class brainpy.integrators.fde.GL.GLShortMemory(f, alpha, inits, num_memory=None, num_step=None, dt=None, name=None, state_delays=None)[source]#

Efficient Computation of the Short-Memory Principle in Grünwald-Letnikov Method 1.

According to the explicit numerical approximation of Grünwald-Letnikov, the fractional-order derivative \(q\) for a discrete function \(f(t_K)\) can be described as follows:

\[{{}_{k-\frac{L_{m}}{h}}D_{t_{k}}^{q}}f(t_{k})\approx h^{-q} \sum\limits_{j=0}^{k}C_{j}^{q}f(t_{k-j})\]

where \(L_{m}\) is the memory lenght, \(h\) is the integration step size, and \(C_{j}^{q}\) are the binomial coefficients which are calculated recursively with

\[C_{0}^{q}=1,\ C_{j}^{q}=\left(1- \frac{1+q}{j}\right)C_{j-1}^{q},\ j=1,2, \ldots k.\]

Then, the numerical solution for a fractional-order differential equation (FODE) expressed in the form


can be obtained by

\[x(t_{k})=f(x(t_{k-1}))h^{q}- \sum\limits_{j=1}^{k}C_{j}^{q}x(t_{k-j}).\]

for \(0 < q < 1\). The above expression requires infinity memory length for numerical solution since the summation term depends on the discritized time \(t_k\). This implies relatively high simulation times.

To reduce the computational time, the upper bound of summation needs to be modified by \(k=v\), where

\[\begin{split}v=\begin{cases} k, & k\leq M,\\ L_{m}, & k > M. \end{cases}\end{split}\]

This is known as the short-memory principle, where \(M\) is the memory window with a width defined by \(M=\frac{L_{m}}{h}\). As was reported in 2, the accuracy increases by increaing the width of memory window.


>>> import brainpy as bp
>>> a, b, c = 10, 28, 8 / 3
>>> def lorenz(x, y, z, t):
>>>   dx = a * (y - x)
>>>   dy = x * (b - z) - y
>>>   dz = x * y - c * z
>>>   return dx, dy, dz
>>> integral = bp.fde.GLShortMemory(lorenz,
>>>                                 alpha=0.96,
>>>                                 num_step=500,
>>>                                 inits=[1., 0., 1.])
>>> runner = bp.integrators.IntegratorRunner(integral,
>>>                                          monitors=list('xyz'),
>>>                                          inits=[1., 0., 1.],
>>>                                          dt=0.005)
>>> import matplotlib.pyplot as plt
>>> plt.plot(runner.mon.x.flatten(), runner.mon.z.flatten())
  • f (callable) – The derivative function.

  • alpha (int, float, jnp.ndarray, bm.ndarray, sequence) – The fractional-order of the derivative function. Should be in the range of (0., 1.).

  • num_step (int) –

    The length of the short memory.

    ..versionadded:: 2.1.11

  • num_memory (int) –

    The length of the short memory.

    Changed in version 2.1.11: Please use num_step instead. num_memory will be no longer supportted since 2.2.0.

  • inits (sequence) – A sequence of the initial values for variables.

  • dt (float, int) – The numerical precision.

  • name (str) – The integrator name.



Clemente-López, D., et al. “Efficient computation of the Grünwald-Letnikov method for arm-based implementations of fractional-order chaotic systems.” 2019 8th International Conference on Modern Circuits and Systems Technologies (MOCAST). IEEE, 2019.


M. F. Tolba, A. M. AbdelAty, N. S. Soliman, L. A. Said, A. H. Madian, A. T. Azar, et al., “FPGA implementation of two fractional order chaotic systems”, International Journal of Electronics and Communications, vol. 78, pp. 162-172, 2017.

__init__(f, alpha, inits, num_memory=None, num_step=None, dt=None, name=None, state_delays=None)[source]#


__init__(f, alpha, inits[, num_memory, ...])

load_states(filename[, verbose])

Load the model states.

nodes([method, level, include_self])

Collect all children nodes.




Reset function of the delay variables.

save_states(filename[, variables])

Save the model states.


Set the integral function.

train_vars([method, level, include_self])

The shortcut for retrieving all trainable variables.

unique_name([name, type_])

Get the unique name for this object.

vars([method, level, include_self])

Collect all variables in this node and the children nodes.




All arguments when calling the numer integrator of the differential equation.



The numerical integration precision.


The integral function.



The parameters defined in the differential equation.


State delays.


The variables defined in the differential equation.