Image of the energy-momentum map (in gray). The singular line is represented by the horizontal solid line. The small full dot indicates the position of the image of the pinched-curled torus. A loop transversally crossing is depicted by dashed lines.
Schematic representation on the torus of the cycles and depicted, respectively, by dashed and solid lines.
Local bifurcation diagram in the neighborhood of the origin (, ) for the resonant system. The singular locus is represented by the large solid line. The small full dot indicates the position of the origin. The dashed line depicts a loop used to calculate the fractional monodromy matrix.
Intersections of the reduced phase space (in large solid lines) with the level sets for , , and in the plane . The energy-momentum map corresponding to this diagram is given by Eq. (36).
Same as Fig. 3 but for the resonant system.
Transport of the cycle along a path around line . The path is a semicircle of radius . The solid lines without arrows represent arbitrary branch cuts of the Riemann surfaces, and the full dots the ramification points (see text). The parts in solid and dashed lines of the loop lie, respectively, in the upper and lower leaves of the Riemann surface. For the first figure, the ramification points are from left to right , , , and .
Definition of the cycles and for the resonance. The position of the pole of is represented by a cross.
Complex discriminant locus (solid lines) of the energy-momentum map of Eq. (1) for and . The gray plane corresponds to the real bifurcation diagram. The different branches of are given by Eq. (15) near the origin. The six branches intersect at the origin. Four of the branches are real and lie in the gray plane. The dashed lines represent the loop locally deformed near to the complex domain. The semicircle which corresponds to (see text) is in a complex -plane with fixed.
Decomposition of the loop into the paths and .
Schematic representation of three of the roots of the polynomial as a function of and . These roots are the roots of the principal part of defined by Eq. (14). The polynomial has another root larger in module which is not represented here as it undergoes no bifurcation. The small insets depict the graph of as a function of for different values of and . The position of the insets gives the corresponding values of and . The solid lines are real lines of singularities of .
Evolution of three of the roots (see text) of the polynomial from the resonance and for a semicircle going from to ( fixed). Numerical values are taken to be and . The radius of the semicircle is . The energy-momentum map is given by Eq. (36). The full and open dots represent, respectively, the roots for the starting and the ending points of the path. The dashed line is a circle of radius , which corresponds to the leading term of the expansion of the roots as .
Definition of the cycles , , and for the resonance. The cross indicates the position of the pole of .
Transport of the cycle along a semicircle around line . The radius of the semicircle is . The last three figures are equivalent but with different cuts. The cross indicates the position of the pole of .
Same as Fig. 13 but for three semicircles around line .
Riemann surface of the function arctan. The solid and dashed lines, respectively, lie in the upper and the lower leaves. The ramification points correspond to the complex numbers and .
Schematic representation of the continuous transport of the basic cycles when the singular line is crossed. and are, respectively, represented by dashed and solid lines.
Schematic representation of the basis cycles for . Representatives of and are, respectively, represented by dashed and solid lines.
Same as Fig. 16 but for the resonance.
Same as Fig. 17 but for the resonance.
Semiclassical bifurcation diagram for the resonance and the energy-momentum map of Eq. (36). The line of singularities is represented by a solid line. The open dot indicates the position of the origin of the bifurcation diagram. The final cell after a counterclockwise closed loop around the origin is depicted by dashed lines.
Same as Fig. 20 but for the resonance. The corresponding energy-momentum map is given by Eq. (B5).
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