Fixed points & Periodicity

Fixed points

ChaosTools.fixedpoints — Function

fixedpoints(ds::DynamicalSystem, box, p = ds.p; kwargs...) → fp, eigs, stable

Return all fixed points fp of the given ds that exist within the state space subset box for parameter configuration p. Fixed points are returned as a Dataset. For convenience, a vector of the Jacobian eigenvalues of each fixed point, and whether the fixed points are stable or not, are also returned.

box is an appropriate IntervalBox from IntervalRootFinding.jl. E.g. for a 3D system it would be something like

v, z = -5..5, -2..2   # 1D intervals
box = v × v × z       # `\times = ×`, or use `IntervalBox`

Internally IntervalRootFinding.jl is used and as a result we are guaranteed to find all fixed points that exist in box, regardless of stability. Since IntervalRootFinding.jl returns an interval containing a unique fixed point, we return the midpoint of the interval as the actual fixed point. Naturally, limitations inherent to IntervalRootFinding.jl apply here.

The output of fixedpoints can be used in the BifurcationKit.jl as a start of a continuation process. See also periodicorbits.

Keywords

method = IntervalRootFinding.Krawczyk configures the root finding method, see the docs of IntervalRootFinding.jl for all posibilities.
tol = 1e-15 is the root-finding tolerance.

Stable and Unstable Periodic Orbits of Maps

Chaotic behavior of low dimensional dynamical systems is affected by the position and the stability properties of the periodic orbits of a dynamical system.

Finding unstable (or stable) periodic orbits of a discrete mapping analytically rapidly becomes impossible for higher orders of fixed points. Fortunately there is a numeric algorithm due to Schmelcher & Diakonos which allows such a computation. Notice that even though the algorithm can find stable fixed points, it is mainly aimed at unstable ones.

The functions periodicorbits and lambdamatrix implement the algorithm:

ChaosTools.periodicorbits — Function

periodicorbits(ds::DiscreteDynamicalSystem,
               o, ics [, λs, indss, singss]; kwargs...) -> FP

Find fixed points FP of order o for the map ds using the algorithm due to Schmelcher & Diakonos^{[Schmelcher1997]}. ics is a collection of initial conditions (container of vectors) to be evolved.

Optional Arguments

The optional arguments λs, indss, singss must be containers of appropriate values, besides λs which can also be a number. The elements of those containers are passed to: lambdamatrix(λ, inds, sings), which creates the appropriate $\mathbf{\Lambda}_k$ matrix. If these arguments are not given, a random permutation will be chosen for them, with λ=0.001.

Keyword Arguments

maxiters::Int = 100000 : Maximum amount of iterations an i.c. will be iterated before claiming it has not converged.
disttol = 1e-10 : Distance tolerance. If the 2-norm of a previous state with the next one is ≤ disttol then it has converged to a fixed point.
inftol = 10.0 : If a state reaches norm(state) ≥ inftol it is assumed that it has escaped to infinity (and is thus abandoned).
roundtol::Int = 4 : The found fixed points are rounded to roundtol digits before pushed into the list of returned fixed points FP, if they are not already contained in FP. This is done so that FP doesn't contain duplicate fixed points (notice that this has nothing to do with disttol). Turn this to typemax(Int) to get the full precision of the algorithm.

Description

The algorithm used can detect periodic orbits by turning fixed points of the original map ds to stable ones, through the transformation

\[\mathbf{x}_{n+1} = \mathbf{x}_n + \mathbf{\Lambda}_k\left(f^{(o)}(\mathbf{x}_n) - \mathbf{x}_n\right)\]

with $f$ = eom. The index $k$ counts the various possible $\mathbf{\Lambda}_k$.

Performance Notes

All initial conditions are evolved for all $\mathbf{\Lambda}_k$ which can very quickly lead to long computation times.

ChaosTools.lambdamatrix — Function

lambdamatrix(λ, inds::Vector{Int}, sings) -> Λk

Return the matrix $\mathbf{\Lambda}_k$ used to create a new dynamical system with some unstable fixed points turned to stable in the function periodicorbits.

Arguments

λ<:Real : the multiplier of the $C_k$ matrix, with 0<λ<1.
inds::Vector{Int} : The ith entry of this vector gives the row of the nonzero element of the ith column of $C_k$.
sings::Vector{<:Real} : The element of the ith column of $C_k$ is +1 if signs[i] > 0 and -1 otherwise (sings can also be Bool vector).

Calling lambdamatrix(λ, D::Int) creates a random $\mathbf{\Lambda}_k$ by randomly generating an inds and a signs from all possible combinations. The collections of all these combinations can be obtained from the function lambdaperms.

Description

Each element of inds must be unique such that the resulting matrix is orthogonal and represents the group of special reflections and permutations.

Deciding the appropriate values for λ, inds, sings is not trivial. However, in ref.^[Pingel2000] there is a lot of information that can help with that decision. Also, by appropriately choosing various values for λ, one can sort periodic orbits from e.g. least unstable to most unstable, see^{[Diakonos1998]} for details.

ChaosTools.lambdaperms — Function

lambdaperms(D) -> indperms, singperms

Return two collections that each contain all possible combinations of indices (total of $D!$) and signs (total of $2^D$) for dimension D (see lambdamatrix).

Standard Map example

For example, let's find the fixed points of the Systems.standardmap of order 2, 3, 4, 5, 6 and 8. We will use all permutations for the signs but only one for the inds. We will also only use one λ value, and a 21×21 density of initial conditions.

First, initialize everything

using DynamicalSystems, PyPlot, StaticArrays

ds = Systems.standardmap()
xs = range(0, stop = 2π, length = 21); ys = copy(xs)
ics = [SVector{2}(x,y) for x in xs for y in ys]

# All permutations of [±1, ±1]:
singss = lambdaperms(2)[2] # second entry are the signs

# I know from personal research I only need this `inds`:
indss = [[1,2]] # <- must be container of vectors!!!

λs = 0.005 # <- only this allowed to not be vector (could also be vector)

orders = [2, 3, 4, 5, 6, 8]
ALLFP = Dataset{2, Float64}[];

Dataset{2, Float64}[]

Then, do the necessary computations for all orders

for o in orders
    FP = periodicorbits(ds, o, ics, λs, indss, singss)
    push!(ALLFP, FP)
end

Plot the phase space of the standard map

iters = 1000
dataset = trajectory(ds, iters)
for x in xs
    for y in ys
        append!(dataset, trajectory(ds, iters, SVector{2}(x, y)))
    end
end
fig = figure()
m = Matrix(dataset)
PyPlot.scatter(view(m, :, 1), view(m, :, 2), s= 1, color = "black")
PyPlot.xlim(xs[1], xs[end])
PyPlot.ylim(ys[1], ys[end]);

(0.0, 6.283185307179586)

and finally, plot the fixed points

markers = ["D", "^", "s", "p", "h", "8"]
colors = ["b", "g", "r", "c", "m", "grey"]

for i in 1:6
    FP = ALLFP[i]
    o = orders[i]
    PyPlot.plot(columns(FP)...,
    marker=markers[i], color = colors[i], markersize=10.0 + (8-o), linewidth=0.0,
    label = "order $o", markeredgecolor = "yellow", markeredgewidth = 0.5)
end
legend(loc="upper right", framealpha=0.9)
xlabel("\$\\theta\$")
ylabel("\$p\$")
fig.tight_layout(pad=0.3); fig

Okay, this output is great, and we can easily tell that it is correct for many reasons:

It is the same fig. 12 of this publication.
Fixed points of order $n$ are also fixed points of order $2n, 3n, 4n, ...$
Besides fixed points of previous orders, original fixed points of order $n$ come in (possible multiples of) $2n$-sized pairs (see e.g. order 5). This is a direct consequence of the Poincaré–Birkhoff theorem.

Estimating the Period

The function estimate_period offers ways for estimating the period (either exact for periodic timeseries, or approximate for near-periodic ones) of a given timeseries. We offer five methods to estimate periods, some of which work on evenly sampled data only, and others which accept any data. The figure below summarizes this:

ChaosTools.estimate_period — Function

estimate_period(v::Vector, method, t=0:length(v)-1; kwargs...)

Estimate the period of the signal v, with accompanying time vector t, using the given method.

If t is an AbstractArray, then it is iterated through to ensure that it's evenly sampled (if necessary for the algorithm). To avoid this, you can pass any AbstractRange, like a UnitRange or a LinRange, which are defined to be evenly sampled.

Methods requiring evenly sampled data

These methods are faster, but some are error-prone.

:periodogram or :pg: Use the fast Fourier transform to compute a periodogram (power-spectrum) of the given data. Data must be evenly sampled.
:multitaper or mt: The multitaper method reduces estimation bias by using multiple independent estimates from the same sample. Data tapers are then windowed and the power spectra are obtained. Available keywords follow: nw is the time-bandwidth product, and ntapers is the number of tapers. If window is not specified, the signal is tapered with ntapers discrete prolate spheroidal sequences with time-bandwidth product nw. Each sequence is equally weighted; adaptive multitaper is not (yet) supported. If window is specified, each column is applied as a taper. The sum of periodograms is normalized by the total sum of squares of window.
:autocorrelation or :ac: Use the autocorrelation function (AC). The value where the AC first comes back close to 1 is the period of the signal. The keyword L = length(v)÷10 denotes the length of the AC (thus, given the default setting, this method will fail if there less than 10 periods in the signal). The keyword ϵ = 0.2 (\epsilon) means that 1-ϵ counts as "1" for the AC.

Methods not requiring evenly sampled data

These methods tend to be slow, but versatile and low-error.

:lombscargle or :ls: Use the Lomb-Scargle algorithm to compute a periodogram. The advantage of the Lomb-Scargle method is that it does not require an equally sampled dataset and performs well on undersampled datasets. Constraints have been set on the period, since Lomb-Scargle tends to have false peaks at very low frequencies. That being said, it's a very flexible method. It is extremely customizable, and the keyword arguments that can be passed to it are given in the documentation.
:zerocrossing or :zc: Find the zero crossings of the data, and use the average difference between zero crossings as the period. This is a naïve implementation, with only linear interpolation; however, it's useful as a sanity check. The keyword line controls where the "crossing point" is. It deffaults to mean(v).

For more information on the periodogram methods, see the documentation of DSP.jl and LombScargle.jl.

Example

Here we will use a modified FitzHugh-Nagumo system that results in periodic behavior, and then try to estimate its period. First, let's see the trajectory:

using DynamicalSystems, PyPlot

function FHN(u, p, t)
    e, b, g = p
    v, w = u
    dv = min(max(-2 - v, v), 2 - v) - w
    dw = e*(v - g*w + b)
    return SVector(dv, dw)
end

g, e, b  = 0.8, 0.04, 0.0
p0 = [e, b, g]

fhn = ContinuousDynamicalSystem(FHN, SVector(-2, -0.6667), p0)
T, Δt = 1000.0, 0.1
v = trajectory(fhn, T; Δt)[:, 1]
t = 0:Δt:T

fig = figure()
plot(0:Δt:T, v)
fig.tight_layout(pad=0.3); fig

Examining the figure, one can see that the period of the system is around 91 time units. To estimate it numerically let's use some of the methods:

estimate_period(v, :autocorrelation, t)

91.0

estimate_period(v, :periodogram, t)

91.62720091627202

estimate_period(v, :zerocrossing, t)

91.07000000000001

Schmelcher1997P. Schmelcher & F. K. Diakonos, Phys. Rev. Lett. 78, pp 4733 (1997)
Pingel2000D. Pingel et al., Phys. Rev. E 62, pp 2119 (2000)
Diakonos1998F. K. Diakonos et al., Phys. Rev. Lett. 81, pp 4349 (1998)