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Chaos in limit cycle systems with external periodic excitations
ht. 1. Non-Luuar Printed
Mechanics, Vol. u. No. 5. pp. 349-361. 1987
@X0-7462/87 13.00 + 0.00 Q 1987 Pcrgamoa Journals Ltd
in GreatBriIaia
CHAOS IN LIMIT CYCLE SYSTEMS WITH EXTERNAL PERIODIC EXCITATIONS W.-H. STEEB Department of Physics, Rand Afrikaans University, PO Box 524, Johannesburg 2000, Republic of South Africa and A. KUNICK Kraftwerk Union, Rechenzentrum, D-8520 Erlangen, West Germany (Received 30 May 1985; received for publication 2 February 1987) Abstract-Various limit cycle systems in the plane with an external periodic excitation are investigated from the point of view of chaotic behavior. Numerical studies are performed and a singular point analysis is carried out. 1. INTRODUCTION
Limit cycle systems appear in the mathematical analysis of many phenomena. Among these are the theory of laser, biochemical oscillators, circadian rhythms, and many engineering applications in particular in electronics. The best known example is the socalled Van der Pol equation dZx/dt2 -
p(i -
x*)dx/dt + x = 0
(1)
(,u > 0) which arises in the study of oscillatory vacuum tube circuits [1,2]. It is clear that for x < 1 the damping is negative and for x > 1 positive. From graphical considerations Van der Pol obtained the solutions for various values of p and in each case the result was a rapid approach to a steady state oscillation (limit cycle behavior). Another dynamical system which also arises in electronics takes the form [3] d*x/dt* - ~(1 - x*)dx/dt + x3 = 0.
(2)
Studying in the phase plane (x,dx/dt = y) we also find a stable limit cycle (p > 0). Both equations belong to a larger class which we discuss in Section 2. The dynamical system d*x/dt* + psin(dx/dt) + x = 0 furnishes an approximate description of the behavior of a second order phase-locked an electrical circuit used extensively in a variety of applications, notably in communication. Equation (3) has an infinite number of limit cycles in the phase (x,dx/dt = y) [4-73. Also in chemical kinetics stable limit cycle systems arise. A simple model system Brusselator
(3) loop, space plane is the
dx/dt = x*y - Bx - x + A dyjdt = -x*y + Bx,
(4)
where x, y denote concentrations and the quantities A, B are positive constants. Another chemical model system [8] is given by dx/dt = x(p - x2 - y*) + 2y dy/dt = y(p - x2 - y*) - 2x, where ~1> 0. The stable limit cycle is given by x2 -I- yf = p. 349
(5)
350
W.-H. STEEB and A. KUNICK
In the present paper we study limit cycle systems numerically under the influence of an external periodic perturbation. To be precise, we investigate d’x/dt* - ~(1 - x*)dx,‘dt + x” = kcos(Qt)
(6)
d2x/dt2 + psin(dx/dt) + x = kcos@t),
(7)
and
where n = 1, 3, 5 and k is the amplitude of the external perturbation and R its frequency. In particular, we are interested in whether or not there is a chaotic response. In addition we perform a singular point analysis for equation (6). The Brusselator (4) with an external periodic excitation has been widely discussed by Tomita and collaborators [9- 111. They found chaotic behavior in dependence of the parameters k and R. The Van der Pol equation with an external periodic perturbation has been investigated by various authors. Periodic solutions to this equation, and their stability properties, have been extensively studied in the past fifty years. The regular case (p << 1) has been investigated with averaging methods by Krylov and Bogoliuboff [12-j, and with analytic and topological methods by Littlewood and others [13,14]. The case with ,u >> 1 has been studied by Flaherty and Hop~nsteadt [lS] and others [16-181. Linkens [19] studied a coupled system of two Van der Pol oscillators for p << 1. Ueda (33 investigated equation (6) with n = 3 numerically. Depending on the bifurcation parameters p, k and R he found chaotic behavior. In particular he emphasized that equation (5) with n = 3 can show chaotic behavior, whereas equation (5) with n = 1 (Van der Pol case) does not show chaotic behavior in the range 0 < g < I. To our knowledge equation (5) with n = 5 and equation (6) have not been investigated so far. The limit cycle system (5) coupled with diffusion (reaction diffusion equation) has been studied by Strampp et al. [20]. Here chaotic behavior occurs (space-time chaos). 2. LIMIT
CYCLE
SYSTEMS
Before considering the influence of the external perturbation we briefly discuss the behavior of the limit cycle systems. Equation (6) (k = 0) with n odd belongs to a class of equations which show limit cycle behavior [2f, namely d2x/dt2 i- f(x)dx/dt
-I-g(x) = 0,
(8)
where the functions f and g satisfy the following conditions in order that equation (7) has a stable limit cycle: (i)/is even, g is odd, both continuous for all x, andf(0) < 0; (ii) xg(x) > 0 X for x # 0; (iii)g is Lipschitzian; (iv) F(x) + + cio as x + + co, where F(x) = f(x’)dx’; (II)F i^0 has a single positive zero at x = a and is monotone increasing for x B a. For example, the function g(x) = x” (n odd) satisfies the conditions given above. Consider now the equation d2xjdt2 - ~(1 - x*)dx/dt + X” = 0
(9)
in the phase plane (x,dx/dt = y), i.e. dxjdt = y,
dy/dt = ~(1 - x2)y - x”
(10)
where n is odd. The only time independent solution is the origin. Observe that a periodic solution (and therefore a limit cycle) in the (x,y) plane must enclose at least one critical point (time independent solution). Since we assume that p > 0 we find that the origin is unstable. The limit cycle of equation (9) is stable (p > 0). This means all trajectories lying inside or outside the limit cycle tend to the limit cycle. For g small the limit cycle itself is
Limit cycle systems
351
nearly a circle around the origin. If p = 0 we only have periodic orbits (no limit cycles) and the equations of motion can be derived from a Hamiltonian function. The periodic orbits are given by y2/2 + x”+ ‘/(n + 1) = C’, where Cz is determined by the initial values. Consider now equation (7) (k = 0) in the phase plane, i.e. dxfdt = y,
dyfdt = -psiny
- x.
(11)
Here, too, only one critical point exists, namely the origin. The origin is stable if p > 0 and unstable for p < 0. Thus if p > 0, then there is a neighborhood of the origin where all trajectories tend to the origin. Equation (11) admits an infinite number of limit cycles. The stability of the limit cycles has been studied by D’Heedene [S]. 3. NUMERICAL
INVESTIGATIONS
For studying equations (6) and (7) numerically we put dx/dt = y and obtain dxfdt = y,
dyldt = ~(1 - x2)y - x” + kcos(nt)
(12)
and dx/dt = y,
dyldt = -psiny
- x + kcos(Rt).
(13)
Equations (12) and (13) are invariant under f + t + 2nn/R (n EZ). When we put z = Rt, it follows that dx/dt = y,
dy/dc = p( 1 - x2)y - x” + kcos(z),
dzldt = R,
(14)
and dx/dt = y,
dyldr = -psiny
- x + kcos(z),
dzldt = 52,
(15)
where z(t = 0) = 0. Equations (14) and (15) are defined on R’xS’. For calculating the phase portraits we apply equations (14) and (15). Also for calculating the autocorrelation functions C,, and C,, we use equations (14) and (15). The autocorrelation function C,, is given by
(16)
Chaotic motion for equations (13) and (14) is indicated if the autocorrelation functions C,, and C,,, decay. Due to Ueda [3,21] we introduce a diffeomorphism on the (x, y) plane into itself, which can also serve to decide numerically whether or not there is chaotic behavior. The diffeomorphism is introduced as follows: Assume that (t, x,,, yO) --) x(t. x0, y,J, (t, x,,, y,,) + y(t, x0, yO)is a solution to equation (12) [or equation (13)] starting from a point p0 = (x,, yO) at t = 0. Let p1 = (x,,y,) be the point of this solution at t = 211/R.This means x1 = x(2+& x0, yO), y, = y(27r/Q, x,,, yO). Thus we have defined a diffeomorphism f;: R2 + R2, pO + pl, where i. = (p, k,R). The steady state motion of equations (12) or (13) is represented by an attractor of the diffeomorphism f’. If an attractor is composed of a single periodic group, then the corresponding motion is periodic and hence deterministic or regular. This means a periodic solution is represented by a fixed or n-periodic point of fi, i.e. p = f:(p) (p E R2, n E Z’). However, if the attractor is composed of a closed, invariant set of fi, containing infinitely many unstable periodic groups, then chaotic motion occurs. In addition, we also calculate the one dimensional Lyapunov exponent i., where we select the biggest rate. If i., > 0, then chaotic motion is indicated.
352
W.-H. STEEB and A. KUNICK
Now let us give our numerical results for equations (12) and (13). Throughout we have studied the range where p < 1. We shall let digital time integrations run for a long time so that all transients have decayed and then allow a single trajectory to wander over the final attractor. Thus the phase portrait is the orbit x(t), y(r) after the decay of transients (i.e. the projection of the (x(t), y(t), z(r)) orbit onto the (x,y)-plane. Then the attractor of the diffeomorphism (which is nothing but a period-advanced mapping) is a subset of this orbit. Notice that for a better visualization we use a different scale for the phase portrait and the attractor, respectively. First of all let us give some selected numerical results. Consider first equation (12). Let p = 0.2, k = 1, R = 4 and n = 1, 3, 5. In Fig. 1 we have plotted the phase portrait for n = 1. Figure 2 shows the phase portrait for n = 3. The autocorrelation functions C,, and
Y
Fig. 1. Phase portrait
of equation
(12) for n = 1, p = 0.2. k = 1. R = 4
Fig. 2. Phase
of equation
(12) for n = 3, p = 0.2, k = 1, R = 4.
portrait
Limit cycle systems
353
C,, do not decay in any of the three cases. In addition we have calculated the attractor due to the technique of Ueda [3,2 11. In all cases the one-dimensional Lyapunov exponent is equal to zero within the numerical accuracy. These numerical results indicate that there is no chaotic behavior in any of the three cases. Consider now the case with the values p = 0.2, k = 17, and R = 4. In Fig. 3 we have plotted the phase portrait for n = 1 and Fig. 4 shows the attractor. We find that the autocorrelation functions do not decay. Hence the numerical results indicate that there is no chaotic motion. In Fig. 5 the phase portrait is depicted for the case where n = 3 and Fig. 6 shows the attractor. For n = 3 we find that the autocorrelation functions decay. The attractor has been calculated by Ueda [3]. The one-dimensional Lyapunov exponent is
Fig. 3. Phase portrait
Fig. 4. Attractor
of equation
of equation
(12) for n = 1, p = 0.2, k = 17, R = 4.
(12) for n = 1, /1 = 0.2, k = 17, R = 4.
354
W.-H.
STEEBand
A.
KIJNICK
given by i, = 0.34. These numerical results indicate that we have chaotic behavior. The strange attractor is identica! with a closure of unstable manifolds of a saddle point of the diffeomorphism fi: RZ -+ R’, which is defined by using the solutions to equations (12). The unstable manifolds of these cases are infinitely long but are confined within the bounded region and present homoclinic structure intersecting with stable manifolds. Womoclinic structure causes the existence of infinitely many periodic points. When we consider the same parameter values but n = 5 the numerical results do not indicate chaotic motion. A case for n = 5 in which we find chaotic behavior is given by p = 0.2, k = 3.2 and ir = 4. The attractor is given by Fig. 7. The autocorrelation functions decay (Fig. 8). The one-dimensional Lyapunov exponent is given by i., = 0.28.
X Fig. 5. Phase portrait or equation (12) for n = 3. p = 0.2, k = 17, Q = 4 (chaotic motion).
Fig. 6. Attractor of equation (12) for n = 3, p = 0.2, k = 17, L? = 4 (chaotic motion).
Limit cycle systems
355
Our numerical studies in the range 0 c p c 1,O < R < 10, and 0 < k < 20 indicate that we can find chaotic behavior for n = 3 and n = 5. Obviously, we can also find chaotic motion for n = 7, 9 and so on. For n = 1 (Van der Pol oscillator with a periodic external perturbation) we do not find chaotic behavior. This result has also been found by Ueda
c31.
For n = 3, p = 0.2, R = 4 we have studied the range 10 < k < 20 in more details. Figure 9 shows the behavior in dependence of the amplitude k. The dots indicate regular behavior and the crosses indicate chaotic behavior. Consider now equation (13). Here we study the range Ip((< 1. In Fig. 10 we have plotted
L + ++
++
c+ :+* l+
YI
l l
f,
+
+
c
+t++
=++++++ + ++ c‘,’ +C++++ + +
-I + i
B +; ++
‘++
+ t
*+:
4
+
++
+
+++ +
++ +
l+++
l
t++++ +&~++++ t_ +; +q*+ +
l+
+++++
c
*
&
f’ +
+++b ++ + + + + ++ + +++ + *+ +++++ + +
+
l
++ +*+++ ++++++ ++ +++ ++ +++++ + ++ *+:+ * ; =; ++++++ett++++++* +++ +++++ + ++-*+++ +*+ + +
+
‘+
+ f *+ +; ;: ++*. + *+*c’ + + ++* L++ * *
l ++ +
+
+*++
+i :
+ +* +‘:
+
l
*+ *
.+ tt+:
.++++
+
X Fig. 7. Attractor
Fig. 8. Autocorrelation
of equation
(12) for n = 5, p = 0.2, k = 3.2, R = 4 (chaotic
motion).
function C, of equation (12) for n = 5, p = 0.2, k = 3.2, R = 4 (chaotic motion).
356
W.-H.
STEEBand
A. KUNICK
10
15
20
15
15.5
16
15.05
15.1
15.015
15.020
15.01G5
t 15.OI50
K
f K
7
15
K
L
15.010
15.ouo
K
L
K
Fig. 9. Behavior of equation (12) for II = 3. p = 0.2. R = 4. 10 < k Q 20
X Fig. 10. Phase portrait of equation (13) for p = 0.5. k = 0.532. n = 0.8.
the phase portrait for p = 0.5, k = 0.532, and R = 0.8. Figure 11 shows the phase portrait for p = -0.5, k = 0.532, and R = 0.8. The autocorrelation functions do not decay. Figures 12 and 13 show the phase portraits for k = 17 and R = 4 where in Fig. 12 the damping coefficient is ~1= 0.2 and in Fig. 13 p = -0.2. Here, too, we do not have chaotic motion. From further numerical studies we infer that there is no chaotic motion in the range IA < 1. 4. SINGULAR
POINT
ANALYSIS
is desirable to have a simple approach for deciding whether a dynamical system is integrable or not. In classical mechanics [i.e. for ordinary differential equations (odes)] the It
Limit cycle systems
Fig. 11. Phase portrait for equation (13) for p = -0.5, k = 0.532, SZ= 0.8.
Fig. 12. Phase portrait of equation (13) for p = 0.2, k = 17, Q = 4.
so-called singular point analysis (sometimes called Painleve test) can serve to decide whether a dynamical system is algebraically integrable. Notice, however, that the Painleve test fails if the first integrals are transcendental functions. The structure of the singularities of the solutions of the ode is studied in the complex plane [22]. We look for representations with so-called psi series. In particular we are interested in finding out whether a given ode has the so-called Painleve property. An ode (or a system of odes) in the complex domain is said to be of Painleve type (or has the PainlevC property) if the only movable singularities of all its solutions are poles. This means that there be no movable branch points or movable essential singularities.
358
W.-H.
Fig. 13. Phase portrait
STEEB
of equation
and A. KL’NICK
(13) for ;i = -0.2.
k = 17, R = 4
A necessary condition for an nth order ode of the form @w/d? = F(r, w,. . . , d”- ‘w,‘dz”- ‘f
(17)
where F is rational in w,. . . , d”- rw/dzn- ’ and analytic in z, to have Painlevi property is that there is a Laurent expansion
w(t) = (z -
ZJk f aiz - Zlv’ j=O
with (n - 1) arbitrary expansion coefficients (besides the pole position which is arbitrary). Notice that more than one branch may arise. Each branch must be a Laurent series. We find that none of the differential equations given in Section 1 satisfy this necessary condition. In some cases it is useful to extend the Painlevt property to the so-called quasi Painleve property, since if this property is fulfilled we can also find integrable systems. The autonomous system dx/dt = g(x) where g is rational in x1,. . . ,x, is said to be of quasi Painlevi type if it admits only expansions of the form
Wi(.z)= (z -
z*)l” f a,{2 - Zlv‘
(19)
j=O
where (n - 1) expansion
coefficients are arbitrary
and the exponents
ki are given by
ki = m/n (a, tt~E Z\(O)).
We find that equation (5) has the quasi-Painieve property. This is due to the fact that this equation can be solved by quadrature introducing polar coordinates. For all other equations this property is not fulfilled. Nevertheless we may well ask whether there are a psi series. Psi series for systems which show chaotic behavior have been studied by various authors [23-25f. For exampIe, for the Lorenz model [23] and the Duffing equation [25] so-called logarithmic psi series arise. Let us study now the Van der Pol equation with a periodic external perturbation. We
359
Limit cycle systems
consider this equation in the complex domain, i.e. d2wJdz’ - ~(1 - w2)dw/dz + w = kcos(Qz,)cos(R(z - zr)) - ksin@z,)sin(Q(z - zr)).
(20)
The singular point analysis has been widely described in the literature [22-263. Thus we give only the results. We obtain the psi series W(Z)
=
(Z -
Z1)-1’2
~
(Uj(Z
-
Z*y’
+
bj(Z
-
(21)
Z1~+3’2).
j=O
where b. can be chosen arbitrarily and the first few coefficients are given by a, = Pao/2,
u; = 3/(2~), b, = -(1/7)(pb,
(22)
- 2kcos(Rz,)).
When we consider equation (6) with n = 3 in the complex domain, we obtain w(z) = (z - zl)-
“’ f (aj(z - z$ +
z#+~‘~)
bj(z -
j=O
(23)
where b. can be chosen arbitrarily and a; = 3/(2~), _ a 1 = ~a,/2 + d, b, = -(1/7)(pb,
+ 6a;b, - Zkcos(Rz,)).
(24)
In order to decide for which parameter values chaotic behavior occurs we have to study the location of the singularities in the complex plane. Such a numerical study has been performed by Tabor and Weiss [23] for the Lorenz mode1 and for the non-linear Langevin equation by Frisch and Morf [27]. Let us study system (6) (n = 3) considered in the complex plane. Putting dw/dz = u and u = Rz, we obtain dw/dz = v dv/dz = p( 1 - w2)v - w3 + kcosu du/dz = R.
(25)
Now we set z = t + is, w(z) = w,(t,r) + iw,(c,r) and so on. Since dw,/dt = dw,/dr, aw,/at = -dw,/dr and so on (Cauchy-Riemann condition) we find
aw,/at = vl, ah/at = ~h(i
h,/at
= v2
- W: + W;) +
av,iat = Au2(1 - w: + w:) -
ih,/at = R,
2U2WlW2)
-
W,(W:
2vlw,w,)
-
W#Wf
-
3~;)
-
w:)
+
kcosP,coshu2
-
ksinu,sinhu,
h,fat = 0
(264
and
awl/a7 = -v2,
aw2jd7 = v1
a07
= -pb2(l
%/dr
= ~(Vr(i - W; + W:) +
au,/a7 = 0,
- 4 + 4) - Zv,w,w,) + c?u,/dr = i-2.
2V2WlW2)
-
W2(3Wf
W,(W;
-
-
3~:)
w$)
+
+
ksinwrsinhu,
kcosu,coshu,
CW
36a
W.-f-f.STEEB and A. KUPUICK
We have determined numerically the pattern of singularities in the complex plane nearest to the real t axis. If we choose the parameter values in the chaotic regime we have an irregular pattern. If we choose the parameter values in the region where there is no chaos we have a regufar pattern (compare also Tabor and Weiss [23] and Frisch and Morf f27f). Consider the equation f = -x3 (or .t = I’, j = -x3). Then we know that the solution is given by an elliptic function, namely x(c) = Ccn(C(r - co), l/a, where C and t, are the constants of integration. The elliptic functions are periodic functions. Consider now the equation given above in the complex time pIane (z = t + is), i.e. d’w/d? = --v’. Again the solution is given by the elliptic function en but with a complex argument. fts singularities are simple poles of order one and are distributed doubly periodically in the whole complex z-plane. This regular distribution of singularities (in the present case poles) reflects faithfully on periodicity of the solutions. Taking into account the damping term and the external perturbation the singularities are redistributed in the complex plane. The real solution is mainly determined by the singularities located nearest to the real axis, As mentioned above we find an irregular pattern of these singularities when the system shows chaotic behaviour. 5. CONCLUSlON
Our numerical analysis indicates that equation (6) does not show chaotic behavior for I( < 1 and n = 1. For n = 3, 5,. +. we can find chaotic motion for FL< 1. The different concepts for characterising chaotic motion (auto~orrelation functions, Poincare map, Lyapunov exponents) coincide. For equation (7) we do not find chaotic behavior for igi < t. If we consider the equation d2s,dt2 $ /tsin(ds/dr) -I- X” = kcos(f&)
(27)
fn = 3,5, __.f then also chaotic motion can arise for [pi < 1. Thus the non-linearity X” (n = 3,5,. . .f is responsible for the occurrence of chaotic motion for {jr1-z I. It is worthwhile to compare our results given above with those for the Duffing equation which is given by
where p > 0, c 3 0. This equation has been studied by various authors (compare Steeb e? al. [28] and references therein). As described above we can write equation (26) as dx/dt = y1
dl:idt = -/.L~L’ - bs - c.y3 + kcos(Qr).
Equation (29) is invariant under t -, r + 2ntt,Q (n E Z). The di~~omorphism _&(; =; (it7 tt, c, k, f2) defined by Ueda [3,2IJ is of the contracting type in the given case, whereas for equation (6) it is not of the contracting type. This is due to the fact that the equation (12) is self-oscillatory, whereas the Duffing equation is quiescent unless an external (periodic) force is applied. When we perform a singuiar point anafysis for equation (29) considered in the complex domain we obtain a logarithmic psi series
where the expansion coefficient n,, can be chosen arbitrarily, Thus also from the singular point analysis the behavior is quite different. In the present case we have logarithmic branch points, whereas for equation (6) (n = 1 or II = 3) considered in the complex domain we have algebraic branch points. For equation (29) (considered in the complex domain) we also find from numerical studies in the complex plane a regular pattern of the singularities when parameter vatues are studied where there is no chaotic motion For parameter values with chaotic behavior of the dynamical system (29) the pattern of the singularities is irregular.
Limit cycle systems
361
REFERENCES 1. B. Van der Pol, On relaxation oscillations. Phil. Mug. 2, 978-992 (1926). 2. N. Levinson and 0. K. Smith, A general equation for relaxation oscillations. Duke Mach. J. 9, 382-403 (1942). 3. Y. Ueda and N. Akamatsu. Chaotically transitional phenomena in the forced negative resistence oscillator. IEEE Trans. Circuits Sysr. 28, 217-224 (1981). 4. H. Hochstadt and B. H. Stephan. On the limit cycle of i + bsini + x = 0. Archs. Rat. Mech. Anal. 23. 368370 (1967). 5. R. N. D’Heedene, For all real ~1,.< + psinl + I = 0 has an infinite number of limit cycles. J. Oifi Equat. 5, 564-571 (1969). 6. P. J. Ponzo and N. Wax, On periodic solutions of the system i = y - F(x), g = -g(x). J. Difl Equat. 10, 262-269 (1971). 7. W.-H. Steeb and A. Kunick. Limit cycle oscillations and a perturbed harmonic oscillator. Int. J. Non-linear Mech. 17, 41-45 (1982). 8. J. Dreitlein and M. Smoes, Model for oscillatory kinetic systems. J. Theor. Eiol. 46, 559-564 (1974). 9. K. Tomita and T. Kai, Chaotic response of a limit cycle. J. Srat. Phys. 21, 65-86 (1979). 10. K. Tomita, T. Kai and F. Hikami, entrainment of a limit cycle by a periodic external excitation. frog. Theor. Phys. 57. 1159-1177 (1977). 1I. T. Kai and K. Tomita, Stroboscopic phase portrait of a forced non-linear oscillator, Prog. Theor. Phys. 61, 54-73 (1979). 12. N. Krylov and N. Bogoliuboff, Introducrion to Non-linear Mechanics. Ann. of Math. Ser. No. II, Princeton U.P. 1947. 13. J. E. Littlewood, On non-linear differential equations of the second order. Acta Math. 97, 268-308 (1957). 14. J. J. Stoker, Non-linear Vibrations. Interscience, New York (1950). 15. J. E. Flaherty and F. C. Hoppensteadt, Frequency entrainment of a forced Van der Pol oscillator. Srud. Appl. Math. 58, 5-15 (1978). 16. J. Gugenheimer, Symbolic dynamics and relaxation oscillations. Physica lD, 227-235 (1980). 17. M. Levi, Periodic forced relaxation oscillation. In Lecture Notes in Mathematics, (Edited by A. Dold and B.
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Mechanics, Vol. u. No. 5. pp. 349-361. 1987
@X0-7462/87 13.00 + 0.00 Q 1987 Pcrgamoa Journals Ltd
in GreatBriIaia
CHAOS IN LIMIT CYCLE SYSTEMS WITH EXTERNAL PERIODIC EXCITATIONS W.-H. STEEB Department of Physics, Rand Afrikaans University, PO Box 524, Johannesburg 2000, Republic of South Africa and A. KUNICK Kraftwerk Union, Rechenzentrum, D-8520 Erlangen, West Germany (Received 30 May 1985; received for publication 2 February 1987) Abstract-Various limit cycle systems in the plane with an external periodic excitation are investigated from the point of view of chaotic behavior. Numerical studies are performed and a singular point analysis is carried out. 1. INTRODUCTION
Limit cycle systems appear in the mathematical analysis of many phenomena. Among these are the theory of laser, biochemical oscillators, circadian rhythms, and many engineering applications in particular in electronics. The best known example is the socalled Van der Pol equation dZx/dt2 -
p(i -
x*)dx/dt + x = 0
(1)
(,u > 0) which arises in the study of oscillatory vacuum tube circuits [1,2]. It is clear that for x < 1 the damping is negative and for x > 1 positive. From graphical considerations Van der Pol obtained the solutions for various values of p and in each case the result was a rapid approach to a steady state oscillation (limit cycle behavior). Another dynamical system which also arises in electronics takes the form [3] d*x/dt* - ~(1 - x*)dx/dt + x3 = 0.
(2)
Studying in the phase plane (x,dx/dt = y) we also find a stable limit cycle (p > 0). Both equations belong to a larger class which we discuss in Section 2. The dynamical system d*x/dt* + psin(dx/dt) + x = 0 furnishes an approximate description of the behavior of a second order phase-locked an electrical circuit used extensively in a variety of applications, notably in communication. Equation (3) has an infinite number of limit cycles in the phase (x,dx/dt = y) [4-73. Also in chemical kinetics stable limit cycle systems arise. A simple model system Brusselator
(3) loop, space plane is the
dx/dt = x*y - Bx - x + A dyjdt = -x*y + Bx,
(4)
where x, y denote concentrations and the quantities A, B are positive constants. Another chemical model system [8] is given by dx/dt = x(p - x2 - y*) + 2y dy/dt = y(p - x2 - y*) - 2x, where ~1> 0. The stable limit cycle is given by x2 -I- yf = p. 349
(5)
350
W.-H. STEEB and A. KUNICK
In the present paper we study limit cycle systems numerically under the influence of an external periodic perturbation. To be precise, we investigate d’x/dt* - ~(1 - x*)dx,‘dt + x” = kcos(Qt)
(6)
d2x/dt2 + psin(dx/dt) + x = kcos@t),
(7)
and
where n = 1, 3, 5 and k is the amplitude of the external perturbation and R its frequency. In particular, we are interested in whether or not there is a chaotic response. In addition we perform a singular point analysis for equation (6). The Brusselator (4) with an external periodic excitation has been widely discussed by Tomita and collaborators [9- 111. They found chaotic behavior in dependence of the parameters k and R. The Van der Pol equation with an external periodic perturbation has been investigated by various authors. Periodic solutions to this equation, and their stability properties, have been extensively studied in the past fifty years. The regular case (p << 1) has been investigated with averaging methods by Krylov and Bogoliuboff [12-j, and with analytic and topological methods by Littlewood and others [13,14]. The case with ,u >> 1 has been studied by Flaherty and Hop~nsteadt [lS] and others [16-181. Linkens [19] studied a coupled system of two Van der Pol oscillators for p << 1. Ueda (33 investigated equation (6) with n = 3 numerically. Depending on the bifurcation parameters p, k and R he found chaotic behavior. In particular he emphasized that equation (5) with n = 3 can show chaotic behavior, whereas equation (5) with n = 1 (Van der Pol case) does not show chaotic behavior in the range 0 < g < I. To our knowledge equation (5) with n = 5 and equation (6) have not been investigated so far. The limit cycle system (5) coupled with diffusion (reaction diffusion equation) has been studied by Strampp et al. [20]. Here chaotic behavior occurs (space-time chaos). 2. LIMIT
CYCLE
SYSTEMS
Before considering the influence of the external perturbation we briefly discuss the behavior of the limit cycle systems. Equation (6) (k = 0) with n odd belongs to a class of equations which show limit cycle behavior [2f, namely d2x/dt2 i- f(x)dx/dt
-I-g(x) = 0,
(8)
where the functions f and g satisfy the following conditions in order that equation (7) has a stable limit cycle: (i)/is even, g is odd, both continuous for all x, andf(0) < 0; (ii) xg(x) > 0 X for x # 0; (iii)g is Lipschitzian; (iv) F(x) + + cio as x + + co, where F(x) = f(x’)dx’; (II)F i^0 has a single positive zero at x = a and is monotone increasing for x B a. For example, the function g(x) = x” (n odd) satisfies the conditions given above. Consider now the equation d2xjdt2 - ~(1 - x*)dx/dt + X” = 0
(9)
in the phase plane (x,dx/dt = y), i.e. dxjdt = y,
dy/dt = ~(1 - x2)y - x”
(10)
where n is odd. The only time independent solution is the origin. Observe that a periodic solution (and therefore a limit cycle) in the (x,y) plane must enclose at least one critical point (time independent solution). Since we assume that p > 0 we find that the origin is unstable. The limit cycle of equation (9) is stable (p > 0). This means all trajectories lying inside or outside the limit cycle tend to the limit cycle. For g small the limit cycle itself is
Limit cycle systems
351
nearly a circle around the origin. If p = 0 we only have periodic orbits (no limit cycles) and the equations of motion can be derived from a Hamiltonian function. The periodic orbits are given by y2/2 + x”+ ‘/(n + 1) = C’, where Cz is determined by the initial values. Consider now equation (7) (k = 0) in the phase plane, i.e. dxfdt = y,
dyfdt = -psiny
- x.
(11)
Here, too, only one critical point exists, namely the origin. The origin is stable if p > 0 and unstable for p < 0. Thus if p > 0, then there is a neighborhood of the origin where all trajectories tend to the origin. Equation (11) admits an infinite number of limit cycles. The stability of the limit cycles has been studied by D’Heedene [S]. 3. NUMERICAL
INVESTIGATIONS
For studying equations (6) and (7) numerically we put dx/dt = y and obtain dxfdt = y,
dyldt = ~(1 - x2)y - x” + kcos(nt)
(12)
and dx/dt = y,
dyldt = -psiny
- x + kcos(Rt).
(13)
Equations (12) and (13) are invariant under f + t + 2nn/R (n EZ). When we put z = Rt, it follows that dx/dt = y,
dy/dc = p( 1 - x2)y - x” + kcos(z),
dzldt = R,
(14)
and dx/dt = y,
dyldr = -psiny
- x + kcos(z),
dzldt = 52,
(15)
where z(t = 0) = 0. Equations (14) and (15) are defined on R’xS’. For calculating the phase portraits we apply equations (14) and (15). Also for calculating the autocorrelation functions C,, and C,, we use equations (14) and (15). The autocorrelation function C,, is given by
(16)
Chaotic motion for equations (13) and (14) is indicated if the autocorrelation functions C,, and C,,, decay. Due to Ueda [3,21] we introduce a diffeomorphism on the (x, y) plane into itself, which can also serve to decide numerically whether or not there is chaotic behavior. The diffeomorphism is introduced as follows: Assume that (t, x,,, yO) --) x(t. x0, y,J, (t, x,,, y,,) + y(t, x0, yO)is a solution to equation (12) [or equation (13)] starting from a point p0 = (x,, yO) at t = 0. Let p1 = (x,,y,) be the point of this solution at t = 211/R.This means x1 = x(2+& x0, yO), y, = y(27r/Q, x,,, yO). Thus we have defined a diffeomorphism f;: R2 + R2, pO + pl, where i. = (p, k,R). The steady state motion of equations (12) or (13) is represented by an attractor of the diffeomorphism f’. If an attractor is composed of a single periodic group, then the corresponding motion is periodic and hence deterministic or regular. This means a periodic solution is represented by a fixed or n-periodic point of fi, i.e. p = f:(p) (p E R2, n E Z’). However, if the attractor is composed of a closed, invariant set of fi, containing infinitely many unstable periodic groups, then chaotic motion occurs. In addition, we also calculate the one dimensional Lyapunov exponent i., where we select the biggest rate. If i., > 0, then chaotic motion is indicated.
352
W.-H. STEEB and A. KUNICK
Now let us give our numerical results for equations (12) and (13). Throughout we have studied the range where p < 1. We shall let digital time integrations run for a long time so that all transients have decayed and then allow a single trajectory to wander over the final attractor. Thus the phase portrait is the orbit x(t), y(r) after the decay of transients (i.e. the projection of the (x(t), y(t), z(r)) orbit onto the (x,y)-plane. Then the attractor of the diffeomorphism (which is nothing but a period-advanced mapping) is a subset of this orbit. Notice that for a better visualization we use a different scale for the phase portrait and the attractor, respectively. First of all let us give some selected numerical results. Consider first equation (12). Let p = 0.2, k = 1, R = 4 and n = 1, 3, 5. In Fig. 1 we have plotted the phase portrait for n = 1. Figure 2 shows the phase portrait for n = 3. The autocorrelation functions C,, and
Y
Fig. 1. Phase portrait
of equation
(12) for n = 1, p = 0.2. k = 1. R = 4
Fig. 2. Phase
of equation
(12) for n = 3, p = 0.2, k = 1, R = 4.
portrait
Limit cycle systems
353
C,, do not decay in any of the three cases. In addition we have calculated the attractor due to the technique of Ueda [3,2 11. In all cases the one-dimensional Lyapunov exponent is equal to zero within the numerical accuracy. These numerical results indicate that there is no chaotic behavior in any of the three cases. Consider now the case with the values p = 0.2, k = 17, and R = 4. In Fig. 3 we have plotted the phase portrait for n = 1 and Fig. 4 shows the attractor. We find that the autocorrelation functions do not decay. Hence the numerical results indicate that there is no chaotic motion. In Fig. 5 the phase portrait is depicted for the case where n = 3 and Fig. 6 shows the attractor. For n = 3 we find that the autocorrelation functions decay. The attractor has been calculated by Ueda [3]. The one-dimensional Lyapunov exponent is
Fig. 3. Phase portrait
Fig. 4. Attractor
of equation
of equation
(12) for n = 1, p = 0.2, k = 17, R = 4.
(12) for n = 1, /1 = 0.2, k = 17, R = 4.
354
W.-H.
STEEBand
A.
KIJNICK
given by i, = 0.34. These numerical results indicate that we have chaotic behavior. The strange attractor is identica! with a closure of unstable manifolds of a saddle point of the diffeomorphism fi: RZ -+ R’, which is defined by using the solutions to equations (12). The unstable manifolds of these cases are infinitely long but are confined within the bounded region and present homoclinic structure intersecting with stable manifolds. Womoclinic structure causes the existence of infinitely many periodic points. When we consider the same parameter values but n = 5 the numerical results do not indicate chaotic motion. A case for n = 5 in which we find chaotic behavior is given by p = 0.2, k = 3.2 and ir = 4. The attractor is given by Fig. 7. The autocorrelation functions decay (Fig. 8). The one-dimensional Lyapunov exponent is given by i., = 0.28.
X Fig. 5. Phase portrait or equation (12) for n = 3. p = 0.2, k = 17, Q = 4 (chaotic motion).
Fig. 6. Attractor of equation (12) for n = 3, p = 0.2, k = 17, L? = 4 (chaotic motion).
Limit cycle systems
355
Our numerical studies in the range 0 c p c 1,O < R < 10, and 0 < k < 20 indicate that we can find chaotic behavior for n = 3 and n = 5. Obviously, we can also find chaotic motion for n = 7, 9 and so on. For n = 1 (Van der Pol oscillator with a periodic external perturbation) we do not find chaotic behavior. This result has also been found by Ueda
c31.
For n = 3, p = 0.2, R = 4 we have studied the range 10 < k < 20 in more details. Figure 9 shows the behavior in dependence of the amplitude k. The dots indicate regular behavior and the crosses indicate chaotic behavior. Consider now equation (13). Here we study the range Ip((< 1. In Fig. 10 we have plotted
L + ++
++
c+ :+* l+
YI
l l
f,
+
+
c
+t++
=++++++ + ++ c‘,’ +C++++ + +
-I + i
B +; ++
‘++
+ t
*+:
4
+
++
+
+++ +
++ +
l+++
l
t++++ +&~++++ t_ +; +q*+ +
l+
+++++
c
*
&
f’ +
+++b ++ + + + + ++ + +++ + *+ +++++ + +
+
l
++ +*+++ ++++++ ++ +++ ++ +++++ + ++ *+:+ * ; =; ++++++ett++++++* +++ +++++ + ++-*+++ +*+ + +
+
‘+
+ f *+ +; ;: ++*. + *+*c’ + + ++* L++ * *
l ++ +
+
+*++
+i :
+ +* +‘:
+
l
*+ *
.+ tt+:
.++++
+
X Fig. 7. Attractor
Fig. 8. Autocorrelation
of equation
(12) for n = 5, p = 0.2, k = 3.2, R = 4 (chaotic
motion).
function C, of equation (12) for n = 5, p = 0.2, k = 3.2, R = 4 (chaotic motion).
356
W.-H.
STEEBand
A. KUNICK
10
15
20
15
15.5
16
15.05
15.1
15.015
15.020
15.01G5
t 15.OI50
K
f K
7
15
K
L
15.010
15.ouo
K
L
K
Fig. 9. Behavior of equation (12) for II = 3. p = 0.2. R = 4. 10 < k Q 20
X Fig. 10. Phase portrait of equation (13) for p = 0.5. k = 0.532. n = 0.8.
the phase portrait for p = 0.5, k = 0.532, and R = 0.8. Figure 11 shows the phase portrait for p = -0.5, k = 0.532, and R = 0.8. The autocorrelation functions do not decay. Figures 12 and 13 show the phase portraits for k = 17 and R = 4 where in Fig. 12 the damping coefficient is ~1= 0.2 and in Fig. 13 p = -0.2. Here, too, we do not have chaotic motion. From further numerical studies we infer that there is no chaotic motion in the range IA < 1. 4. SINGULAR
POINT
ANALYSIS
is desirable to have a simple approach for deciding whether a dynamical system is integrable or not. In classical mechanics [i.e. for ordinary differential equations (odes)] the It
Limit cycle systems
Fig. 11. Phase portrait for equation (13) for p = -0.5, k = 0.532, SZ= 0.8.
Fig. 12. Phase portrait of equation (13) for p = 0.2, k = 17, Q = 4.
so-called singular point analysis (sometimes called Painleve test) can serve to decide whether a dynamical system is algebraically integrable. Notice, however, that the Painleve test fails if the first integrals are transcendental functions. The structure of the singularities of the solutions of the ode is studied in the complex plane [22]. We look for representations with so-called psi series. In particular we are interested in finding out whether a given ode has the so-called Painleve property. An ode (or a system of odes) in the complex domain is said to be of Painleve type (or has the PainlevC property) if the only movable singularities of all its solutions are poles. This means that there be no movable branch points or movable essential singularities.
358
W.-H.
Fig. 13. Phase portrait
STEEB
of equation
and A. KL’NICK
(13) for ;i = -0.2.
k = 17, R = 4
A necessary condition for an nth order ode of the form @w/d? = F(r, w,. . . , d”- ‘w,‘dz”- ‘f
(17)
where F is rational in w,. . . , d”- rw/dzn- ’ and analytic in z, to have Painlevi property is that there is a Laurent expansion
w(t) = (z -
ZJk f aiz - Zlv’ j=O
with (n - 1) arbitrary expansion coefficients (besides the pole position which is arbitrary). Notice that more than one branch may arise. Each branch must be a Laurent series. We find that none of the differential equations given in Section 1 satisfy this necessary condition. In some cases it is useful to extend the Painlevt property to the so-called quasi Painleve property, since if this property is fulfilled we can also find integrable systems. The autonomous system dx/dt = g(x) where g is rational in x1,. . . ,x, is said to be of quasi Painlevi type if it admits only expansions of the form
Wi(.z)= (z -
z*)l” f a,{2 - Zlv‘
(19)
j=O
where (n - 1) expansion
coefficients are arbitrary
and the exponents
ki are given by
ki = m/n (a, tt~E Z\(O)).
We find that equation (5) has the quasi-Painieve property. This is due to the fact that this equation can be solved by quadrature introducing polar coordinates. For all other equations this property is not fulfilled. Nevertheless we may well ask whether there are a psi series. Psi series for systems which show chaotic behavior have been studied by various authors [23-25f. For exampIe, for the Lorenz model [23] and the Duffing equation [25] so-called logarithmic psi series arise. Let us study now the Van der Pol equation with a periodic external perturbation. We
359
Limit cycle systems
consider this equation in the complex domain, i.e. d2wJdz’ - ~(1 - w2)dw/dz + w = kcos(Qz,)cos(R(z - zr)) - ksin@z,)sin(Q(z - zr)).
(20)
The singular point analysis has been widely described in the literature [22-263. Thus we give only the results. We obtain the psi series W(Z)
=
(Z -
Z1)-1’2
~
(Uj(Z
-
Z*y’
+
bj(Z
-
(21)
Z1~+3’2).
j=O
where b. can be chosen arbitrarily and the first few coefficients are given by a, = Pao/2,
u; = 3/(2~), b, = -(1/7)(pb,
(22)
- 2kcos(Rz,)).
When we consider equation (6) with n = 3 in the complex domain, we obtain w(z) = (z - zl)-
“’ f (aj(z - z$ +
z#+~‘~)
bj(z -
j=O
(23)
where b. can be chosen arbitrarily and a; = 3/(2~), _ a 1 = ~a,/2 + d, b, = -(1/7)(pb,
+ 6a;b, - Zkcos(Rz,)).
(24)
In order to decide for which parameter values chaotic behavior occurs we have to study the location of the singularities in the complex plane. Such a numerical study has been performed by Tabor and Weiss [23] for the Lorenz mode1 and for the non-linear Langevin equation by Frisch and Morf [27]. Let us study system (6) (n = 3) considered in the complex plane. Putting dw/dz = u and u = Rz, we obtain dw/dz = v dv/dz = p( 1 - w2)v - w3 + kcosu du/dz = R.
(25)
Now we set z = t + is, w(z) = w,(t,r) + iw,(c,r) and so on. Since dw,/dt = dw,/dr, aw,/at = -dw,/dr and so on (Cauchy-Riemann condition) we find
aw,/at = vl, ah/at = ~h(i
h,/at
= v2
- W: + W;) +
av,iat = Au2(1 - w: + w:) -
ih,/at = R,
2U2WlW2)
-
W,(W:
2vlw,w,)
-
W#Wf
-
3~;)
-
w:)
+
kcosP,coshu2
-
ksinu,sinhu,
h,fat = 0
(264
and
awl/a7 = -v2,
aw2jd7 = v1
a07
= -pb2(l
%/dr
= ~(Vr(i - W; + W:) +
au,/a7 = 0,
- 4 + 4) - Zv,w,w,) + c?u,/dr = i-2.
2V2WlW2)
-
W2(3Wf
W,(W;
-
-
3~:)
w$)
+
+
ksinwrsinhu,
kcosu,coshu,
CW
36a
W.-f-f.STEEB and A. KUPUICK
We have determined numerically the pattern of singularities in the complex plane nearest to the real t axis. If we choose the parameter values in the chaotic regime we have an irregular pattern. If we choose the parameter values in the region where there is no chaos we have a regufar pattern (compare also Tabor and Weiss [23] and Frisch and Morf f27f). Consider the equation f = -x3 (or .t = I’, j = -x3). Then we know that the solution is given by an elliptic function, namely x(c) = Ccn(C(r - co), l/a, where C and t, are the constants of integration. The elliptic functions are periodic functions. Consider now the equation given above in the complex time pIane (z = t + is), i.e. d’w/d? = --v’. Again the solution is given by the elliptic function en but with a complex argument. fts singularities are simple poles of order one and are distributed doubly periodically in the whole complex z-plane. This regular distribution of singularities (in the present case poles) reflects faithfully on periodicity of the solutions. Taking into account the damping term and the external perturbation the singularities are redistributed in the complex plane. The real solution is mainly determined by the singularities located nearest to the real axis, As mentioned above we find an irregular pattern of these singularities when the system shows chaotic behaviour. 5. CONCLUSlON
Our numerical analysis indicates that equation (6) does not show chaotic behavior for I( < 1 and n = 1. For n = 3, 5,. +. we can find chaotic motion for FL< 1. The different concepts for characterising chaotic motion (auto~orrelation functions, Poincare map, Lyapunov exponents) coincide. For equation (7) we do not find chaotic behavior for igi < t. If we consider the equation d2s,dt2 $ /tsin(ds/dr) -I- X” = kcos(f&)
(27)
fn = 3,5, __.f then also chaotic motion can arise for [pi < 1. Thus the non-linearity X” (n = 3,5,. . .f is responsible for the occurrence of chaotic motion for {jr1-z I. It is worthwhile to compare our results given above with those for the Duffing equation which is given by
where p > 0, c 3 0. This equation has been studied by various authors (compare Steeb e? al. [28] and references therein). As described above we can write equation (26) as dx/dt = y1
dl:idt = -/.L~L’ - bs - c.y3 + kcos(Qr).
Equation (29) is invariant under t -, r + 2ntt,Q (n E Z). The di~~omorphism _&(; =; (it7 tt, c, k, f2) defined by Ueda [3,2IJ is of the contracting type in the given case, whereas for equation (6) it is not of the contracting type. This is due to the fact that the equation (12) is self-oscillatory, whereas the Duffing equation is quiescent unless an external (periodic) force is applied. When we perform a singuiar point anafysis for equation (29) considered in the complex domain we obtain a logarithmic psi series
where the expansion coefficient n,, can be chosen arbitrarily, Thus also from the singular point analysis the behavior is quite different. In the present case we have logarithmic branch points, whereas for equation (6) (n = 1 or II = 3) considered in the complex domain we have algebraic branch points. For equation (29) (considered in the complex domain) we also find from numerical studies in the complex plane a regular pattern of the singularities when parameter vatues are studied where there is no chaotic motion For parameter values with chaotic behavior of the dynamical system (29) the pattern of the singularities is irregular.
Limit cycle systems
361
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