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Discrete Fourier transform represents data by a superposition of sines and cosines that have various amplitudes and frequencies. With time series of length N, the range of frequencies that can be considered goes from 1/N to 1/2. Powers of individual Fourier component are called power spectral density (power spectrum). For self-affine signals the power spectrum P( f ) decays via a power law [Mandelbrot, 1977]. Beta= - log p(f) /log f The value of the power-law decay can be obtained as a slope of linear regression applied to the power spectrum in log–log coordinate system. After calculating the spectral decay coefficient (Beta) , we use the relationship introduced above to obtain the fractal dimension: D= (5 - Beta) / (2) The power decay was estimated from the whole accessible frequency domain.

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  • $\begingroup$ Hey, your question is a bit unclear. For instance, you want to calculate the dimension of which object? And why do you think you can do that from bifurcations? $\endgroup$
    – stafusa
    Apr 22, 2020 at 18:19
  • $\begingroup$ The Rossler system, a system of three non-linear ordinary differential equations originally studied by Otto Rossler. These differential equations define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor. The defining equations was about the Rössler system : dx / dt = - y - z dy / dt = x + ay dz / dt = b + z(x - c) $\endgroup$
    – S.moh
    Apr 24, 2020 at 16:16
  • $\begingroup$ So you want to measure the dimension of the Rössler attractor? And what's the reference for the equation you include in the question? Is it for the dimension of the attractor or of the time series? How would they be related? $\endgroup$
    – stafusa
    Apr 24, 2020 at 16:18
  • $\begingroup$ In mathematics of dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the system. In the Rossler system bifurcation occur if any of the three parameter value is varied while other two parameter are fixed. I want to connect the subject(Bifurcation) to power spectral of the rossler system. $\endgroup$
    – S.moh
    Apr 26, 2020 at 10:01
  • $\begingroup$ OK, but none of this answers my questions. What's your reference, or is that a research topic? $\endgroup$
    – stafusa
    Apr 26, 2020 at 10:52

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