Ergodic theory and visualization. II. Fourier mesochronic plots visualize (quasi)periodicsetsZoran Levnajić and Igor Mezić Citation: Chaos 25, 053105 (2015); doi: 10.1063/1.4919767 View online: http://dx.doi.org/10.1063/1.4919767 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/25/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in On the use of Fourier averages to compute the global isochrons of (quasi)periodic dynamics Chaos 22, 033112 (2012); 10.1063/1.4736859 Ergodic theory and visualization. I. Mesochronic plots for visualization of ergodic partition and invariant sets Chaos 20, 033114 (2010); 10.1063/1.3458896 Discrete dynamical systems embedded in Cantor sets J. Math. Phys. 47, 022705 (2006); 10.1063/1.2171518 Chaos in periodically driven systems AIP Conf. Proc. 548, 31 (2000); 10.1063/1.1337757 Quasi-periodicity, global stability and scaling in a model of Hamiltonian round-off Chaos 7, 49 (1997); 10.1063/1.166240

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Ergodic theory and visualization. II. Fourier mesochronic plots visualize(quasi)periodic sets

Zoran Levnajic1,2 and Igor Mezic2

1Faculty of Information Studies in Novo mesto, 8000 Novo mesto, Slovenia2Department of Mechanical Engineering, University of California Santa Barbara, Santa Barbara,California 93106, USA

(Received 30 November 2014; accepted 23 April 2015; published online 6 May 2015)

We present an application and analysis of a visualization method for measure-preserving dynamical

systems introduced by I. Mezic and A. Banaszuk [Physica D 197, 101 (2004)], based on frequency

analysis and Koopman operator theory. This extends our earlier work on visualization of ergodic

partition [Z. Levnajic and I. Mezic, Chaos 20, 033114 (2010)]. Our method employs the concept of

Fourier time average [I. Mezic and A. Banaszuk, Physica D 197, 101 (2004)], and is realized as a

computational algorithms for visualization of periodic and quasi-periodic sets in the phase space.

The complement of periodic phase space partition contains chaotic zone, and we show how to iden-

tify it. The range of method’s applicability is illustrated using well-known Chirikov standard map,

while its potential in illuminating higher-dimensional dynamics is presented by studying the

Froeschl�e map and the Extended Standard Map. VC 2015 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4919767]

Equations modeling most of the real-world systems are

usually impossible to solve. A researcher must therefore

resort to computational methods approximating the

actual solution, or at least identifying relevant details. Of

particular interest is the detection of (almost) cyclic sets

that have periodic or quasi-periodic structure. The classi-

cal notion of Fourier averages—when interpreted in

phase space—was shown to identify these. Our method

relies on two well established methodologies: frequency

analysis, a tool for decomposing the motion into separate

easily tractable components, and ergodic theory, a

powerful mathematical framework for statistical analysis

of motion. For illustrating the method’s implementation,

we employ Chirikov standard map, a widely studied and

well understood chaotic system. We show that our

method is able to detect and graphically visualize any

periodic set within the phase space, that is, a set com-

posed of several disjoint parts that are periodically vis-

ited. Our method is applicable to systems of arbitrary

dimensionality and works regardless of their integrabil-

ity. In fact, we foresee the main application exactly in the

domain of high-dimensional system.

I. INTRODUCTION

Many problems in modern science and technology can

be formalized as dynamical systems.3,4 Yet, models of natu-

ral and engineering phenomena are often formulated as a

high-dimensional system of equations.5 This calls for the de-

velopment and improvement of computational methods

aimed at analyzing dynamical systems, with or without solv-

ing them explicitly. Over the past decades, parallel to the fas-

cinating growth of the computational power, the number of

analytic tools for complex dynamical systems also grew,6 to

include applications as diverse as diagnostics of oil spill

movement.7

Dynamical systems are still typically studied via direct

integration and visual representation of trajectories, taking

into account the ratio between precision and numerical cost.8

Some methods are based on statistical analysis of the dynam-

ics, such as return time plots9 and exit time plots.10 Others

revolve around looking for invariant sets and measures11 or

stable and unstable manifolds.12,13

Phase-space visualization methods usually work as algo-

rithms for dividing the phase space into subsets according

to a prescribed property. In our previous work,2 we analyzed

a method for visualization of invariant sets, based on the

ergodic partition theory.14,15 By computing the dynamical

time averages of chosen functions, we constructed

Mesochronic Plots (MP) allowing to graphically identify all

invariant sets and thus obtain the invariant phase space parti-

tion. Our method relied on applying the ergodic theory,16

whose power in the statistical modeling of dynamical sys-

tems can be seen in both theory17 and experiment.18 In this

paper, we extend our earlier work, and present a novel visu-

alization method, able to detect periodic phase space subsets

or any periodicity (or frequency).

In fact, frequency analysis, except being widely used for

data analysis,19,20 is also a powerful tool for studying

motion. It is applied primarily by decomposing the dynamics

into harmonic components, which is elegantly done via spec-

tral decomposition.21 The earliest implementation involves

estimating the system’s fundamental frequencies,22 which

eventually opened new ways of quantifying chaos. Another

approach relying on wavelets allows the detection of reso-

nance trappings and transitions,23 specifically useful for ex-

amination of weakly chaotic orbits. Frequency analysis is

shown to provide insights into the difference between noisy

flows and the colored noise.24 And of course, this framework

1054-1500/2015/25(5)/053105/12/$30.00 VC 2015 AIP Publishing LLC25, 053105-1

CHAOS 25, 053105 (2015)

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has interesting applications other than dynamical systems,

notably in neuroscience25 and in the study of sound and

music.26

There are several ways to “marry” the frequency analy-

sis with the ergodic theory, some of which have been

explored in the past.27 One of them is based on defining the

Fourier time average (FTA), as a generalization of the usual

functional time average known from ergodic theory.1 It can

be shown that the constancy of the absolute value of FTA

(FTA is complex-valued) is related to the invariance of the

underlying phase space portion, in a way analogous to the

usual time average.28,29 In this paper, we present a new com-

putational visualization method, constructed by computing

FTA over the grid of initial phase space points. Such 2D

plot, computed for a prescribed frequency x (or periodicity

p ¼ 1x), we call Mesochronic Fourier Plot (MFP), in analogy

with MP from Ref. 2. Periodic sets of periodicity p are

revealed as joint level sets of the absolute value of FTA.

When multiple linearly independent functions are consid-

ered, one obtains Mesochronic Fourier Scatter Plot (MFSP),

in analogy with MSP from Ref. 2. This can be used to iden-

tify the periodic partition of the phase space (meaning, the

partitioning of the phase space into a union of disjoint peri-

odic sets of prescribed periodicity). Such partition can then

be graphically visualized by coloring the separate subsets

differently. The precision of the obtained partition depends

on the number of linearly independent functions considered.

As we also show in what follows, our method is closely

related with the spectral decomposition of Koopman opera-

tor, while it is complementary to the analysis via classical

Fourier transform.30 Also, note that out method is fully sup-

ported for arbitrary measure-preserving system, regardless of

its integrability (which contrasts many of the above men-

tioned methods).

Upon exposing the mathematical details of our method,

we first present a simple implementation via Chirikov stand-

ard map,31 whose dynamical properties have been widely

studied and are well understood. Once the working and the

applicability range of our method is clear, we demonstrate

its power by analyzing higher-dimensional dynamical sys-

tems, in particular, Froeschl�e map33 and extended standard

map.32 In fact, the full potential of our method lies in the

study of high-dimensional dynamical systems, which can be

most easily done by examining MFPs of lower-dimensional

surfaces/planes.

II. THE VISUALIZATION METHOD

We begin by presenting the mathematical basis of our

method. Since our interest here is exposing the method’s

applicability, we will skip the rigorous proofs and refer the

reader to Refs. 1 and 16.

Consider a measure-preserving map T on a compact

metric phase space A evolving in discrete time t

xtþ1 ¼ Txt; t 2 Z; xt 2 A: (1)

An invariant set B � A for the dynamics T is a set such that

each trajectory that starts in it stays in it forever3

x 2 B() Ttx 2 B 8t 2 Z:

We are here interested in the specific class of invariant sets

called periodic sets. A period-p set Bp is an invariant set

composed of p disjoint subsets; Bp ¼ [pk¼1B

ðkÞp , with the fol-

lowing property:

x2BðkÞp )Tx2Bðkþ1Þp ; T2x2Bðkþ2Þ

p ;…Tpx2BðkþpÞp �BðkÞp :

(2)

Therefore, a trajectory starting in BðkÞp visits the entire cycle

Bðkþ1Þp ;B

ðkþ2Þp … visiting each subset once, before coming

back to BðkÞp after p iterations. Periodic set can be seen as a

generalization of periodic orbit, while period-1 set is of

course the usual invariant set.

In Paper I,2 we studied a method based on ergodic

theory aimed at identifying the invariant sets. Below, we

present the extension of that study, to the method for visual-

ization of (quasi)periodic sets originally introduced in Ref. 1.

The time average f* of a function f under the map equa-

tion (1) is defined as

f � xð Þ ¼ limt!1

1

t

Xt�1

k¼0

f Tkxð Þ; (3)

for almost every (a.e.) x 2 A and for every f 2 L1(A).16 The

algorithm presented2 relies on computation of the time aver-

ages and operates by decomposing the phase space into the

ergodic partition, which is identified as partitioning into a

union of invariant sets. The extension hereby utilizes the

concept of Fourier average (FTA).

Consider the map in Eq. (1) and a real-valued function

f2 L2(A). For a given frequency x 2 ½0; 12� and x 2 A, we

define the FTA f �x as

f �x xð Þ ¼ limt!1

1

t

Xt�1

k¼0

ei2pkxf Tkxð Þ: (4)

Fourier average generalizes the concept of time average

since for x¼ 0 we have f �x ¼ f �. In contrast to Eq. (3), FTA

f �x is complex-valued, and is not an invariant function, since

f �xðTtxÞ ¼ e�i2pxtf �xðxÞ:

However, the absolute value (radius) of f �x is an invariant

function

jf �xðTtxÞj ¼ jf �xðxÞj: (5)

If jf �xj is non-zero and constant over some subset of the phase

space, that indicates the invariance and (quasi)periodicity of

that subset. The subset is periodic if x is rational, and quasi-

periodic if x is irrational. We adopt the notation hx ¼ jf �xj.The Fourier average for a point (initial condition) belong-

ing to period-p set can be non-zero only if the frequency x“resonates” with the (quasi)periodicity of the set, i.e.,

nX¼mx, where n, m are non-zero integers. In all other cases,

the FTA eventually averages out to zero. To illustrate this, we

note that the expression averaged in Eq. (4) depends on the

discrete time k through the radial part f(Tkx) and the phase

053105-2 Z. Levnajic and I. Mezic Chaos 25, 053105 (2015)

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part ei2pkx. The latter directly relates to the selected frequency

x. If the selected frequency (periodicity) coincides with X of

a given set Bp, the sampling of complex values will exhibit a

(quasi)periodic pattern, allowing for Fourier average to con-

verge to a non-zero (complex) value. Otherwise, the sampled

complex values will be randomized in phase, and, regardless

of the choice of f, will always average out to zero.

We implement our method by selecting a frequency xand computing hx for a grid of initial phase space points.

This computation divides the phase space into two regions:

one composed of periodic sets resonating with x and the

remaining non-resonating region, with zero Fourier averages.

The final value of hx also depends on the choice of f, and in

fact, for some choices of f, hx can average out to zero despite

the correct frequency. Hence, in order to visualize all peri-

odic sets in their entirety, one needs to consider for each fre-

quency more linearly independent functions f.In addition to hx, we also study the argument arg f �x of

the FTA. The argument is not an invariant function, and

depicts the “phase” of the set inside the (quasi)periodic sub-

set of A.

Crucial for visualization is whether the frequency x is

rational or irrational. In addition to the resonating periodic

sets, each choice of rational periodicity also visualizes the sets

whose periodicities are integer multiples of p (for example,

visualizing period-3 sets also reveals period-6, period-9 sets,

etc.). The most meaningful choices for the frequencies are

thus mutually prime values. We call period-n partition the di-

vision (partitioning) of phase space into a union of disjoint pe-

riod-n sets. On the other hand, an irrational frequency will

visualize quasi-periodic sets such as Kolmogorov-Arnold-

Moser (KAM) curves with irrational winding numbers.

Such analysis related to the eigen-decomposition of the

Koopman operator U, defined for the map equation (1) via

composition

ðUf Þx ¼ ðf � TÞx:

U acts as an evolution operator in the functional space as

ðUf Þx ¼ f ðTxÞ, enabling an alternative representation of the

dynamics.1 It follows that, the f �x is an eigenfunction of Uwith the eigenvalue e�i2px:

Uf �x ¼ e�i2pxf �x:

Note also that frequency decomposition of a selected

trajectory of a system computed via Fourier transform

reveals all the frequencies that compose that trajectory. In

contrast, Fourier averages that we deploy here are computed

for a large set of trajectories (initial points), but only for the

single chosen frequency.

III. SINGLE FUNCTION PLOTS

In this section, we present the implementation our

method relying two-dimensional (2D) Chirikov standard

map.31 Its dynamical properties are well known, and its

extensions are still widely studied.34 It is a measure-

preserving (symplectic) map, mapping the square ½0; 1��½0; 1� � ½0; 1�2 onto itself

x0 ¼ xþ yþ e sinð2pxÞ ½mod 1�y0 ¼ yþ e sinð2pxÞ ½mod 1� (6)

(the usual standard map’s parameter k is k¼ 2pe). The map

possesses a very rich structure of periodic sets changing with

the parameter e. We set up an orthogonal grid of D�Dpoints on the square [0, 1]2 using D¼ 800. We then compute

the FTA for each grid point after T iterations. Thus, we

obtain a grid of D�D values of hx, which is our MFP,

visualized by assigning a color-value to each grid-point. We

employ on the log-scale since the obtained values range over

six decades. The number T is set in accordance with the con-

vergence properties, which are studied in detail in Sec. IV.

Here, we mostly use T¼ 30 000. For simplicity, we mostly

employ a single Fourier function f ðx; yÞ ¼ sinðpxÞ.In Fig. 1, we show four plots for e¼ 0.12 and different

frequency values. The known periodic sets of all periodici-

ties are correctly visualized. In the x ¼ 12

(p¼ 2) plot, the

central period-2 set is clearly pronounced. As expected, all

sets of higher even periodicity (4, 6…) are also pronounced

(standard map equation (6) is on torus). In the plot for x ¼ 13

(p¼ 3), two largest period-3 sets centered around period-3

orbits are visible, along with period-6 sets and period-12

sets. Similarly, four period-5 sets and a single period-10 set

can be seen in x ¼ 15

plot. Many periodic sets of higher

periodicities can be also recognized (due to limited resolu-

tion) as horizontal curves. These periodic sets resonate with

the selected frequency, and have the hx-values in range of

O(10�1). In contrast, non-resonating periodic sets in all

three plots have hx-values in range of O(10�5), which

clearly distinguishes them from the resonating periodic sets.

Interestingly, the chaotic zone around the hyperbolic fixed

point (0, 0) can be also recognized in these three plots, since

its hx-values are between the resonating and the non-

resonating value range, in range of O(10�3). As pointed out

before, f �x for all non-resonating points converge to zero.

However, the convergence for the non-resonating periodic

sets is much faster than for chaotic zone. In fact, the chaotic

zone weakly resonates at all frequencies, which, in principle,

allows its visualization.35 We develop this argument later.

The last plot in Fig. 1 is done for x¼ 0 (equivalent to

x¼ 1), and thus represents time average. Recall that now all

invariant sets resonate, regardless of their periodicity, includ-

ing the chaotic zone. This means that all hx values are in

range of O(10�1), as clear from the plot. This is the only fre-

quency where the period-1 sets resonate (around fixed point

at 12; 0

� �). While useful in detecting the invariant sets, the

time average Eq. (3) cannot reveal their periodicity, since all

periods are multiples of 1. For this reason, considering x¼ 0

is not particularly useful in our analysis here, but it clarifies

the difference with the usual time average. We also note that

all periodic sets visualized in Fig. 1 have internal structure

involving smaller periodic sets with diverse periodicities,

which are not visible due to limited resolution.

As discussed at length in Ref. 2, the choice of function

used for time averaging determines which details of the

invariant sets will be revealed. The same holds for FTA: in

order to capture all sets periodic at a given frequency, we

053105-3 Z. Levnajic and I. Mezic Chaos 25, 053105 (2015)

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need to consider more linearly independent functions. To

illustrate this, in Fig. 2 we show MFP for x ¼ 12

at e¼ 0.12

using four different functions. Each function identifies differ-

ent phase space features belonging to the period-2 phase

space partition. Two plots on the left isolate further period-2

structures within large-scale periodic sets. Two plots on the

right fail to detect the large central period-2 set due to func-

tion being even in x.

Given that each function identifies a potentially new per-

iodic set within a given periodic partition, it is necessary to

consider the information from multiple linearly independent

functions. To illustrate this, we construct the MFSP by plot-

ting the values of hx for one function against the values for

another function. In Fig. 3, top and bottom, we show two

scatter plots each done using two functions, with frequencies

x ¼ 13

and x ¼ 15, respectively. Each point in a scatter plot is

defined by its x-coordinate (hx for the first function) and by

its y-coordinate (hx for the second function). For both plots,

the same patterns observed earlier are visible. Scatter plot

points belonging to non-resonating periodic set, chaotic

FIG. 2. Single-function plots of hx for

e¼ 0.12 and x ¼ 12, using various func-

tion as indicated.

FIG. 1. Single-function plots of hx for

e¼ 0.12 and various frequencies x.

053105-4 Z. Levnajic and I. Mezic Chaos 25, 053105 (2015)

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zone, and resonating periodic sets are identified in the figure.

Interestingly, between regions, there are almost continuous

transitions, which correspond to chaotic phase space points

evolving in the vicinity of periodic sets.

Next, we investigate the visualization of periodic set in

relation to various values of e in the map (6). In Fig. 4, we

show four MFPs for x ¼ 13

and different e-values. In the plot

for e¼ 0, the lines y ¼ 13

and y ¼ 23

entirely consisting of

period-3 orbits are clearly visible. The phase space regions

near the periodic sets weakly resonate, since non-resonating

trajectories, whose structure is close to periodic, converge to

zero more slowly, similar to the chaotic zone. This effect is

also present in plot for e¼ 0.06—where, as expected, two

main period-3 sets can be observed, in addition to many

minor periodic sets of higher periodicity that is a multiple of

3. Plot for e¼ 0.18 exhibits period-3 and period-6 sets of var-

ious shapes. Observe the secondary period-6 sets located

around primary period-3 sets and central period-2 set. At this

e-value, the chaotic zone can be clearly seen in a sharp con-

trast with both resonating and non-resonating periodic sets.

Similar contrast is found for e¼ 0.24, where the chaotic zone

is even more uniformly colored.

Selection of rational frequencies up to 12

allows identifi-

cation of periodic sets with integer periodicities (note that

choosing, e.g., x ¼ 25

still yields period-5 sets). However,

standard map possesses an even richer variety of quasi-

periodic orbits with irrational rotation numbers, which are

expected to resonate at irrational frequencies. Chaotic zone

is expected to resonate at all frequencies, including irrational

ones. To examine this, we consider the golden mean fre-

quency xg ¼ffiffi5p�1

2. In Fig. 5 we show another four MFPs

with increasing values of e, computed for x¼xg. The invar-

iant KAM curve for the corresponding frequency is clearly

pronounced in e¼ 0.06 plot. In the plot for e¼ 0.12, the

FIG. 3. Scatter plots of two functions (indicated on axes) for e¼ 0.12 and

x ¼ 13; x ¼ 1

5. Phase space regions with different dynamical behaviors are

indicated.

FIG. 4. Single-function plots of hx for

x ¼ 13

and various values of e.

053105-5 Z. Levnajic and I. Mezic Chaos 25, 053105 (2015)

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curve is still visible, together with the weakly resonating

chaotic zone. Break-up of the golden mean KAM curve

marks the chaotic transition in dynamical systems such as

standard map.3 Our method can, in principle, be used for

examining this transition. In the last two plots for e¼ 0.18

and e¼ 0.24, the chaotic zone is the only resonating phase

space region, in addition to the secondary KAM golden

mean curve visible in e¼ 0.24 plot. The resonating and non-

resonating hx-values in all plots are again in the same value

range as discussed above. The irrational frequency allows

for easier visualization of the chaotic zone.

We found that periodic and quasi-periodic sets can ei-

ther resonate or not resonate, depending on the selected fre-

quency. In contrast, the chaotic zone resonates for allfrequencies, although it always does so weakly. We show

this quantitatively via histograms of hx-values, computed for

different frequencies and reported in Fig. 6. The parameter

e¼ 0.24 is chosen so that the chaotic and regular regions

occupy roughly similar portions of the phase space.

Histograms for x ¼ 12

and x ¼ 13

each consist of three well

pronounced (groups of) peaks. Each corresponds to one

phase space region, as indicated in the figure. The peaks are

well separated; their distinction, in fact, grows with final iter-

ation number T (see Sec. IV). While all the peaks for peri-

odic sets are very sharp, the chaotic peak displays a more

smeared shape. In the histogram for x¼xg, the resonating

peaks are very faint, while the other two peaks lie in the

range of the corresponding peaks for rational frequencies.

Chaotic zone consists of trajectories with a broad Fourier

spectrum, which is reflected in our findings. This indicates

that for any selection of frequency, FTA indeed divides the

phase space into three disjoint regions: periodic sets resonat-

ing at the selected frequency x, periodic sets not resonating

at the frequency x (i.e., set of periodicity incommensurable

with x), and the chaotic region (resonating at all

frequencies).

We finish this section by studying the behavior of the

complex phase of Fourier averages. Recall that f �x is not an

invariant function, since each iteration rotates it by the factor

e�i2px. In Fig. 7, we show the values of f �x in the complex

plane. Radial coordinate is in log-scale for consistency with

other plots. Along the radial coordinate, we immediately rec-

ognize three peaks considered in Fig. 6: dense cluster around

jf �xj Oð10�5Þ (non-resonating periodic sets), circular ring

at jf �xj Oð10�3Þ (chaotic region), and outer points for

jf �xj Oð10�1Þ (resonating periodic sets). The phase of f �x in

the chaotic zone is uniformly randomized over the circle.

In contrast, resonating periodic sets are consistent in phases;

in profile for x ¼ 13, we find six radially coherent groups of

f �x-values symmetrically organized around the center, each

group identifying a separate subset of period-3 and period-6

sets. Similarly, five radially coherent groups of f �x-values are

corresponding to the resonating period-5 sets in profile for

FIG. 5. Single-function plots of hx for

irrational frequency xg ¼ffiffi5p�1

2and

various values of e.

FIG. 6. Distribution of hx-values for e¼ 0.24 and frequencies x ¼ 12; 1

3;xg.

Peaks corresponding to different phase space regions are indicated.

053105-6 Z. Levnajic and I. Mezic Chaos 25, 053105 (2015)

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x ¼ 15, along with a rim of scattered points representing

smaller sets of higher periodicity. The complex phase of

FTA reflects the periodicity of the phase space region.

To illustrate the behavior of complex phase in the phase

space, we show, in Fig. 8, four plots of argf �x, pictured

through the cyclic colorbar. As noted earlier and observed in

Fig. 7, all resonating periodic sets exhibit coherent phase val-

ues, symmetrically organized among the subsets. In x ¼ 13

plot, three subsets of large period-3 sets have, respectively,

arg f �x ’ 0; arg f �x ’ p3, and arg f �x ’ � p

3. Similarly, each sub-

set of period-5 sets in x ¼ 15

plot has a defined phase value

p6 2pn5

for n¼ 0, 1, 2. On the other hand, chaotic zone exhib-

its a completely randomized phase values, uniformly distrib-

uted in (�p, p), which is another way to discern it from the

regular region. Non-resonating periodic sets are relatively

uniform in phase value.

Finally, we examine, in Fig. 9, the evolution of phase

space with increase of e, through the lenses of arg f �x for x¼ 1

3(cf. Fig. 4). The complex phases of the three subsets of

each period-3 set are 2p3

apart from one another. Note that the

effect in its weak form is also present in the vicinity of

the period-3 sets, as clearly visible in e¼ 0 plot. In contrast,

the chaotic zone is uniformly randomized in values of argf �x.

The phase of non-resonating periodic sets is zero (but this is

irrelevant since absolute value jf �xj ¼ hx is anyhow zero).

IV. THE CONVERGENCE PROPERTIES

In practical measurements, obviously, a finite time aver-

age must be used, leading to finite time FTA. In this section,

we examine how hx converge to their limit values. As we

show, the convergence regime directly depends on the dy-

namical nature of the considered trajectory, and can thus be

used to establish the number of iterations T.

We take f ðx; yÞ ¼ sinðpxÞ for x ¼ 13

at e¼ 0.12, and

consider the evolution of hx as function of time. In Fig. 10,

we show the value of hx obtained after t iterations, for three

different points representing different dynamical regions.

While the convergence of hx is fast for both resonating and

non-resonating phase space point, it becomes very irregular

FIG. 7. Values of f �x in complex plane for f ðx; yÞ ¼ sinðpxÞ; e ¼ 0:12 and

x ¼ 13; 1

5(log-scale in radial coordinate). For better visibility, we used T¼ 105.

FIG. 8. Complex phase values argf �xfor e¼ 0.12 and various frequencies x.

053105-7 Z. Levnajic and I. Mezic Chaos 25, 053105 (2015)

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for the chaotic point. In fact, the convergence properties are

equivalent to those for the usual time averages.2 For regular

trajectories, the convergence rate is given by const:t , which

refers to both resonating and non-resonating periodic sets. In

chaotic region, the convergence is much less predictable; the

error can, in general, be bounded from below by 1ffitp . This also

depends on the strength of chaotic motion—for very strong

chaos (mixing), the convergence rate approaches 1ffitp . These

properties remain the same if the irrational frequencies are

considered: for quasi-periodic orbits, whether resonating or

not, the convergence exhibits const:t rate. In contrast, for cha-

otic zone, the convergence is again at best given by 1ffitp .

We now examine the applicability of our method in

detecting the chaotic zone. In Fig. 11, we show the conver-

gence colorplot, indicating the fraction of phase space

points with a given hx at a certain time. Three peaks evolve

with time, corresponding to resonating, chaotic, and non-

resonating part of the phase space (cf. Fig. 6). The resonat-

ing peak does not move. In contrast, both chaotic and non-

resonating peaks move towards zero, but with different

speeds. The speed of the non-resonating peak is roughly

double the speed of the chaotic peak. This shows that the

convergence rate for the chaotic zone is approximately 1ffitp ,

meaning that for a sufficiently long time, it can be clearly

distinguished from the non-resonating region.36 Note also

that the shape of peaks does not change significantly as they

move. Fig. 11 considers e¼ 0.24 where the chaos is strong;

while this result does hold for a general e-value, the separa-

tion of peaks might be slower for smaller e values, or might

be less clear due to smaller chaotic zone.

FIG. 9. Complex phase values argf �xfor x ¼ 1

3and various values of e.

FIG. 10. The value of hx as function of time, for e¼ 0.12 and x ¼ 13, and for

three different initial points, belonging to different dynamical regions.

FIG. 11. The fraction of phase space points (colorbar), as function of time tand hx (log-scale on x and y coordinate, but not in colorbar) for e¼ 0.24 and

x ¼ 13.

053105-8 Z. Levnajic and I. Mezic Chaos 25, 053105 (2015)

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V. THE FROESCHL�E MAP

So far we illustrated the applicability of our visualiza-

tion method relying on well-known standard map. In this

section and the next one, we demonstrate the method’s full

potential by studying more complicated higher-dimensional

dynamical systems.

We begin with the Froeschl�e map, which is a measure-

preserving mapping of 4-dimensional cube [0, 1]4 onto itself,

introduced in the context of celestial dynamics.33 It is defined as

x01 ¼ x1 þ y1 þ e1 sinð2px1Þ þ g sinð2px1 þ 2px2Þ ½mod 1�y01 ¼ y1 þ e1 sinð2px1Þ þ g sinð2px1 þ 2px2Þ ½mod 1�x02 ¼ x2 þ y2 þ e2 sinð2px2Þ þ g sinð2px1 þ 2px2Þ ½mod 1�y02 ¼ y2 þ e2 sinð2px2Þ þ g sinð2px1 þ 2px2Þ ½mod 1�;

(7)

and it can be seen as two symplecticly coupled standard

maps. We apply our method to examine the evolution and

eventual destruction of periodic sets as the coupling parame-

ters are increased. To this end, we fix e1¼ 0.12 and e2¼ 0,

which amounts to examining a standard map with e¼ 0.12

(studied at length above) coupled to a shear flow. Our

investigation is carried out by varying the parameter g which

regulates the interaction between the two standard maps.

As in Ref. 2, we begin by the MFPs of 2D phase space slice

(2D section in 4D phase space). We construct a grid of 500

� 500 points in (x1, y1)-space on the section given by (x2, y2)

¼ (0, 0). We consider the function f ðx1;y1;x2;y2Þ¼2cosð2py1Þþcosð2py1Þcosð2py2Þþcosð12px2Þþcosð12px1Þ and run the

dynamics for 2�105 iterations. Three frequencies are consid-

ered: rational x¼ 12

and 13, and irrational xg. For each frequency,

three values of parameters g are examined. The resulting

MFPs are shown in Fig. 12. For g¼0.01, the phase space

similar to the known standard map can be recognized, with

all periodic sets as expected. As g increases, the periodic sets of

higher multiples of the base period disintegrate, while the cha-

otic zone spreads, eventually covering most of the phase space.

This effect is consistently found in plots for all frequencies.

Interestingly, in the xg plot for g¼0.02, a quasi-periodic curve

with irrational winding number appears to resonate strongly.

As before, the values of hx are again in the same intervals

corresponding to the phase space regions. Remember that in

general, invariant sets are here four dimensional, meaning

that only their 2D projections can be visible in these section

plots.

FIG. 12. MFPs of function f ðx1; y1; x2; y2Þ ¼ 2 cosð2py1Þ þ cosð2py1Þ cosð2py2Þ þ cosð12px2Þ þ cosð12px1Þ under the dynamics of Froeschl�e map with

e1¼ 0.12 and e2¼ 0, for x ¼ 12; 1

3and xg, computed on the phase space section (x2¼ 0, y2¼ 0) over the grid of 500� 500 points. The dynamics was run for

T¼ 2� 105 iterations. The values of g are indicated in each plot.

053105-9 Z. Levnajic and I. Mezic Chaos 25, 053105 (2015)

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Next, we examine the global evolution of the periodic

sets with variations of g for e1¼ 0.12 and e2¼ 0. We start

with 104 976 phase space points, which we pick at random

from [0, 1]4 in order to avoid hitting too many resonances,

since they may lead to trivial behavior. For each initial

point, we compute hx after T¼ 2� 105 iterations for the

same function as in Fig. 12, and for the same three frequen-

cies. For each value of g, the histogram of hx values is

computed, and the results are shown in Fig. 13. Histograms

for all frequencies consist of the usual three groups of

peaks: two corresponding to the resonating and non-

resonating periodic sets and one to the chaotic zone. The

histograms are a clear illustration of spreading of the cha-

otic motion with the increase of the coupling parameter g.

Initially, the chaotic peak is rather small, while the peri-

odic peaks are both large. Increase of g systematically

enlarges the chaotic peak at the expense of periodic peaks,

eventually comprising most of the phase space. Actually,

as discussed in Ref. 2, the chaotic transition for Froeschl�emap is qualitatively similar to that of standard map (break-

up of invariant KAM tori37), although it appears to involve

more complicated mechanisms. Interestingly, the non-

resonating periodic peak is considerably larger than the

resonating one, for all three frequencies. This indicates

that a majority of the periodic sets in the Froeschl�e map are

of high periodicity, incommensurable with the three con-

sidered frequencies.

VI. EXTENDED STANDARD MAP

In this section, we study the extended standard

map using Fourier averages. The map has been proposed in

Ref. 32 as a three-dimensional generalization of standard

map. It is a measure-preserving action-action-angle mapping

of the cube [0, 1]3 onto itself, defined by

x0 ¼ xþ e sinð2pzÞ þ d sinð2pyÞ ½mod 1�y0 ¼ yþ e sinð2pzÞ ½mod 1�z0 ¼ zþ xþ e sinð2pzÞ þ d sinð2pyÞ ½mod 1�: (8)

When either e or d are zero, the phase space can be divided

into parallel 2D planes, such that dynamics on each of plane

is isolated from the dynamics on other planes. On the other

hand, when both e and d are non-zero, this dynamical sys-

tem is argued to be ergodic over the entire phase space.32 In

fact, no invariant surface persists once both perturbations

are simultaneously positive. On top of the numerical evi-

dence already provided via ergodic partitioning in Ref. 2,

we here provide an additional argument supporting this

claim.

We set a three-dimensional grid of 50� 50� 50 points

covering the phase space of extended standard map with

e¼ 0.01 and d¼ 0.001. We run the Fourier averages of func-

tion f ðx; y; zÞ ¼ sinð4pyÞ cosð4pxÞ cosð12pzÞ for T¼ 2� 105

iterations, and for three frequencies x ¼ 12; 1

3;xg. We show

the histograms of the obtained values of hx in Fig. 14. For

all three histograms, both the range of values and the bell-

shape of the curves indicate that all grid points lie in the cha-

otic region of the phase space. This suggests that there are no

periodic sets of the considered periodicities of any size. In

fact, it implies that there are no periodic sets of any periodic-

ity, since any periodic set of frequency different from the

considered would be visible as non-resonating. Note that

irrational frequency xg agrees with this result. This clearly

confirms the claim of the extended standard map being er-

godic. This point also serves to note the usefulness of our

method: we employed FTA analysis to study higher-

FIG. 13. Histograms of hx values computed for f ðx1; y1; x2; y2Þ ¼ 2 cosð2py1Þ þ cosð2py1Þ cosð2py2Þ þ cosð12px2Þ þ cosð12px1Þ under the dynamics of

Froeschl�e map for x ¼ 12; 1

3and xg for a 4D grid of 104 976 random phase space points. The dynamics was run for T¼ 2� 105 iterations. The g value is indi-

cated in each histogram.

053105-10 Z. Levnajic and I. Mezic Chaos 25, 053105 (2015)

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dimensional dynamical systems, much more complicated

than the standard map.

VII. DISCUSSION AND CONCLUSIONS

We exposed a new method of analysis of measure-

preserving dynamical systems based on Fourier averages

proposed in Ref. 1, with roots in both frequency analysis and

ergodic theory, as a continuation of our earlier work on er-

godic partition theory.2,15 We showed the method’s imple-

mentation using the known properties of Chirikov standard

map. Of course, our method is in no way limited to discrete-

time dynamical systems (maps) considered here. Except for

bigger computational demand, the method is fully imple-

mentable for continuous-time dynamical systems. The distri-

bution of values of hx was analyzed, resulting in the

identification of histogram peaks that correspond to different

dynamical regions, in particular, resonating periodic set,

non-resonating periodic sets, and chaotic region. The conver-

gence issues have been considered in relation to estimating

the precision of the obtained hx. The full potential of the

method was demonstrated in an application to higher-

dimensional dynamical systems, 4D Froeschl�e map, and 3D

extended standard map. Known dynamical features of these

systems were readily found, in addition to certain novel

insights.

Our method is applicable regardless of the system’s inte-

grability or other dynamical properties. The key generaliza-

tion to follow is the development of clustering algorithms, in

analogy with MSPs examined in Ref. 2, aimed at visualizing

the entire periodic partition for a given frequency.40,41 This

is done by considering Fourier averages of multiple func-

tions with the same frequency and optimally clustering the

scatter plot points (cf. Fig. 3). The periodic partition is

obtained upon identifying the phase space part that corre-

sponds to each cluster of scatter plot points. When a suffi-

cient number of linearly independent functions is considered,

this clustering algorithm will reveal the periodic partition

with any prescribed precision. Of course, this opens the

question of optimal clustering, which may depend on the dy-

namical system and the corresponding scatter plot (e.g.,

shape/volume of the clustering cells). Furthermore, the

relationships between the convergence properties and the na-

ture of the underlying trajectory are to be further investi-

gated. While the convergence slope gives a rough estimate

of the type of the orbit, a detailed analysis separating

between rational and irrational frequencies might yield fur-

ther insights.

The full applicative extent of our method lies in high-

dimensional systems, which can be investigated in various

ways, for instance via MFPs done over 2D phase space sec-

tions. In fact, the method can be easily extended to distrib-

uted systems, such as complex networks,38 and, in particular,

networks that model extended dynamical systems.34,39 This

could reveal novel details related to the collective dynamical

behavior, usually displayed by such systems.

ACKNOWLEDGMENTS

Work supported by the Republic of Slovenia via

Creative Core FISNM-3330-13-500033 (also funded by the

European Union and the European Regional Development

Fund), in addition to the ARRS Program Nos. P1-0383 and

P1-0044, and Project No. J1-5454. Work also supported by

the AFOSR Grant Nos. F49620-03-1-0096 and FA9550-09-

1-0141 and ARO MURI W911NF-14-1-0359. Part of this

work was done during Z.L.’s stay at University of

California, Santa Barbara, another part during his stay at

Jo�zef Stefan Institute, and another during his stay at Potsdam

University. We thank the colleagues Umesh Vaidya, Arkady

Pikovsky, Dima Shepelyansky, and Toma�z Prosen for useful

comments and discussions.

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