 C (i.e., we do not consider

singularity at 1),  2 (C, 0). Then for  6¼ 0the

system is Fuchsian – it has two logarithmic poles at



1=2

whose confluence for  = 0givesasaresulta

pole at 0 which might be of order 2 if B(0) 6¼ 0or1

if B(0) = 0.

In this section we consider only the situation

when the family producing the confluence is

isomonodromic for values of the parameters before

the confluence.

Example 7 This is the case of family [5] with B  0

and A being a constant nonresonant n  n matrix.

Indeed, the change of time t 7!

1=2

t () transforms the

family into the family (t

 1)dX=dt = tAX

(independent of ) which is a Fuchsian system (at 1

as well).

Suppose now that t 2 CP

(i.e., we consider the

singularity at 1 as well). Hence, the monodromy

operator M

around 1 is independent of  up to

conjugacy (it is conjugate to exp(2iA)). On the

other hand, consider the monodromy operator M

defined by a contour circumventing counterclock-

wise both poles at 

1=2

(one can choose as such

a contour a circumference centered at the origin

and of sufficiently large radius). It equals M

1

and it is well defined for  = 0aswell.(Thisis

not the case of the monodromy operators defined

by contours circumventing only one of the poles

at 

1=2

.) Hence, up to conjugacy M

is indepen-

dent of .AsM

is in a sense the only

monodromy operator that can be defined by

a contour depending continuously on  for all

 2 (C, 0) and not passing through a pole of the

system, one can say that the family is strongly

isomonodromic.

Example 8 Consider now family [5] with n = 2,

A ¼

d 0

0 d



; B ¼

0 



where d 2 C.For 6¼ 0 the family is isomonodromic –

the change of time () followed by the change of

variables

X 7!



1=2



XðÞ

brings the family to the form

ðt

 1Þ

d 0

0 d



t þ



which is independent of , hence, isomonodromic.

However, the change of variables () is not defined

for  = 0. The monodromy operator M

(defined as

above) is scalar for  = 0 and conjugate to a Jordan

block of size 2 for  6¼ 0. Hence, the family is not

strongly isomonodromic.

The following example is closely connected

to singularity theory. It has been suggested by

F Pham.

Example 9 Consider the Abelian integrals

dx=ðx

þ sx þ tÞ and

x dx=ðx

þ sx þ tÞ

taken over a closed contour  belonging to a

nonsingular fiber of the function f (x) = x

þ sx þ t.

Suppose that x

þ sx þ t 6¼ 0on. Obviously, I

and I

depend only on [], the class of homotopy

equivalence of . Set

þ sx þ t ¼ðx  x

Þðx  x

Þ;

¼ x

ðs; tÞ

Then one has

¼ 2i

j¼1



k;j

k1

= 3x

þ s



; k ¼ 1; 2

where the integers 

k, j

depend only on [] (the

contour  is homotopy equivalent to a linear

combination with integer coefficients of small

loops around the roots of f; the integral along such

a loop is computed using residua). Note that

:¼ dx

=dt ¼1= 3x

þ s



An easy computation shows that the integrals I

, I

satisfy the following Picard–Fuchs system of differ-

ential equations:

t

 2s

=3 ¼ 2I

=9  t

¼ I

The system admits also a presentation of the form



2t=32s=9

4s

=27 t = 3



Here the unknown variables form a vector column

of length 2; to obtain a 2  2 matrix, one has to

choose another contour 

(linearly independent

Isomonodromic Deformations 175

with  as a linear combination of the loops around

the roots x

) which gives the second column of the

matrix. The system is strongly isomonodromic – its

matrix-residuum at 1 equals diag(2=3, 1=3); hence,

the monodromy operator M

up to conjugacy equals

diag(exp(4i=3), exp(2i=3)).

A A Bolibrukh has considered the possibility of

confluence of poles in Schlesinger’s equation

(i.e., the possibility to have equalities of the form

= a

in system [1]). He has considered the so-called

normalized isomonodromic confluences, that

is, isomonodromic confluences defined by Pfaffian

systems with coefficient forms ! = !

þ !

alone

(see the previous section). He has shown that

a normalized isomonodromic confluence of singular

points of Fuchsian systems of linear differential

equations on Riemann’s sphere can only lead to

a system with regular singular points. This is a

partial answer to a problem stated by V I Arnol’d:

how to express a system with regular singular

points as a limit of Fuchsian systems?

Other Results

In the case of a linear system with irregular singular

point, isomonodromy means that the formal mono-

dromy and the Stokes multipliers do not change

throughout the deformation. The formal mono-

dromy can be computed from the formal normal

form (the latter can be found algorithmically; this is

due to H Turrittin). Consider, for simplicity, the

nonresonant case, that is, the case when the leading

matrix in the Laurent series of the system at the

singular point has distinct eigenvalues (this defini-

tion differs from the one in the case of a Fuchsian

singular point). The Stokes multipliers are linear

operators acting on the solution space. They are

defined as follows: there exist sectors of maximal

opening centered at the singular point on each of

which the solution is uniquely defined by its

asymptotic development. Two solutions X

, X

having one and the same asymptotic development

in two overlapping sectors are related by X

= X

where C is a Stokes multiplier. The monodromy

operator is expressed as a product of the operator

of formal monodromy and the Stokes multipliers.

Isomonodromic deformations of systems with irregu-

lar singular points have been constructed by B

Malgrange. Isomonodromic deformations have been

usedbyYSibuyaandCHLinandbyYSibuyaand

T J Tabara to investigate Stokes multipliers.

At the beginning of the twentieth century,

P Painleve´ and B Gambier have classified the

differential equations of second order,

¼ Rðx; u; u

Þ½6

(where R is analytic in x and rational in u and u

)

whose solutions do not have branch-type movable

singularities. From the 50 equations (up to local

transformation) discovered by them only six are not

reduced to linear ones. These are the so-called

Painleve´ equations. They appear often as isomo-

nodromy conditions for families of linear differen-

tial equations and this has given the idea to

develop the isomonodromic deformation method. It

consists in associating with eqn [6] a linear system

d=d ¼ Að; x; u; u

Þ ½7

with matrix-valued coefficients rational in .

The deformation of the coefficients in x is described

by eqn [6] in such a way that the monodromy data of

system [7] remain the same. Thus, the monodromy

data of system [7] are first integrals of eqn [6].

Example 10 The Painleve´ II equation

 xu  2u

¼ 

is associated with the system

d

4i

 ix  2iu

4iu  2u



i



4iu  2u

i



4i

þ ix þ 2iu



The idea to present the Painleve´ equations as

isomonodromy conditions originate from the works

of Fuchs (1907) and Garnier (1912). It has been

used, for example, in the papers of Flaschka and

Newell (1980), Jimbo and Miwa (1981),andIts and

Novokshenov (1986).

See also: Holonomic Quantum Fields; Integrable

Systems: Overview; Painleve

Equations;

Riemann–Hilbert Problem; WDVV Equations and

Frobenius Manifolds.