Illustration of action on a plane for the map suggested in the original paper of Plykin (a), and for the version of the map with the Plykin–Newhouse attractor (b). Each of them may be associated with a map defined on the sphere, say, by means of the stereographic projection.
The unit sphere with marked points A, B, C, D, neighborhoods of which in the further construction will correspond to the holes not visited by trajectories on the attractor. The north and south poles are indicated with N and S, respectively. The angular coordinates are shown for some point M, and axes of the Cartesian coordinates , , are depicted.
Attractor of the map (13) at and in three-dimensional space on the unit square (a), its representation in the angular coordinates (b), and a portrait on the plane in stereographic projection (c). Panel (d) shows for comparison the portrait of the Plykin–Newhouse attractor reproduced from Ref. 24 for the Hunt model (Ref. 22). The orientation is selected especially to see better the correspondence of the structure of the filaments with that in diagram (c).
A graphical illustration for verification of the hyperbolic nature of the attractor for the map (13), with and . A positive result of the test follows from existence of the gap between the line and the black and gray sets of points, which represent, respectively, the upper and lower edges of the intervals of , which met the verified conditions.
Plots of the real amplitudes and vs time in the transient process obtained from numerical solution of the differential equations (8) at and .
(a) Portrait of attractor for the system (8) in three-dimensional space . In the cross section with the horizontal plane (mod6) observe the object identical to that shown in Fig. 3(b). (b) Portrait of the attractor in the plane of real amplitudes relating to one of the oscillators and separated by time interval . Technique of representation in gray scales is used: Brighter tones correspond to higher probability of visiting the pixels by the representative points. The parameter values are , .
Four Lyapunov exponents vs parameter at (a) and vs parameter at (b) for the Poincaré map represented in terms of complex amplitudes. A zero exponent occurs due to invariance of the equations with respect to the overall phase shift.
The power spectral density for one of the coupled oscillators vs the frequency computed by processing a realization obtained from numerical integration of Eqs. (8) at and .
Cross sections of the cones for the case of the three-dimensional map with one expanding and two contracting directions. A picture of proper inclusion is shown (a) for expanding cones and (b) for contracting cones . A circle circumscribed around the ellipse is located inside the unit disk if condition (A4) is valid.
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