A classical mechanical system is analysed which exhibits complicated scattering behaviour. In the set of all incoming asymptotes there is a fractal subset on which the scattering angle is singular. Though in the complement of this Cantor set the deflection function is regular, one can choose impact parameter intervals leading to arbitrarily complicated trajectories. We show how the complicated scattering behaviour is caused by unstable periodic orbits having homoclinic and heteroclinic connections. Thereby a hyperbolic invariant set is created leading to horseshoe chaos in the flow. This invariant set contains infinitely many unstable localised orbits (periodic and aperiodic ones). The stable manifolds of these orbits reach out into the asymptotic region and create the singularities of the scattering function.

The dynamics of a billiard in a gravitational field between a vertical wall and an inclined plane depends strongly on the angle 0 between wall and plane. Most conspicuously, the relative amount of chaotic versus regular parts of the energy surface shows pronounced oscillations as a function of 0, with distinct minima for 0 near 90"/n (n = 2, 3, . . .). This brearhing is also seen in the Lyapunov exponents. It reflects a repetetive pattern in the linear stability properties of families of periodic orbits. To study these orbits and their stability, Birkhoffs decomposition of the Poincari map into the product of two involutions is employed. The breathing in the amount of chaos can then be discussed in terms of the topology of symmetry lines, and of the corresponding directions of reflection.

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The strange attractor of the Lorenz model is found to be well approximated by suitably chosen two-dimensional invariant mmufolds through the three stationary points of' the flow in phase space. The stationary probability density, defined by the two~ensiona1 flow on the invariant manifolds, is determined in the vicinity of the origin of the phase space in terms of two parameters and compared with the numerically determined stationary distribution on the Lorenz attractor.

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The origin of the golden angle γ = 137.5° found in most leaf arrangements is investigated by an inhibitor diffusion model. A recursive iteration scheme is set up generating inhibitor concentrations on the meristematic ring and angles xn between succeeding leaves. The universality of γ is reflected by existence of a broad parameter range with lim xn ≈ γ. Likewise, the golden angle is shown to occur in a modified Gierer‐Meinhardt model of nonlinear activator and inhibitor reactions.