Volume 85A, number 4 PHYSICS LETTERS 28 September 1981

ON THE ANALYTIC STRUCTURE OF THE HENON-HEILES SYSTEM *

Y.F. CHANG ‘, M. TABOR and J. WEISS Center for Studies of Nonlinear Dynamics, La Jolla Institute, La Jolla, CA 92037, USA

and

G. CORLISS Department of Mathematics and Statistics, Marquette University, Milwaukee, WI 53233, USA

Received 5 May 1981 Revised manuscript received 13 July 1981

Solutions of the Henon-Heiles hamiltonian that are analytically continued into the complex time domain are found to possess a neutral boundary with a self-similar structure.

Recent results suggest that there is an intimate connection between the integrability and analytic structure of dynamical systems. By looking for solu- tions that exhibited the “Painleve property”, i.e. the only movable singularities are poles, we were able to identify those sets of adjustable parameter values for which the Lorenz system is integrable. Furthermore a detailed analysis of the nonintegrable regimes showed them to be characterized by a very rich multisheeted structure [ I]. The use of the Painleve’ property to determine integrability was suggested by Ablowitz et al. [2] in the context of partial differential equa-

tions and also applied to the Lorenz system by Segur

[3]. Here we report the preliminary results of a study of the analytic structure of the Henon-Heiles hamil-

tonian. The Painleve property would again appear to be successful in determining when the system is inte- grable; however, our main result is to show the exis-

tence and structure of a natural boundary in the non- integrable regimes. The existence of such a structure in nonintegrable systems has recently been discussed by Greene and Percival [5]. The theory of differential equations in the complex domain and the concept of

* Supported by ONR Contract N-00014-79-C-0537. 1 On leave from the Department of Computer Science, Uni-

versity of Nebraska, Lincoln, NE 68588, USA.

a natural boundary are clearly discussed in the stan- dard text by Hille [4].

We write the Henon-Heiles hamiltonian in the gener,

al form

H=:(~+~+X2ty2)tDx2y-:~3, (1)

and the equations of motion as

;;‘=-x-2Dxy, jL_y-&t@. (2)

The leading-order behaviour of the solutions at a singu-

larity at time t = t, is determined by making the sub- stitution

x=a(t-t,)a, y=b(t-r,)P,

and equating most singular terms. It is important to note that there are two possible behaviours

a=-2 9 p=-2, (3)

and

cu=; +;(l-48D/C)y2, p=-2, (4)

Following the procedure of Ablowitz et al., it is fairly easy to show that the solutions to (1) are Painleve, i.e. expressible as simple Laurent series, for the parameter values

D=-1, C=l,

and

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Volume 85A, number 4 PHYSICS LETTERS 28 September 1981

D=-1, C=6.

These results have also been reported by Bountis et al. [6] who provide numerical evidence supporting the conjecture that the system is integrable in the latter case. (In the former case the system is separable and hence integrable and in the latter case Greene has

recently found the second integral of motion [7] .) Here we examine the analytic structure of the solu- tions of (1) with D = C = 1 for which the system is

nonintegrable and displays a well studied mixture of regular and irregular trajectories. For those parameter values the leading-order behaviours are

cr=-2, p=-2, (9

and

(u=;&ffl, p=-2. (6)

Our numerical results are obtained using the recent-

ly developed ATSMCC integrator [8] modified to inte- grate along paths in the complex plane. This is a highly efficient Taylor-series method that yields detailed in- formation about the position and order of the singu- larity nearest the point of integration.

For clarity of discussion, we define the “level-one” singularities to be those that are closest to the real (time) axis. For a regular trajectory, the level-one singularities are positioned in a narrow band parallel to the real axis. For an irregular trajectory, the level- one singularities are less evenly distributed. For many initial conditions examined, corresponding to both

regular and irregular motion, the sequence of level- one singularities is as follows:

-2,;,-2,-2,;,-2,-24-2 )...,

where the numbers refer to the real parts of the leading-

order behaviour (Y given in (5) and (6). A typical se- quence of level-one singularities and the associated

real time solution is shown in fig. 1. Occasionally other sequences of singularities are found but in all cases both leading-order behaviours are present.

,Beyond the level-one singularities, a remarkable singularity structure develops with which we can asso- ciate the following geometrical construction. (The following description should be read with close refer- ence to fig. 2.) The “level-two” singularities are found to be positioned at the apex of an (almost perfect) isosceles triangle with base angles equal to 20-23”;

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Fig. 1. Real time solutions [solid lines: x(t) and dotted lines : y(t)] and the corresponding level-one singularities for (a) a regular trajectory at E = 0.125 and (b) an irregular trajectory at E = 0 .125. Open dots: singularities of leading order -2 and crosses: singularities of leading order i + ii@.

the base being the line joining any two adjacent level- one singularities. If the level-one singularities are both or order -2, the level-two singularity is of order l/2.

If the base is composed of one order -2 and one order l/2 singularity, the corresponding level-two singularity is of order -2. The “level-three” singularities are found at the apex of. an isosceles triangle with base angles of about 25”; the base now being the line joining a level- one to a level-two singularity. This construction is then

Fig. 2. The geometrical construction of the natural boundary. The numbers correspond to the level of singularity.

Volume 85 A, number 4 PHYSICS LETTERS 28 September 1981

Im(t)

I 1

0 1.0 20 Re(t)

Fig. 3. Some singularity spirals associated with the regular trajectory in fig. la. The numbers refer to the level of singu- larity; open dots: singularities of order -2, crosses: singulari- ties of order f + ii&i.

continued ad infinitum; the base angle of the isosceles

triangles remain constant at approximately 2.5”. In general, each level-n singularity will be found at the apex of an isosceles triangle with a level+ - 1) singularity in one base corner and any level-m (m < II - 1) in the other. There will be twice as many level-n singularities as level -(n - 1) singularities,. As one fol- lows successive levels of singularities, it is as though one is winding up a “spiral staircase”. By “level-ten”,

the singularity would appear to be nearer the real axis than the original level-one singularity but it must be remembered that one is now on a higher portion of the riemannian manifold from which the real axis is no longer directly accessible.

The consequence of this construction is that a double spiral (one clock-wise, and one anti-clockwise) of singularities emanates from evev singularity. A typical example is shown in fig. 3. At each level, the

number of singularities doubles and the total number thus approaches infinity in a very short time in the

imaginary direction, thereby leading to a natural bound- ary. That is, given this structure, it is impossible to continue the solution beyond a fixed finite distance in the imaginary direction of the independent variable, t.

This self-similar structure is reminiscent of the “fractal” objects described by Mandelbrot 191. Fol- lowing Mandelbrot, we calculate, using our geometrical construction (fig. 2), the fractal dimensional of the natural boundary to be

D = ln(2 cosec cp)/ln(2) = 1.1419... ,

where here we take cp to be 25”. For bounded motion, the structure of the natural

boundary does not change with energy. All that hap- pens is that it moves slightly closer to the real axis as the energy is increased. However, as the system param- eters D and Care varied the structure of the natural

boundary undergoes some remarkable changes. These results will be described in detail elsewhere [lo].

Recently, Frisch and Morf [l 1 ] have shown that

the structure of the singular set is directly related to the high-frequency, intermittent behaviour of a par- ticular nonlinear dynamical system. Our results indi-

cate that it is possible to obtain a rather precise char- acterization of the singular set of the Henon-Heiles

system. Furthermore, this structure, a multi-sheeted natural boundary determined by a self-similar construc- tion, appears to be unique in the mathematical litera- ture of differential equations [4]. Finally, we remark that a rigorous characterization of the singular set of a dynamical system may provide a means of obtaining

insight into its various properties.

References

[l] M. Tabor and J. Weiss, Analytic structure of the Lorenz system (1980), to be published in Phys. Rev. A.

[ 21 M.J. Ablowitz, A. Ramani and H. Segur, J. Math Phys. 21 (1980) 715; see also: S. Kowalevsky, Acta Math. 14 (1980) 81.

[ 31 H. Segur, SoIitons and the inverse scattering transform, Lectures Intern. School of Physics “Enrico Fermi” (Varema, Italy, July 1980).

[4] E. Hille, Ordinary differential equations in the complex domain, Wiley-Interscience Series in Pure and applied mathematics (Wiley, New York, 1976).

[S] J.M. Greene and J.C. Percival, Hamiltonian maps in the complex plane, Princeton Plasma Physics Laboratory report DPPL-1744 (January 1981).

[6] T. Bountis, H. Segur and F. Vivaldi, Integrable hamil- tonian systems and the PainIeve property, preprint (1981).

[ 7 ] J. Greene, private communication. [8] Y.F. Chang and G. Corliss, J. Inst. Math. Appl. 25 (1980)

349. [9] B. Mandelbrot, Fractals: form, chance and dimension

(Freeman, San Francisco, 1977). [lo] Y.F. Chang, M. Tabor and J. Weiss, in preparation. [ll] U. Frisch and R. Morf, Phys. Rev. A 23 (1981) 2673.

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