Periodic orbit theory in fractal drums

Periodic orbit theory in fractal drums

The level statistics of pseudointegrable fractal drums is studied numerically using periodic orbit theory. We find that the spectral rigidity ∆3(L), which is a measure for the correlations between the eigenvalues, decreases to quite small values (as compared to systems with only small boundary roughness), thereby approaching the behavior of chaotic systems. The periodic orbit results are in goo...

Eigenfrequencies of Fractal Drums Eigenfrequencies of Fractal Drums

A method for the computation of eigenfrequencies and eigenmodes of fractal drums is presented. The approach involves first mapping the unit disk to a polygon approximating the fractal and then solving a weighted eigenvalue problem on the unit disk by a spectral collocation method. The numerical computation of the complicated conformal mapping was made feasible by the use of the fast multipole m...

Periodic Orbit Theory

Beyond the periodic orbit theory

The global constraints on chaotic dynamics induced by the analyticity of smooth flows are used to dispense with individual periodic orbits and derive infinite families of exact sum rules for several simple dynamical systems. The associated Fredholm determinants are of particularly simple polynomial form. The theory developed suggests an alternative to the conventional periodic orbit theory appr...

Localizations in Fractal Drums: An Experimental Study

The low-frequency eigenmodes of a fractal drum are studied through experimental observation of the acoustical resonances of a fractal-shaped liquid crystal film. The resonance frequencies agree with the numerical predictions for the Laplacian eigenvalues in the fractal domain with Dirichlet boundary conditions. The amplitude distribution of the modes is detected by scanning the excitation posit...