Finding the Popular Frequency! Time Series Data with the Fourier Transformation Olivia B. Hoff Hirophysics.com
→ Mathematical description: Logistic Map → Mathematical description: Xn+1= rXn(1-Xn) →Polynomial mapping of degree 2. →Complex and chaotic behavior can occur with the parameter, r. →Gives us the various time series by changing r through its simplicity. Hirophysics.com
Logistic Map cont... → A property of the logistic map is that it becomes chaotic around the value 3.57 and reaches complete chaos at 4. →The stretching and folding of the mapping results in gradual divergence of data points and exponential divergence. →Shown by the complexity and unpredictability of the time-related data. Hirophysics.com
Logistic Map cont... Examples: Some of regular (non-chaotic) series with Logistic map. Hirophysics.com
Logistic Map cont... Cases for multiple periods Hirophysics.com
Logistic Map cont... Chaotic time series, r = 3.7 Hirophysics.com
Fourier Transformation → Transforms one valued function into another. → The domain of the original function is usually time and the domain of the new function is frequency. → Describes which frequencies are present in the original function. → The Fourier transformation measures how much of an individual frequency is present in a function f(t). Hirophysics.com
Some of examples with regular time series Hirophysics.com
Some of examples with regular time series 2 Hirophysics.com
Some of examples with chaotic time series Hirophysics.com
Completely Chaotic Cases Hirophysics.com
Other Cases Hirophysics.com
Conclusions The logistic map is expressed with a simple iteration, yet the time series are complicated by only changing the parameter. The Fourier Transformation gives a qualitative index to distinguish chaotic from non-chaotic time series. Hirophysics.com