Fractal Dimension Of Strange Attractor Research Articles

Numerical Research on Self-Sustained Oscillation and Chaos of Natural Convection in Horizontal Fluids Layer

Numerical simulation was conducted to study self-sustained oscillation and chaos of natural convection in horizontal fluids layer heated from below. The numerical simulation was conducted using SIMPLE algorithm with a QUICK scheme. Discrete time series was analyzed to study stability problem. Correlation dimension was calculated, their frequency characteristics were analyzed and chaos attractor was extracted. The results showed that oscillation appeared when Ra was over 2×104 and chaos appeared when Ra was over 5×106. When parameter was varied, chaos appeared. Strange attractor was used to describe chaos system. Fractal dimension of strange attractor was 1.19. Saturated embedding dimension of phase space reconstruction was 5. The time delay was selected for 5.

Ulam method and fractal Weyl law for Perron-Frobenius operators

We use the Ulam method to study spectral properties of the Perron-Frobenius operators of dynamical maps in a chaotic regime. For maps with absorption we show that the spectrum is characterized by the fractal Weyl law recently established for nonunitary operators describing poles of quantum chaotic scattering with the Weyl exponent $\nu=d-1$, where $d$ is the fractal dimension of corresponding strange set of trajectories nonescaping in future times. In contrast, for dissipative maps we find the Weyl exponent $\nu=d/2$ where $d$ is the fractal dimension of strange attractor. The Weyl exponent can be also expressed via the relation $\nu=d_0/2$ where $d_0$ is the fractal dimension of the invariant sets. We also discuss the properties of eigenvalues and eigenvectors of such operators characterized by the fractal Weyl law.