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On the analytic structure of the Henon-Heiles system
Volume 85A, number 4
ON THE ANALYTIC STRUCTURE
28 September 1981
PHYSICS LETTERS
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 solutions 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 equations 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 hamiltonian. The Painleve property would again appear to be successful in determining when the system is integrable; however, our main result is to show the existence and structure of a natural boundary in the nonintegrable 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, University of Nebraska, Lincoln, NE 68588, USA.
a natural boundary are clearly discussed in the standard text by Hille [4]. We write the Henon-Heiles hamiltonian in the gener, al form H=:(~+~+X2ty2)tDx2y-:~3, and the equations
(1)
of motion as
;;‘=-x-2Dxy,
jL_y-&t@.
(2)
The leading-order behaviour of the solutions at a singularity at time t = t, is determined by making the substitution x=a(t-t,)a,
y=b(t-r,)P,
and equating most singular terms. It is important note that there are two possible behaviours a=-2
9
p=-2,
to
(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 211
Volume 85A, number 4
D=-1,
PHYSICS LETTERS
28 September 1981
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 solutions 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 recently developed ATSMCC integrator [8] modified to integrate along paths in the complex plane. This is a highly efficient Taylor-series method that yields detailed information about the position and order of the singularity 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 levelone singularities are less evenly distributed. For many initial conditions examined, corresponding to both regular and irregular motion, the sequence of levelone singularities is as follows: -2,;,-2,-2,;,-2,-24-2
)...,
where the numbers refer to the real parts of the leadingorder behaviour (Ygiven in (5) and (6). A typical sequence 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 associate the following geometrical construction. (The following description should be read with close reference 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”; 212
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 levelone 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 levelone 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.
PHYSICS LETTERS
Volume 85 A, number 4
D = ln(2 cosec cp)/ln(2) = 1.1419...
Im(t)
I
0
1.0
1
20
Re(t)
Fig. 3. Some singularity spirals associated with the regular trajectory in fig. la. The numbers refer to the level of singularity; open dots: singularities of order -2, crosses: singularities 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 follows 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 boundary. 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. Following Mandelbrot, we calculate, using our geometrical construction (fig. 2), the fractal dimensional of the natural boundary to be
28 September 1981
,
where here we take cpto be 25”. For bounded motion, the structure of the natural boundary does not change with energy. All that happens is that it moves slightly closer to the real axis as the energy is increased. However, as the system parameters 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 particular nonlinear dynamical system. Our results indicate that it is possible to obtain a rather precise characterization of the singular set of the Henon-Heiles system. Furthermore, this structure, a multi-sheeted natural boundary determined by a self-similar construction, appears to be unique in the mathematical literature 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 hamiltonian 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.
213
ON THE ANALYTIC STRUCTURE
28 September 1981
PHYSICS LETTERS
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 solutions 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 equations 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 hamiltonian. The Painleve property would again appear to be successful in determining when the system is integrable; however, our main result is to show the existence and structure of a natural boundary in the nonintegrable 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, University of Nebraska, Lincoln, NE 68588, USA.
a natural boundary are clearly discussed in the standard text by Hille [4]. We write the Henon-Heiles hamiltonian in the gener, al form H=:(~+~+X2ty2)tDx2y-:~3, and the equations
(1)
of motion as
;;‘=-x-2Dxy,
jL_y-&t@.
(2)
The leading-order behaviour of the solutions at a singularity at time t = t, is determined by making the substitution x=a(t-t,)a,
y=b(t-r,)P,
and equating most singular terms. It is important note that there are two possible behaviours a=-2
9
p=-2,
to
(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 211
Volume 85A, number 4
D=-1,
PHYSICS LETTERS
28 September 1981
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 solutions 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 recently developed ATSMCC integrator [8] modified to integrate along paths in the complex plane. This is a highly efficient Taylor-series method that yields detailed information about the position and order of the singularity 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 levelone singularities are less evenly distributed. For many initial conditions examined, corresponding to both regular and irregular motion, the sequence of levelone singularities is as follows: -2,;,-2,-2,;,-2,-24-2
)...,
where the numbers refer to the real parts of the leadingorder behaviour (Ygiven in (5) and (6). A typical sequence 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 associate the following geometrical construction. (The following description should be read with close reference 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”; 212
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 levelone 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 levelone 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.
PHYSICS LETTERS
Volume 85 A, number 4
D = ln(2 cosec cp)/ln(2) = 1.1419...
Im(t)
I
0
1.0
1
20
Re(t)
Fig. 3. Some singularity spirals associated with the regular trajectory in fig. la. The numbers refer to the level of singularity; open dots: singularities of order -2, crosses: singularities 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 follows 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 boundary. 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. Following Mandelbrot, we calculate, using our geometrical construction (fig. 2), the fractal dimensional of the natural boundary to be
28 September 1981
,
where here we take cpto be 25”. For bounded motion, the structure of the natural boundary does not change with energy. All that happens is that it moves slightly closer to the real axis as the energy is increased. However, as the system parameters 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 particular nonlinear dynamical system. Our results indicate that it is possible to obtain a rather precise characterization of the singular set of the Henon-Heiles system. Furthermore, this structure, a multi-sheeted natural boundary determined by a self-similar construction, appears to be unique in the mathematical literature 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 hamiltonian 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.
213