CHAOS AND STRANGE ATTRACTOR OF PARALLEL-PUMPED MAGNONS IN YIG

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CHAOS AND STRANGE ATTRACTOR OF

PARALLEL-PUMPED MAGNONS IN YIG

H. Yamazaki, M. Mino

To cite this version:

JOURNAL DE PHYSIQUE

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

CHAOS AND STRANGE ATTRACTOR OF PARALLEL-PUMPED MAGNONS IN

YIG

H. Yamazaki and M. Mino

Department of Physics, Faculty of Science, Okayama University, Tsushima, Okayama 700, Japan

Abstract. - Above the spin-wave instability threshold under parallel-pumping, period-doubling, chaos, period-halving and periodic window are observed in YIG. The successive Poincare sections of the strange attractor exhibit the stretching and folding effects of sheet of trajectories which confirms the fractal characteristics of the strange attractor.

As a microwave magnetic field applied parallel to the magnetization is increased beyond some critical power, the number of magnons in a very narrow re- gion in wave number space grows exponentially, while the others stay almost at the thermal equilibrium lev- els. By increasing the driving power further, nonequi- librium stationary state is occasionally broken by oc- currence of oscillations of magnon amplitude. These auto-oscillations proceed to period-doubling bifurca- tions, chaos [l-31 and periodic window. Theoretical studies of chaos caused by four-magnon interaction term have been numerically performed [4-51. In this paper, we will present fractal characteristics of strange attractor observed in YIG. Three-dimensional portrait of strange attractor, fractal dimension and the largest Lyapunov exponent are obtained.

Experiments were performed at a pumping fre- quency of 8.9 GHz and a t a temperature of 4.2 K. A disk-shaped YIG, 1.28 mm in diameter and 0.40 mrn thick, is mounted on the bottom of the TElol cavity. Both the microwave and the static magnetic fields are applied along the [ I l l ] direction which is perpendicu- lar to the disk.

In order to understand a dynamical system, it is useful to know a trajectory in the multi-dimensional phase space. An instantaneous state of dynamical sys- tem is described by point which moves along a curve, a phase-space trajectory. The three-dimensional time series data are generated from the single time series data by a procedure of time delay [6]. Trajectories are constructed by taking time delay of 3 psec, which is about one fifth of the fundamental period. Fig- ure 1 is the two-dimensional projection of phase por- trait at various driving power where the power P is measured from the threshold for spin-wave instability. At P = 1.66 dB, the oscillations are regular and phase portrait for period-2 is shown at P = 2.20 dB. Phase portraits from P = 2.73 dB to P = 3.60 dB show chaotic ones. At P = 4.08 dB, period-halving transi- tion from chaos to period-4, to period-2 a t P = 4.32 dB and t o period-1 a t P = 5.25 dB occurs.

Figure 2 is the view of the two-dimensional projec- tion of strange attractor a t P = 3.60 dB. Figure 3

Fig. 1. - Phase space portraits.

presents the Poincare sections constructed by the in- tersection of positively directed trajectories with the plane, which is normal to the page passing through the (A-J) of figure 2. As is clearly seen in this figure, the trajectories form a two-dimensional sheet. By exam- ining the evolution of the trajectories in the Poincare sections at successive points along the strange attrac- tor, the sequence of stretching (H --, I --, J -+ A + B)

C8 - 1610 JOURNAL DE PHYSIQUE

I I

V ( t )

Fig. 2. - Two dimensional projection of the strange attrac- tor.

and folding (C --, D -+ E -+ F --, G -+ H) is evidently observed. Since infinitely repeated stretching and fold- ing process with the evolution of trajectories causes a fractal structure to the strange attractor, the frac- tal dimension is one of the characteristic measures of chaos.

Fractal dimension of the strange attractor is esti- mated by a procedure of the correlation integral [7]. Correlation exponent d is given by C ( r ) r d , where C(r) is the correlation integral and r i s the distance be- tween data points in the multidimensional phase space. The exponent d is obtained with varying the embed- ding dimension n. With increasing n, correlation expo- nent d approaches 2.0 which is concluded as the fractal dimension of the strange attractor in YIG.

Another quantitative measure of a strange attractor are positive Lyapunov exponents which characterize the average rate of exponential divergence of nearby trajectories within the attractor. Negative Lyapunov exponents characterize the average rate of exponen- tial convergence of trajectories onto the attractor. The largest Lyapunov exponent can be conventionally com- puted by using the empirical one-dimensional return

Fig. 3.

-

map [I]. The largest Lyapunov exponent for YIG is obtained as 0.34. The positive Lyapunov exponent confirms the chaotic behavior in YIG.

[I] Mino, M. and Yarnazaki, H., J. Phys. Soc. Jpn

55 (1986) 4168.

[2] de Aguiar, F. M. and Rezende, S. M., Phys. Rev.

Lett. 56 (1986) 1070.

[3] Bryant, P., Jeffries, C. and Nakamura, K., Phys.

Rev. Lett. 60 (1988) 1185.

[4] Zakharov, V. E., L'vov, V. S. and Starobinets, S. S., Sou. Phys.-Usp. 17 (1975) 896.

[5] Ohta, S. and Nakarnura, K., J. Phys.

C

[6] Fraser, A. M. and Swinney, H. L., Phys. Rev.

A 33 (1986) 134.