2.1.2.1. Maps

2.1.2.1.1. Main Functions

teaspoon.MakeData.DynSysLib.maps.logistic_map(parameters=[3.5], dynamic_state=None, InitialConditions=None, L=1000.0, fs=1, SampleSize=500)[source]

The logistic map [1] was generated as

\[x_{n+1} = rx_n(1-x_n)\]

where we chose the parameters \(x_0 = 0.5\) and \(r = 3.6\) for a chaotic state. You can set \(r = 3.5\) for a periodic response. We solve this system for 1000 data points and keep the second 500 to avoid transients.

../../../_images/Logistic_map.png
Parameters:

  • parameters (Optional[float]) – Array of 1 float [r]

  • dynamic_state (Optional[string]) – Dynamic state (‘periodic’ or ‘chaotic’)

  • L (Optional[int]) – Number of map iterations.

  • fs (Optional[int]) – sampling rate for simulation.

  • SampleSize (Optional[int]) – length of sample at end of entire time series

Returns:

Array of the time indices as t and the simulation time series ts

Return type:

array

References

teaspoon.MakeData.DynSysLib.maps.henon_map(parameters=[1.25, 0.3, 1.0], dynamic_state=None, InitialConditions=None, L=1000.0, fs=1, SampleSize=500)[source]

The Hénon map [2] was solved as

\[ \begin{align}\begin{aligned}x_{n+1} &= 1 -ax_n^2+y_n,\\y_{n+1}&=bx_n\end{aligned}\end{align} \]

where we chose the parameters \(a = 1.20\), \(b = 0.30\), and \(c = 1.00\) for a chaotic state with initial conditions \(x_0 = 0.1\) and \(y_0 = 0.3\). You can set \(a = 1.25\) for a periodic response. We solve this system for 1000 data points and keep the second 500 to avoid transients.

../../../_images/Henon_map.png
Parameters:

  • parameters (Optional[floats]) – Array of 3 floats [a,b,c].

  • dynamic_state (Optional[string]) – Dynamic state (‘periodic’ or ‘chaotic’)

  • L (Optional[int]) – Number of map iterations.

  • fs (Optional[int]) – sampling rate for simulation.

  • SampleSize (Optional[int]) – length of sample at end of entire time series

Returns:

Array of the time indices as t and the simulation time series ts

Return type:

array

References

teaspoon.MakeData.DynSysLib.maps.sine_map(parameters=[0.8], dynamic_state=None, InitialConditions=None, L=1000.0, fs=1, SampleSize=500)[source]

The Sine map is defined as

\[x_{n+1} = A\sin{(\pi x_n)}\]

where we chose the parameter \(A = 1.0\) for a chaotic state with initial condition \(x_0 = 0.1\). You can also change \(A = 0.8\) for a periodic response. We solve this system for 1000 data points and keep the second 500 to avoid transients.

../../../_images/Sine_map.png
Parameters:

  • parameters (Optional[floats]) – Array of 1 float [A].

  • dynamic_state (Optional[string]) – Dynamic state (‘periodic’ or ‘chaotic’)

  • L (Optional[int]) – Number of map iterations.

  • fs (Optional[int]) – sampling rate for simulation.

  • SampleSize (Optional[int]) – length of sample at end of entire time series

Returns:

Array of the time indices as t and the simulation time series ts

Return type:

array

teaspoon.MakeData.DynSysLib.maps.tent_map(parameters=[1.05], dynamic_state=None, InitialConditions=None, L=1000.0, fs=1, SampleSize=500)[source]

The Tent map [3] was solved as

\[x_{n+1} = A\min{([x_n, 1-x_n])}\]

where we chose the parameter \(A = 1.50\) for a chaotic state with initial condition \(x_0 = 1/\sqrt{2}\). You can also change \(A = 1.05\) for a periodic response. We solve this system for 1000 data points and keep the second 500 to avoid transients.

../../../_images/Tent_map.png
Parameters:

  • parameters (Optional[float]) – Array of 1 float [A].

  • dynamic_state (Optional[string]) – Dynamic state (‘periodic’ or ‘chaotic’)

  • L (Optional[int]) – Number of map iterations.

  • fs (Optional[int]) – sampling rate for simulation.

  • SampleSize (Optional[int]) – length of sample at end of entire time series

Returns:

Array of the time indices as t and the simulation time series ts

Return type:

array

References

teaspoon.MakeData.DynSysLib.maps.linear_congruential_generator_map(parameters=[0.9, 54773, 259200], dynamic_state=None, InitialConditions=None, L=1000.0, fs=1, SampleSize=500)[source]

The Linear Congruential Generator map is defined as

\[x_{n+1}=(ax_n+b)\mod c\]

where we chose the parameter \(a = 1.1\) for a chaotic state with initial condition \(x_0 = 0.1\). You can also change \(a = 0.9\) for a periodic response. \(b\) and \(c\) are set to 54,773 and 259,200 respectively for both dynamic states. We solve this system for 1000 data points and keep the second 500 to avoid transients.

../../../_images/Linear_Congruential_Generator_map.png
Parameters:

  • parameters (Optional[float]) – Array of 3 floats [a,b,c].

  • dynamic_state (Optional[string]) – Dynamic state (‘periodic’ or ‘chaotic’)

  • L (Optional[int]) – Number of map iterations.

  • fs (Optional[int]) – sampling rate for simulation.

  • SampleSize (Optional[int]) – length of sample at end of entire time series

Returns:

Array of the time indices as t and the simulation time series ts

Return type:

array

teaspoon.MakeData.DynSysLib.maps.rickers_population_map(parameters=[13], dynamic_state=None, InitialConditions=None, L=1000.0, fs=1, SampleSize=500)[source]

The Ricker’s Population map is defined [4] as

\[x_{n+1}=ax_n e^{-x_n}\]

where we chose the parameter \(a = 20\) for a chaotic state with initial condition \(x_0 = 0.1\). You can set \(a = 13\) for a periodic response. We solve this system for 1000 data points and keep the second 500 to avoid transients.

../../../_images/Rickers_Popoulation_map.png
Parameters:

  • parameters (Optional[float]) – Array of 1 parameter [a].

  • dynamic_state (Optional[string]) – Dynamic state (‘periodic’ or ‘chaotic’)

  • L (Optional[int]) – Number of map iterations.

  • fs (Optional[int]) – sampling rate for simulation.

  • SampleSize (Optional[int]) – length of sample at end of entire time series

Returns:

Array of the time indices as t and the simulation time series ts

Return type:

array

References

teaspoon.MakeData.DynSysLib.maps.gauss_map(parameters=[6.2, -0.2], dynamic_state=None, InitialConditions=None, L=1000.0, fs=1, SampleSize=500)[source]

The Gauss map is defined [5] as

\[x_{n+1}=e^{-\alpha x_n^2}+\beta\]

where we chose the parameters \(\alpha = 6.20\) and \(\beta = -0.35\) for a chaotic state with initial condition \(x_0 = 0.1\). You can set \(\beta = -0.20\) for a periodic response. We solve this system for 1000 data points and keep the second 500 to avoid transients. Taken from https://en.wikipedia.org/wiki/Gauss_iterated_map

../../../_images/Gauss_map.png
Parameters:

  • parameters (Optional[floats]) – Array of parameters [alpha, beta].

  • beta (Optional[float]) – System parameter.

  • dynamic_state (Optional[string]) – Dynamic state (‘periodic’ or ‘chaotic’)

  • L (Optional[int]) – Number of map iterations.

  • fs (Optional[int]) – sampling rate for simulation.

  • SampleSize (Optional[int]) – length of sample at end of entire time series

Returns:

Array of the time indices as t and the simulation time series ts

Return type:

array

References

teaspoon.MakeData.DynSysLib.maps.cusp_map(parameters=[1.1], dynamic_state=None, InitialConditions=None, L=1000.0, fs=1, SampleSize=500)[source]

The Cusp map is defined [6] as

\[x_{n+1} = 1 - a\sqrt{|x_n|}\]

where we chose the parameter \(a = 1.2\) for a chaotic state with initial condition \(x_0 = 0.5\). You can set \(a = 1.1\) for a periodic response. We solve this system for 1000 data points and keep the second 500 to avoid transients.

../../../_images/Cusp_map.png
Parameters:

  • parameters (Optional[float]) – Array of system parameters [a].

  • dynamic_state (Optional[string]) – Dynamic state (‘periodic’ or ‘chaotic’)

  • L (Optional[int]) – Number of map iterations.

  • fs (Optional[int]) – sampling rate for simulation.

  • SampleSize (Optional[int]) – length of sample at end of entire time series

Returns:

Array of the time indices as t and the simulation time series ts

Return type:

array

References

teaspoon.MakeData.DynSysLib.maps.pinchers_map(parameters=[1.3, 0.5], dynamic_state=None, InitialConditions=None, L=1000.0, fs=1, SampleSize=500)[source]

The Pincher’s map is defined [7] as

\[x_{n+1} = |\tanh{(s(x_n-c))}|\]

where we chose the parameters \(s = 1.6\) and \(c = 0.5\) for a chaotic state with initial condition \(x_0 = 0.0\). You can set \(s = 1.3\) for a periodic response. We solve this system for 1000 data points and keep the second 500 to avoid transients.

../../../_images/Pinchers_map.png
Parameters:

  • parameters (Optional[float]) – Array of system parameters [s,c].

  • dynamic_state (Optional[string]) – Dynamic state (‘periodic’ or ‘chaotic’)

  • L (Optional[int]) – Number of map iterations.

  • fs (Optional[int]) – sampling rate for simulation.

  • SampleSize (Optional[int]) – length of sample at end of entire time series

Returns:

Array of the time indices as t and the simulation time series ts

Return type:

array

References

teaspoon.MakeData.DynSysLib.maps.sine_circle_map(parameters=[0.5, 1.5], dynamic_state=None, InitialConditions=None, L=1000.0, fs=1, SampleSize=500)[source]

The Sine Circle map is defined [8] as

\[x_{n+1} = x_n + \omega -\left[\frac{k}{2\pi}\sin{(2\pi x_n)}\right]\]

where we chose the parameters \(\omega = 0.5\) and \(k = 2.0\) for a chaotic state with initial condition \(x_0 = 0.0\). You can set \(k = 1.5\) for a periodic response. We solve this system for 1000 data points and keep the second 500 to avoid transients.

../../../_images/Sine_Circle_map.png
Parameters:

  • parameters (Optional[float]) – Array of system parameters [omega, k].

  • dynamic_state (Optional[string]) – Dynamic state (‘periodic’ or ‘chaotic’)

  • L (Optional[int]) – Number of map iterations.

  • fs (Optional[int]) – sampling rate for simulation.

  • SampleSize (Optional[int]) – length of sample at end of entire time series

Returns:

Array of the time indices as t and the simulation time series ts

Return type:

array

References

teaspoon.MakeData.DynSysLib.maps.lozi_map(parameters=[1.5, 0.5], dynamic_state=None, InitialConditions=None, L=1000.0, fs=1, SampleSize=500)[source]

The Lozi map is defined [9] as

\[ \begin{align}\begin{aligned}x_{n+1} &= 1 - a|x_n| +by_n,\\y_{n+1} &= x_n\end{aligned}\end{align} \]

where we chose the parameters \(a = 1.7\) and \(b = 0.5\) for a chaotic state with initial conditions \(x_0 = -0.1\) and \(y_0 = 0.1\). You can set \(a = 1.5\) for a periodic response. We solve this system for 1000 data points and keep the second 500 to avoid transients.

../../../_images/Lozi_map.png
Parameters:

  • parameters (Optional[float]) – Array of system parameters [a,b].

  • dynamic_state (Optional[string]) – Dynamic state (‘periodic’ or ‘chaotic’)

  • L (Optional[int]) – Number of map iterations.

  • fs (Optional[int]) – sampling rate for simulation.

  • SampleSize (Optional[int]) – length of sample at end of entire time series

Returns:

Array of the time indices as t and the simulation time series ts

Return type:

array

References

teaspoon.MakeData.DynSysLib.maps.delayed_logstic_map(parameters=[2.2], dynamic_state=None, InitialConditions=None, L=1000.0, fs=1, SampleSize=500)[source]

The Delayed Logistic map is defined [10] as

\[ \begin{align}\begin{aligned}x_{n+1} &= ax_n(1-y_n)\\y_{n+1} &= x_n\end{aligned}\end{align} \]

where we chose the parameter \(a = 2.27\) for a chaotic state with initial conditions \(x_0 = 0.001\) and \(y_0 = 0.001\). You can set \(a = 2.20\) for a periodic response. We solve this system for 1000 data points and keep the second 500 to avoid transients.

../../../_images/Delayed_Logistic_map.png
Parameters:

  • parameters (Optional[float]) – Array of system parameters [a].

  • dynamic_state (Optional[string]) – Dynamic state (‘periodic’ or ‘chaotic’)

  • L (Optional[int]) – Number of map iterations.

  • fs (Optional[int]) – sampling rate for simulation.

  • SampleSize (Optional[int]) – length of sample at end of entire time series

Returns:

Array of the time indices as t and the simulation time series ts

Return type:

array

References

teaspoon.MakeData.DynSysLib.maps.tinkerbell_map(parameters=[0.7, -0.6, 2, 0.5], dynamic_state=None, InitialConditions=None, L=1000.0, fs=1, SampleSize=500)[source]

The Tinkerbell map is defined [11] as

\[ \begin{align}\begin{aligned}x_{n+1} &= x_n^2 - y_n^2 + ax_n + by_n,\\y_{n+1} &= 2x_ny_n + cx_n + dy_n\end{aligned}\end{align} \]

where we chose the parameters \(a = 0.9\), \(b = -0.6\), \(c = 2.0\), and \(d = 0.5\) for a chaotic state with initial conditions \(x_0 = 0.0\) and \(y_0 = 0.5\). You can set \(a = 0.7\) for a periodic response. We solve this system for 1000 data points and keep the second 500 to avoid transients.

../../../_images/Tinkerbell_map.png
Parameters:

  • parameters (Optional[float]) – Array of system parameters [a,b,c,d].

  • dynamic_state (Optional[string]) – Dynamic state (‘periodic’ or ‘chaotic’)

  • L (Optional[int]) – Number of map iterations.

  • fs (Optional[int]) – sampling rate for simulation.

  • SampleSize (Optional[int]) – length of sample at end of entire time series

Returns:

Array of the time indices as t and the simulation time series ts

Return type:

array

References

teaspoon.MakeData.DynSysLib.maps.burgers_map(parameters=[0.75, 1.6], dynamic_state=None, InitialConditions=None, L=3000.0, fs=1, SampleSize=500)[source]

The Burger’s map is defined [12] as

\[ \begin{align}\begin{aligned}x_{n+1} &= ax_n - y_n^2,\\y_{n+1} &= by_n + x_ny_n\end{aligned}\end{align} \]

where we chose the parameters \(a = 0.75\) and \(b = 1.75\) for a chaotic state with initial conditions \(x_0 = -0.1\) and \(y_0 = 0.5\). You can set \(b = 1.60\) for a periodic response. We solve this system for 3000 data points and keep the second 500 to avoid transients.

../../../_images/Burgers_map.png
Parameters:

  • parameters (Optional[float]) – Array of system parameters [a,b].

  • dynamic_state (Optional[string]) – Dynamic state (‘periodic’ or ‘chaotic’)

  • L (Optional[int]) – Number of map iterations.

  • fs (Optional[int]) – sampling rate for simulation.

  • SampleSize (Optional[int]) – length of sample at end of entire time series

Returns:

Array of the time indices as t and the simulation time series ts

Return type:

array

References

teaspoon.MakeData.DynSysLib.maps.holmes_cubic_map(parameters=[0.27, 2.77], dynamic_state=None, InitialConditions=None, L=3000.0, fs=1, SampleSize=500)[source]

The Holme’s Cubic map is defined [13] as

\[ \begin{align}\begin{aligned}x_{n+1} &= y_n,\\y_{n+1} &= -bx_n + dy_n - y_n^3\end{aligned}\end{align} \]

where we chose the parameters \(b = 0.20\) and \(d = 2.77\) for a chaotic state with initial conditions \(x_0 = -0.1\) and \(y_0 = 0.5\). You can set \(b = 0.27\) for a periodic response. We solve this system for 3000 data points and keep the second 500 to avoid transients.

../../../_images/Holmes_Cubic_map.png
Parameters:

  • parameters (Optional[float]) – Array of system parameters [b,d].

  • dynamic_state (Optional[string]) – Dynamic state (‘periodic’ or ‘chaotic’)

  • L (Optional[int]) – Number of map iterations.

  • fs (Optional[int]) – sampling rate for simulation.

  • SampleSize (Optional[int]) – length of sample at end of entire time series

Returns:

Array of the time indices as t and the simulation time series ts

Return type:

array

References

teaspoon.MakeData.DynSysLib.maps.kaplan_yorke_map(parameters=[-1.0, 0.2], dynamic_state=None, InitialConditions=None, L=1000.0, fs=1, SampleSize=500)[source]

The Kaplan Yorke map is defined [14] as

\[ \begin{align}\begin{aligned}x_{n+1} &= [ax_n](\mod 1),\\y_{n+1} &= by_n + \cos{(4\pi x_n)}\end{aligned}\end{align} \]

where we chose the parameters \(a = -2.0\) and \(b = 0.2\) for a chaotic state with initial conditions \(x_0 = -0.1\) and \(y_0 = 0.5\). You can set \(a = -1.0\) for a periodic response. We solve this system for 1000 data points and keep the second 500 to avoid transients.

../../../_images/Kaplan_Yorke_map.png
Parameters:

  • parameters (Optional[float]) – Array of system parameters [a,b].

  • dynamic_state (Optional[string]) – Dynamic state (‘periodic’ or ‘chaotic’)

  • L (Optional[int]) – Number of map iterations.

  • fs (Optional[int]) – sampling rate for simulation.

  • SampleSize (Optional[int]) – length of sample at end of entire time series

Returns:

Array of the time indices as t and the simulation time series ts

Return type:

array

References

teaspoon.MakeData.DynSysLib.maps.gingerbread_man_map(parameters=[1.0, 1.0], dynamic_state=None, InitialConditions=None, L=2000, fs=1, SampleSize=500)[source]

The Gingerbread Man Map is defined [15] [16] as

\[ \begin{align}\begin{aligned}x_{n+1} &= 1 - ay_n + n|x_n|,\\y_{n+1} &= x_n\end{aligned}\end{align} \]

where we chose the parameters \(a = 1.0\) and \(b = 1.0\). For a chaotic state, initial conditions \(x_0 = 0.5\) and \(y_0 = 1.8\), and for a periodic response \(x_0 = 0.5\) and \(y_0 = 1.5\). We solve this system for 2000 data points and keep the last 500 to avoid transients.

../../../_images/Gingerbread_Man_map.png
Parameters:

  • a (Optional[float]) – System parameter.

  • b (Optional[float]) – System parameter.

  • dynamic_state (Optional[string]) – Dynamic state (‘periodic’ or ‘chaotic’)

  • L (Optional[int]) – Number of map iterations.

  • fs (Optional[int]) – sampling rate for simulation.

  • SampleSize (Optional[int]) – length of sample at end of entire time series

Returns:

Array of the time indices as t and the simulation time series ts

Return type:

array

References