External field induced chaos in an infinite square well potential

Physica 19D (1986) 145-152 North-Holland, Amsterdam

EXTERNAL FIELD INDUCED CHAOS IN AN INFINITE SQUARE WELL POTENTIAL W.A. LIN and L.E. REICHL Center for Studies in Statistical Mechanics, University of Texas, Austin, TX 78712, USA Received 28 December 1984 Revised manuscript received 6 September 1985

We study the motion of a particle in an infinite square well potential in the presence of a monochromatic external field. The equations of motion of this system have a particularly simple structure compared to other driven nonlinear systems, and yet the system exhibits a transition to chaotic behavior. The critical amplitudes of the external field where 'breakdown' of KAM invariants occurs and large scale chaos sets in are computed by numerical experiment. They are found to be in good agreement with predictions based on a renormalization group scheme.

1. Introduction One of the most important recent developments for understanding how large scale chaos occurs in nonlinear conservative dynamical systems has been the renormalization group techniques introduced by Escande and Doveil [1]. In order to confirm the basic validity of the renormalization group approach, it is important to find dynamical systems on which the renormalization group methods can be tested unambiguously. In a recent series of papers L.E. Reichl and W.M. Zheng [2] and W.A. Lin and L.E. Reichl [3] have tested the renormalization group methods on the conservative Duffing system [2] and for a pendulum perturbed by a monochromatic dynamic external field [3]. For those systems the authors found that the renormalization group methods which use the so-called 'pendulum approximation' to generate the renormalization group transformation do not give good predictions for the 'breakdown' of the phase space, because of the existence of a separatrix in the unperturbed system. At present there is no known renormalization group scheme that can give accurate predictions for these systems. In this paper we wish to describe a dynamic system for which renormalization group methods,

based on the pendulum approximation, give extremely accurate predictions for the breakdown of the phase space and the onset of global chaos. The original work of Escande and Doveil was based on a very simple Hamiltonian which they constructed to model the behavior of a charged particle in the presence of plasma waves. They also showed that renormalization group methods work well for the standard map Hamiltonian. For both of these systems the starting Hamiltonian contains resonance terms with constant coefficients. The system we consider here consists of a particle trapped in an infinite square well potential and perturbed by a dynamic monochromatic external field. For this system, as we shall show, the resonance terms in the starting Hamiltonian again have constant coefficients, and renormalization group methods work well. We begin our discussion in section 2 with the unperturbed systems and we obtain the transformation to action and angle variables. In section 3 we turn on the perturbation and estimate the locations of induced primary resonances. In section 4 we reduce the many-resonance Hamiltonian into the paradigm Hamiltonian with only two resonance terms. In section 5 we compute the onset of breakdown numerically and compare with

0167-2789/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

146

W.A. Lin and L.E. Reichl/ External field induced chaos

P

theoretical estimates. In section 6 we make some concluding remarks. In the appendix we give some details of how the numerical experiment was performed.

2. Unperturbed system We consider a particle of mass ½ oscillating in an infinite square well potential. The Hamiltonian can be written as

Ho=P2+ b [ * l ( x - 1) + ~ ( - x -

1)1,

(b--' oo),

Fig. 1. A typical phase space trajectory for the Hamiltonian

H0.

(2.1) where x is position of the particle, p is the momentum, 71 is the unit step function ( ~ ( x ) = 1 for x > 0 and 71(x)=0 for x < 0), and b is the height of finite square well. The position and momentum are periodic functions of time with period • = 2/IP[ and can be written as x(t)=

- 1 + 2tE~0 sign [sin(~rt E~o)],

p ( t ) = ~ o sign[sin(~rt~o)],

(2.2)

3. Perturbed system Let's now turn on the perturbation and consider the following Hamiltonian: / 4 - - ~ /•2 +

a)+~l(-x-

a)]

(b--,

(3.1)

(2.3)

over the time interval [-~'/2, r/2], with proper initial conditions, where E 0 is the energy and sign [ ] means sign of the argument. A typical phase space trajectory is shown in fig. 1. We may perform a canonical transformation to action and angle variables (I, •): 2~ x = - 1 + rr s i g n [ s i n a i ,

(2.4)

~rI P=-U

(2.5)

where m is the mass,/3 is the momentum, ~ is the position of the particle, a is used to adjust the width of the well, ? is the time, ~ and ~2 are amplitude and frequency respectively of the external field. We can write this Hamiltonian in terms of dimensionless coordinates with which we introduce an arbitrary unit of energy c. After a canonical transformation:

f l = cH, sign [sin ~ ],

b[~(~-

c ~=--e,a

Yc= ax, b=bc,

~ = 2v~-mcp, ?=a

2~/-2~t,

~2= a1 2 ~ m I2,

with the Hamiltonian assumes the following form: I=

~r

'

(2.6)

7r2It

H = p 2 + b [*l(x - 1) + * l ( - x - 1)] + excos(~2t),

(2.7)

(3.2)

For ~ ~ [-~r, ~r]. x and p are periodic functions of # with period 2~r.

where, now, all quantities are dimensionless. Substituting eqs. (2.4) and (2.5) into eq. (3.2), we

v~- T '

W.A. Lin and L.E. Reichl/ External field induced chaos

147

obtain ~.212

H = - - 4 - + e x ( I , v~) cos ~2t.

(3,3)

Since x is a periodic function of #, we can expand it in terms of Fourier series and obtain H = 4~r2I2

,tt2 V 4 Eo~e n ~

1 cos (n@ - ~ t ) .

(3.4)

--00

n odd

We see that an infinite number of resonance zones are induced into the system by the presence of the external field. Unlike previous cases considered, resonances for this system have amplitudes which are independent of the action variable, I. For any e, resonances occur at energies E,~ determined by

-

-

~.'

d#=.W~ ~ = n-dT

(3.5)

P

? "~.~

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•

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a

Fig. 2 plots the resonance energies as a function of frequency. N o t e that all resonance zones converge to E o = 0. Therefore there is a stochastic layer near E o = 0. These resonance zones and the stochastic

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F i g . 2. R e s o n a n c e energies as a [unction o f the f r e q u e n c y of the e x t e r n a l field.

0 X

F i g . 3. P h a s e s p a c e trajectories for the H a m i l t o n i a n 1 2 = 5 , e = 0.05.

H

at

W.A. Lin and L.E. Reichl/ External field induced chaos

148

Table I A comparison between Ip2l obtained from theory and that from numerical experiments for 12= 5, e = 0.05

n

~2 IP,~I = ~

IP~I (fromplot)

1 3 5 7 9

1.5915 0.5305 0.3183 0.2274 0.1768

1.586 0.526 0,318 0,227 0,174

Table II A comparison between values of the full width, Ap, of primary resonance zones, n = 1,3,5, obtained from theory and obtained from numerical experiment for 12 ~ 5, E = 0.05 n

4 5p =~n2~-

5 p (from plot)

1 3 5

0.4026 0.1342 0.08053

0.400 0.133 0.0824

F = - ( I / n ) ( x l + ~2t), where x 1 is the new coordinate, the new Harniltonian would be a constant of motion. It is then easy to show that the full width of the separatrix is A I = (8/rr2n) 2 ~ . Making use of eq. (2.6), this corresponds to a maximum separation in p of Ap = (4/¢rn) 2 ~ . Table II gives a comparison of Ap's as given by this formula with those directly measured from the plot. The agreements between them are good.

4. Reduction to the paradigm Hamiltonian Suppose we wish to know the critical value of e where breakdown between n = ~ and n = ~ + 2 zones begins to occur. To a good approximation, we can make two resonance approximations on the Hamiltonian and obtain ~2/2

H - T layer are confirmed by our strobe plot based on the Hamiltonian, eq. (3,2) (see the appendix for details of computation), as shown in fig. 3 for 12 --- 5, e = 0.05. There are 23 different trajectories in the plot. Each trajectory contains 1000 points. Starting from high energy we see chains of one, three, five islands and their separatrices. Then chains of seven and nine islands, and at low energy we find a stochastic sea. Some daughter islands and K A M trajectories are also shown. The locations of primary resonances are fairly accurately given by eq. (3.5) as shown in table I, where we give a comparison of IpCl = ~ as given by eq. (3.5) with those directly measured from the plot. T o check if the widths of the resonance zone are accurately represented by the amplitudes of the resonance terms, one can make one-resonance approximation with Hamiltonian

H(X)=~r212

4

4~ 1 c o s ( n 0 - I 2 t ) , ¢r2n 2

+ 1 cos [(h + 2 ) t ~ - ~2t]}. (h + 2) 2

(4.1)

We then make a canonical transformation with generating function I F ( I, xx) = - -~( x x + ~2t ).

(4.2)

The old and new canonical variables are related through a = - -~- =

( x t + I2t),

(4.3~

aF I p, = _ ~ = ~.

(4-41

The new Hamiltonian is now

OF H 1 = H -~ 8t

(3.6)

where n is some positive odd integer. By making a canonical transformation with generating function

4e ( _ ~ cos( h ~ _ ~2t )

7r 2

-

rr2h2p21 4

×oos

4e{1 1 ~r2 h E C ° S X l + ( h + 2 ) 2

2

1/

(4:

W.,4. Lin and L.E. Reichl/ External fieM induced chaos

A n additional canonical transformation with: x ' = x 1,

t'=-

~

Table IIl A comparison between theory and numerical experiment for the critical e.

(h+2)h2~r2 h+2 412 Pl +

v' =

h=l

12t

Methods

(h + 2)2h2~r 2 8ga

(h + 2) 2 /-I1+

h=3

12 = 1

12 = 2

12 = 3

,12 = 5

Overlap criterion 0.0312 0.125 0.0703 0.195 Renormalization group 0.0176 0.0703 0.0360 0.100 Numerical experiment 0.018-0.019 0,072-0.075 0.035-0.038 0.095-0.1

and

H'

149

8

will bring the H a m i l t o n i a n to the form of the p a r a d i g m H a m i l t o n i a n of ref. 1:

H'

v '~

=

2

(~+2)2E COS X

(a)

2122

,

,q

v

,

t

I

,

|

t (4.6) L Oa

T h e half-width of the zone h is h+2 X = -----if- 2 ~

(4.7)

P

a n d that for zone h + 2 is ^

Y = -~ 2v~-.

(4.8) |

5. Comparison between numerical results and theoretical estimates T a b l e I I I lists the critical values of e. The colu m n s for h = 1 correspond to b r e a k d o w n between n = 1 and n = 3 zones, while those for h --- 3 corres p o n d to b r e a k d o w n between n = 3 and n = 5 zones. T h e values for the overlap criterion [4, 5] are d e t e r m i n e d f r o m the condition X + Y = 1. F o r the r e n o r m a l i z a t i o n group scheme, the values corres p o n d to h -- 1 are obtained f r o m the k = 3 curve in fig. 8 of ref. 1 (J. Stat. Phys.). F o r h = 3, a curve with k = 35- are interpolated from the 4 curves of the s a m e figure. T h e critical e is then determined f r o m this curve.

E0 0

t~

t( 104) Fig. 4. a) A trajectory with initial conditions: x ( t = O)= 1, p ( t = 0 ) = - 0 . 2 7 7 at 12= 3, e =0.035. There is a total of 5000 points, b) Energy as a function of time for the same trajectory.

150

W . A . L i n a n d L, E. R e i c h l / E x t e r n a l f i e M i n d u c e d chaos

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Fig. 5. a) A trajectory with initial conditions: x ( t = 0 ) = 1, p ( t = 0) = - 0 . 2 9 5 at 12 = 3, r = 0.038. There is a total of 5000

points, b) E n e r g y as a function of time for the same trajectory.

N u m e r i c a l solutions of the phase space trajectories are o b t a i n e d at discrete times with interval 2~r/~2. T o find whether there is b r e a k d o w n between n = h and n = h + 2 zones, we start the trajectories in the " r o t a t i o n " part of the n = h separatrix layer near the hyperbolic fixed point at x = 1. W e try to start the trajectories at points as close as possible to the n = h + 2 zone as long as we k n o w that these points are connected with the separatrix layer of n = h zone. I n this way we can speed up the trajectory to enter n = h + 2 zone if b r e a k d o w n has already occurred. T o illustrate h o w we obtained the critical e's, let us take h = 3 and ~2 = 3 as an example. Fig. 4a

o

o'.s

t t(lO

4)

Fig. 6. a) A trajectory with initial conditions: x ( t = 0 ) = 1, p(t=0)= - 0 . 3 at ~2=3, e = 0 . 0 5 . There is a total of 5000 points, b) E n e r g y as a function of time for the same trajectory.

shows the trajectory at e = 0.035. It starts at p = - 0 . 2 7 7 and x = 1. After 5000 points the traject o r y remains attached to n ~ 3 zone. Fig. 4b gives p l o t of energy as a function of time. The part witl~ larger oscillations in energy corresponds to motior inside the separatrix. The part with smaller oscilla. tion in energy corresponds to m o t i o n outside th~ separatrix. W e see the particle spends a long tim~ outside the separatrix without entering n = 5 zone T h e r e seems to be a barrier that forbids it to do so C h a n g i n g initial conditions a small a m o u n t doe: n o t alter the situation (not shown). This gives g o o d indication that there is no breakdown at thi

151

W.A. Lin and L.E. Reichl/ External field induced chaos

value of e. Increasing e to 0.038, fig. 5a shows that the trajectory is able to enter the domain of the n = 5 zone. Starting from the n = 3 zone the trajectory first makes contact with the secondary zones, then it connects with the n = 5 zone. It is apparent that there is no overlap between the separatrices of the n = 3 and n = 5 main resonances, but breakdown has already occurred. The energy plot, fig. 5b, gives us some idea about the slow diffusion of the trajectory to lower energy. Increasing e further to e = 0.05, fig. 6a, b tells us that the trajectory quickly reaches the domain of n = 5 zone, indicating breakdown has occurred. The above analysis shows that the critical e is somewhere between 0.035 and 0.038. Other critical values of e are obtained in the sam e way.

6. Conclusion The existence and location of the resonance zones and the stochastic layer at low energy induced by the dynamical external field as predicted by the theory are confirmed by the numerical experiment. For cases considered we have found that the renormalization group predictions for the onset of breakdown are in good agreement with numerical experiments. The primary reason for this is that for the square well system the amplitudes of the resonance zones are independent of action and therefore the Hamiltonian from the outset is in pendulum-like form, whereas for our previous cases, one needs to introduce the pendulum approximation to reduce the Hamiltonian to the standard Hamiltonian of Escande and Doveil. As we found in previous papers, in systems for which the amplitude of the resonance zones depend on action one must check, at a given frequency of external field, whether or not the pendulum approximation gives a good estimate for both the size and position of each resonance zone. If so, the techniques of Escande and Doveil should give good results. It is interesting that even though our system has an infinite number of primary resonances, whereas

the model system of Escande and Doveil has only two, the renormalization analysis continues to give excellent results. This indicates, as noted in our previous paper, that the behavior of a given region of phase space is determined almost entirely by the two neighboring primary resonances. Finally, it appears that the square well system provides a unique system for understanding the extent, to which chaotic-like behavior remains in quantum systems. The perturbed quantum square well system is amenable to both analytic and numerical calculation. Work on the quantum version is currently in progress by the authors.

Appendix In this appendix we give some details of computation for the numerical experiment. Consider the motion of the particle before it hits the wall. The equation of motion is .~ = - 2ecos $2t.

(A.1)

This has a general solution

x ( t ) = c l + c2t + p ( t ) -- ~

2~

- ~ sin I2t,

cos

(A.2)

where C 1 and C 2 are constants determined from the initial conditions. We see from eq. (A.2) that for t = l(2¢r/D) with l being an integer, p = constant. This is the reason why at low energies we see many equally spaced clots at the same momentum. Once it hits the wall i t has to flip the m o m e n t u m and move with a new set of C1 and C 2. It is then necessary to determine when it's going to hit the wall. This is computed numerically by using IMSL subroutine ZBRENT. When 112C2/2e1 < 1 the particle makes forward and backward motions on its way toward the wall. So in addition to x at Dt = 21~r, one must calculate

152

W.A. Lin and L.E. Reichl/ External fieM induced chaos

x at

Ot~2) = ~r -

precision. T h e subroutine Z B R E N T is modified to d o u b l e precision and achieves 28 significant figures for the times at the walls.

(oc l] ' sin- 1 ~~

for

C 2 > O,

Acknowledgements or

(

I2t~° = ~r - sin- 1 ~ 2e

),

( oc2 ), ~2t~2)= 2~r + sin -1 ~ --)--i-e

for

C 2_< O,

to check if the position h a s gone outside of the walls, t~1) a n d t~2) are the times when the particle m a k e s turns. If the particle has reached the dom a i n outside the walls, the time at this point and the time at the previous point are supplied as input to Z B R E N T to find the time at the wall. We then calculate the m o m e n t u m at this time, flip the m o m e n t u m a n d continue to the next point with a new set of C 1 and C 2. N u m e r i c a l c o m p u t a t i o n s are p e r f o r m e d with C D C dual c y b e r 170/750 with 29 digits for double

T h e authors wish to thank U.S, Naval Air Systems C o m m a n d Contract No. MDA903-85-C-0029 for partial s u p p o r t of this work. Author W.A. Lin wishes to t h a n k the Welch F o u n d a t i o n of Texas for partial support, L.E. Reichl wishes to thank G. S c h m i d t for a useful conversation.

References [1] D.F. Escande and F. Doveil, J. Stat. Phys. 26 (1981) 257; Phys. Lett. A83 (1981) 307; D.F. Escande, Phys. Scripta T2/1 (1982) 126. [2] L.E. Reichl and W.M. Zheng, Phys. Rev. A29 (1984) 2186; A30 (1984) 1068. [3] W.A. Lin and L.E. Reichl, Phys. Rev. A31 (1985) 1136. [4] B.V. Chirikov, Phys. Reports 52 (1979) 263. [5] See also L.H. Walker and J. Ford, Phys. Rev. 188 (1969) 416.