CHARACTERIZATION OF CHAOS IN SPIN WAVE TURBULENCE

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CHARACTERIZATION OF CHAOS IN SPIN WAVE

TURBULENCE

S. Rezende, F. de Aguiar, A. Azevedo

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C8, Suppl6ment au no 12, Tome 49, dkembre 1988

CHARACTERIZATION OF CHAOS IN SPIN WAVE TURBULENCE

S. M. Rezende, F. M. de Aguiar and A. Azevedo

Departamento de Fisica, Universidade Federal de Pernambuco 50739, Recife, Brazil

Abstract. - We study transitions to chaos of spin wave turbulence in YIG with microwave parallel and perpendicular pumping. The dimension of the strange attractors at the onset of chaos is in all cases less than 2, indicating that few modes participate in the spin-wave chaotic dynamics.

Spin-wave instabilities in magnetic materials dis- play a variety of nonlinear dynamic phenomena such as self-oscillations, period multiplication, irregular period oscillations, intermittency and chaos [I-61. In

the usual experimental set-up to investigate spin-wave turbulence, the sample is driven by a microwave rf field either parallel or perpendicular to the static magnetic field. The nonlinear dynamic effects are observed in the amplitude modulation which appears in the microwave field returning from the sample when the pumping exceeds a threshold value. This modulation arises from the dynamic interplay between parametric spin-wave modes, but there is no direct experimental evidence of the nature or the number of such modes. Although a two-mode model has successfully explained many features of the.experimenta1 results [3, 71it is expected that an entire manifold of magnon modes pumped above the threshold be involved in the pro- cess [8, 91.

In order t o gain understanding about the number of modes involved in spin-wave chaos, we report in this paper the characterization of strange attractors of two routes t o chaos previously observed in microwave pumping experiments in yttrium iron garnet (YIG), both with the parallel pumping and the subsidiary resonance configurations. The experiments are performed at frequencies ip the range 9.2-9.4 GHz and we mea- sure the low frequency (50-500 kHz) self-oscillations that develop in the amplitude of the signal reflected from the resonant cavity. At conveniently selected crystal orientation and static field values, the low- frequency modulation shows period multiplication and chaos as the pumping is increased, and two clearly dis- tinct scenarios have been observed both in the parallel and transverse pumping processes. The first scenario, which we call F henceforth, is a typical Feigenbaum route to chaos. The original self-oscillation with period T bifurcates t o periods 2 T, 4

T,

8 T, etc. with increasing microwave power before it becomes chaotic. In the other scenario ( C ) there is only one period dou-

bling before the onset of chaos, with the period 2 T signal going through a narrow region of aperiodic oscillations and no evidence of periods 4 T o r 8 T as in scenario F.

Time series data are obtained from the low- frequency amplitude modulation a t intervals of 0.2 ,us and used to characterize the chaotic attractors by means of the embedding technique. Two relevant quantities that can be calculated from the data are the fractal dimension and the Kolmogorov entropy. The dimension characterizes the attractor static properties and it is also a lower bound on the number of inde- pendent variables needed to model the dynamics. The Kolmogorov entropy K measures the loss of information on the initial conditions per unit time. Both q u a - tities can be used to discriminate between stochastic and deterministic dynamical systems. In order t o ob- tain a reliable characterization of the attractors we have used two algorithms to calculate the dimensions, one proposed by Grassberger and Procaccia (GP) several years ago

[lo]

and another introduced recently I l l ] by Badii and Politi (BP).

In the GP algorithm one considers a time series {Xi)

,,,,...,,

of points on the attractor and defines the correlation integral

C (T) = _N-03lim

1

_{N 2}

C

B (T

-

\Xi

-

X i / ) (1)

i#j

where 0 is the Heaviside function. C (T) may be cal-

culated from a time series of a single physical quan- tity x (t) (the voltage of the microwave diode detec- tor in our case) with the embedding technique. E-

dimensional vectors Xi (t) are constructed with coor- dinates x (t)

,

x (t

+

T )

, ...,

x (t

-

(E

-

1) r)

,

where T

is a fixed delay time and E the embedding dimension.

For small r the correlation inegral scales as

where the correlation exponent d = lim log Cllog r is

T-0

close to the fractal dimension of the attactor.

In the method of Badii and Politi the central quan- tity is the moment of order y calculated from the probability distribution p (6, n ) of nearest-neighbor dis- tances

S

among n randomly chosen points on the attractor

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C8 - 1606 JOURNAL DE PHYSIQUE From the scaling form of this as n --+ oo one defines a

y dependent dimension function 7 log n

('1 =

-

2%

log [M, (%)I.

The dimension function yields, for specific 7 values, several dimensions previously defined [ll]. In partic- ular, the fixed point Do = D ( 7 ) = 7 is the fractal dimension and Dl = D (7 = 0) is the information dimension, typically Do

2

D l . The generalized metric entropies K (7) can be obtained from D (7)

.

For 7 = 0 it satisfies the relation

where (6 (n, E)) is the moment calculated from the E- dimensional embedded vectors and T is the sampling

time.

Strange attractors obtained from scenarios F and C described previously, both in the parallel pumping and subsidiary resonance processes, have been studied with the GP and BP methods. Figure 1 shows the results at the onset of chaos with scenario C in the subsidiary resonance experiments described in reference 161. Plots of log (6) for y = 0 versus log n for E = 1,2,

...,

15 are obtained from 2048 data points. For E

>

6 the slopes a t the higher log n end converge t o a value for the infor- mation dimension D l

=

1.5, in good agreement with the value obtained with the GP method. The same procedure has been used t o calculate the dimension of the attractors with scenarios F in subsidiary resonance and C in parallel pumping. In all cases we find 1.4

<

d

<

1.8, which is close to the value for scenario C in YIG under parallel pumping of reference [12].

From the time series data we have also calculated the entropy K (0 ) as a function of the embedding dimension. We have found in both scenarios F and C, that for large E the entropy converges to a constant values Kc a! 0.05, which confirms the deterministic

nature of chaos in our experiments.

The above results show that the broad power spec- t r a observed in spin-wave turbulence experiments is indeed a manifestation of deterministic chaos. Differ-

25 30 35 40

log n

Fig. 1.

-

log 6 us. log n for 7 = 0 and varying embedding dimension E, for the attractor in scenario C of subsidiary resonance in YIG.

ent routes to chaos are obtained depending on the experimental configuration, all with strange attractors a t the onset of chaos with dimension 1.4

<

d

<

1.8. This result indicates that despite the fact that a large number of magnon modes is excited the number of degrees of freedom is small, strongly suggesting that very few collective modes are involved in the chaotic dynamics, in agreement with recent theoretical results [13, 141. The nature of these modes, however, remains to be clarified.

Acknowledgments

The authors would like to acknowledge M. Warden and G. Broggi of the Physics Institute, University of Zurich, for providing the software to calculate 6 (n) and invaluable discussions. One of us (S.M.R.) acknowl- edges Prof. F. Waldner for stimulating discussions and the hospitality in Zurich. This work has been sup- ported by the Brazilian agencies FINEP, CNPq and CAPES.

[l] Gibson, G. and Jeffries, C., Phys. Rev. A 29 (1984) 811.

[2] Waldner, F., Barberis, D. R. and Yamazaki, H., Phys. Rev. A 31 (1985) 420.

[3] De Aguiar, F. M. and Rezende, S. M., Phys. Rev. Lett. 56 (1986) 1070.

[4] Smirnov, A. I., Sov. Phys. JETP 63 (1986) 222. [5] Bryant, P., Jeffries, C. and Nakamura, K., of the

Int. Conf. on the Physics of Chaos and Systems far from Equilibrium, Monterrey, California, 1987 (North-Holland, Amsterdam) 1987, p. 25. [6] Rezende, S. M., De Aguiar, F. M. and Azevedo,

A., Magnetic Excitations and Fluctuations 11, Edis. U. Balucani, S. W. Lovesey, M. G. Rasetti, and V. Tognetti (Springer-Verlag) 1987, p. 79. [7] Nakamura, K., Ohta, S. and Kawasaki, K., J.

Phys. C 15 (1982) L143.

[8] Suhl, H. and Zhang, X. Y., Phys. Rev. Lett. 57 (1986) 1480.

[9] Lim, S. P. and Huber, D. L., Phys. Rev. B'37 (1988) 5426.

[lo] Grassberger, P. and Proccacia, I., Phys. Rev. Lett. 50 (1983) 346.

[Ill Badii, R. and Politi, A., Phys. Rev. Lett. 52 (1984) 1661; J. Stat. Phys. 40 (1985) 725. [12] Mino, M. and Yamaaki, H., J. Phys. Soc. Jpn

55 (1986) 4168.

[13] Gill, T. L. and Zachary, W. W., J. Appl. Phys. 6 1 (1987) 4130.