- Email: [email protected]
Evidence for homoclinic orbits as a precursor to chaos in a magnetic pendulum
Physica 24D (1987) 383-390 North-Holland, Amsterdam
EVIDENCE FOR HOMOCLINIC ORBITS AS A PRECURSOR TO CHAOS IN A MAGNETIC PENDULUM* F.C. M O O N , J. C U S U M A N O and P.J. H O L M E S Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, New York, USA
Received 15 October 1985 Revised manuscript received 11 July 1986
Experimental evidence is presented which supports the theory that homoclinic orbits in a Poincar6 map associated with a phase space flow are precursors of chaotic motion. A permanent magnet rotor in crossed steady and time-varyingmagnetic fields is shown to satisfy a set of third order differential equations analogous to the forced pendulum or to a particle in a combined periodic and traveling wave force fidd. Critical values of magnetic torque and forcing frequency are measured for chaotic oscillations of the rotor and are found to be consistent with a lower bound for the existence of homoclinic orbits derived by the method of Melnikov. The fractal nature of the strange attractor is revealed by a Poincar6 map triggered by the angular position of the rotor. Numerical simulations using the model also agree well with both theoretical and experimental criteria for chaos.
1. ~ u ~ o n The existence of homoclinic orbits in Poincar6 m a p s of various equations of dynamical systems has been shown to be a precursor for horseshoe m a p s and subharmonic motions [1, 2]. The method of Melnikov has been used to develop a criterion for the existence of homoclinic orbits [2, 3]. This criterion is believed to provide a lower bound in the p a r a m e t e r space for the region of chaotic behavior. This has been demonstrated in numerical experiments on the two-well potential problem [4] as well as in physical experiments on the chaotic vibrations of a buckled beam [5]. The existence of homoclinic orbits has also been shown to provide an indicator of fractal basin boundary properties in the initial condition space [4]. (See also M c D o n a l d et al. [6] for a discussion of fractal basin boundaries.) ....
*Supported in part by grants from the Air Force Office of Scientific Research, Mathematical and Aerospace Sciences Divisions and the National Science Foundation, Mechanical Systems Program, Engineering Directorate.
The motion of a particle in both space periodic and time periodic force fields has served as a model for several problems in physics. These problems include the classical pendulum, a charged particle in a moving electric force field, synchronous rotors and the Josephson junction. The general f o r m of the equation of motion common to m a n y of these studies is given by
+
+ a[1 + f($2t)] sinx
= ,8 g(k
- ,,,t) + & h ( k x
+ ,,,t) + '!,
(1)
where f ( ) , g ( ) , h ( ) are periodic functions. The nonlinear dynamics of a particle in a travelling electric force field ( f ( ) = 0, f12 = 7 / = 0) have been studied b y Zaslavskii and Chirikov [7]. Experiments on chaotic motions of a magnetic rotor in a rotating magnetic field (y = a = 7 / = 0) were p e r f o r m e d b y Croquette and Poitou [8]. The dynamics of a Josephson junction ( f ( ) = k = 0 , fll = f12) have been studied by Odyniec and Chua [9] and Salam and Sastry [10] and the equivalent p r o b l e m for a forced pendulum has received study b y Hockett and Holmes [11] and Gwinn and
0167-2789/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
384
F.C. Moon et aL / Homoclinic orbits as a precursor to chaos
Westervelt [12] who reported on fractal basin boundaries for this problem. Also Koch and Levin [13] and McLaughlin [14] have studied the parametric excitation of the pendulum equation ( , / =
fll = f12 = 0). In this paper we present experimental and analytical data on a problem related to (1) of the form
initial conditions. Horseshoes occur in all the known candidates for strange attractors including the Lorenz system and the Htnon map [3]. The effect of homoclinic orbits of the type detected in the present analysis, is to cause oscillatory (librating) and running (rotating) motions of the rotor to mix in a chaotic fashion (cf. [12]).
O"+ 70 + sin 0
2. Brief review of analysis
= ½ft [cos (0 - tot) + cos(0 + to/)] +f0 = fl cos 0 cos o~t + f0,
(2)
where 0 will represent the rotation of a magnetic dipole magnet in both static and time-periodic magnetic fields about an axis orthogonal to the magnetic moment vector. Like the two-well potential problem, the undamped, unforced pendulum has a saddle point in the phase plane whose separatrices intersect in a dosed loop. Thus for small damping and forcing, one expects a saddle point to exist in the Poincar6 map based on periodic time sampling. (See refs. [2, 3] for details.) The existence of this saddle point leads one to explore the role of homoclinic orbits (which occur when the separatrices intersect) as harbingers of chaotic dynamics. In this paper we present experimental and numerical evidence for the thesis that homoclinic orbits in the Poincar6 map of forced motion in a periodic potential field are a precursor for chaotic motions. In addition, we derive a sufficient condition for the existence of such homoclinic orbits in the solution of (2) using a method due to Melnikov. The method has been used previously by the third author to derive a precursor criterion for chaos in the two-well potential strange attractor [1, 2]. This theory is based on the theorem (see Guckenheimer and Holmes [3]) that the presence of homoclinic orbits in the Poincar6 map implies that the map has a Smale horseshoe. A horseshoe map contains a countable infinity of unstable periodic orbits, an uncountable set of bounded nonperiodic orbits and a dense orbit. The existence of horseshoes also implies a sensitive dependence on
For the Melnikov analysis we assume that "¢, fo, fl in (2) are all small; specifically "/= ~7, fo = ' f 0 , f l = ' f l , where 0 < , << 1 and 7, f0, fl are of order one. The Hamiltonian for the undamped, unforced problem is given by H = ½02 + (1 - cos 8), and H is constant ( H = 2) on the homoclinic (saddle separatrix) orbit emanating from the saddle point (8 = v = 0) in the phase plane (fig. 1). When the system is perturbed, the equations take the form
OH
OH
d=~+~gl,
0=-~+Eg2,
(3)
where c << 1 and gx, g2 are periodic time-depen, dent perturbation functions. The flow takes place in a three-dimensional phase space (8, v, tot) where the phase ~t of the periodic force is taken modulo 2~r. Using a Poincar6 map synchronized with the
Ws I
Fig. 1. Sketch of stable and unstable manifolds of the Poincar6 map.
F.C. Moon et aL/ Hornoclinicorbitsas a precursor to chaos phase ~0t, it has been shown [2, 14] that the Poincar6 map of the suspended, three-dimensional flow has a saddle point and stable and unstable manifolds (separatrices) W ~ , W ~, close to the original Hamiltonian homoclinic orbit (fig. 1). In general the Poincar6 map manifolds W ~, W u may not intersect. However, when they intersect transversely a theorem of Poincar6 implies that they do so an infinite number of times which leads to the horseshoe map properties described above. The Melnikov function provides a measure of the separation between W s, W u as a function of the distance along the unperturbed homoclinic orbit (this distance is parameterized by the phase of the Poincar6 map ~to). This function can be shown to be given by (see [3]), .Xf(to) =
oog(Oo,v o , t + t o ) ' W H ( O o , vo)dt. (4)
385
(4). We remark that simple zeros in ..¢¢(t0) imply that d(to) changes sign with non-zero speed as t o varies and hence that the manifolds W s, W u intersect transversely. The time dependence of g, leading to the "phase" t o in ..g, is crucial h e r e - t h e unperturbed homoclinic orbits are not transverse and no chaos can occur in that case. For the present system, gl = 0, g2 = - ~ / v + f o + f l c o s O c o s O t and 0 o = 2 t a n - l ( s i n h t ) , v0= 2 sech t is a solution on the unperturbed homoclinic orbit. Evaluation of (4) yields = - 87 + 2*rf0 + 2*rfl~02 sech (eros/2) cos 60t0. (7) (The details leading from (4) to (5) are similar to those found in [3].) ~ has a simple zero when the following inequality is satisfied:
4V f l > --;- - fo
cosh (*ro~/2) ~2
(8)
This formula is obtained by writing the distance d(to) = c~¢l(t0) + 0(¢ 2) between W u and W s, projected onto the normal to the unperturbed orbit, as the scalar
where we have cancelled the c factors. In our experiments fo = 0, so that the critical value for the forcing torque is given by
d(t, to) = (xU(t, to) - xS(t, to) ) " wH(00, Vo)
flc-
(5) (fig. 1). Here x u's = (0)u,s denote solutions in the -0 / unstable and stable manifolds respectively. We expand x u's in power series in c, the evolution of the first order terms being governed by the variational equation 02H OOOv 02H
002
OZH Ov 2 0 2H
4~/ c o s h ( - ~ ) . ,/'g0~ 2
(9)
3. Equation of motion for a magnetic pendulum While the gravitational pendulum is a natural system to test the homoclinic orbit criterion for
,0o
OOOO
[gl(Oo, v0' t-- to) + [g2(0o, V0' t - t0)
(6)
where (0o, v0) is the solution of the unperturbed homoclinic orbit. A short computation yields eq.
Fig. 2. Sketch of magnetic dipole in crossed steady and time periodic magnetic fields.
386
F.C. Moon et al./ Homoclinic orbits as a precursor to chaos
the periodic potential, we found it more convenient to use a dipole rotor in crossed steady and oscillating magnetic fields. The specific technical details of the experiment are described below. In this section we derive the equation of motion for the rotary motion of the dipole. Referring to fig. 2, we assume that a permanent magnetic dipole of strength M is free to rotate about an axis orthogonal to M in a uniform magnetic field. The vertical field component is assumed to be constant, B~, while the horizontal field is assumed to be harmonic in time, B0 cos 12t. (We neglect the small gradient in B d due to the time rate of change of the field.) The magnetic torque in the rotor is given by M × B. If we assume that there is some linear viscous damping in the bearings of the rotor, the equation for the angular rotation O(t) is
given by J # + c~ + M B s sin 0 = M B d cos 0 cos ~2t,
where J is the second moment of mass of the rotor about the axis of rotation and c is the damping constant. To nondimensionalize this equation, we choose the reciprocal of the small angle natural frequency ( M B J J ) - I / 2 as the unit of time. The equation then takes the form (2) analyzed above (with fo = 0): O"+ ",/t~+ sin 0 = f l cos 0 cos oat.
TRIGGER
TAiOMETER
ER 0",
RAMP
GENERATOR
l
) Y SLIT IN
/-I SINELLATOR OSCI
(11)
Here "t, o~ are nondimensional damping and frequency constants, respectively, and f l =- B o / B s . In the actual experiment the coils that create Bd, B S are identical. Thus the field ratio in (11) is
DIGITAL OSCI LLOSCOPE
bJ
(10)
MOTOR WITHCOILS FOURFIELD
Fig. 3. Diagram of experimentalsetup to measure position triggered Poincar6 maps.
F.C. Moon et a l . / Homoclinic orbits as a precursor to chaos
equal to the ratio of currents in the coils, i.e., fl = Id/Is" Finally, we assume that the frequency is low enough that inductive effects in the coil are small and that the back emf in the coils from the rotor is also small so that we can replace the current ratio with the voltage ratio across the coils.
387
velocity and the phase of the driving signal, modulo 2¢r, was then stored and plotted on a digital oscilloscope. The driving phase was measured by synchronizing a periodic/linear ramp signal with the harmonic drive voltage. The amplitude of the ramp signal was then linearly proportional to the phase o~t (fig. 3). A linear tachometer was used to measure the angular velocity of the rotor.
4. Description of experiment 5. Experimental results The magnetic pendulum consisted of a stepper motor permanent magnet rotor with four stator coils ¢r/2 radians apart. One pair of coils was given a direct current bias to produce a steady magnetic field while the remaining pair was connected to a harmonic current source. With the dc voltage alone, the measured torque versus angle was approximately modeled by a sinusoidal function of angle. There were small cogging torques due to the ferromagnetic stator poles, but it is believed that their effect was not significant. The measured rotor inertia was 74 g cm 2. With a bias current of 2 V, the natural frequency of the rotor was 8.7 Hz and the nondimensional damping was 7--0.5. The damping was estimated by measuring the amplitude decay envelope of the unforced system for small displacements. Care was taken to warm the rotor bearings by running it prior to obtaining the damping data. The L / R time constant for one pair of coils was 2.2 ms. The driving frequency ranged from 4-20 Hz. In some experiments the frequency was fixed while the driving amplitude was raised and in other measurements the driving amplitude was fixed and the frequency increased. The goal of these tests was to determine the region in the f - ~ parameter space where the motion changed from periodic to chaotic. The criterion used was to observe Poincar6 maps of the motion and look for a fractal-like attractor. In these experiments it was more convenient to obtain an unconventional Poincar6 map triggered when the rotor crossed O = 0 (fig. 3). The angular
Periodic and chaotic time histories of the rotor velocity are shown in fig. 4a, b. A Poincar6 map triggered by the position O -- 0 is shown in figs. 5,
.I///////A/////A//////A///// -IVVVVV/VVVVV/VVVVVV/VVVV
AAAAAAAAAAAAAAAAA. VVVVVVVVVVVVVVVVVV Fig. 4. Velocity history of periodic and chaotic motions of magnetic rotor.
388
F.C. Moon et aL / Homoclinic orbits as a precursor to chaos
, :!. .
"
•k~
.,
.:... ':
.
,,, . .
. •
..-
•
.'"
. <"
% ,'~.,
.,¢.
~'~!~ ~
"
" ",":., "'i'.
• :.~ " ,
k~', .~' ..' (I-L. " ::, ~,.:
,,
I ~ ;
~,.
"
';"k
,~" ~,~
" . ~ , : . ' .:. ~.
"
~,~
~" ~
Fig. 5. Poincar6 map of chaotic motion; velocity vs. phase of forcing torque.
Fig. 6. Polar representation of Poincar~ map in fig. 5.
20
r I nJ
~t exp. data, f increasing Irun 1 I o exp. data, f increasing [run 2] ~ exp. data, ¢= increasing Irun 11
/I(~CHAOTI C O ~ / .-
g exp. data, ~ increasing [run 21
15
77
IU
Y
i
=E 10 0
Z 0
\
/
CHAOTIC
/
o tL
11¢=0.51 I
0.5
I
1.0 FREQUENCY
I
1.5 [w]
I
2.0
Fig. 7. Comparison of Melnikov criterion with experimentally measured regions of parameter space.
,
I
2.5
389
F.C. Moon et a l . / Homoclinic orbits as a precursor to chaos
6. In fig. 5 we show the cylindrical phase space Poincar6 map. Note that the right and left edges match as required by theory. The small portion missing from the edges of the Poincar6 map are due to a 10 ° fiat section at the beginning of each periodic ramp cycle used to measure the phase tot. This fiat section of the ramp function also caused Poincar6 points in the first 10 ° phase to be mapped on one vertical line on the left edge in fig. 5. This presentation of the Poincar6 map is not convenient if one were to vary the Poincar6 plane 0 = O0. In fig. 6 we have mapped the cylinder in fig. 5 onto a cross-section of a thick torus. In this diagram the velocity is linear in the radius and the phase ~0t runs in the circumferential direction. If many 0 Poincar6 sections were obtained, the attractor would lie in a thick torus where 0 would represent the azimuthal torus direction. Finally, in fig. 7 we show the driving torque-frequency plane and the measured boundaries of the chaotic
parameter basins. Some points represent increasing amplitude at fixed frequency while for others the frequency was increased at a fixed amplitude. The values of nondimensional torque f in (11) were found by taking the ratio of ac to dc voltage amplitudes on the two crossed field coils. The frequency data are normalized by the small amplitude natural frequency of 8.7 Hz.
6. Comparison of experiment, theory and simulation The experimental data in fig. 7 shows a very good agreement between the criterion for homoclinic orbits using Melnikov theory and the experimental values of torque and frequency at which strange attractors occurred. It appears that for moderate damping 3' - 0.5, the Melnikov formula (9) not only is a lower bound, but could serve as a
20
I /
/
,,-
// //
15
U.l l..i a. =E 1 0
o R-K s i m u l a t i o n , f o R-K s i m u l a t i o n ,
boundaries for exp. pts.
//
C3 m
~ / ~ CHAOTIC
<
/
z
unO IJ.
increasing
~ increasing
PERIODIC
/
/
,
/
5
CHAOTIC / ~ j PERIODIC
MELNIKOV
MET HOD
O0
I
0.5
I
1.0 FREQUENCY
I
1.5 (w]
I
[~=O.S]
2 0
Fig. 8. Comparisonof Melnikovcriterion with data from Runge-Kutta numericalsimulation.
I
2.5
390
F.C. M o o n et a l . / Homoclinic orbits as a precursor to chaos
practical criterion for the critical torque for chaos. The observed regions of chaos appeared to lie in two peninsulas in the f-to plane. These fingers of chaotic motion, however, are believed to contain periodic motions as well. While we have drawn a smooth b o u n d a r y connecting the experimental points, we have no guarantee that this boundary is actually smooth; it may in fact be fractal, as has been observed in other experiments [15]. There m a y also be further chaotic regions in the f-~o plane, especially in the upper right hand comer, but we did not have the voltage available to explore this regime. In addition to the experiments, fourth order R u n g e - K u t t a numerical integrations of eq. (2) were performed. Poincar6 maps based on the time or phase ~0t showed a similar structure to the experimental maps based on rotary position O. Finally, critical values of the forcing torque amplitude f versus frequency were obtained (fig. 8) and the numerically simulated data showed good agreement with both theory and experiment. The good agreement with the experimental data is important because it justifies the neglect of small cogging torques in the motor in deriving eq. (2). The successful application of the Melnikov method to both the magnetic pendulum and the buckled elastica [1, 5] in predicting a lower bound
on the chaotic regime suggests that use of the homoclinic orbit criterion be explored in other low-dimensional chaotic systems.
References [1] P.J. Holmes, Phil Trans. Roy. Soc. 292 (1979) 419. [2] B.D. Greenspan and P.J. Holmes, Nonlinear Dynamics and Turbulence. G. Barenblatt, G. Iooss and D.D. Joseph, eds (Pitman, London, 1983), chap. 10. [3] J. Guckenheimer and P.J. Holmes,Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, Berlin, 1983). [4] F.C. Moon and G-X Li, Phys. Rev. Lett. 55 (1985) 1439. [5] F.C. Moon, J. Appl. Mech. 47 (1980) 638. [6] S.W. McDonald, C. Grebogi, E. Ott and J.A. Yorke, Physica 17D (1985) 125. [7] G.M. Zaslavskii and B.V. Chirikov, Sov. Phys. Uspekhi 14 (1972) 549. [8] V. Croquette and C. Poitou, J. Phys. Lett. 42 (1981) L-537. [9] M. Odyniec and L.O. Chua, IEEE Trans. Circuits and Systems CAS-30 (1983) 308; CAS-32 (1985) 34. [10] F.M.A. Salam and S.S. Sastry, IEEE Trans. on Circuits and Systems, to appear. [11] K.G. Hockett and P.J. Holmes, Ergod. Th. Dynam. Sys. 6 (1986) 205. [12] E.G. Gwinn and R.M. Westervelt, Phys. Rev. Lett. 54 (1985) 1613. [13] B.P. Koch and R.W. Leven, Physica 16D (1985) 1. [14] J.B. McLaughlin, J. Stat. Phys. 24 (1981) 375. [15] F.C. Moon, Phys. Rev. Lett. 53 (1984) 962.
EVIDENCE FOR HOMOCLINIC ORBITS AS A PRECURSOR TO CHAOS IN A MAGNETIC PENDULUM* F.C. M O O N , J. C U S U M A N O and P.J. H O L M E S Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, New York, USA
Received 15 October 1985 Revised manuscript received 11 July 1986
Experimental evidence is presented which supports the theory that homoclinic orbits in a Poincar6 map associated with a phase space flow are precursors of chaotic motion. A permanent magnet rotor in crossed steady and time-varyingmagnetic fields is shown to satisfy a set of third order differential equations analogous to the forced pendulum or to a particle in a combined periodic and traveling wave force fidd. Critical values of magnetic torque and forcing frequency are measured for chaotic oscillations of the rotor and are found to be consistent with a lower bound for the existence of homoclinic orbits derived by the method of Melnikov. The fractal nature of the strange attractor is revealed by a Poincar6 map triggered by the angular position of the rotor. Numerical simulations using the model also agree well with both theoretical and experimental criteria for chaos.
1. ~ u ~ o n The existence of homoclinic orbits in Poincar6 m a p s of various equations of dynamical systems has been shown to be a precursor for horseshoe m a p s and subharmonic motions [1, 2]. The method of Melnikov has been used to develop a criterion for the existence of homoclinic orbits [2, 3]. This criterion is believed to provide a lower bound in the p a r a m e t e r space for the region of chaotic behavior. This has been demonstrated in numerical experiments on the two-well potential problem [4] as well as in physical experiments on the chaotic vibrations of a buckled beam [5]. The existence of homoclinic orbits has also been shown to provide an indicator of fractal basin boundary properties in the initial condition space [4]. (See also M c D o n a l d et al. [6] for a discussion of fractal basin boundaries.) ....
*Supported in part by grants from the Air Force Office of Scientific Research, Mathematical and Aerospace Sciences Divisions and the National Science Foundation, Mechanical Systems Program, Engineering Directorate.
The motion of a particle in both space periodic and time periodic force fields has served as a model for several problems in physics. These problems include the classical pendulum, a charged particle in a moving electric force field, synchronous rotors and the Josephson junction. The general f o r m of the equation of motion common to m a n y of these studies is given by
+
+ a[1 + f($2t)] sinx
= ,8 g(k
- ,,,t) + & h ( k x
+ ,,,t) + '!,
(1)
where f ( ) , g ( ) , h ( ) are periodic functions. The nonlinear dynamics of a particle in a travelling electric force field ( f ( ) = 0, f12 = 7 / = 0) have been studied b y Zaslavskii and Chirikov [7]. Experiments on chaotic motions of a magnetic rotor in a rotating magnetic field (y = a = 7 / = 0) were p e r f o r m e d b y Croquette and Poitou [8]. The dynamics of a Josephson junction ( f ( ) = k = 0 , fll = f12) have been studied by Odyniec and Chua [9] and Salam and Sastry [10] and the equivalent p r o b l e m for a forced pendulum has received study b y Hockett and Holmes [11] and Gwinn and
0167-2789/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
384
F.C. Moon et aL / Homoclinic orbits as a precursor to chaos
Westervelt [12] who reported on fractal basin boundaries for this problem. Also Koch and Levin [13] and McLaughlin [14] have studied the parametric excitation of the pendulum equation ( , / =
fll = f12 = 0). In this paper we present experimental and analytical data on a problem related to (1) of the form
initial conditions. Horseshoes occur in all the known candidates for strange attractors including the Lorenz system and the Htnon map [3]. The effect of homoclinic orbits of the type detected in the present analysis, is to cause oscillatory (librating) and running (rotating) motions of the rotor to mix in a chaotic fashion (cf. [12]).
O"+ 70 + sin 0
2. Brief review of analysis
= ½ft [cos (0 - tot) + cos(0 + to/)] +f0 = fl cos 0 cos o~t + f0,
(2)
where 0 will represent the rotation of a magnetic dipole magnet in both static and time-periodic magnetic fields about an axis orthogonal to the magnetic moment vector. Like the two-well potential problem, the undamped, unforced pendulum has a saddle point in the phase plane whose separatrices intersect in a dosed loop. Thus for small damping and forcing, one expects a saddle point to exist in the Poincar6 map based on periodic time sampling. (See refs. [2, 3] for details.) The existence of this saddle point leads one to explore the role of homoclinic orbits (which occur when the separatrices intersect) as harbingers of chaotic dynamics. In this paper we present experimental and numerical evidence for the thesis that homoclinic orbits in the Poincar6 map of forced motion in a periodic potential field are a precursor for chaotic motions. In addition, we derive a sufficient condition for the existence of such homoclinic orbits in the solution of (2) using a method due to Melnikov. The method has been used previously by the third author to derive a precursor criterion for chaos in the two-well potential strange attractor [1, 2]. This theory is based on the theorem (see Guckenheimer and Holmes [3]) that the presence of homoclinic orbits in the Poincar6 map implies that the map has a Smale horseshoe. A horseshoe map contains a countable infinity of unstable periodic orbits, an uncountable set of bounded nonperiodic orbits and a dense orbit. The existence of horseshoes also implies a sensitive dependence on
For the Melnikov analysis we assume that "¢, fo, fl in (2) are all small; specifically "/= ~7, fo = ' f 0 , f l = ' f l , where 0 < , << 1 and 7, f0, fl are of order one. The Hamiltonian for the undamped, unforced problem is given by H = ½02 + (1 - cos 8), and H is constant ( H = 2) on the homoclinic (saddle separatrix) orbit emanating from the saddle point (8 = v = 0) in the phase plane (fig. 1). When the system is perturbed, the equations take the form
OH
OH
d=~+~gl,
0=-~+Eg2,
(3)
where c << 1 and gx, g2 are periodic time-depen, dent perturbation functions. The flow takes place in a three-dimensional phase space (8, v, tot) where the phase ~t of the periodic force is taken modulo 2~r. Using a Poincar6 map synchronized with the
Ws I
Fig. 1. Sketch of stable and unstable manifolds of the Poincar6 map.
F.C. Moon et aL/ Hornoclinicorbitsas a precursor to chaos phase ~0t, it has been shown [2, 14] that the Poincar6 map of the suspended, three-dimensional flow has a saddle point and stable and unstable manifolds (separatrices) W ~ , W ~, close to the original Hamiltonian homoclinic orbit (fig. 1). In general the Poincar6 map manifolds W ~, W u may not intersect. However, when they intersect transversely a theorem of Poincar6 implies that they do so an infinite number of times which leads to the horseshoe map properties described above. The Melnikov function provides a measure of the separation between W s, W u as a function of the distance along the unperturbed homoclinic orbit (this distance is parameterized by the phase of the Poincar6 map ~to). This function can be shown to be given by (see [3]), .Xf(to) =
oog(Oo,v o , t + t o ) ' W H ( O o , vo)dt. (4)
385
(4). We remark that simple zeros in ..¢¢(t0) imply that d(to) changes sign with non-zero speed as t o varies and hence that the manifolds W s, W u intersect transversely. The time dependence of g, leading to the "phase" t o in ..g, is crucial h e r e - t h e unperturbed homoclinic orbits are not transverse and no chaos can occur in that case. For the present system, gl = 0, g2 = - ~ / v + f o + f l c o s O c o s O t and 0 o = 2 t a n - l ( s i n h t ) , v0= 2 sech t is a solution on the unperturbed homoclinic orbit. Evaluation of (4) yields = - 87 + 2*rf0 + 2*rfl~02 sech (eros/2) cos 60t0. (7) (The details leading from (4) to (5) are similar to those found in [3].) ~ has a simple zero when the following inequality is satisfied:
4V f l > --;- - fo
cosh (*ro~/2) ~2
(8)
This formula is obtained by writing the distance d(to) = c~¢l(t0) + 0(¢ 2) between W u and W s, projected onto the normal to the unperturbed orbit, as the scalar
where we have cancelled the c factors. In our experiments fo = 0, so that the critical value for the forcing torque is given by
d(t, to) = (xU(t, to) - xS(t, to) ) " wH(00, Vo)
flc-
(5) (fig. 1). Here x u's = (0)u,s denote solutions in the -0 / unstable and stable manifolds respectively. We expand x u's in power series in c, the evolution of the first order terms being governed by the variational equation 02H OOOv 02H
002
OZH Ov 2 0 2H
4~/ c o s h ( - ~ ) . ,/'g0~ 2
(9)
3. Equation of motion for a magnetic pendulum While the gravitational pendulum is a natural system to test the homoclinic orbit criterion for
,0o
OOOO
[gl(Oo, v0' t-- to) + [g2(0o, V0' t - t0)
(6)
where (0o, v0) is the solution of the unperturbed homoclinic orbit. A short computation yields eq.
Fig. 2. Sketch of magnetic dipole in crossed steady and time periodic magnetic fields.
386
F.C. Moon et al./ Homoclinic orbits as a precursor to chaos
the periodic potential, we found it more convenient to use a dipole rotor in crossed steady and oscillating magnetic fields. The specific technical details of the experiment are described below. In this section we derive the equation of motion for the rotary motion of the dipole. Referring to fig. 2, we assume that a permanent magnetic dipole of strength M is free to rotate about an axis orthogonal to M in a uniform magnetic field. The vertical field component is assumed to be constant, B~, while the horizontal field is assumed to be harmonic in time, B0 cos 12t. (We neglect the small gradient in B d due to the time rate of change of the field.) The magnetic torque in the rotor is given by M × B. If we assume that there is some linear viscous damping in the bearings of the rotor, the equation for the angular rotation O(t) is
given by J # + c~ + M B s sin 0 = M B d cos 0 cos ~2t,
where J is the second moment of mass of the rotor about the axis of rotation and c is the damping constant. To nondimensionalize this equation, we choose the reciprocal of the small angle natural frequency ( M B J J ) - I / 2 as the unit of time. The equation then takes the form (2) analyzed above (with fo = 0): O"+ ",/t~+ sin 0 = f l cos 0 cos oat.
TRIGGER
TAiOMETER
ER 0",
RAMP
GENERATOR
l
) Y SLIT IN
/-I SINELLATOR OSCI
(11)
Here "t, o~ are nondimensional damping and frequency constants, respectively, and f l =- B o / B s . In the actual experiment the coils that create Bd, B S are identical. Thus the field ratio in (11) is
DIGITAL OSCI LLOSCOPE
bJ
(10)
MOTOR WITHCOILS FOURFIELD
Fig. 3. Diagram of experimentalsetup to measure position triggered Poincar6 maps.
F.C. Moon et a l . / Homoclinic orbits as a precursor to chaos
equal to the ratio of currents in the coils, i.e., fl = Id/Is" Finally, we assume that the frequency is low enough that inductive effects in the coil are small and that the back emf in the coils from the rotor is also small so that we can replace the current ratio with the voltage ratio across the coils.
387
velocity and the phase of the driving signal, modulo 2¢r, was then stored and plotted on a digital oscilloscope. The driving phase was measured by synchronizing a periodic/linear ramp signal with the harmonic drive voltage. The amplitude of the ramp signal was then linearly proportional to the phase o~t (fig. 3). A linear tachometer was used to measure the angular velocity of the rotor.
4. Description of experiment 5. Experimental results The magnetic pendulum consisted of a stepper motor permanent magnet rotor with four stator coils ¢r/2 radians apart. One pair of coils was given a direct current bias to produce a steady magnetic field while the remaining pair was connected to a harmonic current source. With the dc voltage alone, the measured torque versus angle was approximately modeled by a sinusoidal function of angle. There were small cogging torques due to the ferromagnetic stator poles, but it is believed that their effect was not significant. The measured rotor inertia was 74 g cm 2. With a bias current of 2 V, the natural frequency of the rotor was 8.7 Hz and the nondimensional damping was 7--0.5. The damping was estimated by measuring the amplitude decay envelope of the unforced system for small displacements. Care was taken to warm the rotor bearings by running it prior to obtaining the damping data. The L / R time constant for one pair of coils was 2.2 ms. The driving frequency ranged from 4-20 Hz. In some experiments the frequency was fixed while the driving amplitude was raised and in other measurements the driving amplitude was fixed and the frequency increased. The goal of these tests was to determine the region in the f - ~ parameter space where the motion changed from periodic to chaotic. The criterion used was to observe Poincar6 maps of the motion and look for a fractal-like attractor. In these experiments it was more convenient to obtain an unconventional Poincar6 map triggered when the rotor crossed O = 0 (fig. 3). The angular
Periodic and chaotic time histories of the rotor velocity are shown in fig. 4a, b. A Poincar6 map triggered by the position O -- 0 is shown in figs. 5,
.I///////A/////A//////A///// -IVVVVV/VVVVV/VVVVVV/VVVV
AAAAAAAAAAAAAAAAA. VVVVVVVVVVVVVVVVVV Fig. 4. Velocity history of periodic and chaotic motions of magnetic rotor.
388
F.C. Moon et aL / Homoclinic orbits as a precursor to chaos
, :!. .
"
•k~
.,
.:... ':
.
,,, . .
. •
..-
•
.'"
. <"
% ,'~.,
.,¢.
~'~!~ ~
"
" ",":., "'i'.
• :.~ " ,
k~', .~' ..' (I-L. " ::, ~,.:
,,
I ~ ;
~,.
"
';"k
,~" ~,~
" . ~ , : . ' .:. ~.
"
~,~
~" ~
Fig. 5. Poincar6 map of chaotic motion; velocity vs. phase of forcing torque.
Fig. 6. Polar representation of Poincar~ map in fig. 5.
20
r I nJ
~t exp. data, f increasing Irun 1 I o exp. data, f increasing [run 2] ~ exp. data, ¢= increasing Irun 11
/I(~CHAOTI C O ~ / .-
g exp. data, ~ increasing [run 21
15
77
IU
Y
i
=E 10 0
Z 0
\
/
CHAOTIC
/
o tL
11¢=0.51 I
0.5
I
1.0 FREQUENCY
I
1.5 [w]
I
2.0
Fig. 7. Comparison of Melnikov criterion with experimentally measured regions of parameter space.
,
I
2.5
389
F.C. Moon et a l . / Homoclinic orbits as a precursor to chaos
6. In fig. 5 we show the cylindrical phase space Poincar6 map. Note that the right and left edges match as required by theory. The small portion missing from the edges of the Poincar6 map are due to a 10 ° fiat section at the beginning of each periodic ramp cycle used to measure the phase tot. This fiat section of the ramp function also caused Poincar6 points in the first 10 ° phase to be mapped on one vertical line on the left edge in fig. 5. This presentation of the Poincar6 map is not convenient if one were to vary the Poincar6 plane 0 = O0. In fig. 6 we have mapped the cylinder in fig. 5 onto a cross-section of a thick torus. In this diagram the velocity is linear in the radius and the phase ~0t runs in the circumferential direction. If many 0 Poincar6 sections were obtained, the attractor would lie in a thick torus where 0 would represent the azimuthal torus direction. Finally, in fig. 7 we show the driving torque-frequency plane and the measured boundaries of the chaotic
parameter basins. Some points represent increasing amplitude at fixed frequency while for others the frequency was increased at a fixed amplitude. The values of nondimensional torque f in (11) were found by taking the ratio of ac to dc voltage amplitudes on the two crossed field coils. The frequency data are normalized by the small amplitude natural frequency of 8.7 Hz.
6. Comparison of experiment, theory and simulation The experimental data in fig. 7 shows a very good agreement between the criterion for homoclinic orbits using Melnikov theory and the experimental values of torque and frequency at which strange attractors occurred. It appears that for moderate damping 3' - 0.5, the Melnikov formula (9) not only is a lower bound, but could serve as a
20
I /
/
,,-
// //
15
U.l l..i a. =E 1 0
o R-K s i m u l a t i o n , f o R-K s i m u l a t i o n ,
boundaries for exp. pts.
//
C3 m
~ / ~ CHAOTIC
<
/
z
unO IJ.
increasing
~ increasing
PERIODIC
/
/
,
/
5
CHAOTIC / ~ j PERIODIC
MELNIKOV
MET HOD
O0
I
0.5
I
1.0 FREQUENCY
I
1.5 (w]
I
[~=O.S]
2 0
Fig. 8. Comparisonof Melnikovcriterion with data from Runge-Kutta numericalsimulation.
I
2.5
390
F.C. M o o n et a l . / Homoclinic orbits as a precursor to chaos
practical criterion for the critical torque for chaos. The observed regions of chaos appeared to lie in two peninsulas in the f-to plane. These fingers of chaotic motion, however, are believed to contain periodic motions as well. While we have drawn a smooth b o u n d a r y connecting the experimental points, we have no guarantee that this boundary is actually smooth; it may in fact be fractal, as has been observed in other experiments [15]. There m a y also be further chaotic regions in the f-~o plane, especially in the upper right hand comer, but we did not have the voltage available to explore this regime. In addition to the experiments, fourth order R u n g e - K u t t a numerical integrations of eq. (2) were performed. Poincar6 maps based on the time or phase ~0t showed a similar structure to the experimental maps based on rotary position O. Finally, critical values of the forcing torque amplitude f versus frequency were obtained (fig. 8) and the numerically simulated data showed good agreement with both theory and experiment. The good agreement with the experimental data is important because it justifies the neglect of small cogging torques in the motor in deriving eq. (2). The successful application of the Melnikov method to both the magnetic pendulum and the buckled elastica [1, 5] in predicting a lower bound
on the chaotic regime suggests that use of the homoclinic orbit criterion be explored in other low-dimensional chaotic systems.
References [1] P.J. Holmes, Phil Trans. Roy. Soc. 292 (1979) 419. [2] B.D. Greenspan and P.J. Holmes, Nonlinear Dynamics and Turbulence. G. Barenblatt, G. Iooss and D.D. Joseph, eds (Pitman, London, 1983), chap. 10. [3] J. Guckenheimer and P.J. Holmes,Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, Berlin, 1983). [4] F.C. Moon and G-X Li, Phys. Rev. Lett. 55 (1985) 1439. [5] F.C. Moon, J. Appl. Mech. 47 (1980) 638. [6] S.W. McDonald, C. Grebogi, E. Ott and J.A. Yorke, Physica 17D (1985) 125. [7] G.M. Zaslavskii and B.V. Chirikov, Sov. Phys. Uspekhi 14 (1972) 549. [8] V. Croquette and C. Poitou, J. Phys. Lett. 42 (1981) L-537. [9] M. Odyniec and L.O. Chua, IEEE Trans. Circuits and Systems CAS-30 (1983) 308; CAS-32 (1985) 34. [10] F.M.A. Salam and S.S. Sastry, IEEE Trans. on Circuits and Systems, to appear. [11] K.G. Hockett and P.J. Holmes, Ergod. Th. Dynam. Sys. 6 (1986) 205. [12] E.G. Gwinn and R.M. Westervelt, Phys. Rev. Lett. 54 (1985) 1613. [13] B.P. Koch and R.W. Leven, Physica 16D (1985) 1. [14] J.B. McLaughlin, J. Stat. Phys. 24 (1981) 375. [15] F.C. Moon, Phys. Rev. Lett. 53 (1984) 962.