Analytic Structure of Certain Damped and Driven Nonlinear Oscillators

Indian J. Phys. 67A (6). 477 - 491 (1993)

A n a l y t i c s t r u c t u r e o f c e r t a i n d a m p e d a n d d r i v e n n o n l i n e a r o s c i l l a t o r s

M Lakshmanan

Centre for Nonlinear Dynamics, Deportment of Physics, Bharathidasan University, Tiruchirapalli - 620 024, India

Abstract : There exists a close connection between integrability and singularity structure of solutions of nolinear dynamical systems. One can associate the so called Painlevd property with integmblc systems, including soliton equations, in that the solutions are meromorphic. Then the question arises how the singularity structure of nonintegrable and chaotic dynamical systems is distributed in the complex time plane and what is their role on real time dynamics In this lecture, it is pointed out that for a clas.s of damped and driven nonlinear oscillators the singularities get clustered in a multiarmed infinite Riemann sheeted structure and then their consequences are discussed.

K eyw ord s : Dynamical systems, chaos, singularity structure

PACS Nos. : 02 yO.+p, 03.20.+i

1: Introduction

Our understanding of dynamical systems has undergone a revolution in recent limes.When nonlinearity is present, even complicated systems such as the Korteweg-de Vries equation or Davey-Stewartson equation can admit regular, well-defined and coherent structures like soliions, drornions, etc.[l-3]. On the other hand even very simple and lower dimensional nonlinear dynamical systems such as the damped driven pendulum can admit very complicated and chaotic motions which are highly sensitive to initial conditions [4-6]. Thus one of the most challenging and intriguing problems in contemporary nonlinear dynamics is to identify whether a given dynamical system is regular or chaotic and whether it is integrable or nonintegrable.

It is remarkable to note that such problems have drawn the attention of scientists even during the last century. Particularly there had been considerable activity during 1860-1910 by mathematicians like Fuchs, Painlevd, Gambier, Gamier, etc. in classifying ordinary differential equations as per the singularities exhibited by the solutions in the complex plane of the independent variable [7]. Integrable dynamical systems were then identified with systems whose solutions are meromorphic in the complex time plane as in the work of

© 1993 lACS

Kovalevskaya [8]. However, these developments were to remain dormant for more than 60 years until after the advent of soliton systems, which are completely integrable infinite dimensional systems. Ablowitz et al [9] have noticed a strong connection between soliton systems and meromorphic solutions in that ordinary differential equations (odes) obtained through a reduction of the partial differential equations turn out to be always free from movable critical points (movable branch points and essential singularities). In recent times, this connection has been more strongly established and soliton systems in general turn out to be free from movable critical manifolds [10].

At the same time there has been a revival of interest in applying the singularity structure analysis or the Painlevi analysis to finite dimensional dynamical systems. A large number of finite dimesional integrable systems have been found in this way and categorized (see for e.g. [11,12] ). Then the question arises what the singularity structure of nonintegrable and chaotic systems will turn out to be and what informations can be gleaned out of them.

Recent investigations on nonintegrable systems have identified three important types of singularity structures in the complex-time domain of the solutions. They may be broadly classified as

i) Movable branch points with complex exponents :

They are mainly exhibited by Hamiltonian systems like the Henon-Heiles, coupled quartic oscillator systems, etc. admitting natural boundaries [13,14].

ii) Movable algebraic branch points : »

Such singular points are mainly exhibited by Hamiltonian systems like the Calogero-Moser type system, soft billiards [15] and some non-Hamiltonian systems such as the Duffing- vander Pol oscillator [16].

ill) Movable logarithmic branch points:

These are predominantly exhibited by damped and driven oscillator systems such as the Duffing oscillator, damped and driven pendulum and so on [17-20].

In this article, we briefly analyse the nature of singularities exhibited by a class of damped and driven nonlinear oscillator systems admitting movable logarithmic branch points and investigate their intricate inner and global structure. We show that such nonintegrable systems are characterized by multiarmed, infinite Riemann sheeted structure orsingularities in the neighbourhood of a given singularity. We illustrate this for the case of a quadratic nonlinear oscillator which is acted on by a damping and external periodic force. Then we summarize the results for the other interesting oscillator systems.

The plan of the paper is as follows. To start with in Section 2 we give a brief introduction to the classification of singular points exhibited by the solutions of the nonlinear

A n a l)^ stnicm rectf certain damped and driven nonlinear oscillators 479 odes and their connection to dynamical systems. We also briefly summarize the AUowitz

e t a l's algorithmic procedure [9] to identify the singularity structure. In Section 3, we

investigate the analytic structure of the damped and driven quadratic anharmonic oscillator. It is shovm how clustering of singularities with infinitely Riemann multisheeted structure with multiarms occur. In Section 4, we sununmize the results for typical oscillators and discuss their implications.

2. Singular points, ordinary differential equations and dynamical systems

2 .1 . S in g u la r p o in ts - fix e d a n d m o v a b le :

Considering the solutions of ordinary differential equations in the complex plane of the independent variable r € C , one may distinguish between o r d in a r y p o i n t s (where the solution is regular) and singular points (where the solution ceases to be analytic). These singular points (SP s) can be either fixed or movable.

Considering linear odes of the form

d ^ x

dt"

+

ifl-l

a X

d tn-l + ■••+ P„ ( t ) x = 0. ₍1) one can easily check that the singularities of x always occur at the singularities of P, (f). Thus the SP s of linear differential equations are always fixed by the form of the des. As an example, we may cite the case of the Legendre equation:

y - ^{2 t , n( n} + 1) _

■ y + - r — j f y = 0- (2)

Then the SP s arc fixed at r = + 1 and / = - 1.

On the other hand the SP s of nonlinear odes can be movable (in addition to fixed S P 's). As simple examples, we consider the first order nonlinear odes ;

d x 2 ^ 1

d t

i) i ! i - + x = 0 = > j c =

- *o)’

16 c ', arbitrary (3)

d x , 3 f.

“> I T * ' - "

1

X =

iii) A + = 0 = > x = - log(/ - /o)

(4)

(5) Thus even in the above simple examples we have respectively movable pole, movable algehnw branch point and movable logarithmic branch point singularities. Similarly, we can have movable essential singular points. Thus, the distinguishing feature of nonlinear odes is that the movable singularities exhibited by their solutions can be placed anywhere in the

complex /-plane and so are initial condition dependent. Out of all the movable singularities, the movable branch points (both algebraic and logarithmic) and essential singularities are named as movable critical points.

2.2. Classification o f odes and dynamical systems:

Considering the first ordei odes

— = F ( x j \ t e C ,

d t (6)

where F is rational in x and analytic in t , Fuchs [7] in 1884 had concluded that the only equation free from movable critical points is the generalized Riccati equation

dt (7)

where />o, P\ andp2 analytic in t^. Extending the analysis to second order odes, Painleve Gambier, Gamier and coworkers [7] have concluded that out of the general class

^ = f( ^

d r ^V dt » X, f (8)

where F is rational function in — , algebraic in x and locally analytic in /, only 50d x

canonical equations are free from movable critical points. These include the six Painleve dt transcendental equations

Pi

d.2X , 2

— =- = 6x -I- f,

dt^

d.2X - 3

— ^ ^ l x + /jc + a , d r

P v

d^X f dx\^ X dx a 3 fix ^ ^ s

dt ^ d t J t dt t t

+ 2 ( i - ' - a ) x ^ + P ,

d r 2 \ d t J 2

' '

^ = f — + — 1 f — Y - dt^ v2jc x - l ) \ d t ) t dt

(9)

(10)

(11⁾

(12)

(x -l)^ (' P'] X

— r 7 J " " 7

Sx(x + 1) (^ - 1) '

(13)

j2

Analytic structure c f certain damped and driven nonlinear oscillators 48]

4 , ifi * _L *

d r 2vx x - \ x - t J V d t J

-fU _]--- L-'l f^'l

u t - \ X - t J y d t )

A x - l ) ( x - t ) ^ pt

[ x^

r i t -1 ) ^ S t( t -

(X ix

L l l ) l

- t f

1

Here a, P, S, are all constani parameters. The study of the fuller properties of these odes itself constitutes a field of its own and these systems occur in very many physical situations such as soliton systems, two dimensional quantum gravity, statistical mechanics and so on [21,22].

Finally,attempts have also been made by Bureau [23] and Chazy [24] to classify third and higher order odes though the analysis remains rather incomplete.

2.3 . Rigid body dynamics : Kovalevskaya’s analysis [25} :

The first dynamical problem to be analysed through singularity structure approach appears to be the problem of a heavy rigid body under the influence of gravity. The equations of motion are of the form

dJ , ^ de

---= JaS2 --- = e A l J ,

dt dt

where the angular velocity £2 and the angular momentum / arc given by

£2 - Cf2 3 ^ 3

(15)

(16) with respect to a moving trihedral e^ ,i = 1,2,3 fixed to the body . Then the vertical unit vector e and the centre of mass given by

e = GCffj + J3e2 + )«3, + ^^^3, (17)

where a, j3, and 7 refer to the direction cosines which define the orientation of the heavy rigid body. Eqs. (15) can be rewritten in component form as

= (B - - pz^ + = pi2y -

dt dt

= (C - - yx„ + az^. ^ = yQy - o Q ,,

at ut

. d a . .

= i A - B )Q ^ a ^ - orvo + p x ^ , ^ = o a ^ - p a ^ .

dt dt (18)

Kovalevskaya [8] approached this problem by demanding that integrable cases of (18) should admit only meromorphic solutions,that is solutions free from movable critical points.

She found that only four cases pass this test;

JCo=yo = Zo = 0, Zo>0.

Hi) A = B s Q

iv) yo = Zo*0. A ^ B - 2 C .

(19a) (19b) (19c) (19d) Out of these, the first two were already known to be explicitly integrable in terms of elliptic functions by Euler (1750) and Lagrange (1788) respectively and the third one is strightforward. The fourth choice was entirely new and Kovalevskaya succeeded in obtaining the general solution for this case in terms of hyperelliptic functions. Thus, Kovalevskaya established a strong connection between meromorphic nature of solutions and integrability of nonlinear dynamical systems.

2 .4 . D y n a m ic a l s y s te m s -d a m p e d a n d d riv en sy stem s :

The above type of singularity analysis approach however failed to attract much attention for several decades until it was realized that soliton systems, which constitute infinite dimensional integrable nonlinear systems[l-3], are of Painlevd type. Many finite dimensional dynamical systems were also investigated for their integrability and nonintegrability properties in recent times [11,12] through the Painlevd approach. In this connection it will be quite useful to apply this approach to some of the ubiquitous nonlinear damped and driven oscillators. In faot many such systems turn out to be nonintegrable variants of the Painlevd transcendental equations admitting movable logarithmic branch points. They are as follows :

i) Q u a d ra tic D uffing o sc illa to r [ 2 6 ] :

x-i-ax+P x+ = f cosax, which can be considered as a variant of the Pj eq.(9).

ii) C u b ic D iiffing o sc illa to r [ 6 ] :

X + OCX + Px-t-yx^ cosox,

which is a variant of the Pn (10).

Hi) D a n c e d a n d driven p en du lu m [ 5 ] : j: + o r + <0q sin jc = / cos OX, which under the transfonhation

y = exp(oc),r = -if

(20)

(21)

(22)

(23)

Analytic structure of certain damped and driven nonlinear oscillators 483 becomes a variant of the Pm c q .( ll) :

. . . 2 . 0 ) 3 \ 2

y y - y + iotyy + ~ 2 ^----² ^ cos/io)r = 0.

(iv ) D a m p e d a n d d riven M o rse o sc illa to r [ 2 0 ] :

Similarly the system

jc + oti + j3e'’'( l = /c o s wf, under the transfonnation

(24)

becomes

y = e

XV - + ayy - j 9 y + Py^ + fy^ cos ox = 0.

(25)

(26)

(27) Thus, it is clear that many of the contemporary interesting nonlinear dynamical systems exhibiting chaotic motions and strange attractors [4-6] turn out to be variants of Painlevd equations admitting movable logarithmic branch points. In order to understand the real time dynamic behaviour of these systems, then there in an urgent need to investigate the intricate singularity structure of the singular points in the complex time plane of these systems. In this article, we will carry out such a study for the above systems using the so called ARS algorithm through a procedure developed by Fournier e t a l [17].

2.5. A R S alg o rith m ic p ro c ed u re :

In order to determine whether the solution of a given ode admits movable branch points or not, the Ablowiiz-Ramani-Segur (ARS) algorithmic procedure gives a systematic method to locate them [9]. Locally we look for a Laurent series representation of the solution

V *0) m=0

(28) in a deleted neighborhood 0 < |t-tol < ^ as r-»rg and determine the nature of q and a„'s.

If logarithmic branch points are admitted at the power m = n, then the local representation can be modified in the form of a psi series [27]

x ( f ) = I

(» -^o r

S ^ ’

jstO k ^ o

(29)

>:=0 k^O

and then one can analyse the nature of the series in order to identify the structure of the singularities.

3. Analytic structure of quadratic anharmonic oscillator equation 3, J. The damped -driven oscillator :

The damped and driven quadratic nonlinear oscillator

d'^x- d x 2 n 2

^ + a— — + cOqX + px = ycos ax

dt dt (30)

is a prototypical example of chaotic dynamical systems. It is a model for the engineering systems in noisy or ill-defined environments like ship capsize [26J.

5.2. The generalized P \ :

In order to study the analytic strucrure of (30), we will take a special case of (30) with forcing generalized, namely,

^2

— Y = + /(O . /( O : iinalytic in t . dt

Eq. (31) is a generalization of th P| eq.(9).

In a deleted neighborhood of the SP /q ,

Substituting the leading order in (31), we obtain

= \, q = - 2 .

So the Laurent series can be given in the form

(31)

jr(/) = t'

/=o

- 2

(32)

(33)

(34)

In order that this is a suitable local representation of the solution x(t) of (31) in the neighborhood of tQ ^ (34) should have two arbitrary constants out of which one is fo itself.

Using (34) in (31), and equating the coefficients of we obtain the recursionrrelation ( ; - 2 ) ( y - 3 ) a , = 6 ^

F. =

n\ dt"

(35)

Analytic structure o f certain damped and driven nonlinear oscillators

Solving we get

Oq 1* ” O2 — — 0,

. - r - IT

^ 1 0 ° 6 '

(36a) (36b) (36c) 485

0-^6+'^2 = 0 ’

and Q y j ^ 7are obtained from the preceding coefficients.

From (36c), we find that F ^-O and that the coefficient c/e> is arbitrary, which means

/ ( 0 = / o + / , / (37)

and this corresponds to Pi eq. (9) which is of course free from movable critical points. For all other forms of f{t) . eq. (31) shows an inconsistency at 7 = 6 in the Laurent series (34), requiring the introduction of a logarithmic term, signalling the presence of a movable critical point.

3.3. The psi series :

Thus when /(/) is not linear in we introduce the double infinite series, the so called psi senes [27],

OO QO

xit) = X r = ( ? - / ( , ) -> 0

/-=0 k=0

= X X (38)

/-o k = 0

' / “^6 it 4 “ Substituting (38) in (31) and equating the coelTicicnis of powers of r , we obtain the recursion relation

{j + bk - 2)(y + bk~?i)aji^ + (2j + 1 2 ^ - 5){k +

+ «: + l)(* + 2)fl,.,2j^2 P i - A . k (39)

Now one finds that

"OO " “ IQ /o

fl,o = — ^ /-’i.««) = arbitrary, floi = ^ (40)

As to is also arbitrary we find that the psi series (38) is a suitable local representation ol the solution . However, the recursion relation (39) is too complicated to get any useful information.

3.4. Closed set o f recursion relations and rescaling:

Though the complete analysis of (39) is too difficult, useful information can be obtained by looking at a subset of the recursion relation corresponding to the coefficients (;' = 0).

They read

(6*-2)(6/:-3)aot = 6 ^ flo flo.,- (41)

or

[ b k ( k - 1) + ^ + Go „

Defining now the generating function

9(z) = ^ ^ 0* ^ = T^InT,

(42)

(43) (42) can be rewritten as

(44)

Thus using (43) we have

^ 2 d^G dG , ^ » 6I

6c ---- T— “J" z --- G — G = 0 , c — T InT

dz^ dz (45)

as the diffcrcnlial equation satisfied by the generating function.

More interestingly, cq. (45) is a rescaled version of the original cq» (31) obtained in the closed neighborhood of the singular point fo^^ong a suitable path in the complex f-plane.

In fact with the transfomiaiion

-v(r) = —r 0 ( z \ c = T Inr, t = (/ - rQ)->0 (46) wc have

(36;- + I2cr‘' + r'-) d~9 j , _ 6\ d9

— ^ + (6c + 7 r --- + 6 6

dz' ' *

= 60- + T* fit). (47)

Thus in the limit r-» 0 and along a suitable path in the complex r-plane, (47) reduces to the eq. (45) satisfied by the generating function 6(c). We note that 6(c) is in fact the j = 0 term of the psi series (38) and it eoiresponds to the case where this term dominates over the other terms of the series.

3 .5 . The sin gularity structure :

Interestingly, the rescaled eq. (45) itself admits Painlevd property and its solution is meromoiphic. In fact, eq. (45) can be explicitly integrated. Defining now

0(z) * y ^ g ( y ) , y =

eq. (45) becomes

^ ^ - tin^

— r - o g d y

Analytic structure of certain damped and driven nonlinear oscillators ⁴⁸⁷

(48)

(49) which is obviously integrable in terms of the Weisstrass elliptic function. Integrating (49) once, we get

,2

( 4 - j • < ( • • • ■ ! - )

Comparing with (45), the integration constant C is fixed as

^ 22 ^ ^ 1

C - - 22 Oq] - — F², ^2 — ² ^^2

(50)

(51)

1

Then the solution of (50) can be written as the Weisstrass elliptic function

where

g = p(y) = e j + ‘'1 ‘'3

. rn^fC(y-yo),*] .

(52a)

Cl + «2 + «3 = 0, (52b)

«lC2 +«2«3 +«3«4 = 0. (52c)

«1«2«3 C

2- (52d)

Thus we obtain

(53) The pole singularities of the solution x(f) are obviously the period parallelogram in the y-plane

y,„ = — [1 14 K ( k ) + i m K { k ' ) ] , l , m e Z (54a)

which in the z-plane becomes

Zi. » y l - (54b)

Then each pole in the z-plane can be mapped into the singularities in the t-plane through the polar transfotmations

z = pe^

> = Z ,1/6

f = re

z = r"lnf, (55)

where /q is now taken lo be the origin for convenience. Thus we have,

r = exp [“(5 + 2;r n) cot (6 0 - ^)], (56a)

p = - (f l + 27rn) cosec ( 6 0 - 0 ) exp [-6(0 + 2;rn) cot ( 60-0)], (56b) where n is a nonnegative integer, characterizing the Riemann sheet. Thus for every singular point in the z-plane given by a fixed p and 0, the singularities in the f-plane are given by the nonlinear map (56). It is clear that in the neighborhood of each such singular point there emanates a 6-armed structure with each arm consisting of infinite number of singularities positioned at different Riemann sheets and all of them cluster as they approach the given singular point. Figure 1 shows a numerical evalution of the clustering of singularities in a

Figure 1. Local singularity structure in the complex /-plane in the neighbourhood of a typical singularity (at the origin), determined from the analytic mapping (56).

typical case. We find that such infinite sheeted Riemann structure with multiarms with clustering seems to be a characteristic of nonintegrable chaotic dynamical systems. Note that the effect of damping qualitatively does not alter this picture. One can actually develop a global picture of the singularities by actually integrating the differential eq. (31) in the

complex plane to identify a complicated clustering of singularities with a self-similar structure. The fuller details will be given elsewhere.

3 .6 . R e su n v n a tio n :

As noted earlier the solution of the rescaled version (45) can be actually thought of as the leading order of resummed version of the psi sereis (38);

■*(') =

Analytic structure of certain damped and driven nonlinear oscillators 489

j *

i 6.

k

= [flo(z) + + 62(1)1^ + ■■■]

oe

= X

it=0

Substituting this series expression into the original equation (31), in the limit t -> 0 along a suitable path, we get the following system of des after equating coefficients of different powers of / to zero,

•. + z9q + 9q - o l = 0, 1 (58)

: 6z^fl| + 3z6| + - ^ 0, - 20o6i = 0 (59)

: 6z^02 - 2 9q92 ■= (60)

+ { \ - ^ 2 k ) z 0 k + - r [ ( t- 2 ) ( ) t- 3 ) - llOo]

6

A-I .

= 1 [ e l y 2 + \ 2 z e l ^ + ( 2 k + i ) +

X

iS=l

We note that for it > 1, the above equations arc linear and that the corresponding homogeneous part admits a solution

6*(z) = Vkiy)> y = z‘'"' * > 1

- ,'/<i (62)

where \ir[ s satisfy the Lam^ equation - 12 V'oV'* = 0

dy^ (63)

and v^o the Weisstrass function, From the knowledge of the solutions of the Lam^ equation (63) the general solution of (61) can be explicitly obtained, thereby obtaining a formal representation for the psi series (57). It will be interesting to explore the connection of the real time dynamics with above explicit representation of the psi :eries in the complex time plane.

4. Discussion

The above analysis can be repeated for other damped and driven oscillator systems [17-20].

One obtains a general picture of clustered singularities with multiarmed infinite sheeted Riemann structure. They can be summarized as follows.

System No. of arms in the local

singularity structure

Solutions of the rescaled equation 1. Quadratic Duffing

2. Cubic Duffing 3. Pendulum 4. Morse

6 4 2 2

elliptic function elliptic function elementary function elementary function Jbese results clearly show a strong connection between the complicated singularity structure and the nonintegrability nature of the damped and driven dynamical systems. More work needs to be done to understand the full implication of the singularity structure on the real time dynamics and on the route and onset of chaos. Specifically, one would like to know whether there is an one to one correspondence between the bifurcation and dynamical aspects of chaotic systems, and the singularity structure in a local or a global sense. For example, for the above oscillators as the systems undergo period doubling route to chaos, there does appear to be perceptible qualitative changes in the global structure of singularities in the complex time plane from an oidered lattice structure to a clustered fractal like structure as the system parameter incrca.ses. Similarly, it is not yet clear whether the psi series-<57) can be summed up in some sense and analytically continued to the real axis to obtain the real time behaviour. For integrablc systems, the Laurent series does reveal several interesting dynamical features such as Lax pairs, Backlund transformations, symmetries, etc. So it is conceivable that the psi series should also possess considerable information about the dynamics and it gives a hope to develop analytical approaches of chaotic dynamical systems.

Analytic structure of certain damped and driven nonlinear oscillators ⁴⁹¹

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