homoclinic and heteroclinic orbits in a modified lorenz system

Information Sciences 165 (2004) 235–245

www.elsevier.com/locate/ins

Homoclinic and heteroclinic orbits ina modified Lorenz system q

Zhong Li a,*, Guanrong Chen b,1, Wolfgang A. Halang a

a Faculty of Electrical Engineering, FernUniversit€at Hagen, Hagen 58084, Germanyb Department of Electronic Engineering, City University of Hong Kong, Kowloon,

Hong Kong SAR, PR China

Received 17 March 2002; received in revised form 8 May 2003; accepted 20 June 2003

Abstract

This paper presents a mathematically rigorous proof for the existence of chaos in a

modified Lorenz system using the theory of Shil’nikov bifurcations of homoclinic and

heteroclinic orbits. Together with its dynamical behaviors, which have been extensively

studied, the chaotic dynamics of the modified Lorenz system are now much better

understood, providing a rigorous theoretic foundation to support studies and applica-

tions of this important class of chaotic systems.

� 2003 Elsevier Inc. All rights reserved.

1. Introduction

Over the past three decades, chaos as an interesting nonlinear phenomenonhas been extensively studied within the scientific, engineering, and mathemat-

ical communities. Recently, chaos is found to be useful with great potentials in

many disciplines, including thorough liquid mixing with low power con-

sumption, high-performance circuit design for telecommunication, collapse

q IS-02-3798-A.*Corresponding author. Tel.: +49-2331-987-2383; fax: +49-2331-987-375.

E-mail addresses: [email protected], [email protected] (Z. Li),

[email protected] (G. Chen).1 Tel.: +852-2788-7922; fax: +852-2788-7791.

0020-0255/$ - see front matter � 2003 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2003.06.005

mail to: [email protected],

236 Z. Li et al. / Information Sciences 165 (2004) 235–245

prevention of power systems, and biomedical engineering applications to such

important aspects as diagnosis and treatments of human brain and heart [2,3].

Creating chaos, therefore, becomes a key issue in such applications where

chaos is important and useful. Given a system or process, which may be linearor nonlinear but is originally nonchaotic or even stable, the question of whe-

ther or not one can generate chaos (and, if so, how) by means of designing a

simple and implementable controller (e.g. a parameter tuner or a state feedback

controller) is known as anticontrol of chaos or chaotification. Tremendous ef-

forts have been devoted to achieving this goal, not only via computer simu-

lations for the task but also by developing complete and rigorous mathematical

theories to support the task [4–7]. For instance, a simple linear partial state-

feedback controller was found to be able to derive the Lorenz system, currentlynot in the chaotic region, to be chaotic. It had led to the discovery of a new

chaotic system, which is compatible to the Lorenz system in structure, in the

sense that it is also a 3-D autonomous equation with only two quadratic terms,

but topologically not equivalent (there does not exist a homeomorphism that

can take one system to the other) and yet has even more complex dynamical

behavior than the Lorenz system. This new chaotic system was named as Chen

system by many others [8,9], for its important duality to the Lorenz system in a

sense defined in [10]. By bridging the gap between the Lorenz and the Chensystems, another new chaotic system has also been discovered, by L€u and Chen[11,12], which represents a transition between the Lorenz and Chen systems.

A simple piecewise-linear circuit has been designed, which exhibits a chaotic

attractor similar to that of the Lorenz system [13]. Moreover, a multiplier-free

modified Lorenz system has also been studied [14,15], in which an additional

control parameter is used to verify the compound nature of the resulting

butterfly-shaped attractor. By designing appropriate control gains, it is possible

to confine the chaotic dynamics from one to another butterfly wing of theattractor, forming two simple attractors which, when merged together, form

the whole butterfly-shaped attractor. These observations have been verified

experimentally by designing a novel circuit in [15]. Other than these observa-

tions and the circuitry realization, further analysis on dynamical behaviors of

such a modified Lorenz system has been carried out lately, including some

basic dynamical properties such as bifurcations, routes to chaos, periodic

windows, and some Poincar�e mappings, either analytically or numerically [1].Nevertheless, it is well known that numerical evidence may occasionally

mislead [16], because computer simulations have finite precision and experi-

mental measurements have finite ranges in the time or frequency domain [18].

The behavior witnessed may be an artifact of the observation device due to

physical limitations. Thus, a rigorous proof is often necessary for fully

understanding chaotic dynamics in various nonlinear dynamic systems. Even

for the first discovery of chaos, the Lorenz attractor coined 40 years ago, which

has already been extensively studied, it is only recently rigorously proved [17].

Z. Li et al. / Information Sciences 165 (2004) 235–245 237

Therefore, for the class of Lorenz-like systems, it is of significance to give a

mathematically rigorous proof to the existence of chaos, although numerical

analysis have been shown repeatedly and convincingly.

Following the work of [1], this paper applies the Shil’nikov theorem to abetter understanding of the chaotic behavior in the modified Lorenz system

through finding homoclinic and heteroclinic orbits in it by checking the

Shil’nikov inequality, thereby rigorously proving the existence of chaos in the

modified Lorenz system.

2. The modified Lorenz system

Recently, a butterfly-shaped attractor was observed in [15], from the fol-lowing multiplier-free modified Lorenz system (in the Lorenz system, the two

multipliers xy and xz are responsible for the generation of chaos):

3-0.8-0.6-0.4-0.2

00.20.40.60.8

y

(a)

Fig. 1

m ¼ 0:

_x ¼ aðy � xÞ_y ¼ Kðb� zÞ þ m_z ¼ jxj � cz;

8<: ð1Þ

where a, b, c and m are constants, while K is a bipolar switching constant givenby

K ¼ sgnðxÞ ¼ 1 xP 0

�1 x < 0:

�ð2Þ

Here, constant a controls the condition for oscillation while b acts as anamplitude threshold; parameter c is a damping constant and m is a displace-ment constant. System (1) has a chaotic attractor as shown in Fig. 1 when

a ¼ 0:9, b ¼ 2, c ¼ 0:1, and m ¼ 0 [1].

-0.8-0.6 -0.4 -0.2 0

0.2 0.4 0.6

1.5

2

2.5

xz -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.81.6

1.8

2

2.2

2.4

2.6

2.8

3

x

z

(b)

. Chaotic attractors observed from the modified Lorenz system (1). a ¼ 0:9, b ¼ 2, c ¼ 0:1,(a) 3-D phase space portrait; (b) projection on x–z plane.


The modified Lorenz system (1) has similar qualitative dynamics to the

original one, but is multiplier-free. Some basic dynamical properties of this

system have been investigated in [1], including such as symmetry, invariance

(m ¼ 0), dissipativity, and appearance of attractor. In the following, theShil’nikov theorem is applied to mathematically prove the existence of chaos in

the modified Lorenz system (1).

3. The Shil’nikov theorem

Although computer simulation and physical experiment are the most widely

used techniques for understanding complex dynamical behaviors in nonlinear

systems, they are not completely adequate for full understanding, unless they

are used in conjunction with rigorous analysis [19]. Generally speaking, an

analytical approach is needed to guarantee that chaotic behavior exists in a

precise sense of mathematics.

One of the most important tools for analyzing chaotic behaviors of non-

linear dynamical systems is the fundamental theorem of Shil’nikov bifurcationof homoclinic orbits of flows in R3 [20]. A homoclinic orbit is a trajectory C,which is asymptotic to another distinct trajectory I in both forward and re-

verse directions of time. We are mostly concerned with the case that other

trajectory I is an equilibrium point; but if I is a periodic orbit, C is called a p-homoclinic orbit to avoid possible confusion [16]. Note that homoclinic orbits

are not structurally stable: they may disappear under small perturbations of the

vector field, corresponding to the concerned case of small parameter changes,

which turns out to be important in the context of bifurcations.Shil’nikov proved that the existence of some kind of homoclinic orbits im-

plies that there are arbitrarily small perturbations of the vector field that have

p-homoclinic orbits and are therefore chaotic in a precise sense. It allows us tolook for a bifurcation in which a homoclinic orbit is created and destroyed as

parameters vary, and to guarantee that there is a chaotic attractor nearby.

The Shil’nikov theorem is formally described as following:

Theorem 1 (Shil’nikov). Consider a flow in R3, which has an equilibrium at theorigin, having a real eigenvalue c < 0 and a pair of complex eigenvalues r � jxwith r > 0 and x 6¼ 0. Suppose the flow has a homoclinic orbit C (so C is tangentto the eigenvector of c in forward time and to the eigenspace of r � jx in reversetime).

If jr=cj < 1, the flow can be perturbed into a flow with a countable set ofhorseshoes.

Remark. The case of c > 0 and r < 0 is similar, since one only needs to reversethe time coordinate.


A heteroclinic orbit is a cycle of trajectories fC0;C1; . . . ;Ccg together with aset of distinct trajectories fI0;I1; . . . ;Icg such that

(i) C0 is asymptotic to I0 in forward time and to Ic in reverse time; and(ii) Cj (0 < j6 c) is asymptotic to Ij in forward time and to Ij�1 in reverse

time.

The simplest example is a ‘‘saddle loop’’ where Ij are saddle equilibria and Cj

are separatrices, forming a directed cycle.

As shown in [21], bifurcations of suitable heteroclinic orbits produce the

same features as do bifurcations of homoclinic orbits.

4. Finding homoclinic and heteroclinic orbits

In this section, some elementary algebraic and geometric approaches are

applied to find homoclinic and heteroclinic orbits, thus in terms of the

Shil’nikov theorem the existence of chaos in the modified Lorenz system isproved.

4.1. Eigenvalues and eigenvectors

Assume that the parameters a, b, and c are all positive in system (1). The

equilibria of system (1) can be easily derived by solving the three equations_x ¼ _y ¼ _z ¼ 0. Here, only the case m ¼ 0 is discussed. As to other cases, such asmP b, m < �b, m ¼ �b, or jmj < b, discussions are similar. If m ¼ 0, system(1) has two equilibria, S� ¼ ð�bc;�bc; bÞ, which are symmetrically placed withrespect to the z-axis. As compared to the original Lorenz system, this systemretains the two nontrivial equilibria but abandons the trivial equilibrium point.This means that the two nontrivial equilibria are responsible for generating

chaos in the original Lorenz system.

Since system (1) is piecewise-linear, the state space can be divided into two

subspaces, denoted by Dþ and D�, and the separating plane is x ¼ 0 as shownin Fig. 2. In each of subspace, the system equations are linear, so the system

dynamics in the neighborhood of the equilibria are determined by the eigen-

values of the corresponding system Jacobian, which, when evaluated at the two

equilibria, is

J� ¼�a a 0

0 0 �KK 0 �c

24

35: ð3Þ

Its eigenvalues are the solutions of the following characteristic cubic equa-

tion:

x

y

z

+S

−S

+D

−D

O

l

C

)( +SEs

)( −SEs

Fig. 2. Eigenspaces of the equilibria and other geometric sets.


f ðkÞ ¼ k3 þ ðaþ cÞk2 þ ack þ a ¼ 0: ð4Þ
It is then easy to see that the two equilibria of system (1) have the same sta-
bility. Furthermore, the stability only relies on the parameters a and c, but isindependent of parameter b.Each equilibrium point has a real eigenvalue c and a complex conjugate pair

of eigenvalues r � jx. The symmetry implies that the eigenvalues of Sþ and S�are the same. Obviously, c < 0 and r > 0. Suppose that u�, v�, and w� are the

right-eigenvectors corresponding to c and r � jx at S�, respectively, and letn� ¼ v� � w� be the normal vector to the plane spanned by v� and w�. It is

known that n� is equal to the left-eigenvector of c and can be calculated bynT�ðJ� � cIÞ ¼ 0. It then follows that

uþ ¼cþ c

� 1c

1

264

375; u� ¼

�ðcþ cÞ1c

1

264

375;

nþ ¼

1aþc

�ðcþ cÞ1

264

375; n� ¼

�1aþc

ðcþ cÞ1

264

375:

ð5Þ

4.2. Geometric structure of the state space

In a certain range of parameters, both equilibria are saddle focus type. Thismeans Sþ or S� possesses a 1-D stable manifold corresponding to c (denoted byEsðSþÞ or EsðS�Þ) and a 2-D unstable manifold corresponding to r � jx (de-


noted by EuðSþÞ or EuðS�Þ). The parameter range, in which equilibria exist assaddle foci, is limited by the Hopf-bifurcation of S�.Next, define the following invariant sets:

EsðS�Þ :x� bc

�ðcþ cÞ ¼y � bc�1=c ¼ z� b

1;

EuðS�Þ : ðx� S�Þn� ¼ 0;ð6Þ

where x ¼ ðx; y; zÞ. As the parameters a, b, and c change, the eigenspaces movearound. Only the cases where they remain exist and have the same character isconsidered, that is, the eigenvalues still satisfy c < 0, r > 0, x 6¼ 0, and theShil’nikov inequality jr=cj < 1.The following geometric sets are also defined:

U0 ¼ fðx; y; zÞjx ¼ 0g;

C ¼ EsðSþÞ \ U0 ¼ 0;bc

cðcþ cÞ

�þ bc;

bccþ c

�;

l ¼ EuðS�Þ \ U0 ¼ ðcþ cÞðy þ bcÞ þ ðz� bÞ � bcaþ c

¼ 0;

ð7Þ

where U0 is the separating plane between Dþ and D�, C is the intersection pointof EsðSþÞ with U0, and l is the intersection line of U0 and EsðS�Þ, as shown inFig. 2.

4.3. Homoclinic and heteroclinic orbits

In terms of the Shil’nikov theorem, if one can show the existence of a ho-

moclinic orbit or a heteroclinic orbit in (1), then one will have a deeper

understanding of the nature of the flow at nearby values.

4.3.1. Heteroclinic orbits

If a trajectory linking Sþ and S� can be found, then the symmetry ensuresthat there is another trajectory linking S� to Sþ, so there will be a heteroclinicorbit. This trajectory must diverge from S� on EuðS�Þ, leave D�, and then hit

the intersection point C precisely, and finally come into Sþ along EsðSþÞ. Thatis, it must contain the line segment CSþ. It is easy to recognize that if C lies on lthen there will be a heteroclinic connection. This means (by substituting the

coordinates of C into the equation of l in (7)) that

1þ 2cðcþ cÞc

� 1

cþ c� 1

aþ c¼ 0; ð8Þ

where c must satisfy the characteristic Eq. (4). To actually find a heteroclinicorbit, one need to find at least one pair of fa; bg, so that (4) and (8) are satisfiedsimultaneously, while the Shil’nikov inequality jr=cj < 1 still holds.


Some additional manipulation on (8) and (4) leads to a simpler form:

ða� cÞc2 � a ¼ 0: ð9Þ
Solving (9) then gives
c1;2 ¼ �ffiffiffiffiffiffiffiffiffiffiffia

a� c

r: ð10Þ

Note that c < 0 in this discussion. Substituting c2 ¼ �ffiffiffiffiffiffia

a�c

pinto (4) yields

c4 � 2ac3 þ ða2 � 2Þc2 þ 6ac� 4a2 þ 1 ¼ 0: ð11Þ
Eq. (11) is a transcendent equation, but can be solved numerically. If the
parameters a and c satisfy (11) and also the Shil’nikov inequality jr=cj < 1,then a heteroclinic orbit can be found.By tedious trial-and-error, a set of parameters are found: a ¼ 0:6602,

b ¼ 2:0105, and c ¼ 0:2496, which can build up a heteroclinic connection, asshown in Fig. 3.

The eigenvalues of S� are

c ¼ �1:2189;r � jx ¼ 0:1546� j0:7195:

ð12Þ

It is easy to see that jr=cj is smaller than 1; therefore, the Shil’nikov theoremcan be applied to help understand the truly chaotic nature of the modified

Lorenz system.

Fig. 3. Projection of a heteroclinic orbit on the x–z plane.


4.3.2. Homoclinic orbits

A solution of homoclinic orbit through Sþ must leave Sþ spirally alongEuðSþÞ in Dþ, enter and run across D�, then in a possibly short time hit the

intersection point C to return to Dþ, and finally converge to the equilibrium Sþalong EsðSþÞ. To find such a homoclinic orbit, one can integrate backwardsfrom C. It must hit the intersection line of U0 and EuðSþÞ, and then spiral inreverse time into Sþ.Let s be the smallest positive number such that /�sðCÞ 2 U0, where / is the

flow of system (1) and (2). So, s is a solution of xð�sÞ ¼ 0, where xðtÞ, yðtÞ andzðtÞ are the components of the following flow:

xð�sÞ ¼ c1e�csu� þ c2e�ðrþjxÞsv� þ c3e�ðr�jxÞsw�; ð13Þ

where xð�sÞ ¼ ðxð�sÞ; yð�sÞ; zð�sÞÞ. Since xð0Þ ¼ C, one can easily find thatc2 ¼ c3; thus, Eq. (13) can be rewritten as

xð�sÞ ¼ �c1ðcþ cÞe�cs þ 2c2e�rs½�ðcþ rÞ cosð�xsÞ þ x sinð�xsÞ�;

yð�sÞ ¼ c11

ce�cs þ 2c2e�rs

r2 þ x2½r cosð�xsÞ þ x sinð�xsÞ�;

zð�sÞ ¼ c1e�cs þ 2c2e�rs cosð�xsÞ;ð14Þ

where c1 ¼ ðrþcÞbcðcþcÞðr�cÞ, and c2 ¼ bc

2ðc�rÞ.

After all parameters being fixed, by setting xð�sÞ ¼ 0, one can solve for snumerically. Then, substituting the value of s into (14), the coordinates of thepoint of the flow, which hit the surface of U0, can be derived. If the coordinatessatisfy the equation of the intersection line of U0 and EuðSþÞ, then the homo-clinic orbit can be found.

Similar to [16], there is a side condition. The point /�sðCÞ cannot lie any-where on the line U0 \ EuðSþÞ; it must lie in the interval ðP1; P2Þ, as shown inFig. 4. Here, P1 is the point where the flow in EuðSþÞ is tangent to the lineU0 \ EuðSþÞ, while P2 is the point where the trajectory through P1 meets the lineas it spirals out from Sþ. The trajectory in reverse time from C cannot enter theleft-hand side of P1, and it is not wanted to enter to the right-hand side of P2because then it would reenter Dþ to the left-hand side of P1.Following the typical method described in [16], one can find a set of

parameters for establishing a homoclinic orbit. Keeping b and c unchanged, asin last subsection, and then adjust a to be a ¼ 0:5651. Then, a homoclinic orbitcan be found, as shown in Fig. 5.

The eigenvalues of S� are c ¼ �1:1315 and r � jx ¼ 0:1584� j0:6887. Onecan see that the Shil’nikov inequality jr=cj < 1 is satisfied.

Fig. 5. Projection of a homoclinic orbit on the x–z plane.

)(0 +∩ SEU u

1P 2P

)( +SEu

Fig. 4. Dynamical motion in the plane EuðSþÞ.


5. Concluding remarks

This paper has presented a mathematically rigorous proof to the existence of

chaos in a modified Lorenz system by means of the Shil’nikov theorem through

finding homoclinic and heteroclinic orbits in the system. Consequently, the

very nature of chaos in the modified Lorenz system becomes fully understood,

enhancing the numerical analysis of [1].


Acknowledgement

Z. Li is grateful to the support of the Alexander von Humboldt foundation.

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homoclinic and heteroclinic orbits in a modified lorenz system

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