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Journal of Magnetism and Magnetic Materials 104-107 (1992) 1041-1042 ? North-Holland M

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Bifurcations, chaos and control of chaos in spin-wave instabilities

Antonio Azevedo and Sergio M. Rezende

Departamento de F&a, Unirwsidade Federal de Pernambuco, 50739 Recife, Brazil

An experimental study of microwave pumped spin-wave instabilities in YIG reveals new types of bifurcations and the

possibility of controlling chaos with a small perturbation in a system parameter.

Parametric excitation of spin-waves driven at high

microwave power constitutes one of the most interest-

ing nonlinear dynamical phenomena. Since the earliest

investigations of spin-wave instabilities it has been

known that at high microwave pumping power the

spin-wave system develops low-frequency (10 kHz-1 MHz) oscillations. These so-called auto-oscillations display a rich variety of behavior such as period multi- plication, intermittency, spiking and chaos, which have attracted increasing attention recently [ 11.

In this paper we report two novel features of the dynamical behavior of spin-wave instabilities in two distinct situations: at the onset of the auto-oscillations we show that different types of bifurcations may occur [2]; in a fully chaotic regime we show, for the first time, that the spin-wave chaotic state can be suppressed by a small modulation in a chosen system parameter [4].

The experiments were performed at room tempera- ture with polished YIG spheres, driven by an X-band microwave magnetic field h applied either perpendicu- lar (subsidiary resonance) or parallel to the dc field H,,. The apparatus is described in detail in refs. [2,4].

The qualitative changes in the nonlinear dynamical behavior that occur when one or more system parame- ters are varied are called bifurcations and occur at well defined parameter values. Usually it is stated in the literature that the auto-oscillations originate in a Hopf bifurcation. Here we show that this is not always the case. A Hopf bifurcation is easily obtained in the parallel pumping configuration with the [ill] YIG crystal axis parallel to H,. Fig. 1 shows the experimen- tal results: the auto-oscillation sets in at a critical field h: with a finite frequency f0 but with vanishing ampli- tude. As the microwave field h increases the amplitude increases according to the scaling law A y (h/h: - l)p, p = 0.5, whereas the auto-oscillation frequency stays constant. These are characteristic features of a Hopf bifurcation [2].

Fig. 2 shows .another type of bifurcation that is in marked contrast with the scenario above. It was ob- served in the subsidiary resonance configuration with the field along the [llO] direction. In this case the auto-oscillation sets in with a finite amplitude A and very small frequency. As h increases above the thresh- old, the amplitude stays constant while the frequency

increases rapidly. This behavior can result in homo- clinic recurrence of the “Silnikov type” [3].

The two types of bifurcations described were theo- retically studied by analysing the stability of the eigen- values of the Jacobian matrix for the two-mode spin- wave equations of motion [2]. In fact, ref. [5] shows that the auto-oscillations described by the spin-wave equa- tions may result from a variety of different types of bifurcations.

In a recent paper Ott et al. [6] suggested that the chaotic state can be controlled by a small periodic perturbation of an available system parameter. In this part of the paper we report experimental results with spin-wave instabilities that demonstrate the chaos sup- pression.

The experiments were carried out with a polished YIG sphere (diameter 1.0 mm) in the “subsidiary-reso- nance” configuration. The experimental setup is simi- lar to that presented in ref. [l], except for a small modulation applied in the dc magnetic field. The mod- ulation was produced by a coil (diameter 1.5 cm), inside the microwave cavity. This allows the modula-

Fig. 1. Oscilloscope traces showing the behavior of the auto-

oscillation with increasing microwave field R = h/h, in paral-

lel-pumping with a YIG sphere oriented along [ll I], H,, = 1570

Oe, fp =9.5 GHz. R = 1.224, 1.255, 1.293, 1.321, 1.342 from top to bottom. The constant frequency and the rapid growth

in the amplitude characterizes a Hopf bifurcation.

0312~8853/92/$05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved

1042 A. Azel,edo, SM. Rezende / Bifurcations and chaos in suin-waw instabilities

Fig. 2. Oscilloscope traces of the auto-oscillation obtained

with subsidiary-resonance pumping with increasing microwave

field R = h/h, in YIG sphere oriented along [llO], Ho = 1950

Oe, f, = 9.4 GHz. R = 1.862, 1.872, 1.883, 1.894, 1.905, 1.906

from top to bottom. Note that the auto-oscillation sets in with

a finite amplitude and rapidly increasing frequency, quite

different from the Hopf scenario of fig. I.

tion of the dc field H = H, + AH cos(21~f,t) over a broad frequency range O-10 MHz, typically with 6H/H,, = 10-4.

Windows of controlling chaos were obtained in the parameter space, varying both amplitude SH and fre- quency f, of the harmonic perturbation. Fig. 2 shows the power spectra of the auto-oscillation for (a) chaotic situation with SH = 0 and (b) suppression of chaos with 6H = 435 mOe and f1 = 1480 kHz. The control of chaos is further demonstrated by variation of the infor- mation dimension D, and the metric entropy K of the attractor. These quantities have been obtained with the embedding technique from the time-delayed digitized signals [6]. We employ the method of Badii and Politi [7] to calculate D, = D(0) and K = K(O) from 2048 data points. With increasing field modulation the val- ues of D(O) and K(O) decrease towards the values D(O) = 1 and K(0) = 0 appropriate for periodic signals [4]. This confirms that by using a carefully chosen small parameter perturbation, it is possible to control a chaotic state [8].

The authors thank Prof. Jair Koiller and Dr. FlLivio M. de Aguiar for helpful discussions. This work was

-0 500 1000 1500 2000 2500 f (kHz)

1000 1500 2000 2500 f (kHz)

Fig. 3. Auto-oscillations power spectra. Spectrum (a) corre-

sponds to a fully chaotic regime without parameter system

perturbation and (b) corresponds to a chaotic state sup-

pressed with a small modulation in the dc field. The ampli-

tude and frequency of the modulation are 6 H = 435 mOe and

f, = 1480 kHz respectively.

partially supported by FINEP, CNPq, CAPES and FACEPE (Brazilian agencies).

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