Generalized Lorenz-Type Systems

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

199, 1]13 Ž1996.

0123

Generalized Lorenz-Type Systems Mostafa A. Abdelkader 25 Sh. Champollion, Alexandria 21131, Egypt Submitted by Thanasis Fokas Received July 22, 1994

Exact general solutions of the Lorenz system of differential equations with arbitrary parameters are unknown. We first present exact particular solutions of this system for special values of the parameters, in terms of Jacobian elliptic functions, but no sensitive dependence on the initial conditions Žleading to chaotic behavior. is exhibited by these solutions. We next consider two generalizations of the Lorenz system involving quadratic and cubic terms and obtain their exact general solutions in terms of Bessel and modified Bessel functions, exhibiting sensitive dependence on initial conditions for certain parameter ranges. Finally, a generalized system involving arbitrary powers is reduced to the general Duffing equation with damping, which can be solved exactly when some of the parameters are interconnected. The exact solutions presented, showing precisely how chaotic behavior occurs, are useful for testing conjectures about such differential systems, as well as testing the validity and accuracy of approximate computer solutions, for which numerical errors may grow exponentially fast. Q 1996 Academic Press, Inc.

1. INTRODUCTION Third-order autonomous systems of nonlinear ordinary differential equations of the form

˙x s f 1 Ž x, y, z . ,

˙y s f 2 Ž x, y, z . ,

˙z s f 3 Ž x, y, z . ,

for real variables and with initial values Ž x 0 , y 0 , z 0 . at t s 0, arise in several applications. Their qualitative and numerical studies have been extensive, but their global analysis is notoriously difficult, and exact solutions are rarely, if ever, obtainable. The functions Ž f 1 , f 2 , f 3 . usually contain a number of parameters, and the investigation of the behavior of the solution curves when the parameters are varied is of paramount importance, since the character of the solutions may change drastically, depending on the values of some of the parameters, as well as on the initial values of the variables. The most interesting systems are those for 1 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

2

MOSTAFA A. ABDELKADER

which sensitive dependence on the initial values gives rise to chaotic behavior of the solution trajectories: trajectories which start out with initial values very close together eventually divergew1]4x. This implies that small initial errors may increase exponentially with time, so that long-time behavior becomes completely unascertainable. One of the best known and most extensively studied third-order systems is the Lorenz system

˙x s aŽ y y x . ,

˙y s f Ž x . y ny y xz,

˙z s xy y bz,

Ž 1.

where f Ž x . s rx and n s 1, with the parameters a, b, r being positive. Although the system appears simple, no exact general solutions of it have yet been discovered, but numerical experimentation has shown that for certain parameter ranges the solutions exhibit chaotic behavior, albeit this has not been proved. Referring to this system, Hirsch w1, p. 40x writes: ‘‘Trajectories which start out very close together eventually diverge, with no relationship between their long-run behaviors. But this chaotic behavior has not been pro¨ ed; and in fact, almost nothing has been proved about this particular system, . . . ’’. In this paper we first give an exact particular solution of Ž1. with f Ž x . s rx, b s 2 a, and r a

s

n a

y

2 9

ž

1q

n a

2

/

.

In the original Lorenz system it is usually assumed that a ) b q 1, so our value for b is probably not of interest physically, although mathematically it has some interest. The solution is obtained in terms of Jacobian elliptic functions, and has arbitrary values for x 0 and y 0 , but is restricted to the set of initial values z 0 s x 02r2 a, so it is a particular sub-class of the general solution not exhibiting chaotic behavior. Remarkably, although the exact general solution of the simple-looking Lorenz system, whose nonlinearity is mild, has so far remained undiscovered, when one augments it by adding suitable strongly nonlinear terms exact general solutions can be obtained, and they exhibit chaotic behavior for certain parameter ranges. To exhibit this interesting phenomenon, we consider Ž1. when f Ž x . s rx q Ž x 3r2 a. and b s 2 a, and obtain the exact solutions for arbitrary values of the parameters a, n, r, and initial values x 0 , y 0 , z 0 . The interesting features about these solutions are that: Ži. depending upon the values of a, x 0 , z 0 , they are expressed in terms of either Bessel functions or modified Bessel functions, so that the oscillatory or nonoscillatory character of the solution curves is determined by these values, and Žii. that the order of the Bessel functions is a function of a, n, r. This exact manner of dependence on the parameters and the initial

GENERALIZED LORENZ-TYPE SYSTEMS

3

values, which determine the chaotic behavior, seems to be difficult to deduce rigorously from any qualitative or numerical analysis. We next consider another generalization of Ž1. which has a cubic term in the second equation and a quadratic term in the third equation, the value of b being arbitrary, and we obtain the exact general solutions, which are similar to those of the first generalized system; consequently, they also exhibit chaotic behavior for a certain parameter range. Finally, we consider a more general system having arbitrary powers of x, reduce it to the generalized Duffing equation of nonlinear mechanics, which can be solved exactly, when some of the parameters are interconnected, by a method given in w5x. The exact solutions we present, showing precisely how chaotic behavior occurs, should be of interest to nonlinear dynamical systems theorists for testing conjectures about such differential systems, and to numerical analysts for testing the validity and accuracy of approximate computer solutions, for which numerical errors tend to grow exponentially fast w6x.

2. A SPECIAL LORENZ SYSTEM An exact general solution of the Lorenz system for arbitrary values of the parameters and any initial values seems at present unattainable, so we have endeavored to obtain a particular solution of a special form of the system for which x 0 and y 0 are arbitrary, but z 0 is a certain function of x 0 . We thus consider the system

˙x s aŽ y y x . ,

˙y s rx y ny y xz,

˙z s xy y bz,

Ž 2.

where the parameters are positive and satisfy the relations r

b s 2 a,

a

s

n a

y

2 9

ž

1q

n a

2

/

,

Ž 3.

so that

4

n a

s5"3 1y8

½

r a

1r2

5

,

8

r a

O 1,

1 2

-

n a

- 2.

The general solution of Ž2. in the phase space has a family of projections w Ž x, z . s 0 on the Ž x, z . plane. Our particular solution selects only one member of this family, namely z s x 2r2 a, ergo z 0 s x 02r2 a. This selection

4

MOSTAFA A. ABDELKADER

reduces the third part of Eq. Ž2. to the first, and we are left with the nonlinear phase-plane equation dy

aŽ y y x .

dx

x3

s rx y ny y

,

2a

Ž 4.

with arbitrary initial values x 0 , y 0 . The only singular point of Ž4. is at the origin, since Ž r y n. is negative, by Ž3.. Introducing a variable u by means of u

ys

1

x2 q

'2

3

ž

2y

ž

2y

n a

/

x,

Ž 5.

/ 5

Ž 6.

so that at t s 0 we have u0 s

'2 x 02

½

1

y0 y

3

n a

x0 ,

and substituting from Ž5. into Ž4., taking account of Ž3., we get the linear equation

Ž u2 q h .

dx du

q

1 2

ux s

1 3'2

n

ž

1q

/

H Ž u . q x 0 Ž u 20 q h .

a

/

2 a2 h s 1,

,

which has the solution x Ž u2 q h .

1r4

s

1 3'2

ž

1q

n a

1r4

,

Ž 7.

where H Ž u. s

u

Hu Ž u

2

q h.

y3 r4

du.

Ž 8.

0

From Ž2. and Ž5. we get a

1

˙x s ' ux 2 y Ž a q n . x. 3 2

Ž 9.

Substituting for x from Ž7. into the right-hand side of Ž9., and integrating Ž7. and Ž9. parametrically, we get 1 3'2

ž

1q

n a

/

H Ž u . s x 0 Ž u 20 q h .

1r4

½

exp y

1 3

5

Ž a q n . t y 1 . Ž 10 .

GENERALIZED LORENZ-TYPE SYSTEMS

5

To evaluate H Ž u., we make the substitution u2 q h s

h

u 02 q h s

,

4

cn Ž s .

h 4

cn Ž s0 .

,

Ž 11 .

where cnŽ s . is the cosine Jacobian elliptic function with modulus k s 1r '2 , and we get from Ž8. the evaluation H Ž u . s 2 3r4 'a Ž s y s0 . .

Ž 12 .

From Ž10., Ž11., and Ž12. we obtain for sŽ t . the expression s s s0 q

3 x0

'2 Ž a q n . cn Ž s0 .

½

exp y

1 3

5

Ž a q n. t y 1 ,

and Ž7. gives the solution x Ž t . s x0

cn Ž s . cn Ž s0 .

exp y

1 3

Ž a q n. t ,

Ž 13 .

where, from Ž6. and Ž11., s0 is given by cn Ž s0 . s x 0

x 04

½

q 2 ay 0 y

2 3

2 y1 r4

Ž 2 a y n. x0

5

.

From Ž2. and Ž13. we get yŽ t. s

1

'2 a q

1 3

½ ž

x0 cn Ž s0

2y

n a

2

.5

/

x0

sn Ž s . dn Ž s . exp y cn Ž s . cn Ž s0 .

exp y

1 3

2 3

Ž a q n. t

Ž a q n. t ,

and we have z Ž t . s x 2r2 a, where snŽ s . and dnŽ s . are Jacobian elliptic functions with modulus k s 1r '2 . The above particular solutions of the special Lorenz system Ž2., Ž3. do not exhibit chaotic behavior; the existence of these nonchaotic solutions does not preclude that of chaotic solutions to Ž2..

6

MOSTAFA A. ABDELKADER

3. GENERALIZATIONS OF THE LORENZ SYSTEM The Lorenz system Ž2. has only one critical point at Ž0, 0, 0. when r O n Žfor n - r there are three critical points.. We consider a cubic generalization of the linear term Ž rx ., and set b s 2 a Žin the usual treatment of Ž2., it is assumed that a ) b q 1.. We thus consider the system

˙x s aŽ y y x . ,

½

˙y s rx q

x3 2a

5

y ny y xz,

˙z s xy y 2 az, Ž 14.

which has three independent positive parameters a, n, r. This system has only one critical point at Ž0, 0, 0. for all values of the parameters, just like Ž2. when r O n. To solve Ž14., we start by eliminating y and z, and setting t

 x Ž t . rx 0 4 2 s expH ¨ Ž t .

dt,

Ž 15 .

0

we get for ¨ Ž t . the autonomous Lienard equation ´ 2

¨¨ q Ž ¨ q 3a q n . ¨˙ q a Ž ¨ q a q n . y 4 a3 m2 s 0,

Ž 16 .

where ms

1 2

½

4

r a

q 1y

ž

n a

2 1r2

/5

.

We now rewrite Ž16. in the form 2

¨¨ q Ž ¨ q a q n . ¨˙ q a  2 ¨˙ q Ž ¨ q a q n . y Ž 2 am .

2

4 s 0,

multiply by the integrating factor 2 expŽ2 at . to get exp Ž 2 at .  2 ¨¨ q 2 Ž ¨ q a q n . ¨˙4 q 2 a exp Ž 2 at .  2 ¨˙ 2

q Ž ¨ q a q n . y Ž 2 am .

2

4 s 0,

or d dt

2

exp Ž 2 at .  2 ¨˙ q Ž ¨ q a q n . y Ž 2 am .

2

4

s 0.

A first integral of Ž16. is then given by 2

2

2

2 ¨˙ q Ž 2 as . q Ž ¨ q a q n . y Ž 2 am . s 0,

Ž 17 .

GENERALIZED LORENZ-TYPE SYSTEMS

7

where s s c exp Ž yat . and c is a constant of integration. We now introduce a variable ZŽ s . by means of ¨ q a q n q 2a

s dZ Z ds

s 0,

so that Ž17. goes over into the Bessel differential equation d2Z ds 2

1 dZ

q

s ds

½

q 1y

m2 s2

5

Z s 0.

The general solution of Ž16. is thus given by ¨ q a q n q 2 as

X CJmX Ž s . q Jym Ž s.

CJm Ž s . q Jym Ž s .

s 0,

Ž 18 .

where C is a constant of integration, and we assume that m is not an integer; if m is an integer, we replace Jym Ž s . by the Bessel function of the second kind YmŽ s .. From Ž15. and Ž18. we obtain

½

x Ž t , c . s x 0 exp y

1 2

Ž a q n. t

CJm Ž s . q Jym Ž s .

5

CJm Ž c . q Jym Ž c .

,

Ž 19 .

and from Ž19. and Ž14. we get

½

x 0 exp y yŽ t, c. s

1

5½ ž / 5

Ž a q n. t 2 CJm Ž c . q Jym Ž c . q

1

½ ž 2

1y

n a

1 2

1y

n a

/ 5

q m CJm Ž s .

y m Jym Ž s . y CsJmy1 Ž s . y sJymy1 Ž s . ,

and z Ž t, c. s

x2 2a

½

q z0 y

x 02 2a

5

exp Ž y2 at . ,

8

MOSTAFA A. ABDELKADER

where z0

½

cs

a

y

1r2

x 02

5

2 a2

,

and 1

n

y0

½ž / 5 ½ ž / 5

CC1 s

1y

2

y0

C1 s

x0

a

1

y

ymy

1y

2

n

Jym Ž c . y cJymy1 Ž c . ,

x0

y m Jm Ž c . q cJmy1 Ž c . .

a

If Ž z 0ra. - Ž x 02r2 a2 ., the solution of system Ž14. is given in terms of modified Bessel functions by

½

x Ž t , h . s x 0 exp y 1

½

x 0 exp y y Ž t , h. s

1 2

1

½ ž 2

1y

DIm Ž h . q Iym Ž h .

,

5½ ž / 5 Ž / 5 Ž.

Ž a q n. t 2 DIm Ž h . q Iym Ž h . q

DIm Ž w . q Iym Ž w .

5

Ž a q n. t

n

1

1y

2

n

q m DIm Ž w .

a

y m Iym w y DwImy1 w . y wIymy1 Ž w . ,

a

and x2

z Ž t , h. s

2a

½

q z0 y

x 02 2a

5

exp Ž y2 at . ,

where w s h exp Ž yat . , DD 1 s D1 s

1

n

hs y0

½

½ž / 5 ½ ž / 5 2

y0

x0

1y y

a

1 2

ymy

1y

n a

x0

x 02 2 a2

y

z0 a

1r2

5

,

Iym Ž h . y hIymy1 Ž h . ,

y m Im Ž h . q hImy1 Ž h . .

GENERALIZED LORENZ-TYPE SYSTEMS

9

If c s h s 0, the solutions of Ž14. are given by x0

xŽ t. s yŽ t. s

Ay1 x0

aŽ A y 1.

 A exp Ž 2 amt . y 1 4 exp Ž ys 1 t . ,

 s 2 A exp Ž 2 amt . y s 3 4 exp Ž ys1 t . ,

and z Ž t . s x 2r2 a, where 2 s 1 s a q 2 am q n,

½ž

A a 2

y0 x0

2 s 2 s a q 2 am y n,

/ 5 ž

y 2m y 1 q n s a 2

y0 x0

2 s 3 s a y 2 am y n,

q 2 m y 1 q n.

/

As t ª `, so that s s c expŽyat . ª 0, and w s h expŽyat . ª 0, we have m

m

Ž sr2. Jm Ž s . ª , G Ž 1 q m.

Ž wr2. Im Ž w . ª , G Ž 1 q m.

and

where G Ž1 q m. denotes the gamma function. From Ž19. we have, as t ª `, x Ž t, c. ª

x0 M

½

exp y

1 2

Ž a q n. t

5½

C G Ž 1 q m. q

s

x0 M

½

C Ž cr2 .

m

G Ž 1 q m.

exp

½

y

1 2

Ž ceyatr2.

1 G Ž 1 y m.

m

Ž ceyatr2 .

ym

5

Ž a q n . y am t

m

1 Ž 2rc . q exp y Ž a q n . q am t G Ž 1 y m. 2

½

where M s CJm Ž c . q Jym Ž c . .

5

5

,

5

10

MOSTAFA A. ABDELKADER

Likewise, we have x Ž t , h. ª

x0 N

½

D Ž hr2 .

m

½

exp

G Ž 1 q m.

y

1 2

5

Ž a q n . y am t

m

1 Ž 2rh . q exp y Ž a q n . q am t G Ž 1 y m. 2

½

5

5

,

where N s DIm Ž h . q Iym Ž h . , and so x0

x Ž t , c . y x Ž t , h. ª

G Ž 1 y m. = exp

½

Ž 2rc .

½

m

M

am y

1 2

y

Ž 2rh .

m

N

5

5

Ž a q n. t ,

which diverges, thus establishing chaotic behavior of the solutions of Ž14., only if n - r, since 0 - am y 12 Ž a q n., which follows from ms

1 2

½

4

r a

n

q 1y

ž

a

2 1r2

/5

,

and since expw y 12 Ž a q n. y am4 t x ª 0. 4. A QUADRATIC-CUBIC GENERALIZATION In the system Ž14. we have set b s 2 a; we now assume an arbitrary value for b, but generalize further by adding a quadratic term to the third equation. We thus consider the system

˙x s aŽ y y x . ,

˙y s rx q

˙z s xy y bz q

ž

b

x3

y ny y xz,

2a

y 1 x2,

/

2a

where the coefficient of x 2 is so chosen in order that exact solutions can be obtained. On eliminating y and z, and setting t

 x Ž t . rx 0 4 2 s expH u Ž t . 0

dt,

GENERALIZED LORENZ-TYPE SYSTEMS

11

we get

½

u ¨ q Ž u q a q n . u˙ q b u˙ q

1 2

2

Ž u q a q n. y

b 2n 2 2

5

s 0,

where

ns

1

 4 ar q Ž a y n . 2 4 b

1r2

.

Multiplying the differential equation by the integrating factor expŽ bt ., we get d dt

½

exp Ž bt . u ˙q

1 2

2 Ž u q a q n. y

b 2n 2 2

5

s 0.

Integrating this, evaluating the constant of integration, and making the substitutions bh s  4 az 0 y 2 x 02 4

1r2

b exp y t 2

½ 5

and uqaqnqb

h dY Y dh

s 0,

we get the Bessel equation d2 Y dh 2

q

1 dY

h dh

½

q 1y

n2 h2

5

Y s 0.

We hence obtain

½

x Ž t . s x 0 exp y

1 2

Ž a q n. t

5

C2 Jn Ž h . q Jy n Ž h . C2 Jn Ž d . q Jy n Ž d .

,

where bd s  4 az 0 y 2 x 02 4

1r2

.

The expressions for y Ž t . and z Ž t ., and the value of C2 in terms of the parameters and initial values, readily follow from the system differential equations; for imaginary d, modified Bessel functions replace the Bessel functions as before. The solutions exhibit chaotic behavior only if n - r.

12

MOSTAFA A. ABDELKADER

5. AN ARBITRARY-POWER GENERALIZATION Another more general system than Ž14. is the system

˙x s ay y bx,

˙y s rx q

s 2a

x mq 2 y ny y x m z,

˙z s s xy y 2 bz, where m / l. The first integral zs

s 2a

x 2 q k exp Ž y2 bt . ,

where k s z0 y

s 2a

x 02 ,

reduces the third equation of the system to the first, and we are left with the nonautonomous nonlinear equation

¨x q Ž b q n . ˙x q Ž nb y ar . x q akx m exp Ž y2 bt . s 0. The substitution x s X exp

½

2b my1

t

5

yields the generalized Duffing equation X¨ q a X˙ q b X q akX m s 0, where

as bs

Ž 3 q m. b q n, Ž m y 1.

½

2

2b my1

5

q 2b

Ž b q n. q nb y ar. Ž m y 1.

The method of obtaining the exact solution, in terms of a hyper-elliptic integral, for

b s 2a 2

Ž m q 1. Ž m q 3.

2

GENERALIZED LORENZ-TYPE SYSTEMS

13

is indicated at the end of Section 4 of w5x; the author does not know of any other exact solution of the damped Duffing equation.

REFERENCES 1. M. W. Hirsch, The dynamical systems approach to differential equations, Bull. Amer. Math. Soc. 11 Ž1984., 1]64. 2. C. Grebogi, E. Ott, and J. A. Yorke, Chaos, strange attractors, and fractal basin boundaries in nonlinear dynamics, Science 238 Ž1987., 632]638. 3. E. N. Lorenz, Computational chaos}A prelude to computational instability, Physica D 35 Ž1989., 299]317. 4. D. Ruelle, Deterministic chaos: The science and the fiction, Proc. Roy. Soc. London Ser. A 427 Ž1990., 241]248. 5. M. A. Abdelkader, Sequences of nonlinear differential equations with related solutions, Ann. Mat. Pura Appl. 81 Ž1969., 249]259. 6. J. A. Yorke, Do computer trajectories of chaotic systems represent true trajectories?, in SIAM Conference on Dynamical Systems, May 7]11, 1990, Orlando, FL.