Spiral Hyperbolic

Plot the hyperbolic spiral, rd = a and show that the ratio of the distance of any two points from the pole is inversely proportional to the angles between their radii vectores. 

For parabolic or hyperbolic parameters (r, q), the spiral Sp n) is defined as the proper (r, q)-polycycle with n r-gons obtained by taking an r-gon and adding r-gons in sequence, always rotating in the same direction. A formal definition is difficult to write (see [HaHa76, GreOl]), so we show some examples below ... 

Hyperbolic fixed points also illustrate the important general notion of structural stability. A phase portrait is structurally stable if its topology cannot be changed by an arbitrarily small perturbation to the vector field. For instance, the phase portrait of a saddle point is structurally stable, but that of a center is not an arbitrarily small amount of damping converts the center to a spiral. 

We briefly comment on the basic steps in the proof of theorem 1. For homogeneous lighting of intermediate strength a rigidly rotating spiral wave solution was assumed to be given by u . The manifold is close to the unperturbed normally hyperbolic center manifold SE(2)u given by the... 

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