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Illustrative Session for the Poincare Package

Note: The results in this worksheet were generated on a Pentium-II 400 PC with 128 MB RAM. For testing this package on a computer that has a slower processor, try changing the stepsize, iterations, or even the time interval, as explained in the help pages.

$restart$

>	$with (DEtools, poincare, generate_ic, zoom, hamilton_eqs) &colon;$

The Toda Hamiltonian

Reference: A.J. Lichtenberg and M.A. Lieberman, "Regular and Stochastic Motion", Applied Mathematical Sciences 38 (New York: Springer Verlag, 1994).

>	$H := \frac{1 ({p1}^{2} + {p2}^{2})}{2} + \frac{1 ({&ExponentialE;}^{2 q2 + 2 \sqrt{3} q1} + {&ExponentialE;}^{2 q2 - 2 \sqrt{3} q1} + {&ExponentialE;}^{- 4 q2})}{24} - \frac{1}{8}$

$H := \frac{1}{2} {p1}^{2} + \frac{1}{2} {p2}^{2} + \frac{1}{24} {&ExponentialE;}^{2 q2 + 2 \sqrt{3} q1} + \frac{1}{24} {&ExponentialE;}^{2 q2 - 2 \sqrt{3} q1} + \frac{1}{24} {&ExponentialE;}^{- 4 q2} - \frac{1}{8}$

(1.1)

>	$H, t = - 150 .. 150, \{[0, 0.1, 1.4, 0.1, 0]\} &semi;$

$\frac{1}{2} {p1}^{2} + \frac{1}{2} {p2}^{2} + \frac{1}{24} {&ExponentialE;}^{2 q2 + 2 \sqrt{3} q1} + \frac{1}{24} {&ExponentialE;}^{2 q2 - 2 \sqrt{3} q1} + \frac{1}{24} {&ExponentialE;}^{- 4 q2} - \frac{1}{8}, t = - 150 .. 150, \{[0, 0.1, 1.4, 0.1, 0]\}$

(1.2)

>	$poincare (, stepsize = 0.05, iterations = 5) &semi;$

$_________________________________________________$

$H = .99005020, Initial conditions:, t = 0, p1 = 0.1, p2 = 1.4, q1 = 0.1, q2 = 0$

$Number of points found crossing the (p1,q1) plane: 127$

$Maximum H deviation : .5740000000e-5 %$

$_________________________________________________$

$Time consumed: 4 seconds$

Figure 1.a. shows a 2-D surface-of-section (2PS) over the q2=0 plane, with 127 intersection points lying on smooth curves.

>	$poincare (, stepsize = 0.05, iterations = 5, scene = [p2, q2])$

$_________________________________________________$

$H = .99005020, Initial conditions:, t = 0, p1 = 0.1, p2 = 1.4, q1 = 0.1, q2 = 0$

$Number of points found crossing the (p2,q2) plane: 146$

$Maximum H deviation : .5730000000e-5 %$

$_________________________________________________$

$Time consumed: 4 seconds$

Figure 1.b is a 2PS over the q1=0 plane with 146 intersection points. The smoothness of the curves in both (p,q) planes is related to the integrability of the system.

A Poincare space-of-section corresponding to Figure1.a can be manipulated with the mouse to obtain the following illustrative perspectives:

>	$F2 := poincare (H, t = - 100 .. 100, \{[0, 0.1, 1.4, 0.1, 0]\}, stepsize = 0.1, iterations = 4, scene = [p1 = - 1.5 .. 1.5, q1 = - 1.5 .. 1.5, q2 = - 1.2 .. 1.3], 3) &colon;$

$_________________________________________________$

$H = .99005020, Initial conditions:, t = 0, p1 = 0.1, p2 = 1.4, q1 = 0.1, q2 = 0$

$Maximum H deviation : .3758100000e-3 %$

$_________________________________________________$

$Time consumed: 3 seconds$

(1.3)

$F2$

Figure 2.a - 3-D projection of a surface of section (3PS) showing a KAM surface of regular trajectories. The plot has been manipulated with the mouse to produce a view at $Θ = - 20$ , $Φ = 75$ .

$F2$

Figure 2.b The same figure was manipulated with the mouse to display a plane projection of the 3PS (at $Θ = 0$ , $Φ = - 180$ ) showing how the intersection points are joined outside the 2PS.

Another indication of the integrability of the system is that regular curves exist whatever the value of H. As an example of this, a surface-of-section (one solution curve), and a related 3-D projection, at H=256, can be built as follows:

A set with one list of initial conditions satisfying the Hamiltonian constraint (H0=256):

>	$ics_256 := generate_ic (H, \{t = 0, p2 = 22, q1 = 0, q2 = 0, energy = 256\}, 1) &colon;$

>	$poincare (H, t = - 50 .. 50, ics_256, stepsize = 0.005, iterations = 4, scene = [p2, q2]) &semi;$

$_________________________________________________$

$H = 256.00000, Initial conditions:, t = 0., p1 = 5.291502622, p2 = 22., q1 = 0., q2 = 0.$

$Number of points found crossing the (p2,q2) plane: 342$

$Maximum H deviation : .3289900000e-3 %$

$_________________________________________________$

$Time consumed: 16 seconds$

Figure 3.a shows smooth curves on the 2PS, q1=0 plane.

>	$F3b := poincare (H, t = 0 .. 20, ics_256, stepsize = 0.01, iterations = 4, scene = [p2, q2, q1], 3) &colon;$

$_________________________________________________$

$H = 256.00000, Initial conditions:, t = 0., p1 = 5.291502622, p2 = 22., q1 = 0., q2 = 0.$

$Maximum H deviation : .4215140000e-2 %$

$_________________________________________________$

$Time consumed: 3 seconds$

(1.4)

>	$zoom (F3b, - 30 .. 30, - 3 .. 5, - 4 .. 4)$

Figure 3.b is a 3PS corresponding to Figure 3.a, displaying a KAM surface constituted by just one regular curve. The plot has been manipulated with the mouse to produce a view at $Θ = 100$ , $Φ = 40$ , and the Projection was set to Far (in menu bar when the plot is selected.)

The Henon-Heiles Hamiltonian

Reference: M. Henon and C. Heiles, The Astronomical Journal, 69 (1963) 73.

>	$H := \frac{1 ({p1}^{2} + {p2}^{2} + {q1}^{2} + {q2}^{2})}{2} + {q1}^{2} q2 - \frac{{q2}^{3}}{3}$

$H := \frac{1}{2} {p1}^{2} + \frac{1}{2} {p2}^{2} + \frac{1}{2} {q1}^{2} + \frac{1}{2} {q2}^{2} + {q1}^{2} q2 - \frac{1}{3} {q2}^{3}$

(2.1)

Well-studied surfaces-of-section, presented in several treatises of chaos, with H equal to 1/24, 1/18, 1/12, 1/8, 1/7, and 1/6, are obtained here by using the generate_ic and poincare commands, as follows.

To start with, six sets, related to each value of H respectively, with three different initial conditions each, are generated:

>	$for h in [1 / 24, 1 / 18, 1 / 12, 1 / 8, 1 / 7, 1 / 6] do ics [h] ≔ generate_ic (H, \{t = 0, p2 = 0.1, q2 = - 0.2 .. 0.2, q1 = - 0.2 .. - 0.1, energy = h\}, 3) end do$

${ics}_{\frac{1}{24}} := \{[0., 0.06324555320, 0.1, - 0.2, - 0.2], [0., 0.1570562532, 0.1, - 0.1000000000, 0.2000000000], [0., 0.2254624877, 0.1, - 0.1500000000, 0.]\}$

${ics}_{\frac{1}{18}} := \{[0., 0.1782632261, 0.1, - 0.2, - 0.2], [0., 0.2290075205, 0.1, - 0.1000000000, 0.2000000000], [0., 0.2803767307, 0.1, - 0.1500000000, 0.]\}$

${ics}_{\frac{1}{12}} := \{[0., 0.2955221368, 0.1, - 0.2, - 0.2], [0., 0.3286335345, 0.1, - 0.1000000000, 0.2000000000], [0., 0.3662876829, 0.1, - 0.1500000000, 0.]\}$

${ics}_{\frac{1}{8}} := \{[0., 0.4131182236, 0.1, - 0.2, - 0.2], [0., 0.4374166587, 0.1, - 0.1000000000, 0.2000000000], [0., 0.4663689527, 0.1, - 0.1500000000, 0.]\}$

${ics}_{\frac{1}{7}} := \{[0., 0.4542917041, 0.1, - 0.2, - 0.2], [0., 0.4764951408, 0.1, - 0.1000000000, 0.2000000000], [0., 0.5032040201, 0.1, - 0.1500000000, 0.]\}$

${ics}_{\frac{1}{6}} := \{[0., 0.5039841267, 0.1, - 0.2, - 0.2], [0., 0.5240865070, 0.1, - 0.1000000000, 0.2000000000], [0., 0.5484827558, 0.1, - 0.1500000000, 0.]\}$

(2.2)

After that, surfaces-of-section with around 180 points, calculated in approximately 25 seconds each, with percentile H-deviations ~ $10^{- 5}$ %, can be obtained:

>	$for h in [1 / 24, 1 / 18, 1 / 12, 1 / 8, 1 / 7, 1 / 6] do F4 [h] ≔ poincare (H, t = - 300 .. 300, ics [h], stepsize = .1, iterations = 3, scene = [p2 = - .5 .. 0.5, q2 = - .5 .. 0.5]) &colon; end do$

$_________________________________________________$

$H = .41666667e-1, Initial conditions:, t = 0., p1 = 0.06324555320, p2 = 0.1, q1 = - 0.2, q2 = - 0.2$

$Number of points found crossing the (p2,q2) plane: 184$

$Maximum H deviation : .2087000000e-4 %$

$_________________________________________________$

$H = .41666667e-1, Initial conditions:, t = 0., p1 = 0.1570562532, p2 = 0.1, q1 = - 0.1000000000, q2 = 0.2000000000$

$Number of points found crossing the (p2,q2) plane: 193$

$Maximum H deviation : .3053000000e-4 %$

$_________________________________________________$

$H = .41666667e-1, Initial conditions:, t = 0., p1 = 0.2254624877, p2 = 0.1, q1 = - 0.1500000000, q2 = 0.$

$Number of points found crossing the (p2,q2) plane: 189$

$Maximum H deviation : .2323000000e-4 %$

$_________________________________________________$

$Time consumed: 9 seconds$

${F4}_{\frac{1}{24}} := PLOT (...)$

$_________________________________________________$

$H = .55555556e-1, Initial conditions:, t = 0., p1 = 0.1782632261, p2 = 0.1, q1 = - 0.2, q2 = - 0.2$

$Number of points found crossing the (p2,q2) plane: 187$

$Maximum H deviation : .2400000000e-4 %$

$_________________________________________________$

$H = .55555556e-1, Initial conditions:, t = 0., p1 = 0.2290075205, p2 = 0.1, q1 = - 0.1000000000, q2 = 0.2000000000$

$Number of points found crossing the (p2,q2) plane: 194$

$Maximum H deviation : .3570000000e-4 %$

$_________________________________________________$

$H = .55555556e-1, Initial conditions:, t = 0., p1 = 0.2803767307, p2 = 0.1, q1 = - 0.1500000000, q2 = 0.$

$Number of points found crossing the (p2,q2) plane: 188$

$Maximum H deviation : .2565000000e-4 %$

$_________________________________________________$

$Time consumed: 9 seconds$

${F4}_{\frac{1}{18}} := PLOT (...)$

$_________________________________________________$

$H = .83333333e-1, Initial conditions:, t = 0., p1 = 0.2955221368, p2 = 0.1, q1 = - 0.2, q2 = - 0.2$

$Number of points found crossing the (p2,q2) plane: 181$

$Maximum H deviation : .2751000000e-4 %$

$_________________________________________________$

$H = .83333333e-1, Initial conditions:, t = 0., p1 = 0.3286335345, p2 = 0.1, q1 = - 0.1000000000, q2 = 0.2000000000$

$Number of points found crossing the (p2,q2) plane: 195$

$Maximum H deviation : .4299000000e-4 %$

$_________________________________________________$

$H = .83333333e-1, Initial conditions:, t = 0., p1 = 0.3662876829, p2 = 0.1, q1 = - 0.1500000000, q2 = 0.$

$Number of points found crossing the (p2,q2) plane: 187$

$Maximum H deviation : .3134000000e-4 %$

$_________________________________________________$

$Time consumed: 9 seconds$

${F4}_{\frac{1}{12}} := PLOT (...)$

$_________________________________________________$

$H = .12500000, Initial conditions:, t = 0., p1 = 0.4131182236, p2 = 0.1, q1 = - 0.2, q2 = - 0.2$

$Number of points found crossing the (p2,q2) plane: 164$

$Maximum H deviation : .3097000000e-4 %$

$_________________________________________________$

$H = .12500000, Initial conditions:, t = 0., p1 = 0.4374166587, p2 = 0.1, q1 = - 0.1000000000, q2 = 0.2000000000$

$Number of points found crossing the (p2,q2) plane: 193$

$Maximum H deviation : .5108000000e-4 %$

$_________________________________________________$

$H = .12500000, Initial conditions:, t = 0., p1 = 0.4663689527, p2 = 0.1, q1 = - 0.1500000000, q2 = 0.$

$Number of points found crossing the (p2,q2) plane: 185$

$Maximum H deviation : .4168000000e-4 %$

$_________________________________________________$

$Time consumed: 9 seconds$

${F4}_{\frac{1}{8}} := PLOT (...)$

$_________________________________________________$

$H = .14285714, Initial conditions:, t = 0., p1 = 0.4542917041, p2 = 0.1, q1 = - 0.2, q2 = - 0.2$

$Number of points found crossing the (p2,q2) plane: 166$

$Maximum H deviation : .3943000000e-4 %$

$_________________________________________________$

$H = .14285714, Initial conditions:, t = 0., p1 = 0.4764951408, p2 = 0.1, q1 = - 0.1000000000, q2 = 0.2000000000$

$Number of points found crossing the (p2,q2) plane: 191$

$Maximum H deviation : .5377000000e-4 %$

$_________________________________________________$

$H = .14285714, Initial conditions:, t = 0., p1 = 0.5032040201, p2 = 0.1, q1 = - 0.1500000000, q2 = 0.$

$Number of points found crossing the (p2,q2) plane: 180$

$Maximum H deviation : .4462000000e-4 %$

$_________________________________________________$

$Time consumed: 9 seconds$

${F4}_{\frac{1}{7}} := PLOT (...)$

$_________________________________________________$

$H = .16666667, Initial conditions:, t = 0., p1 = 0.5039841267, p2 = 0.1, q1 = - 0.2, q2 = - 0.2$

$Number of points found crossing the (p2,q2) plane: 190$

$Maximum H deviation : .4426000000e-4 %$

$_________________________________________________$

$H = .16666667, Initial conditions:, t = 0., p1 = 0.5240865070, p2 = 0.1, q1 = - 0.1000000000, q2 = 0.2000000000$

$Number of points found crossing the (p2,q2) plane: 192$

$Maximum H deviation : .5544000000e-4 %$

$_________________________________________________$

$H = .16666667, Initial conditions:, t = 0., p1 = 0.5484827558, p2 = 0.1, q1 = - 0.1500000000, q2 = 0.$

$Number of points found crossing the (p2,q2) plane: 177$

$Maximum H deviation : .4618000000e-4 %$

$_________________________________________________$

$Time consumed: 9 seconds$

${F4}_{\frac{1}{6}} := PLOT (...)$

(2.3)

>	$FF4 := array ([[F4 [\frac{1}{24}], F4 [\frac{1}{18}], F4 [\frac{1}{12}]], [F4 [\frac{1}{8}], F4 [\frac{1}{7}], F4 [\frac{1}{6}]]]) &colon;$

>	${plots}_{} [display] (FF4)$

The figures above reflect the progressive disintegration of the KAM surfaces, occurring with the increase of H up to 1/6. In the plots for H equal to 1/24 and 1/12, invariant curves apparently exist everywhere, but this is not strictly correct. In fact, the model is not integrable, as is reflected by the sequence of figures, and thin resonance layers of stochasticity are densely distributed throughout the 2PS, even for small H.

	References
	For a more complete discussion see: Cheb-Terrab, E.S., and de Oliverira, H.P. "Poincare Sections of Hamiltonian Systems." Computer Physics Communications, Vol. 95, (1996): 171.

For more information, see the following help pages: I ntroduction to the Poincare subpackage, DEtools[generate_ic], DEtools[hamilton_eqs], and DEtools[zoom].

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