Finite-time and exact Lyapunov dimension of the Henon map.

N.V. Kuznetsov,1, 2, ∗ G.A. Leonov,1, 3 and T.N. Mokaev1

1Faculty of Mathematics and Mechanics, St. Petersburg State University, Peterhof, St. Petersburg, Russia2Department of Mathematical Information Technology, University of Jyvaskyla, Jyvaskyla, Finland

3Institute of Problems of Mechanical Engineering RAS, Russia(Dated: December 6, 2017)

This work is devoted to further consideration of the Henon map with negative values of theshrinking parameter and the study of transient oscillations, multistability, and possible existence ofhidden attractors. The computation of the finite-time Lyapunov exponents by different algorithmsis discussed. A new adaptive algorithm for the finite-time Lyapunov dimension computation instudying the dynamics of dimension is used. Analytical estimates of the Lyapunov dimension usingthe localization of attractors are given. A proof of the conjecture on the Lyapunov dimension ofself-excited attractors and derivation of the exact Lyapunov dimension formula are revisited.

I. INTRODUCTION

In 1963, American mathematician and meteorologistEdward Lorenz published an article in which he inves-tigated an approximate model of the fluid convection ina two-dimensional layer and discovered its irregular be-havior [52]. The analysis of this behavior was closelyrelated to the phenomenon of chaos, namely to the sen-sitive dependence on the initial conditions and the exis-tence in the phase space of the system of a chaotic at-tractor with a complex geometrical structure. Later, thisthree-dimensional system was called the Lorenz system,and so far it has great scientific interest [48, 63, 64].

In 1969, French mathematician and astronomer,Michel Henon, showed that the essential properties ofthe Lorenz system (i.e., folding and shrinking of volumes)can be preserved by means of a specially chosen sequenceof approximating maps [32]. The Poincare map of theLorenz system was chosen as the reference, resulting inthe following map ϕ(x, y) : R2 → R2, where

ϕ(x, y) =(1 + y − ax2, bx

), (1)

a > a0 = − (b−1)2

4 (folding parameter), and b ∈ (0, 1)(shrinking parameter) are parameters of mapping. Oneusually considers an equivalent form of the Henon map(e.g. [9, 29, 33])

ϕ(x, y) =(a+ by − x2, x

), (2)

obtained from (1) by changing the coordinates x := ax,y := a

b y. Along with positive values of parameter b,considered in [32], later on one also studies the Henonsystem with negative values b ∈ (−1, 0) [25, 31].

The Henon map and its various generalizations haveattracted the attention of researchers with its compar-ative simplicity and the ability to model its dynam-ics without integrating differential equations (see, e.g.[4, 5, 23, 27–29, 61]). Despite being introduced initially

∗ Corresponding author: nkuznetsov239@gmail.com

as a theoretical transformation, it also has several physi-cal interpretations [7, 31]. Further, we will study Henonmap in the form (2) with b 6= 0, |b| < 1.

II. TRANSIENT OSCILLATIONS,ATTRACTORS AND MULTISTABILITY

Consider a dynamical system with discrete time({ϕt}t≥0, (U ⊆ R2, || · ||)

)generated by the recurrence

equation with map (2)

u(t+ 1) = ϕ(u(t)), u(0) = u0 ∈ U, t ∈ N0, (3)

where ϕt = (ϕ ◦ ϕ ◦ · · · ◦ ϕ)︸ ︷︷ ︸t times

, ϕ0 = idR2 , and u(t) =

(x(t), y(t)) with ||u0|| =√x2

0 + y20 .

The equilibria O± = (x±, x±) of the system (3) existwhen a > a0. Here

x± = 12(b− 1±

√(b− 1)2 + 4a

).

The 2× 2 Jacobian matrix is defined as follows:

J(u0)=J((x0, y0)

)=Dϕ

((x0, y0)

)=(−2x0 b

1 0

), (4)

where |det J(u0)| ≡ |b| < 1, and has the eigenvaluesλ±(u0) = −x0 ±

√b+ x2

0, and eigenvectors ν±(u0) =(−x0 ±

√b+ x2

01

). One can check that equilibrium O−

is always unstable and O+ is unstable when a > a1 =3(b−1)2

4 .In the seminal work [32] for a fixed value b = 0.3 it was

shown numerically that for a < a0 and a > a3 ≈ 1.55any trajectory ϕt(u0) tends to infinity for all u0 ∈ R2.For a0 < a < a3, depending on the initial point u0, thetrajectory ϕt(u0) either tends to infinity, or reaches theattractor that is a stable equilibrium for a0 < a < a1, aperiodic orbit for a1 < a < a2 ≈ 1.06, and a nontrivialchaotic attractor for a2 < a < a3. These numerical ex-periments imply that there is no global attractor in theHenon system for U = R2.

arX

iv:1

712.

0127

0v1

[nl

in.C

D]

3 D

ec 2

017

2

x

y

O+

O−

0.8838955 0.883896 0.8838965 0.8838970.88389

0.883892

0.883894

0.883896

0.883898

0.8839

O+

FIG. 1: Chaotic Henon attractor (blue) for theparameters a = 1.4, b = 0.3 is self-excited with respect toboth equilibria O± (red); the basin of attraction (orange).

Computational errors (caused by finite precision arith-metic) and sensitivity to initial conditions allow one toget a reliable visualization of chaotic attractor by onlyone pseudo-trajectory computed for a sufficiently largetime interval.

Definition 1 ([40, 47, 49, 50]) An attractor is calleda self-excited attractor if its basin of attraction inter-sects with any open neighborhood of a stationary state(an equilibrium), otherwise, it is called a hidden attrac-tor.

For a self-excited attractor its basin of attraction isconnected with an unstable equilibrium and, therefore,self-excited attractors can be localized numerically by thestandard computational procedure in which after a tran-sient process a trajectory, started in a neighborhood ofan unstable equilibrium (e.g., from a point of its unsta-ble manifold), is attracted to the state of oscillation andthen traces it. Thus, self-excited attractors can be easilyvisualized (e.g. the classical Lorenz and Rossler attrac-tors can be visualized by a trajectory from a vicinity ofunstable zero equilibrium).

For a = 1.4, b = 0.3 the equilibria O± are saddles(|λ±(O±)| < 1, |λ∓(O±)| > 1) and one can visualizethe classical Henon attractor [32] (see Fig. 1) from theδ-vicinity of O± using trajectories with the initial data

u0 = O± + δν∓(O±)∗

||ν∓(O±)|| , δ = 0.1.

Similar self-excited attractor can be also obtained fornegative values of parameter b (e.g. for a = 2.1, b = −0.3in [25, 31]).

In [19] the multistability and possible existence of hid-den attractor in the Henon system a = 1.49, b = −0.138were studied by the perpetual point method [18, 57]. Forthese values of parameters |λ+(O−)| > 1, |λ−(O−)| < 1and from the initial point

u0 = O− + δν+(O−)∗

||ν+(O−)|| , δ ≤ 0.01

on the unstable manifold of the saddle O−, defined by theeigenvector ν+(O−), a chaotic attractor in Fig. 2a can bevisualized. Thus, the chaotic attractor obtained in [19]is a self-excited attractor with respect to O−. In addi-tion, a self-excited periodic attractor can be visualizedfrom vicinity of O+. See coexisting self-excited periodicand chaotic attractors and their basins of attraction inFig. 2b.

In [27] the multistability in the Henon system was stud-ied for positive parameters. In Fig. 4a, for parametersa = 0.98, b = 0.4415 there are three co-existing self-excited attractors: period-8 orbit self-excited with re-spect to O± (used initial data u0 = O+ + δ ν−(O+)∗

||ν−(O+)|| ,δ = 10−4), period-12 orbit self-excited with respect toO± (used initial data u0 = O− + δ ν+(O−)∗

||ν+(O−)|| , δ = 0.1),and period-20 orbit self-excited with respect to O± (usedinitial data u0 = O+ − δ ν−(O+)∗

||ν−(O+)|| , δ = 0.1). In Fig. 4b,for parameters a = 0.972, b = 0.4575 there are threeco-existing attractors: period-12 orbit self-excited withrespect to O± (used initial data u0 = O− + δ ν+(O−)∗

||ν+(O−)|| ,δ = 0.01), period-16 orbit self-excited with respect toO± (used initial data u0 = O+ − δ ν−(O+)∗

||ν−(O+)|| , δ = 10−8),and chaotic attractor self-excited with respect to O±

(used initial data u0 = O+ + δ ν−(O+)∗

||ν−(O+)|| , δ = 0.1).Last but no least, in Fig. 4c, for parameters a = 0.97,b = 0.466 there are three co-existing self-excited attrac-tors: period-8 orbit self-excited with respect to O± (usedinitial data u0 = O− + δ ν+(O−)∗

||ν+(O−)|| , δ = 0.01), and twochaotic attractors each one self-excited with respect toO± (used initial data u0 = O+ + δ ν−(O+)∗

||ν−(O+)|| , δ = 10−7,

and u0 = O+ − δ ν−(O+)∗

||ν−(O+)|| , δ = 0.1, respectively). In[23, 24] the coexistence of periodic orbits is studied nearthe critical cases when b → ±1 and the map becomesless and less dissipative. The possible existence of hid-den chaotic attractors in the Henon map requires furtherinvestigation.

In the numerical computation of trajectory over afinite-time interval it is often difficult to distinguish asustained oscillation from a transient oscillation (a tran-sient set in the phase space, e.g. chaotic or quasi periodic,which can nevertheless persist for a long time) [30, 41].Thus, a similar to the above classification can be intro-duced for the transient sets.

3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

y

O+

O−

(a) After the transient process (cyan) the trajectory from theδ-vicinity of the equilibrium O−, δ = 0.01 localizes the chaotic

attractor (blue).

x

y

O+

O−

(b) Self-excited periodic attractor (black) with respect to theunstable equilibria O± (red) and its basin of attraction (green);self-excited chaotic attractor (blue) with respect to the unstable

equilibrium O− (red), its basin of attraction (orange).

FIG. 2: Multistability and coexistence of two local attractors in the Henon system with a = 1.49, b = −0.138.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

y

O+

O−

(a) Transient chaotic set existing for t = 1, . . . , T ≈ 7300.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x

y

O+

O−

(b) Period-7 limit cycle, resulting from the collapsed transient set,t > 7300.

FIG. 3: For the Henon system (3) with parameters a = 1.29915, b = 0.3 the evolution of the self-excited transientchaotic set (blue) with respect to O+.

Definition 2 ([10, 13]) A transient oscillating set iscalled hidden if it does not involve and attract trajec-tories from a small neighborhood of equilibria; otherwise,it is called self-excited.

In the Henon system (3) it is possible to observethe long-lived transient chaotic sets for b = 0.3 anda ∈ (1.29, 1.3). For example, for a = 1.29915 it is pos-

sible to localize a self-excited transient chaotic set1 withrespect to the saddle O+ (|λ+(O+)| < 1, |λ−(O+)| > 1)

1 Similar behavior with hidden transient chaotic set can be ob-tained in various Lorenz-like systems, e.g. in the classical Lorenzsystem [10], in the Rabinovich system [39]

4

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

x

y

O+

O−

(a) Parameters a = 0.98, b = 0.4415: period-8(blue), period-12 (cyan), and period-20 (pink)

orbits.

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

x

y

O+

O−

(b) Parameters a = 0.972, b = 0.4575:period-12 (cyan), period-16 (pink) orbits, and

chaotic attractor (blue).

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

x

y

O+

O−

(c) Parameters a = 0.97, b = 0.466: period-8orbit (cyan), and two chaotic attractors (blue,

pink).

FIG. 4: Multistability in the Henon system (3) with three co-existing self-excited attractors.

using the initial data

u0 = O+ + δν−(O+)∗

||ν−(O+)|| , δ = 0.1,

which persists for ≈ 7300 iterations and after that con-tracts into a period 7 limit cycle.

In order to distinguish an attracting chaotic set (at-tractor) from a transient chaotic set in numerical ex-periments, one can consider a grid of points in a smallneighborhood of the set and check the attraction of corre-sponding trajectories towards the set. Various examplesof hidden transient chaotic sets localization are discussed,e.g., in [10, 13–15, 67].

In [25], for parameters a > 0 and b ∈ (0, 1), it is sug-gested an analytical bounded localization of attractors inthe Henon map by the set B = M \(Q

⋃R1⋃R2), where

M = {x, y | x < m, y < m+ a}, m = a(1+2b)1−b > 0,

Q = {x, y | x < r, y < 0}, r = − 1+√

1+4a2 < 0,

R1 = {x, y | x < −√b(m+ a) + a− r, y ≤ m+ a},

R2 = {x, y | x ≤ m, y < − 1b

(a+√a+ bm− r

)}.

(5)A similar set can be considered for negative values of b.

Further by K we denote a bounded closed invari-ant set, e.g. a maximum attractor with respect to theset of all nondivergent points from B (i.e. u0 ∈ B :lim supt→∞ |ϕt(u0)| 6=∞).

III. FINITE-TIME LYAPUNOV DIMENSIONAND ALGORITHMS FOR ITS COMPUTATION

The concept of the Lyapunov dimension was suggestedin the seminal paper by Kaplan & Yorke [34] and later ithas been developed and rigorously justified in a numberof papers. Nowadays, various approaches to the Lya-punov dimension definition are used (see, e.g. [22, 42]).

Below we consider the concept of the finite-time Lya-punov dimension [35, 39], which is convenient for carry-ing out numerical experiments with finite time.

Let a nonempty closed bounded set K ⊂ R2 beinvariant with respect to dynamical system generatedby (3) {ϕt}t≥0, i.e. ϕt(K) = K for all t ≥ 0(e.g. K is an attractor). Further we use compactnotations for the finite-time local Lyapunov dimension:dimL(t, u0) = dimL(ϕt, u0), the finite-time Lyapunov di-mension: dimL(t,K) = dimL(ϕt,K), and for the Lya-punov dimension: dimL K = dimL({ϕt}t≥0,K).

Consider linearization of system (3) along the solutionu(t, u0) = ϕt(u0), u0 ∈ R2:

v(t+ 1) = J(u(t, u0)

)v(t), v(0) = v0 ∈ R2, t ∈ N0.

(6)Consider a fundamental matrix Φ(t, u0) of solutions oflinearized system (6) such that Φ(0, u0) = I, i.e.

Φ(t, u0) = J(u(t−1, u0))J(u(t−2, u0)) · · · J(u(1, u0))J(u0).

Then for any solution v(t, v0) of (6) with the initial datav(0, v0) = v0 we have

v(t, v0) = Φ(t, u0) v0, u0, v0 ∈ R2. (7)

Let σi(t, u0) = σi(Φ(t, u0)), i = 1, 2 be the singularvalues of Φ(t, u0) (i.e. σi(t, u0) > 0 and σi(t, u0)2 arethe eigenvalues of the symmetric matrix Φ(t, u0)∗Φ(t, u0)with respect to their algebraic multiplicity), ordered sothat σ1(t, u0) ≥ σ2(t, u0) > 0 for any t and u0. Considerthe ordered set of the finite-time Lyapunov exponents atthe point u0 for t > 0:

LE1,2(t,u0)= 1t ln σ1,2(t,u0), LE1(t,u0)≥LE2(t,u0). (8)

Consider the Kaplan-Yorke formula [34] with respect tothe ordered set λ1 ≥ ... ≥ λn:

dKY({λi}ni=1)=j+∑j

i=1λi

|λj+1| , j=max{m :m∑i=1

λi≥0}. (9)

5

For the ordered set of finite-time Lyapunov exponents{LEi(t, u0)}2

i=1 and j(t, u0) = max{m :∑mi=1LEi(t, u0)≥

0} we get

dKY({LEi(t,u0)}2i=1)=

0, j(t, u0)=0

1+ LE1(t,u0)|LE2(t,u0)| , j(t, u0)=1

2 j(t, u0)=2

Then for a certain point u0 and invariant closed boundedset K the finite-time local Lyapunov dimension [35, 39]is defined as

dimL(t, u0) = dKY({LEi(t, u0)}2i=1).

and the finite-time Lyapunov dimension is as follows

dimL(t,K) = supu0∈K

dimL(t, u0). (10)

In this approach the use of Kaplan-Yorke formula (9)with the finite-time Lyapunov exponents can be rig-orously justified by the Douady–Oesterle theorem [17],which implies that for any fixed t > 0 the Lyapunov di-mension of the map ϕt with respect to a closed boundedinvariant set K, defined by (10), is an upper estimate ofthe Hausdorff dimension of the set K:

dimH K ≤ dimL(t,K).

A. Adaptive algorithm for the computation of thefinite-time Lyapunov dimension

To compute the finite-time Lyapunov exponents (8)one has to find the fundamental matrix Φ(t, u0) of (6)from the following variational equation

u(s+1, u0)=ϕ(u(s, u0)), u(0, u0)=u0, s=0, 1, .., t−1Φ(s+1, u0)=J(u(s, u0))Φ(s, u0), Φ(0, u0)=I,

(11)and its Singular Value Decomposition (SVD)

Φ(t, u0) SVD= U(t, u0)Σ(t, u0)V∗(t, u0),

where U(t, u0)∗U(t, u0) ≡ I ≡ V(t, u0)∗V(t, u0),Σ(t, u0) = diag{σ1(t, u0), σ2(t, u0)} is a diagonal ma-trix composed by the singular values of Φ(t, u0), andcompute the finite-time Lyapunov exponents LE1,2(t, u0)from Σ(t, u0) as in (8).

To avoid the exponential growth of values in the com-putation, we use the QR factorization and treppenitera-tion routine:

Φ(t, u0)=J(u(t− 1, u0)) · · · J(u(1, u0)) J(u0) =

=J(u(t−1, u0))· · · J(u(1, u0))Q01 R

01 = · · ·

QR=

Q︷︸︸︷Q0t

R︷ ︸︸ ︷R0t . . .R

01.

Then matrix Σ(t, u0) =U∗(t, u0) Φ(t, u0)V (t, u0) can beapproximated by sequential QR decomposition of theproduct of matrices:

Σ0 :=Φ(t, u0)∗Q0t = (R0

1)∗ . . . (R0t )∗

QR= Q1tR

1t . . . R

11,

Σ1 :=(Q0t )∗Φ(t, u0)Q1

t = (R11)∗ . . . (R1

t )∗QR= Q2

tR2t . . . R

21,

...

Σp :={

(V p)∗Φ(t, u0)∗Up, (p is even)(Up)∗Φ(t, u0) V p, (p is odd) =

= (Rp1)∗. . .(Rpt )∗=(σ1

1 0· σj2

),

where Up := Q0tQ

2t . . . Q

2bp/2ct , V p := Q1

tQ3t . . . Q

2dp/2e−1t ,

and [59, 62]

σpi = Rp1[i, i] . . . Rpt [i, i] −→p→∞

σi(t, u0), i = 1, 2.

For a large t the convergence can be very rapid [62,p. 44] (e.g. p = 1 is taken in [62, p. 44] for the Lorenzsystem with the classical parameters).

For the study of dynamics of the finite-time Lyapunovexponents [39] we can adaptively choose p for s= 1, ..., tso as to obtain a uniform estimate with respect to s:

p(δ)=p(s, δ) : maxi=1,2

| 1s ln σp(s)i (s, u0)− 1

s ln σp(s)−1i (s, u0)|<δ.

(12)Thus, the finite-time Lyapunov exponents can be approx-imated2 as

LEi(t, u0)≈LEp(δ)i (t, u0)= 1

t

t∑s=1

lnRp(δ)s [i, i]. (13)

For the Henon system (3) with canonical parametersa = 1.4, b = 0.3, using the described above adaptivealgorithm with δ = 10−8, we calculate the finite-time

2 In Benettin’s algorithm [6] the so-called finite-time Lya-punov characteristic exponents (LCEs) [53], which are theexponential growth rates of norms of the fundamental ma-trix columns Φ(t, u0) = {v1(t, u0), v2(t, u0)}: LCE1,2(t, u0) =1t

ln ||v1,2(t, u0)||, are computed by (13) with p = 0:LCE1,2(t, u0) ≈ LE0

1,2(t, u0) = 1t

∑t

s=1 lnR0s [i, i]. The fol-

lowing artificial analytical example demonstrates possible dif-ferences between LEs and LCEs: the matrix [35, 36, 39]

R(t) =(

1 eat − e−at

0 1

)has LE1,2(t) = ±|a|, LCE1(t) =

1t

ln((eat− 1

eat )2 + 1) 1

2 ∈ (0, a],LCE2(t) ≡ 0; The approxima-tion by Benettin’s algorithm becomes worse with increasing time:LCE1(t) −→

t→+00 ≡ LE0

1(t), LCE1(t) −→t→+∞

a. Remark that the

notions of LCEs and LEs often do not differ (see, e.g. Eck-mann & Ruelle [20, p.620,p.650], Wolf et al. [65, p.286,p.290-291], and Abarbanel et al. [1, p.1363,p.1364]), e.g. relying onergodicity, however, the computations of LCEs by (13) and LEsby LE0(t, u0) may give non relevant results.

6

0 2000 4000 6000 8000 10000-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 100 200 300 400 500-0.2

0

0.2

0.4

0.6

0.8

1

1.2

9000 9200 9400 9600 9800 100000.011

0.0115

0.012

0.0125

0.013

0.0135

0.014

0.0145

t(a) Simulation follows (13) with p = 0.

0 2000 4000 6000 8000 10000-0.2

0

0.2

0.4

0.6

0.8

1

1.2

9000 9200 9400 9600 9800 100000.011

0.0115

0.012

0.0125

0.013

0.0135

0.014

0.0145

0 100 200 300 400 500-0.2

0

0.2

0.4

0.6

0.8

1

1.2

t(b) Simulation follows (13) with adaptive value p(δ), δ = 10−8.

FIG. 5: Evolution of LE1(t, ua10 ) (red) and LE1(t, ua2

0 ) (blue), computed along the trajectories u(t, ua10 ) and u(t, ua2

0 )for t ∈ [1, 10000] in the Henon system (3) with parameters a = 0.97,b = 0.466.

local Lyapunov dimension dimL(t, u0), where u0 is theinitial point of the trajectory u(t, u0) that localizes theself-excited attractor (see Fig. 1). The comparison of thegraphics for the adaptive algorithm and the algorithmwith p = 0 is presented in Fig. 6. For t = 1000 weobtain dimL(1000, u0) ≈ 1.253. The corresponding nu-merical routine implemented in MATLAB is presentedin Appendix VI.

Applying the statistical physics approach and assum-ing the ergodicity (see, e.g. [26, 34, 42]), the Lyapunovdimension of attractor dimL K is often estimated bythe local Lyapunov dimension dimL(t, u0), correspond-ing to a “typical” trajectory, which belongs to the at-tractor: {u(t, u0), t ≥ 0}, u0 ∈ K, and its limit valuelimt→+∞ dimL(t, u0). However, rigorous check of ergod-icity for the Henon system with a particular value of theparameters is a challenging task (see, e.g. [4, 5]). See,also related discussions in [3][12, p.118][54][66, p.9] [55,p.19], and the works [37, 46] on the Perron effects of thelargest Lyapunov exponent sign reversals. For example,consider parameters a = 0.97,b = 0.466 (see Fig. 4c).In this case for u1

0 = O− + δ1ν+(O−)∗

||ν+(O−)|| , δ1 = 10−4 and

u20 = O+ − δ2

ν−(O+)∗

||ν−(O+)|| , δ2 = 0.1 after a transient pro-cess during [0, Ttrans = 105] we get initial points ua1

0 andua2

0 , respectively and compute finite-time Lyapunov ex-ponents and finite-time local Lyapunov dimension for thetime interval [0, T = 104] by the adaptive algorithm withδ = 10−8: dynamics of the finite-time Lyapunov expo-nents and finite-time local Lyapunov dimension is pre-sented in Fig. 5, finally we have

LE1(T, ua10 )≈0.01198317, dimL(T, ua1

0 )≈1.01545113,LE1(T, ua2

0 )≈0.01405416, dimL(T, ua20 )≈1.01807321.

0 100 200 300 400 500 600 700 800 900 10001.24

1.26

1.28

1.3

1.32

1.34

1.36

1.38

1 2 3 4 51.354

1.356

1.358

1.36

1.362

1.364

1.366

1.368

1.37

1.372

1.374 5

4

33

3

995 996 997 998 999 10001.25295

1.253

1.25305

1.2531

1.25315

1.2532

1.25325

3

33

3

3

3

t

FIG. 6: Evolution of the finite-time local Lyapunovdimension dimL(t, u0) computed with p = 0 (blue) and bythe adaptive algorithm (red). The corresponding values inyellow box shows the number of iterations p(t, δ), which

are necessary to meet the tolerance δ = 10−8.

In one of the pioneering works by Yorke et al. [26,p.190] the exact limit values of finite-time Lyapunov ex-ponents, if they exist and are the same for all u ∈ K, arecalled the absolute ones, and it is noted that the abso-lute Lyapunov exponents rarely exist. Remark that while

7

the time series obtained from a physical experiment areassumed to be reliable on the whole considered time in-terval, the time series, obtained numerically from math-ematical dynamical model, can be reliable on a limitedtime interval only due to computational errors. Also,if the trajectory belongs to a transient chaotic set (see,e.g. Fig. 3), which can be (almost) indistinguishable nu-merically from sustained chaos, then any very long-timecomputation may be insufficient to reveal the limit val-ues of the finite-time Lyapunov exponents and finite-timeLyapunov dimension (see Fig. 7)[39].

0 2000 4000 6000 8000 100000

0.2

0.4

0.6

0.8

1

1.2

1.4

t

FIG. 7: Evolution of the finite-time local Lyapunovdimension dimL(t, u0) (red) computed via the adaptivealgorithm along the transient chaotic set for parametersa = 1.29915, b = 0.3 and t ∈ [1, 10000], and toleranceδ = 10−8. For t ∈ [1, 7300] the behavior seems to be

chaotic.

Thus, to get a reliable numerical estimation of the Lya-punov dimension of attractor K we localize the attractorK ⊂ Kε, consider a grid of points Kε

grid on Kε, andfind the maximum of the corresponding finite-time localLyapunov dimensions for a certain time interval [0, T ]:

dimH K ≤ dimL K ≈ inft∈[0,T ]

maxu∈Kε

grid

dimL(t, u)

= inft∈[0,T ]

maxu∈Kε

grid

(j(t, u) +

LEj(t,u)(t, u)|LEj(t,u)+1(t, u)|

)≤ maxu∈Kε

grid

dimL(T, u) ≈ dimL(T,K).

(14)

Additionally, we can consider a set Kεrnd of N(ε) ran-

dom points in Kε, where N(ε) is the number pointsin Kε

grid. If the maximum of the finite-time local Lya-punov dimensions for Kε

rnd and Kεgrid are different, i.e.

| maxu∈Kε

rnd

dimL(ϕT , u) − maxu∈Kε

grid

dimL(ϕT , u)| > δ, then we

decrease ε. This may help to improve reliability of theresult and at the same time to ensure its repeatability.

IV. THE LYAPUNOV DIMENSION:ANALYTICAL ESTIMATIONS AND EXACT

VALUE

To estimate the Hausdorff dimension of invariantclosed bounded set K, one can use the map ϕt withany time t (e.g., t = 0 leads to the trivial estimatedimH K ≤ 2), and, thus, the best estimation is

dimH K ≤ inft≥0

dimL(t,K).

The following property:

inft≥0

supu0∈K

dimL(t, u0) = lim inft→+∞

supu0∈K

dimL(t, u0), (15)

allows one to introduce the Lyapunov dimension of K as[35]

dimL K = lim inft→+∞

supu0∈K

dimL(t, u0) (16)

and get an upper estimation of the Hausdorff dimension:

dimH K ≤ dimL K.

Recall that a set with noninterger Hausdorff dimensionis referred as a fractal set [20].

In contrast to the finite-time Lyapunov dimension (10),the Lyapunov dimension (16)3 is invariant under smoothchange of coordinates [35, 36]. This property and aproper choice of smooth change of coordinates may sig-nificantly simplify the computation of the Lyapunov di-mension of dynamical system. Consider an effective an-alytical approach, proposed by Leonov [35, 43, 45], forestimating the Lyapunov dimension. In the work [35]it is shown how this approach can be justified by the in-variance of the Lyapunov dimension of compact invariantset with respect to the special smooth change of variablesw = h(u) with Dh(u) = eV (u)(j+s)−1

S, where V (u) is a

3 This definition can be reformulated via the singular value func-tion ωd(Dϕt(u)) = σ1(t, u) · · ·σbdc(t, u)σbdc+1(t, u)d−bdc, whered ∈ [0, n] and bdc is the largest integer less or equal to d:dimL(t, u) = max{d ∈ [0, n] : ωd(Dϕt(u)) ≥ 1} and dimL K =lim inft→+∞

supu∈K

dimL(t, u). Another approach to the introduction

of the Lyapunov dimension of dynamical system was devel-oped by Constantin, Eden, Foias, and Temam [11, 21, 22].They consider

(ωd(Dϕt(u))

)1/tinstead of ωd(Dϕt(u)) and ap-

ply the theory of positive operators to prove the existenceof a critical point ucr

E (which may be not unique), wherethe corresponding global Lyapunov dimension achieves max-imum (see [21]): dimE

L({ϕt}t≥0,K) = inf{d ∈ [0, n] :lim

t→+∞maxu∈K

ln(ωd(Dϕt(u))

)1/t< 0} = inf{d ∈ [0, n] :

lim supt→+∞

ln(ωd(Dϕt(ucr

E )))1/t

< 0} = dimEL({ϕt}t≥0, u

crE ), and,

thus, rigorously justify the usage of the local Lyapunov dimen-sion dimE

L({ϕt}t≥0, u). However this definition does not have aclear sense for finite time.

8

continuous scalar function and S is a nonsingular ma-trix. Let σi(J(u0)), i = 1, 2 be the singular values ofJ(u0) (i.e. the square roots of the eigenvalues of thesymmetrized Jacobian matrix J(u0)∗J(u0)), ordered sothat σ1(J(u0)) ≥ σ2(J(u0)) > 0 for any u0 ∈ K.

Theorem 1 ([35]) If there exist a real s ∈ [0, 1], a con-tinuous scalar function V (u), and a nonsingular 2 × 2matrix S such that

supu0∈K

(ln σ1(J(u0))+s ln σ2(J(u0))+

(V (ϕ(u0))−V (u0)

))<0,

(17)then

dimH K ≤ dimL K < 1 + s.

To avoid numerical localization of attractor, we can con-sider estimation (17) e.g. by the absorbing set or thewhole phase space.

In [8, 56] it is demonstrated how a technique similar tothe above can be effectively used to derive constructiveupper bounds of the topological entropy of dynamicalsystems.

If we consider V = 0 and S = I, then we get theKaplan-Yorke formula with respect to the ordered set oflogarithms of the singular values of the Jacobian matrix:dKY

L ({ln σi(u, S)}2i=1), and its supremum on the set K

gives an upper estimation of the finite-time Lyapunovdimension. This is a generalization of ideas, discussede.g. in [17, 60], on the Hausdorff dimension estimationby the eigenvalues of symmetrized Jacobian matrix.

The Jacobian (4) of the Henon map (2) has the follow-ing singular values:

σ1(x) = 12

(√4x2 + (1 + |b|)2 +

√4x2 + (1− |b|)2

)≥ 1,

σ2(x) = |b|σ1(x) ≤ |b|.

These expressions give the following estimation:

dimH K ≤ dimL K ≤ sup(x,y)∈B

1 + 11− ln |b|

lnσ1(x)

.

The maximum of the right-hand side value is determinedby the maximum value of x2 (or |x|) on B. In [33], forcanonical parameters a = 1.4, b = 0.3 it was consideredthe square (x, y) ∈ [−1.8, 1.8]× [−1.8, 1.8] that gives theestimation

dimH K ≤ dimL K ≤ 1 + 11− ln 0.3

lnσ1(1.8)≈ 1.523

Using Feit’s analytical localization (5) we can get

dimH K ≤ dimL K ≤

≤ 1 +

1− ln b

lnσ1

(max

{−r,m,

√b(m+a)+a−r

})−1

.

For parameters a = 1.4, b = 0.3 we obtain dimH K ≤dimL K ≤ 1.5319.

Remark, that if the Jacobian matrix J(ueq) at one ofthe equilibria has simple real eigenvalues: |λ1(ueq)| ≥|λ2(ueq)|, then the invariance of the Lyapunov dimensionwith respect to linear change of variables implies [35] thefollowing

dimL ueq = dKYL ({ln |λi(ueq)|}2

i=1). (18)

If the maximum of local Lyapunov dimensions on theB-attractor, involving all equilibria, is achieved at equi-librium point: dimL u

creq = maxu0∈K dimL u0, then this

allows one to get analytical formula of the exact Lya-punov dimension4. In general, a conjecture on the Lya-punov dimension of self-excited attractor [35, 38] is thatthe Lyapunov dimension of typical self-excited attractordoes not exceed the Lyapunov dimension of one of unsta-ble equilibria, the unstable manifold of which intersectswith the basin of attraction and visualize the attractor.

Following [44] for the Henon system (3) with parame-ters a > − (b−1)2

4 and |b| < 1 we can consider

S =(

1 00√|b|

), γ = 1

(b−1−2x−)√x2

−+|b|, s ∈ [0, 1).

In this case we have

SJ((x, y)

)S−1 =

(−2x

√|b|√

|b| 0

),

and

σ1((x, y), S)=(√

x2 + |b|+|x|), σ2((x, y), S)= |b|

σ1((x,y),S) .

If we take V ((x, y)) = γ(1 − s)(x + by), then condition(17) with j = 1 and

s > s∗ = 11− ln |b|

lnσ1((x−,x−),S)

is satisfied for all (x, y) ∈ R2 and we do not need anylocalization of the set K in the phase space. By (18) and(9), at the equilibrium point ucreq = (x−, x−) we get

dimL({ϕt}t≥0, (x−, x−)) == dimKY

L ({lnλi(x−, x−)}21) = 1 + s∗.

Therefore, for a bounded invariant set K 3 (x−, x−) (e.g.maximum B-attractor) we have [44]

dimL({ϕt}t≥0,KB) = dimL({ϕt}t≥0, (x−, x−)) =

= 1 + 11− ln |b|

lnσ1((x−,x−),S)

.

4 This term was suggested by Doering et al. in [16].

9

Here for a = 1.4 and b = 0.3 we havedimL({ϕt}t≥0,K

B) = 1.495 ....Embedding of the attractor into three-dimensional

phase space (see attractors of generalized Henon map in[2, 28]) increases the Lyapunov dimension by one.

Using the above approach one can obtain the Lyapunovdimension formulas for invariant sets of other discretesystems (see, e.g. [51, 58]).

V. CONCLUSION

In this work the Henon map with positive and negativevalues of the shrinking parameter is considered and tran-sient oscillations, multistability and possible existence of

hidden attractors are studied. A new adaptive algorithmof the finite-time Lyapunov dimension computation isused for studying the dynamics of the dimension. An-alytical estimate of the Lyapunov dimension using local-ization of attractors is given. The proof of the conjectureon the Lyapunov dimension of self-excited attractors andderivation of the exact Lyapunov dimension formula areextended to negative values of the parameters.

ACKNOWLEDGEMENTS

The work was supported by Russian Science Founda-tion project (14-21-00041).

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12

VI. APPENDIX: MATLAB CODE

Listing 1: henonMap.m – function defining the Henonmap.

1 function out = henonMap ( x, a, b )2 out = zeros (6 ,1);3

4 out (1) = a + b * x(2) - x(1) ˆ2;5 out (2) = x(1);6

7 out (3:6) = [-2 * x(1) , b; 1, 0];8 end

Listing 2: qr pos.m – function implementing the QR de-composition with positive diagonal elements in R.

1 function [Q, R] = qr_pos (A)2

3 [Q, R] = qr(A);4

5 D = diag(sign(diag(R)));6 Q = Q * D; R = D * R;7 end

Listing 3: treppeniterationQR.m – function imple-menting the treppeniteration QR decomposition for productof matrices.

1 function [Q, R] = treppeniterationQR (matFact )

2

3 [˜, dimOde , nFactors ] = size( matFact );4

5 R = zeros (dimOde , dimOde , nFactors );6 Q = eye(dimOde , dimOde );7

8 for jFactor = nFactors : -1 : 19 C = matFact (:, :, jFactor ) * Q;

10 [Q, R(:, :, jFactor )] = qr_pos (C);11 end12 end

Listing 4: computeLEsDiscrTol.m – function imple-menting the LEs numerical computation via the approxima-tion of the singular values matrix with adaptively chosennumber of iterations.

1 function [t, LEs , svdIterations ] =computeLEsDiscrTol (extMap , initPoint ,nFactors , LEsTol )

2

3 % Dimension of the map :4 dimMap = length ( initPoint );5

6 % Dimension of the ext. map (map+var. eq.):7 dimExtMap = dimMap * ( dimMap + 1);8 initCond = initPoint (:);9 fundMat = zeros (dimMap , dimMap , nFactors );

10

11 % Main loop : factorization of thefundamental matrix

12 for iFactor = 1 : nFactors

13

14 extMapSolution = extMap ( initCond );15

16 fundMat (:, :, nFactors - iFactor +1) =reshape ( extMapSolution (( dimMap + 1) :dimExtMap ), dimMap , dimMap );

17

18 initCond = extMapSolution (1 : dimMap );19

20 end21

22 t = 1 : 1 : nFactors ;23 LEs = zeros (nFactors , dimMap );24 svdIterations = zeros(nFactors , 1);25

26 for iFactor = 1 : nFactors27

28 currFactorization = fundMat (:, :,nFactors - iFactor +1 : nFactors );

29

30 currSvdIteration = 1;31

32 LEsWithinTol = false;33

34 while ˜ LEsWithinTol35

36 % Save current iteration number :37 svdIterations ( iFactor ) =

currSvdIteration ;38

39 % Calculate current LEs approximation :40 [˜, R] = treppeniterationQR (

currFactorization );41 for jFactor = 1 : iFactor42 currFactorization (:, :, jFactor ) = R

(:, :, iFactor - jFactor +1) ’;43 end44 accumLEs = zeros (1, dimMap );45 for jFactor = 1 : iFactor46 accumLEs = accumLEs + log(diag(

currFactorization (:, :, jFactor )) ’);47 end48 LEs(iFactor , :) = accumLEs / iFactor ;49

50 % Compare with previous approximation :51 if currSvdIteration > 152 LEsWithinTol = all(abs(LEs(iFactor ,

:) - prevLEs ) < LEsTol );53 end54

55 % Update56 currSvdIteration = currSvdIteration +1;57 prevLEs = LEs(iFactor , :);58

59 end60 end61 end

Listing 5: kaplanYorkeFormula.m – function im-plementing the local Lyapunov dimension calculation viaKaplan-Yorke formula.

1 function LD = kaplanYorkeFormula (LEs)2

13

3 % Initialization of the local Lyupunovdimention :

4 LD = 0;5

6 % Number of LCEs :7 nLEs = length (LEs);8

9 % Sorted LCEs :10 sortedLEs = sort(LEs , ’descend ’);11

12 % Main loop :13 leSum = sortedLEs (1);14 if ( sortedLEs (1) > 0 )15 for i = 1 : nLEs -116 if sortedLEs (i+1) ˜= 017 LD = i + leSum / abs( sortedLEs (i+1) )

;18 leSum = leSum + sortedLEs (i+1);19 if leSum < 020 break;21 end22 end23 end24 end25 end

Listing 6: henonLD.m – script with application of thedescribed numerical procedure for LEs computation to theHenon system.

1 function henonLD2

3 % Canonical parameters4 a = 1.4; b = 0.3;5

6 function out = J( x, a, b )7 out = [-2 * x(1) , b; 1, 0];8 end9

10 % Equilibrium11 S1 = 1/2*((b -1) + sqrt ((b -1) ˆ2 + 4*a));

12

13 [V1 , D1] = eig(J([S1 , S1], a, b));14 D1 = diag(D1);15

16 IX1 = find(abs(D1) > 1);17

18 % Self - excited attractor with respectto S1

19 delta = 1e -3;20 initPoint = [S1 , S1] + delta* V1(:, IX1

(1))’ / norm(V1(:, IX1 (1)));21

22 % Parameters for numerical procedure23 nFactors = 1000;24 LEsTol = 1e -8;25

26 % LEs computation27 [t, LEs , svdIterations ] =

computeLEsDiscrTol (@(x) henonMap (x, a,b), initPoint , nFactors , LEsTol );

28

29 % LD computation30 LD = cellfun ( @kaplanYorkeFormula ,

num2cell (LEs , 2));31

32 % Plotting33 figure (1); hold on;34 plot(t, LEs (:, 1), ’Color ’, ’red ’);35 plot(t, LEs (:, 2), ’Color ’, ’blue ’);36 hold off;37 grid on; axis on;38 xlabel (’t’); ylabel (’LE’)39

40 figure (2); hold on;41 plot(t, LD , ’Color ’, ’green ’);42 hold off;43 grid on; axis on;44 xlabel (’t’); ylabel (’LD’)45 end