6 Self similarity Dimension 6 1 Attractor Dimension

6. Self – similarity : Dimension 6. 1 Attractor Dimension and fractal • System with less than three dimensions cannot be chaotic. Not only the phase space but also the attractor should be more than two dimensions. • Strange attractors with fractal dimensions are common in chaotic system. • Non integer dimensions can be assigned to geometrical objects which exhibit self – similarity. 6. 2 Correlation Dimension • Dimensions and Lyapunov exponents are ways of quantifying the properties of the system. Dimension with its property of invariance is important. ** Correlation dimension is important but not correlation sum. ECE-7000: Nonlinear Dynamical Systems

6. Self – similarity : Dimensions 6. 3: Correlation sum from a time series • Given a time series we first need to reconstruct an auxiliary phase space by an embedding procedure. • Reconstruction procedure involves the choise of delay time (tau). • Once embedding vectors sn are reconstructed, the estimation of correlation dimension is done in two steps. • Step 1: Determine correlation sum for the range of epsilon and for several embedding dimensions m. • Step 2: Inspect correlation sum for signatures of self similarity. • If these signatures are convincing, we compute the value of dimension. We need to extrapolate in order to find correlation sum. Temporal correlations are to be avoided. ECE-7000: Nonlinear Dynamical Systems