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# -*- coding: utf-8 -*-
"""Functions to generate time series of some popular chaotic systems.
This module provides some functions that can be used to generate time
series of some common chaotic systems. Most of the parameters and
initial conditions have been taken from Appendix A of Sprott (2003).
Noise
-----
* falpha -- generates (1/f)^alpha noise.
Deterministic Systems
---------------------
* henon -- generates data using the Henon map.
* ikeda -- generates data using the Ikeda map.
* lorenz -- generates data using the Lorenz equations.
* mackey_glass -- generates data using the Mackey-Glass delay
differential equations.
* roessler -- generates data using the Rössler equations.
"""
from __future__ import absolute_import, division, print_function
import numpy as np
from numpy import fft
from scipy.integrate import odeint
def falpha(length=8192, alpha=1.0, fl=None, fu=None, mean=0.0, var=1.0):
"""Generate (1/f)^alpha noise by inverting the power spectrum.
Generates (1/f)^alpha noise by inverting the power spectrum.
Follows the algorithm described by Voss (1988) to generate
fractional Brownian motion.
Parameters
----------
length : int, optional (default = 8192)
Length of the time series to be generated.
alpha : float, optional (default = 1.0)
Exponent in (1/f)^alpha. Pink noise will be generated by
default.
fl : float, optional (default = None)
Lower cutoff frequency.
fu : float, optional (default = None)
Upper cutoff frequency.
mean : float, optional (default = 0.0)
Mean of the generated noise.
var : float, optional (default = 1.0)
Variance of the generated noise.
Returns
-------
x : array
Array containing the time series.
Notes
-----
As discrete Fourier transforms assume that the input data is
periodic, the resultant series x_{i} (= x_{i + N}) is also periodic.
To avoid this periodicity, it is recommended to always generate
a longer series (two or three times longer) and trim it to the
desired length.
"""
freqs = fft.rfftfreq(length)
power = freqs[1:] ** -alpha
power = np.insert(power, 0, 0) # P(0) = 0
if fl:
power[freqs < fl] = 0
if fu:
power[freqs > fu] = 0
# Randomize complex phases.
phase = 2 * np.pi * np.random.random(len(freqs))
y = np.sqrt(power) * np.exp(1j * phase)
# The last component (corresponding to the Nyquist frequency) of an
# RFFT with even number of points is always real. (We don't have to
# make the mean real as P(0) = 0.)
if length % 2 == 0:
y[-1] = np.abs(y[-1] * np.sqrt(2))
x = fft.irfft(y, n=length)
# Rescale to proper variance and mean.
x = np.sqrt(var) * x / np.std(x)
return mean + x - np.mean(x)
def henon(length=10000, x0=None, a=1.4, b=0.3, discard=500):
"""Generate time series using the Henon map.
Generates time series using the Henon map.
Parameters
----------
length : int, optional (default = 10000)
Length of the time series to be generated.
x0 : array, optional (default = random)
Initial condition for the map.
a : float, optional (default = 1.4)
Constant a in the Henon map.
b : float, optional (default = 0.3)
Constant b in the Henon map.
discard : int, optional (default = 500)
Number of steps to discard in order to eliminate transients.
Returns
-------
x : ndarray, shape (length, 2)
Array containing points in phase space.
"""
x = np.empty((length + discard, 2))
if not x0:
x[0] = (0.0, 0.9) + 0.01 * (-1 + 2 * np.random.random(2))
else:
x[0] = x0
for i in range(1, length + discard):
x[i] = (1 - a * x[i - 1][0] ** 2 + b * x[i - 1][1], x[i - 1][0])
return x[discard:]
def ikeda(length=10000, x0=None, alpha=6.0, beta=0.4, gamma=1.0, mu=0.9,
discard=500):
"""Generate time series from the Ikeda map.
Generates time series from the Ikeda map.
Parameters
----------
length : int, optional (default = 10000)
Length of the time series to be generated.
x0 : array, optional (default = random)
Initial condition for the map.
alpha : float, optional (default = 6.0)
Constant alpha in the Ikeda map.
beta : float, optional (default = 0.4)
Constant beta in the Ikeda map.
gamma : float, optional (default = 1.0)
Constant gamma in the Ikeda map.
mu : float, optional (default = 0.9)
Constant mu in the Ikeda map.
discard : int, optional (default = 500)
Number of steps to discard in order to eliminate transients.
Returns
-------
x : ndarray, shape (length, 2)
Array containing points in phase space.
"""
x = np.empty((length + discard, 2))
if not x0:
x[0] = 0.1 * (-1 + 2 * np.random.random(2))
else:
x[0] = x0
for i in range(1, length + discard):
phi = beta - alpha / (1 + x[i - 1][0] ** 2 + x[i - 1][1] ** 2)
x[i] = (gamma + mu * (x[i - 1][0] * np.cos(phi) - x[i - 1][1] *
np.sin(phi)),
mu * (x[i - 1][0] * np.sin(phi) + x[i - 1][1] * np.cos(phi)))
return x[discard:]
def lorenz(length=10000, x0=None, sigma=10.0, beta=8.0/3.0, rho=28.0,
step=0.001, sample=0.03, discard=1000):
"""Generate time series using the Lorenz system.
Generates time series using the Lorenz system.
Parameters
----------
length : int, optional (default = 10000)
Length of the time series to be generated.
x0 : array, optional (default = random)
Initial condition for the flow.
sigma : float, optional (default = 10.0)
Constant sigma of the Lorenz system.
beta : float, optional (default = 8.0/3.0)
Constant beta of the Lorenz system.
rho : float, optional (default = 28.0)
Constant rho of the Lorenz system.
step : float, optional (default = 0.001)
Approximate step size of integration.
sample : int, optional (default = 0.03)
Sampling step of the time series.
discard : int, optional (default = 1000)
Number of samples to discard in order to eliminate transients.
Returns
-------
t : array
The time values at which the points have been sampled.
x : ndarray, shape (length, 3)
Array containing points in phase space.
"""
def _lorenz(x, t):
return [sigma * (x[1] - x[0]), x[0] * (rho - x[2]) - x[1],
x[0] * x[1] - beta * x[2]]
if not x0:
x0 = (0.0, -0.01, 9.0) + 0.25 * (-1 + 2 * np.random.random(3))
sample = int(sample / step)
t = np.linspace(0, (sample * (length + discard)) * step,
sample * (length + discard))
return (t[discard * sample::sample],
odeint(_lorenz, x0, t)[discard * sample::sample])
def mackey_glass(length=10000, x0=None, a=0.2, b=0.1, c=10.0, tau=23.0,
n=1000, sample=0.46, discard=250):
"""Generate time series using the Mackey-Glass equation.
Generates time series using the discrete approximation of the
Mackey-Glass delay differential equation described by Grassberger &
Procaccia (1983).
Parameters
----------
length : int, optional (default = 10000)
Length of the time series to be generated.
x0 : array, optional (default = random)
Initial condition for the discrete map. Should be of length n.
a : float, optional (default = 0.2)
Constant a in the Mackey-Glass equation.
b : float, optional (default = 0.1)
Constant b in the Mackey-Glass equation.
c : float, optional (default = 10.0)
Constant c in the Mackey-Glass equation.
tau : float, optional (default = 23.0)
Time delay in the Mackey-Glass equation.
n : int, optional (default = 1000)
The number of discrete steps into which the interval between
t and t + tau should be divided. This results in a time
step of tau/n and an n + 1 dimensional map.
sample : float, optional (default = 0.46)
Sampling step of the time series. It is useful to pick
something between tau/100 and tau/10, with tau/sample being
a factor of n. This will make sure that there are only whole
number indices.
discard : int, optional (default = 250)
Number of n-steps to discard in order to eliminate transients.
A total of n*discard steps will be discarded.
Returns
-------
x : array
Array containing the time series.
"""
sample = int(n * sample / tau)
grids = n * discard + sample * length
x = np.empty(grids)
if not x0:
x[:n] = 0.5 + 0.05 * (-1 + 2 * np.random.random(n))
else:
x[:n] = x0
A = (2 * n - b * tau) / (2 * n + b * tau)
B = a * tau / (2 * n + b * tau)
for i in range(n - 1, grids - 1):
x[i + 1] = A * x[i] + B * (x[i - n] / (1 + x[i - n] ** c) +
x[i - n + 1] / (1 + x[i - n + 1] ** c))
return x[n * discard::sample]
def roessler(length=10000, x0=None, a=0.2, b=0.2, c=5.7, step=0.001,
sample=0.1, discard=1000):
"""Generate time series using the Rössler oscillator.
Generates time series using the Rössler oscillator.
Parameters
----------
length : int, optional (default = 10000)
Length of the time series to be generated.
x0 : array, optional (default = random)
Initial condition for the flow.
a : float, optional (default = 0.2)
Constant a in the Röessler oscillator.
b : float, optional (default = 0.2)
Constant b in the Röessler oscillator.
c : float, optional (default = 5.7)
Constant c in the Röessler oscillator.
step : float, optional (default = 0.001)
Approximate step size of integration.
sample : int, optional (default = 0.1)
Sampling step of the time series.
discard : int, optional (default = 1000)
Number of samples to discard in order to eliminate transients.
Returns
-------
t : array
The time values at which the points have been sampled.
x : ndarray, shape (length, 3)
Array containing points in phase space.
"""
def _roessler(x, t):
return [-(x[1] + x[2]), x[0] + a * x[1], b + x[2] * (x[0] - c)]
sample = int(sample / step)
t = np.linspace(0, (sample * (length + discard)) * step,
sample * (length + discard))
if not x0:
x0 = (-9.0, 0.0, 0.0) + 0.25 * (-1 + 2 * np.random.random(3))
return (t[discard * sample::sample],
odeint(_roessler, x0, t)[discard * sample::sample])