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For example, if p = 1 and q = 3, point P makes three rotations around circle R as long as it makes only one rotation around circle r. 9 shows the torus and the Poincaré plane intersecting the torus together with the trajectory on the surface of the torus. The Poincaré plane illustrates its intersection points with the trajectory. When the frequency ratio p/q = 1/3, it is rational. As long as point P rotates once around circle R, it rotates 120° around circle r. 9a) after 1–2–3 rotations around circle R, point P intersects the Ponicaré plane successively at point 1–2–3.

10 The quadratic or logistic iterated map. 18)]. In certain cases, we have a relation between the two coordinates un and vn: vn = Fv(un). 27) one-dimensional map function. 2 in Ref. [4] and page 66 of Ref. [5]) on the wide scope of applications on the one-dimensional discrete map functions. If the dissipation in the system is high enough, then even systems with dimension more than three can be analyzed by one-dimensional map. 28) can be done with numerical or graphical method. The return map or cobweb is a graphical method.

17 Initial conditions (P1, P2, P3, P4) are not placed on any of the invariant manifolds. The trajectories are repelled from Ws(x*) and attracted by Wu(x*). Any of the trajectories (orbits) must remain in the region where it was born. Wu(x*) (or Ws(x*)) are boundaries. of manifolds: stable manifold denoted by Ws and unstable manifold denoted by Wu . If the initial points are on Ws or on Wu, the operation points remain on Ws or Wu forever, but the points on Ws are attracted by x* and the points on Wu are repelled from x*.

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Industrial and Environmental Applns of Direct and Large-Eddy Simulation by S. Biringen, et al.,

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