Chaos, Solitons and Fractals 21 (2004) 803–808

www.elsevier.com/locate/chaos

Bifurcations and chaotic behavior on the Lanford system

Svetoslav Nikolov a,*, Bozhan Bozhkov b

a Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgariab Institute of Solid State Physics, Bulgarian Academy of Sciences, Tsarigradsko Chaussee 72 Blvd., 1784 Sofia, Bulgaria

Accepted 8 December 2003

Abstract

The aim of this article is to investigate in details the bifurcation behavior and show existence of chaotic solutions in

the Lanford system. The regular behavior of the model was thoroughly studied in [Theory and Applications of Hopf

Bifurcation, Cambridge University Press, 1981; Theory of Chaos, Bulgarian Acad. Press, 2001]. Using Lyapunov–

Andronov theory, we define the analytical formulas for the first Lyapunov value (this is not Lyapunov exponent) at the

boundaries of stability. Here, for specific parametric choice, we obtain chaotic behavior of the Lanford system for the

first time to our knowledge. We also calculate the maximal Lyapunov exponent in the parameters space where chaotic

motion of this system exists.

� 2003 Elsevier Ltd. All rights reserved.

1. Introduction

In the past 30 years (since the emergence of chaos in the middle of 1970s), autonomous three-dimensional dynamical

systems play an outstanding role in modern nonlinear dynamics [1] (and references there). These systems can display a

rich diversity of periodic and chaotic solutions dependent upon the specific values of one or more bifurcation (control)

parameters [4,5].

Here, we investigate regular and chaotic behavior of a third-order system of nonlinear ordinary differential equa-

tions

* Co

E-m

0960-0

doi:10.

_x ¼ ðv� 1Þx� y þ xz;

_y ¼ xþ ðv� 1Þy þ yz;

_z ¼ vz� ðx2 þ y2 þ z2Þ;ð1Þ

which is obtained for the first time in [2] by Hopf. Following [3], the system (1) (in this form) was suggested by Lanford

in a private report. That is why, following [3,4], the system (1) is called the Lanford system (LS). For different choice of

dimensionless parameter v only the regular motion of LS has been investigated by [3,4]. Also, Hassard [3] gave a good

qualitative analysis and proofed that LS have a very rich bifurcation behavior. The system (1) has two equilibrium

(steady state) points with coordinates

�x1 ¼ �y1 ¼ �z1 ¼ 0; ð2Þ

�x2 ¼ �y2 ¼ 0; �z2 ¼ v: ð3Þ

rresponding author. Tel.: +35-92-979-6428; fax: +35-92-870-7498.

ail address: s.nikolov@imbm.bas.bg (S. Nikolov).

779/$ - see front matter � 2003 Elsevier Ltd. All rights reserved.

1016/j.chaos.2003.12.040

804 S. Nikolov, B. Bozhkov / Chaos, Solitons and Fractals 21 (2004) 803–808

Now, we continue the investigation (qualitatively and numerically) of the dynamics of LS and we show for the first

time to our knowledge that it is a rich dynamical system possessing a vast number of regular and chaotic solutions.

Following [6], we calculate the first Lyapunov value (this is not Lyapunov exponent––see appendix in [7] or for a

detailed discussion [8]) at the boundary of stability regions R ¼ r ¼ 0 of the system (1). In accordance with Lyapunov–

Andronov theory we have: (i) the sign of Lyalunov’s value determines the character (stable or unstable) of equilibrium

state at R ¼ 0; (ii) at the boundary of stability r ¼ 0, two cases occur––(a) if the first Lyapunov’s value is different from

zero, then the equilibrium state is unstable, (b) if the first Lyapunov’s value is zero, then the equilibrium state is stable;

(iii) the character of equilibrium state, at R ¼ r ¼ 0, qualitatively determines the reconstruction of phase portrait

(including stability or instability of limit cycle) at transition from R < 0 (r < 0) to R > 0 (r > 0) [6,8]. We also calculate

the maximal Lyapunov exponent where in phase space LS is chaotic.

The scheme of the present paper is as follows: In Sections 2 and 3 we present analytical and numerical results

concerning the system (1) for different values of parameter v. In Section 4 we discuss and summarize our results.

2. Analytical and numerical analysis

In this section, we consider the system (1), which presents an autonomous dynamical model. The constant v of thismodel is real and can be negative, zero or positive.

In order to determine the character of first and second fixed points (Eqs. (2) and (3)) we make the following sub-

stitutions into (1)

x ¼ �x1;2 þ x1 ¼ x1; y ¼ �y1;2 þ x2 ¼ x2; z ¼ �zþ x3; ð4Þ

where �z can be �z1 or �z2. Hence, after accomplishing some transformations the system (1) (in local coordinates) has the

form

_x1 ¼ ðv� 1þ �zÞx1 � x2 þ x1x3;

_x2 ¼ x1 þ ðv� 1þ �zÞx2 þ x2x3;

_x3 ¼ ðv� 2�zÞx3 � ðx21 þ x22 þ x23Þ:ð5Þ

The divergence of the flow (5) is

D3 ¼o _x1ox1

þ o _x2ox2

þ o _x3ox3

¼ 3ðv� �zÞ � 2: ð6Þ

The system (5) is dissipative and has attractor when D3 < 0.

According to [6], the Routh–Hurwitz conditions for stability of (2) and (3) can be written in the form

p ¼ 2� 3v > 0; ð7Þ

q ¼ 1þ ðv� 1þ �zÞð3v� 3�z� 1Þ > 0; ð8Þ

r ¼ �ðv� 2�zÞ½ðv� 1þ �zÞ2 þ 1� > 0; ð9Þ

R ¼ pq� r > 0: ð10Þ

Here the notations p, q, r and R are taken from [6]. When conditions (9) or (10) are not valid, the steady states (2) and

(3) become unstable. In order to define the type of stability loss of steady states (2) and (3) it is to calculate the so called

first Lyapunov value [6,7]. In case of three first order differential equations, this value can be determined analytically (at

the boundary of stability R ¼ 0) by the formula in [6]

L1ðk0Þ ¼p4q

2 Að2Þ33 A

ð3Þ33

�h� Að2Þ

22 Að3Þ22

�þ 2Að2Þ

23 Að2Þ22

�þ Að2Þ

33

�

� 2Að3Þ23 Að3Þ

22

�þ Að3Þ

33

�þ 3

ffiffiffiq

pAð2Þ222

�þ Að3Þ

333 þ Að2Þ233 þ Að3Þ

223

�i

þ p4p

ffiffiffiq

p ðp2 þ 4qÞ p2 2Að1Þ22 3Að2Þ

12

�hnþ Að3Þ

13

�þ 2Að1Þ

33 Að2Þ12

�þ 3Að3Þ

13

�þ 4Að1Þ

23 Að2Þ13

�þ Að3Þ

12

�i

þ 4pffiffiffiq

pAð1Þ22

�h� Að1Þ

33

�Að2Þ13

�þ Að3Þ

12

�þ 2Að1Þ

23

�Að3Þ13 � Að2Þ

12

�iþ 16q Að1Þ

22

�þ Að1Þ

33

�Að2Þ12

�þ Að3Þ

13

�o;

ð11Þ

S. Nikolov, B. Bozhkov / Chaos, Solitons and Fractals 21 (2004) 803–808 805

where k0 is defined as a value of v for which the relation R ¼ 0 takes place. Here we note that for first fixed point (Eq.

(2)) R ¼ 0 when

v3 � 2v2 þ 3

2v� 1

2¼ 0; ð12Þ

i.e. when v1 ¼ 1 and v2;3 ¼ 0:5 0:5i, and for second fixed point (Eq. (3)) R ¼ 0 when

v3 � 5

2v2 þ 3v� 1 ¼ 0; ð13Þ

i.e. when v1 ¼ 0:5 and v2;3 ¼ 1 i. For the system (5)

Að2Þ22 ¼ Að3Þ

22 ¼ Að2Þ33 ¼ Að3Þ

33 ¼ Að2Þ23 ¼ Að3Þ

23 ¼ 0: ð14Þ

After substitution of (14) into (11) and accomplishing some transformations and algebraic operations for the first

Lyapunov value L1ðk0Þ (for the system (5)) we obtain

L1ðk0Þ ¼ � p2p

ffiffiffiq

p ðp2 þ 4qÞ ½p2ð2B1 þ 2B2 þ B3Þ þ 8qðB1 þ B2Þ�; ð15Þ

where

B1 ¼ ðv� 2�zÞ2 þ ½1þ ðv� 1þ �zÞ2 þ ðv� 2�zÞðv� 1þ �zÞ�2; ð16Þ

B2 ¼ q½ð2v� 1� �zÞ2 þ 1�; ð17Þ

B3 ¼ �2ffiffiffiq

p2�z

n� vþ ½1þ ðv� 1þ �zÞ2 þ ðv� 2�zÞðv� 1þ �zÞ�

o: ð18Þ

The first Lyapunov value (in (15)) can be negative or positive. If L1ðk0Þ is negative, then in case of transition through the

boundary R ¼ 0 from positive values to negative ones, a stable limit cycle (self-oscillations) emerges. Inversely, in case

of a transition from negative values to positive ones the stable limit cycle disappears, i.e. the self-oscillations cease [6]. In

theory of dynamic system the type of bifurcation behavior near the boundary R ¼ 0 is often called soft loss of stability,

i.e. when the bifurcation parameter k0 changes, the system has reversible behavior. If L1ðk0Þ is positive, then in case of

transition through the boundary R ¼ 0 from positive values to negative ones, an unstable limit cycle emerges. Inversely,

in case of transition from negative values to positive ones, the unstable limit cycle disappears. This type of bifurcation

behavior near the boundary R ¼ 0 is often called hard loss of stability, i.e. the system has irreversible behavior and the

boundary R ¼ 0 is dangerous.

2.1. Investigation of the first fixed point (Eq. (2))

In this case, the equilibrium (steady state) value (2) of the system (1) is the zero one. After substitution of �z ¼ 0 and

v ¼ 1 into (7)–(9) we obtain that p and r are negative. Therefore, the first Lyapunov value at the boundary R ¼ 0 cannot

be calculated. But for v ¼ 0 the boundary r is zero. Following [6], the first Lyapunov value lðk0Þ at the boundary r ¼ 0

has the form

lðk0Þ ¼ aAð1Þ þ bAð2Þ þ cAð3Þ; ð19Þ

where the coefficients a, b, c and AðiÞ (i ¼ 1–3) are also defined by corresponding formulas presented by [6]. For the

system (5) (when �z ¼ 0) a ¼ b ¼ 0. Hence, after accomplishing some transformations, we obtain for lðk0Þ

lðk0Þ ¼ �0:5 6¼ 0; ð20Þ

where c ¼ 2 and Að3Þ ¼ �0:25. According to [6], if lðk0Þ is different from zero, then in case of a transition from negative

values to positive ones, the equilibrium state becomes unstable double point, the system has irreversible behavior and

the boundary r ¼ 0 is dangerous. Also, from sign of the added condition

D� ¼ �p2q2 þ 4q3; ð21Þ

we can have two cases: (i) if D� < 0, then the equilibrium state becomes saddle-knot; (ii) if D� > 0, then the equilibrium

state becomes saddle-focus [6]. Here, D� ¼ 16 > 0, therefore (2) is from saddle-focus type.

806 S. Nikolov, B. Bozhkov / Chaos, Solitons and Fractals 21 (2004) 803–808

2.2. Investigation of the second fixed point (Eq. (3))

After substitution of �z ¼ v ¼ 0:5 and accomplishing some algebraic operations for the first Lyapunov value L1ðk0Þ(for the system (5)) we have

L1ðk0Þ ¼ �13:6682 < 0: ð22Þ

Therefore, following [6,7] the boundary R ¼ 0 is undangerous and the system has reversible behavior, i.e. a stable limit

cycle (self-oscillations) emerges or ceases. This result is in accordance with these obtained by [3,4].

In the following section, we demonstrate numerically these types of behavior. Also, we obtain that in some intervals

of variation of the parameter v the system (1) has chaotic behavior.

3. Numerical results

In previous section we introduced the analytical tools that we will use in our numerical analysis of the system (1).

In Fig. 1(a) the case when v ¼ 0:49 is shown. It is seen that here the system is stable. After v ¼ 0:5 the stable limit

cycle occur (see Fig. 1(b)) and the system (1) has periodic solution. These results are in accordance with results obtained

in our study in previous section.

Fig. 2 shows a bifurcation diagram for the system (1): values of xn are plotted against v regarded as a continuously

varying control parameter. What could one observe in the figure? When v 2 ½0:635; 0:655�, the system has regular

solution with period 1. As v increased further, there is a period-doubling bifurcation that results in a double-loop

attractor (see Fig. 3(a)). We see that chaos occurs there after vP 0:6666. A confirmation of our conclusions is the

Fig. 1. Stable (a) and periodic (b) solution of the system (1) at v ¼ 0:49 and v ¼ 0:51.

Fig. 2. Bifurcation diagram xn versus v generated by computer solution of system (1). Note that v 2 ½0:635; 0:66669�.

Fig. 3. Phase portrait of the system (1) at (a) v ¼ 0:657; (b) v ¼ 0:66668.

Fig. 4. Trajectories of the system (1) with two different initial conditions at v ¼ 0:6666. For dotted line x0 ¼ 0:1 and for solid line

x0 ¼ 0:1.

S. Nikolov, B. Bozhkov / Chaos, Solitons and Fractals 21 (2004) 803–808 807

strange attractor shown in Fig. 3(b), solutions with exponential sensitivity to initial conditions and obtained for this

case maximal Lyapunov exponent. The maximal Lyapunov exponent kmax shows the kind of motion on the phase space:

(i) if kmax < 0 the motion is a stable fixed point; (ii) if kmax ¼ 0 the motion is a stable limit cycle; (iii) if 0 < kmax < 1 the

motion is chaotic and (iv) if kmax ¼ 1 the motion is noise [9]. Following [9], the maximal Lyapunov exponent for a

given data set can be calculated by means of the sum

SðDnÞ ¼ 1

N

XNn0

ln1

jWðbn0ÞjX

bn02W

jsn0þDn

0@ � snþDnj

1A; ð23Þ

where reference points bn0 are embedding vectors, Wðbn0Þ is the neighborhood of bn0 with diameter e, sn0 is the last

element of bn0 and sn0þDn is outside the time span covered by the delay vector bn0 .

For the numerical calculation of kmax we use the TISEAN software package [10]. The obtained maximal Lyapunov

exponent (per unit time) is: kmax ¼ 0:1139 0:0076, when v ¼ 0:66668.In Fig. 4 we show that after v ¼ 0:6666 the chaotic behavior of system takes place. When we have different initial

conditions (which are very close) for x, the system (1) has different trajectories. Here we note that strange attractor for

these two cases is topological one and the same.

4. Summary and conclusions

The paper presents a study of the behavior of Lanford system. Using Lyapunov–Andronov’s theory, we find new

analytical formulas for the first Lyapunov’s value at the stability limit. It enables one to study in detail the bifurcation

behavior of the dynamic system (1). The approach (for first Lyapunov value) proposed here has a basic advantage,

which consist in the following: we can answer the question for structural stability (respectively unstability) of the

Lanford system. Here we note that for all simulations the initial conditions were x0 ¼ y0 ¼ 0:1, z0 ¼ 0:07.

808 S. Nikolov, B. Bozhkov / Chaos, Solitons and Fractals 21 (2004) 803–808

Generalizing the results obtained in Sections 2 and 3, we can conclude that:

1. The emergence of a stable limit cycle with period 1 takes place for a value of the bifurcation parameter v ¼ 0:5 under

a soft stability loss.

2. The boundary of stability r ¼ 0 (for first fixed point Eq. (2)) is dangerous and the Lanford system (1) has irreversible

behavior for v > 0.

3. The first fixed point (Eq. (2)) after v > 0 becomes unstable double point from saddle-focus type.

4. The first Lyapunov value at the boundary R ¼ 0 (for first fixed point Eq. (2)) cannot be calculated.

5. For values of the bifurcation parameter v, larger than v ¼ 0:6666, the system (1) is in a chaotic state (till v ¼ 0:6667).

Acknowledgements

This work was supported by the National Science Fund of the Ministry of Education and Science (Bulgaria), project

MM 1302/2003.

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