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Spatial stability of homogeneous Feigenbaum scenarios in the FMR
Physica D 59 (1992) 193-199 North-Holland
Spatial stability of homogeneous Feigenbaum scenarios in the FMR Wolfgang Weitzel Theoretische Festk6rperphysik, TH Darmstadt, Darmstadt, Germany Received 10 February 1992 Revised manuscript received 3 June 1992 Accepted 3 June 1992 Communicated by F.H. Busse
The dissipative Landau-Lifshitz equation serves as a model for describing the nonlinear dynamics of magnetization inside a transversally driven ferromagnet with uniaxial anisotropy. The consideration of the internal exchange field leads to a partial differential equation possessing spatially homogeneous solutions. The isolated evolution of the uniform mode can exhibit chaotic time-dependence; two Feigenbaum scenarios have been detected. The application of linear stability-analysis to the time-periodic homogeneous solutions shows that spatial instabilities occur before the chaotic state at k = 0 is established.
1. Introduction
The phenomenological equations of motion which are used to describe the dynamics of the magnetization field have proved to be a successful model because they reflect experimental results. An important example are Suhl's spinwave instabilities [1], which occur above a certain parameter threshold and destroy the homogeneous precession. This destabilization of the uniform mode has been verified in numerous measurements of ferromagnetic resonance (FMR) by the detection of changes in the absorption spectrum. Generally, the excitation of spinwaves leads to a rich variety of nonlinear phenomena, which usually exhibit low frequencies (kHzMHz) and which have been observed in many experiments [2]. Effects of this kind occur in the region above the instability threshold when the uniform mode has become unstable, and when the nonlinear interactions between different modes have become effective. Subject of this paper are theoretical examina-
tions of a basic dynamical equation for the magnetization. With the consideration of an uniaxial anisotropy field (essentially: transversal adjustment of the crystal-axis) even the isolated evolution of the uniform mode implies complicated behaviour. In an earlier paper [3], it was shown by Melnikov's method that the uniform mode exhibits homoclinic chaos which is connected with the existence of a strange repellor. This was regarded as a strong hint for the existence of a "measurable" chaos which does not only effect transients but also appears in long-term computer simulations. In this paper, two spatially homogeneous Feigenbaum scenarios are presented and their constituents are now interpreted to be exact solutions of the partial differential equation which is derived from the consideration of the internal exchange field. The possibility of this interpretation determines the outstanding role of the uniform mode and allows the application of linear stability analysis, the framework of which we shall develop in section 3. Treating a local equation, we do not consider boundary
0167-2789/92/$05.00 (~) 1992 - Elsevier Science Publishers B.V. All rights reserved
194
W. Weitzel / Spatial stability of homogeneous Feigenbaum scenarios in the FMR
effects. T h e r e f o r e our results refer to extended samples only, the linear dimensions of which are large compared with a typical magnetic length (e.g. the wavelength of the first excited spinwave). The application of linear stability analysis to the homogeneous subharmonics of the Feigenbaum scenarios was motivated by the question of the existence of a temporal chaos which maintains spatial coherence. In contrast to Suhl's problem we do not expect to find a parameter value, above which a large class of homogeneous solutions becomes spatially unstable. Because of the complexity of our uniform dynamics, it is necessary to consider the behaviour of selected homogeneous solutions.
2. The model-equations
dipolar-interaction has been neglected. Because of the exchange-term we are dealing with a partial differential equation for the magnetization field. Eq. (1) possesses the property of conserving the modulus of magnetization,
IM(r, t){
=
IM(r, 0)l.
(3)
Initially seven parameters appear, but two of them can be removed immediately by a time scaling t--* 3"Ht which means an extraction of the typical Larmor-frequency of the system. For ferromagnets, the exchange constant J has to be positive to ensure minimal energy in the h o m o g e n e o u s case. Of course, the system (1), (2) possesses spatially homogeneous solutions. The (non-autonomous) dynamics of the uniform m o d e occurs on the (two-dimensional) surface of a sphere.
Starting point is the well-known L a n d a u Lifshitz equation [4] 3. Framework of the stability analysis d
cgt M ( r , t) = - 3 ' M × H~ff - AM × (M × Heff),
(1) which serves as a theoretical model for the d a m p e d motion of magnetization inside a ferromagnetic material. We added a sign to the first term, so that for the considered electronic systems the constant 3' represents the modulus of the gyromagnetic ratio. The first term in eq. (1) conserves volume in the phase space while the second term constitutes a dissipative flow describing the damping. The effective field [5] has been chosen in the form Hef f = H e z + h sin t o t e x + A M x e x + J A M ,
In the following we always refer to the timescaled problem. In order to examine the stability of a solution M ~ ( t ) we make the ansatz M(r, t) = M ~ ( t ) + re(r, t) .
(4)
After inserting this ansatz into eqs. (1), (2), bearing in mind that M S ( t ) is an exact solution, one obtains a partial differential equation for the deviation re(r, t), which of course depends on the considered solution M ~ ( t ) . The next step is the linearization of the obtained equation with respect to m. With the following dimensionless quantities:
(2) E~(t)=M~(t)/M
that is, a static field parallel to e~, a sinusoidal driving field along the X-axis and as internal contributions an uniaxial anisotropy-field also adjusted in the x-direction and the exchange field. The anisotropy is assumed to be of crystalline origin only; the nonlocal effect of the
a= h/H , w =- w / T H ,
e,
n(r,t)=m(r,t)/M
b = AM~/3" ,
~,
p = AM~/H ,
(5)
the linearized equation of motion for the deviation reads
195
W. Weitzel / Spatial stability of homogeneous Feigenbaum scenarios in the FMR
0
Ot n(r, t) = H 2 × E ~ + H 1 × n + b E ~ × [H 2 × E ~] + b E ~ × [H I × n] + bn × [ H 1 × E ~ ] ,
(6) where we have used the abbreviations
Accordingly the characteristic matrix can be obtained by integration of the linear system over the range of one period with the initial value ~ ( t 0, t o ) = 1. It can be shown that the eigenvalues of ~ do not depend on the special choice of t o. The condition for asymptotic stability reads now:
H a = e z + a sin totex + p E t e x ,
v=1,2,3. H2
=
Pnxe x + An.
The factor in front of the Laplace operator has been eliminated by a scaling of the length r--->
r.
(7)
An expansion in plane waves n ( r , t) = f d3k c(k, t) e ik'r
(8)
changes eq. (6) into the convenient form
L c(k, t) = ~t[k 2, E~(t), t] c(k, t) 0t
(9)
Of course, in both cases the application to the systems (9) demands an examination of all wavenumbers k which may give rise to a varying stability behaviour. Instability with respect to a particular wave-number k c is sufficient to destroy the homogeneous state and may lead to the excitation of many k-modes when the nonlinear couplings become effective. The special property (3) has an influence on the stability characteristics: There is no effect of the Landau-Lifshitz dynamics on deviations directed parallel to M ~ ( t ) . More precisely, it is possible to show that
which represents an ordinary linear system of dimension three for each wave-number k. The stability of the trivial solution c = 0 and thus the general stability of the linear system, is the subject of linear stability-analysis. In two important cases the asymptotic time dependence is well defined [6]: (i) The matrix ~ does not depend on time. In this case the stability of the linear system is determined by the eigenvalues of ~/. The condition for asymptotic stability reads:
O__Ot[n(r, t ) . E~(t)] = ~0 n . E ~ + n . / ~
Re A ~ ( ~ ) < 0 ,
Al[6~(k)] = 0 .
v=1,2,3.
(ii) The matrix M is time-periodic (Floquetcase): M ( t + r ) = M(t). Then the stability depends on the eigenvalues of a characteristic matrix ~ which is defined through the evolutionmatrix ~g(t, to): c(t) = ~ ( t , to)e(to)--> ~ = ~ ( t o + r, t o ) .
=0
.
(10)
Thus the corresponding component of any Fourier coefficient c(k, t) is a constant: 0
Ot [c(k, t ) . E ~ ( t ) l : 0
Vk.
(11)
In the time-independent case ( d M / d t = 0 ) this result is connected with the existence of an eigenvalue (12)
In the time-periodic case (~/(t + r ) = ~ ( t ) ) eq. (11) gives rise to a characteristic eigenvalue X l [ ~ ( k ) ] = 1.
(13)
The occurrence of these trivial eigenvalues is connected with the structure .of eq. (1) and does
196
W. Weitzel / Spatial stability of homogeneous Feigenbaum scenarios in the FMR
not depend on parameter-values or on the special choice of the solution M~(t). As a consequence asymptotic stability of the linear systems (9) is not admitted. In any case the linear m e t h o d allows the detection of instability. It can proved that the characteristic eigenvalues h ~ ( ~ ) do not change if one turns to a t i m e - d e p e n d e n t tripod ei(t ) (i -- 1, 2, 3) provided that it exhibits the same period as the matrix M(t). In the following we expand the vector c(k, t) in the tripod of spherical coordinates
E~(t),
eo(t),
e~(t),
(14)
the linear stability of the spatially h o m o g e n e o u s solution M~(t), which is characterized by the constant modulus M s and the angles O(t) and ~b(t). The modulus does not a p p e a r explicitly, but is hidden in the p a r a m e t e r s p and b (see eqs. (5)). T h e equation of motion for M~(t) is given by Ms=0, 0 = - p sin 0 sin ~b cos qb - b sin O
+ bp sin 0 cos 0 cos:4~ - a sin w t s i n th + ab sin w t c o s 0 cos ~b,
where we regard E~(t) as the radial unit vector. T h e first c o m p o n e n t of c will be a constant producing the trivial eigenvalue. The remaining two eigenvalues are determined by the reduced dynamics which is created by setting c - E ~ equal to zero. T h e reduced dynamics reads
Ot
c~,
c~
= 1 - p cos 0 cose~b - bp sin th cos ~b - a sin tot cot 0 cos ~b
- a b sin tot sin 10 sin ~b.
(17)
Solutions M " ( t ) touching the z-axis have been treated analogously in Cartesian coordinates, because for 0 = 0 resp. -rr the tripod of spherical coordinates is not defined. H o w e v e r , the representation given above is m o r e compact and illustrates in which way the trivial eigenvalue can be separated.
'
with the matrix elements RI1 = - p sin th cos ~b cos 0 - bk 2 - b cos 0
+ bp cos2~b(1 - 2 sinE0) - a b sin tot sin 0 cos th,
4. Spatial stability of Feigenbaum scenarios
R12 = p ( 1 - 2 cosZth) - k 2 - 2bp sin 4~ cos ~b cos 0 - a sin tot sin-~0 cos ~b
- a b sin wt sin-10 cos 0 sin ~b, RE~ = p sin20 c0s24~ + k 2 + a sin tot sin-~0 cos ~b
+ab sin tot sin-10 cos 0 sin ~b, R : : = p sin ~b cos ~b cos 0 - bk 2 - b cos 0
+ bp sin24~ - bp sin20 cos2~b - a b sin tot sin 0 cos th •
(16)
This two-dimensional linear system determines
T o begin with we want to give some remarks concerning the behaviour of analytically accessible solutions MS(t). In some special cases it is possible to find analytical expressions solving the eqs. (17). Neglecting the driving field by setting a = 0, we can determine the fixed point solutions of the system (17). After insertion of the expressions for E ~ into the matrix ~/ (resp. ~ ) , the stability analysis reduces to a calculation of the eigenvalues, which of course depend on the w a v e - n u m b e r k. It turns out that for our system the normal adjustment of stationary magnetization is 0 = 0. Only in the case of a strong "easyaxis" anisotropy ( p > 1) the spatially stable
w. Weitzel / Spatial stability of homogeneous Feigenbaum scenarios in the FMR equilibrium will be located outside the z-axis (cos 0 = p - 1, ~b = 0 resp. at). The autonomous system without damping (a = b = 0), which we call the unperturbed problem, proves to be completely integrable. In addition to the fixed points we find periodic solutions expressed by Jacobian elliptic functions. With the knowledge of E~(t) the time dependence of the system (15) is well defined and the stability analysis can be carried out by a numerical calculation of the characteristic eigenvalues. It appears that all periodic solutions of the unperturbed problem are spatially unstable. This result has an important consequence on the perturbation theory: Periodic orbits of the unperturbed case (a = b = 0), which are conserved under the influence of a small driving field and a small damping, will be spatially unstable in the same way in the presence of the perturbation, because the characteristic eigenvalues will shift slightly only. We proved the existence of such "resistant" orbits by application of the subharmonic Melnikov-technique [7]. In the general case it is not possible to find analytical' solutions of eq. (17). However, the homogeneous solution M~(t) can be established with the help of a computer. As we want to treat time-periodic systems (9), M~(t) has to be a subharmonic solution of period 2~rm/to (m = 1, 2, 3 . . . . ). It appears that solutions of this kind can be found very easily, since usually the preceding transients die away quickly. Preferably one detects solutions of period 1 (m---1), but also subharmonics of higher order appear. In order to avoid the transients completely, we determine suitable initial values in a precursory computation. With these correct initial values for the system (17), which belong to a well-defined subharmonic solution Me(t), it is possible to integrate the systems (15) and (17) simultaneously regarding them as a four-dimensional system for the dynamical variables c0, c6, 0 and ~b. For a given wave-number k two numerical integrations of this kind over the range of one period 2~rm/to are sufficient to construct the
197
characteristic matrix of the Floquet system (15) (referring to the reduced system (15) we presuppose that Me(t) does not touch the z-axis). In contrast to the unperturbed problem we now can find many periodic solutions Me(t) which do not exhibit spatial instability. In order to study the behaviour for varying parameter values, we proceed as follows: The parameter is changed by small steps and as initial value of each moment we take the adjusted initial value of the preceding step, so that we always stay in the basin of attraction which belongs to the considered subharmonic solution. By this method two Feigenbaum scenarios [8] have been detected and even the period 32 was clearly shown in the computer data. Fig. 1 shows the Feigenbaum scenario 1 up to the period 8. We made use of the stroboscopic Poincar6 map (distance of sectional planes: -0.35
x --0.45 ~
-(
-0.55
I
0.90
0.97
-( 1.04
0.06
fl
-0.02-
..< -0.10
I
0.90
0.97
1.04
1/w Fig. 1. Feigenbaumscenario 1 (p =2, a=0.5, b =0.05).
198
W. Weitzel / Spatial stability of homogeneous Feigenbaum scenarios in the FMR
2~r/~o, location: t o = 0), so that a subharmonic solution of period 2,rrm/oJ appears as a discrete cycle consisting of m points. Furthermore we applied a stereographic projection given by the transformation x=tan½0cos~b,
y=tan½0sin&.
(18)
The varying parameter of scenario 1 is the (reciprocal) frequency of the driving field. The remaining three parameters have been fixed. The examination of the spatial stability showed that the scenario is divided in two parts: The subharmonics of the left part exhibit linear stability with respect to any wave-number k, while all subharmonics of higher order, forming the right part, prove to be spatially unstable. Our numerical investigation of spatial stability reached the period 16. In fig. 1 the critical value (1/oo c = 1.019) is marked by a vertical line. As expected, all subharmonics exhibit linear stability at k = 0, a fact which has been exploited in the numerical construction of M~(t). Clearly, this stability vanishes if one approaches a bifurcation point of the Feigenbaum scenario. The effect of period doubling on the stability behaviour is a squaring of the characteristic matrices ~ ( k ) and thus a squaring of the characteristic eigenvalues. This can be proved by simple considerations if one takes into account that period doubling means a microscopic splitting of the orbit E~(t) only. This statement is confirmed quantitatively by our numerical results. As a consequence, qualitative changes of spatial stability do not occur at the bifurcation points. The detected spatial instability interrupts the route to temporal chaos and leads to the excitation of spin-waves. The nonlinear couplings between different wave-numbers become effective and the dynamics of the uniform mode is changed. Therefore the restriction to the ordinary system (17), which describes the isolated evolution of the uniform mode, is not realistic in the region above the threshold. This range demands an investigation of the complete partial
differential equation, which is not a subject of this paper. Fig. 2 shows another Feigenbaum scenario with the varying parameter a. Here the origin of period 1 is a fixed point solution at a = 0. The stability behaviour proves to be very similar to the first scenario. Again we find a threshold above which spatial instability occurs and once more the critical value (a c = 0.08) is marked by a vertical line. The fact that two scenarios perform a very similar stability behaviour when they are embedded in the partial differential equation suggests strongly that we are facing a typical feature of our magnetic system. In real experiment, the instability might be identified by the disappearance of the subharmonic signal and by the onset of low frequency auto-oscillations which indicate the participation of additional modes.
z o.32-t
\
0.15 0.00
0.13
0.26
a
fil/ -0.02 0.00
0.13
0.26
a
Fig. 2. F e i g e n b a u m scenario 2 ( p = 1.6, b = 0 . 0 8 , ¢o = 1.2).
w. Weitzel / Spatial stability of homogeneous Feigenbaum scenarios in the FMR The dotted line in fig. 2 represents the homoclinic threshold a 0 = 0 . 1 3 2 1 6 . . . , which has been computed by Melnikov's method and which can be regarded as a good approximation since the given perturbation (a < 0 . 3 , b = 0.08) is rather small [3]. Thus for a > a 0 the uniform mode (17) exhibits homoclinic chaos, provided that p, b and to have the values as given in scenario 2. Obviously there are regular subharmonic solutions above the homoclinic threshold. After all they are elements of a Feigenbaum scenario. The p a r a m e t e r values of scenario 1 are located far above the theoretical threshold and probably the whole sequence is embedded in the homoclinic chaos.
199
and the other part, which contains the subharmonics of higher order, being spatially unstable. Therefore one cannot approach the chaotic state without the excitation of spin-waves, the growth of which changes the dynamics of the uniform mode (because of the nonlinear couplings between different wave-numbers). It is possible, that the spatial instabilities interrupt the route to chaos and lead to an orderly state in space-time. On the other hand the excitation of additional modes might be the onset of spin-wave turbulence, which is characterized by a great number of degrees of freedom [9].
Acknowledgement 5. Conclusion The partial differential equation (1), (2) possesses spatially homogeneous solutions MS(t), the linear stability of which we investigated in this paper. For this purpose we linearized the equation of motion for the deviation re(r, t) and then expanded this perturbation in plane waves in order to obtain independent equations of motion for each Fourier coefficient Ck(t ). Thus we examined the stability of homogeneous solutions with respect to spin-wave disturbances of waven u m b e r k. Because of the nonintegrability of the uniform mode (in the general case) we considered special solutions M~(t) established with the help of a computer. We detected two spatially homogeneous Feigenbaum scenarios which consist of subharmonic solutions appearing a f t e r short-lived transients. For solutions of this kind linear stability analysis can be carried out with little numerical effort. It turned out that both scenarios are divided in two parts, one part exhibiting linear stability for all wave-numbers k,
This work was performed within a program of the Sonderforschungsbereich 185 "Nichtlineare D y n a m i k " Frankfurt/Darmstadt. The author wants to express his gratitude to Professor Dr. E. Fick for several helpful discussions.
References [1] H. Suhl, J. Phys. Chem. Solids 1 (1957) 209. ]2] P. Bryant, C. Jeffries and K. Nakamura, Phys. Rev. Lett. 60 (1988) 1185. [3] W. Weitzel and H. Turschner, Phys. Lett. A 140 (1989) 185. [4] L.D. Landau and E.M.-Lifshitz, Phys. Z. Sowjetunion 8 (1935) 153 [see collected papers of L.D. Landau, ed. ter Haar (Pergamon, Oxford, 1965)]. [5] A.I. Akhiezer, V.G. Bar'yakhtar and I.V. Peletminskii, Spin Waves (North-Holland, Amsterdam, 1968). [6] S. Fenyr, T. Frey, Moderne Mathematische Methoden in der Technik, Bd. 1 (Birkh~iuser, Basel, 1967) p. 329. [7] J. Guckenheimer and Ph. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, Berlin, 1983). ]8] M.J. Feigenbaum, J. Stat. Phys. 19 (1978) 25. I9] T.L. Gill and W.W. Zachary, Phys. Lett. A 128 (1988) 419.
Spatial stability of homogeneous Feigenbaum scenarios in the FMR Wolfgang Weitzel Theoretische Festk6rperphysik, TH Darmstadt, Darmstadt, Germany Received 10 February 1992 Revised manuscript received 3 June 1992 Accepted 3 June 1992 Communicated by F.H. Busse
The dissipative Landau-Lifshitz equation serves as a model for describing the nonlinear dynamics of magnetization inside a transversally driven ferromagnet with uniaxial anisotropy. The consideration of the internal exchange field leads to a partial differential equation possessing spatially homogeneous solutions. The isolated evolution of the uniform mode can exhibit chaotic time-dependence; two Feigenbaum scenarios have been detected. The application of linear stability-analysis to the time-periodic homogeneous solutions shows that spatial instabilities occur before the chaotic state at k = 0 is established.
1. Introduction
The phenomenological equations of motion which are used to describe the dynamics of the magnetization field have proved to be a successful model because they reflect experimental results. An important example are Suhl's spinwave instabilities [1], which occur above a certain parameter threshold and destroy the homogeneous precession. This destabilization of the uniform mode has been verified in numerous measurements of ferromagnetic resonance (FMR) by the detection of changes in the absorption spectrum. Generally, the excitation of spinwaves leads to a rich variety of nonlinear phenomena, which usually exhibit low frequencies (kHzMHz) and which have been observed in many experiments [2]. Effects of this kind occur in the region above the instability threshold when the uniform mode has become unstable, and when the nonlinear interactions between different modes have become effective. Subject of this paper are theoretical examina-
tions of a basic dynamical equation for the magnetization. With the consideration of an uniaxial anisotropy field (essentially: transversal adjustment of the crystal-axis) even the isolated evolution of the uniform mode implies complicated behaviour. In an earlier paper [3], it was shown by Melnikov's method that the uniform mode exhibits homoclinic chaos which is connected with the existence of a strange repellor. This was regarded as a strong hint for the existence of a "measurable" chaos which does not only effect transients but also appears in long-term computer simulations. In this paper, two spatially homogeneous Feigenbaum scenarios are presented and their constituents are now interpreted to be exact solutions of the partial differential equation which is derived from the consideration of the internal exchange field. The possibility of this interpretation determines the outstanding role of the uniform mode and allows the application of linear stability analysis, the framework of which we shall develop in section 3. Treating a local equation, we do not consider boundary
0167-2789/92/$05.00 (~) 1992 - Elsevier Science Publishers B.V. All rights reserved
194
W. Weitzel / Spatial stability of homogeneous Feigenbaum scenarios in the FMR
effects. T h e r e f o r e our results refer to extended samples only, the linear dimensions of which are large compared with a typical magnetic length (e.g. the wavelength of the first excited spinwave). The application of linear stability analysis to the homogeneous subharmonics of the Feigenbaum scenarios was motivated by the question of the existence of a temporal chaos which maintains spatial coherence. In contrast to Suhl's problem we do not expect to find a parameter value, above which a large class of homogeneous solutions becomes spatially unstable. Because of the complexity of our uniform dynamics, it is necessary to consider the behaviour of selected homogeneous solutions.
2. The model-equations
dipolar-interaction has been neglected. Because of the exchange-term we are dealing with a partial differential equation for the magnetization field. Eq. (1) possesses the property of conserving the modulus of magnetization,
IM(r, t){
=
IM(r, 0)l.
(3)
Initially seven parameters appear, but two of them can be removed immediately by a time scaling t--* 3"Ht which means an extraction of the typical Larmor-frequency of the system. For ferromagnets, the exchange constant J has to be positive to ensure minimal energy in the h o m o g e n e o u s case. Of course, the system (1), (2) possesses spatially homogeneous solutions. The (non-autonomous) dynamics of the uniform m o d e occurs on the (two-dimensional) surface of a sphere.
Starting point is the well-known L a n d a u Lifshitz equation [4] 3. Framework of the stability analysis d
cgt M ( r , t) = - 3 ' M × H~ff - AM × (M × Heff),
(1) which serves as a theoretical model for the d a m p e d motion of magnetization inside a ferromagnetic material. We added a sign to the first term, so that for the considered electronic systems the constant 3' represents the modulus of the gyromagnetic ratio. The first term in eq. (1) conserves volume in the phase space while the second term constitutes a dissipative flow describing the damping. The effective field [5] has been chosen in the form Hef f = H e z + h sin t o t e x + A M x e x + J A M ,
In the following we always refer to the timescaled problem. In order to examine the stability of a solution M ~ ( t ) we make the ansatz M(r, t) = M ~ ( t ) + re(r, t) .
(4)
After inserting this ansatz into eqs. (1), (2), bearing in mind that M S ( t ) is an exact solution, one obtains a partial differential equation for the deviation re(r, t), which of course depends on the considered solution M ~ ( t ) . The next step is the linearization of the obtained equation with respect to m. With the following dimensionless quantities:
(2) E~(t)=M~(t)/M
that is, a static field parallel to e~, a sinusoidal driving field along the X-axis and as internal contributions an uniaxial anisotropy-field also adjusted in the x-direction and the exchange field. The anisotropy is assumed to be of crystalline origin only; the nonlocal effect of the
a= h/H , w =- w / T H ,
e,
n(r,t)=m(r,t)/M
b = AM~/3" ,
~,
p = AM~/H ,
(5)
the linearized equation of motion for the deviation reads
195
W. Weitzel / Spatial stability of homogeneous Feigenbaum scenarios in the FMR
0
Ot n(r, t) = H 2 × E ~ + H 1 × n + b E ~ × [H 2 × E ~] + b E ~ × [H I × n] + bn × [ H 1 × E ~ ] ,
(6) where we have used the abbreviations
Accordingly the characteristic matrix can be obtained by integration of the linear system over the range of one period with the initial value ~ ( t 0, t o ) = 1. It can be shown that the eigenvalues of ~ do not depend on the special choice of t o. The condition for asymptotic stability reads now:
H a = e z + a sin totex + p E t e x ,
v=1,2,3. H2
=
Pnxe x + An.
The factor in front of the Laplace operator has been eliminated by a scaling of the length r--->
r.
(7)
An expansion in plane waves n ( r , t) = f d3k c(k, t) e ik'r
(8)
changes eq. (6) into the convenient form
L c(k, t) = ~t[k 2, E~(t), t] c(k, t) 0t
(9)
Of course, in both cases the application to the systems (9) demands an examination of all wavenumbers k which may give rise to a varying stability behaviour. Instability with respect to a particular wave-number k c is sufficient to destroy the homogeneous state and may lead to the excitation of many k-modes when the nonlinear couplings become effective. The special property (3) has an influence on the stability characteristics: There is no effect of the Landau-Lifshitz dynamics on deviations directed parallel to M ~ ( t ) . More precisely, it is possible to show that
which represents an ordinary linear system of dimension three for each wave-number k. The stability of the trivial solution c = 0 and thus the general stability of the linear system, is the subject of linear stability-analysis. In two important cases the asymptotic time dependence is well defined [6]: (i) The matrix ~ does not depend on time. In this case the stability of the linear system is determined by the eigenvalues of ~/. The condition for asymptotic stability reads:
O__Ot[n(r, t ) . E~(t)] = ~0 n . E ~ + n . / ~
Re A ~ ( ~ ) < 0 ,
Al[6~(k)] = 0 .
v=1,2,3.
(ii) The matrix M is time-periodic (Floquetcase): M ( t + r ) = M(t). Then the stability depends on the eigenvalues of a characteristic matrix ~ which is defined through the evolutionmatrix ~g(t, to): c(t) = ~ ( t , to)e(to)--> ~ = ~ ( t o + r, t o ) .
=0
.
(10)
Thus the corresponding component of any Fourier coefficient c(k, t) is a constant: 0
Ot [c(k, t ) . E ~ ( t ) l : 0
Vk.
(11)
In the time-independent case ( d M / d t = 0 ) this result is connected with the existence of an eigenvalue (12)
In the time-periodic case (~/(t + r ) = ~ ( t ) ) eq. (11) gives rise to a characteristic eigenvalue X l [ ~ ( k ) ] = 1.
(13)
The occurrence of these trivial eigenvalues is connected with the structure .of eq. (1) and does
196
W. Weitzel / Spatial stability of homogeneous Feigenbaum scenarios in the FMR
not depend on parameter-values or on the special choice of the solution M~(t). As a consequence asymptotic stability of the linear systems (9) is not admitted. In any case the linear m e t h o d allows the detection of instability. It can proved that the characteristic eigenvalues h ~ ( ~ ) do not change if one turns to a t i m e - d e p e n d e n t tripod ei(t ) (i -- 1, 2, 3) provided that it exhibits the same period as the matrix M(t). In the following we expand the vector c(k, t) in the tripod of spherical coordinates
E~(t),
eo(t),
e~(t),
(14)
the linear stability of the spatially h o m o g e n e o u s solution M~(t), which is characterized by the constant modulus M s and the angles O(t) and ~b(t). The modulus does not a p p e a r explicitly, but is hidden in the p a r a m e t e r s p and b (see eqs. (5)). T h e equation of motion for M~(t) is given by Ms=0, 0 = - p sin 0 sin ~b cos qb - b sin O
+ bp sin 0 cos 0 cos:4~ - a sin w t s i n th + ab sin w t c o s 0 cos ~b,
where we regard E~(t) as the radial unit vector. T h e first c o m p o n e n t of c will be a constant producing the trivial eigenvalue. The remaining two eigenvalues are determined by the reduced dynamics which is created by setting c - E ~ equal to zero. T h e reduced dynamics reads
Ot
c~,
c~
= 1 - p cos 0 cose~b - bp sin th cos ~b - a sin tot cot 0 cos ~b
- a b sin tot sin 10 sin ~b.
(17)
Solutions M " ( t ) touching the z-axis have been treated analogously in Cartesian coordinates, because for 0 = 0 resp. -rr the tripod of spherical coordinates is not defined. H o w e v e r , the representation given above is m o r e compact and illustrates in which way the trivial eigenvalue can be separated.
'
with the matrix elements RI1 = - p sin th cos ~b cos 0 - bk 2 - b cos 0
+ bp cos2~b(1 - 2 sinE0) - a b sin tot sin 0 cos th,
4. Spatial stability of Feigenbaum scenarios
R12 = p ( 1 - 2 cosZth) - k 2 - 2bp sin 4~ cos ~b cos 0 - a sin tot sin-~0 cos ~b
- a b sin wt sin-10 cos 0 sin ~b, RE~ = p sin20 c0s24~ + k 2 + a sin tot sin-~0 cos ~b
+ab sin tot sin-10 cos 0 sin ~b, R : : = p sin ~b cos ~b cos 0 - bk 2 - b cos 0
+ bp sin24~ - bp sin20 cos2~b - a b sin tot sin 0 cos th •
(16)
This two-dimensional linear system determines
T o begin with we want to give some remarks concerning the behaviour of analytically accessible solutions MS(t). In some special cases it is possible to find analytical expressions solving the eqs. (17). Neglecting the driving field by setting a = 0, we can determine the fixed point solutions of the system (17). After insertion of the expressions for E ~ into the matrix ~/ (resp. ~ ) , the stability analysis reduces to a calculation of the eigenvalues, which of course depend on the w a v e - n u m b e r k. It turns out that for our system the normal adjustment of stationary magnetization is 0 = 0. Only in the case of a strong "easyaxis" anisotropy ( p > 1) the spatially stable
w. Weitzel / Spatial stability of homogeneous Feigenbaum scenarios in the FMR equilibrium will be located outside the z-axis (cos 0 = p - 1, ~b = 0 resp. at). The autonomous system without damping (a = b = 0), which we call the unperturbed problem, proves to be completely integrable. In addition to the fixed points we find periodic solutions expressed by Jacobian elliptic functions. With the knowledge of E~(t) the time dependence of the system (15) is well defined and the stability analysis can be carried out by a numerical calculation of the characteristic eigenvalues. It appears that all periodic solutions of the unperturbed problem are spatially unstable. This result has an important consequence on the perturbation theory: Periodic orbits of the unperturbed case (a = b = 0), which are conserved under the influence of a small driving field and a small damping, will be spatially unstable in the same way in the presence of the perturbation, because the characteristic eigenvalues will shift slightly only. We proved the existence of such "resistant" orbits by application of the subharmonic Melnikov-technique [7]. In the general case it is not possible to find analytical' solutions of eq. (17). However, the homogeneous solution M~(t) can be established with the help of a computer. As we want to treat time-periodic systems (9), M~(t) has to be a subharmonic solution of period 2~rm/to (m = 1, 2, 3 . . . . ). It appears that solutions of this kind can be found very easily, since usually the preceding transients die away quickly. Preferably one detects solutions of period 1 (m---1), but also subharmonics of higher order appear. In order to avoid the transients completely, we determine suitable initial values in a precursory computation. With these correct initial values for the system (17), which belong to a well-defined subharmonic solution Me(t), it is possible to integrate the systems (15) and (17) simultaneously regarding them as a four-dimensional system for the dynamical variables c0, c6, 0 and ~b. For a given wave-number k two numerical integrations of this kind over the range of one period 2~rm/to are sufficient to construct the
197
characteristic matrix of the Floquet system (15) (referring to the reduced system (15) we presuppose that Me(t) does not touch the z-axis). In contrast to the unperturbed problem we now can find many periodic solutions Me(t) which do not exhibit spatial instability. In order to study the behaviour for varying parameter values, we proceed as follows: The parameter is changed by small steps and as initial value of each moment we take the adjusted initial value of the preceding step, so that we always stay in the basin of attraction which belongs to the considered subharmonic solution. By this method two Feigenbaum scenarios [8] have been detected and even the period 32 was clearly shown in the computer data. Fig. 1 shows the Feigenbaum scenario 1 up to the period 8. We made use of the stroboscopic Poincar6 map (distance of sectional planes: -0.35
x --0.45 ~
-(
-0.55
I
0.90
0.97
-( 1.04
0.06
fl
-0.02-
..< -0.10
I
0.90
0.97
1.04
1/w Fig. 1. Feigenbaumscenario 1 (p =2, a=0.5, b =0.05).
198
W. Weitzel / Spatial stability of homogeneous Feigenbaum scenarios in the FMR
2~r/~o, location: t o = 0), so that a subharmonic solution of period 2,rrm/oJ appears as a discrete cycle consisting of m points. Furthermore we applied a stereographic projection given by the transformation x=tan½0cos~b,
y=tan½0sin&.
(18)
The varying parameter of scenario 1 is the (reciprocal) frequency of the driving field. The remaining three parameters have been fixed. The examination of the spatial stability showed that the scenario is divided in two parts: The subharmonics of the left part exhibit linear stability with respect to any wave-number k, while all subharmonics of higher order, forming the right part, prove to be spatially unstable. Our numerical investigation of spatial stability reached the period 16. In fig. 1 the critical value (1/oo c = 1.019) is marked by a vertical line. As expected, all subharmonics exhibit linear stability at k = 0, a fact which has been exploited in the numerical construction of M~(t). Clearly, this stability vanishes if one approaches a bifurcation point of the Feigenbaum scenario. The effect of period doubling on the stability behaviour is a squaring of the characteristic matrices ~ ( k ) and thus a squaring of the characteristic eigenvalues. This can be proved by simple considerations if one takes into account that period doubling means a microscopic splitting of the orbit E~(t) only. This statement is confirmed quantitatively by our numerical results. As a consequence, qualitative changes of spatial stability do not occur at the bifurcation points. The detected spatial instability interrupts the route to temporal chaos and leads to the excitation of spin-waves. The nonlinear couplings between different wave-numbers become effective and the dynamics of the uniform mode is changed. Therefore the restriction to the ordinary system (17), which describes the isolated evolution of the uniform mode, is not realistic in the region above the threshold. This range demands an investigation of the complete partial
differential equation, which is not a subject of this paper. Fig. 2 shows another Feigenbaum scenario with the varying parameter a. Here the origin of period 1 is a fixed point solution at a = 0. The stability behaviour proves to be very similar to the first scenario. Again we find a threshold above which spatial instability occurs and once more the critical value (a c = 0.08) is marked by a vertical line. The fact that two scenarios perform a very similar stability behaviour when they are embedded in the partial differential equation suggests strongly that we are facing a typical feature of our magnetic system. In real experiment, the instability might be identified by the disappearance of the subharmonic signal and by the onset of low frequency auto-oscillations which indicate the participation of additional modes.
z o.32-t
\
0.15 0.00
0.13
0.26
a
fil/ -0.02 0.00
0.13
0.26
a
Fig. 2. F e i g e n b a u m scenario 2 ( p = 1.6, b = 0 . 0 8 , ¢o = 1.2).
w. Weitzel / Spatial stability of homogeneous Feigenbaum scenarios in the FMR The dotted line in fig. 2 represents the homoclinic threshold a 0 = 0 . 1 3 2 1 6 . . . , which has been computed by Melnikov's method and which can be regarded as a good approximation since the given perturbation (a < 0 . 3 , b = 0.08) is rather small [3]. Thus for a > a 0 the uniform mode (17) exhibits homoclinic chaos, provided that p, b and to have the values as given in scenario 2. Obviously there are regular subharmonic solutions above the homoclinic threshold. After all they are elements of a Feigenbaum scenario. The p a r a m e t e r values of scenario 1 are located far above the theoretical threshold and probably the whole sequence is embedded in the homoclinic chaos.
199
and the other part, which contains the subharmonics of higher order, being spatially unstable. Therefore one cannot approach the chaotic state without the excitation of spin-waves, the growth of which changes the dynamics of the uniform mode (because of the nonlinear couplings between different wave-numbers). It is possible, that the spatial instabilities interrupt the route to chaos and lead to an orderly state in space-time. On the other hand the excitation of additional modes might be the onset of spin-wave turbulence, which is characterized by a great number of degrees of freedom [9].
Acknowledgement 5. Conclusion The partial differential equation (1), (2) possesses spatially homogeneous solutions MS(t), the linear stability of which we investigated in this paper. For this purpose we linearized the equation of motion for the deviation re(r, t) and then expanded this perturbation in plane waves in order to obtain independent equations of motion for each Fourier coefficient Ck(t ). Thus we examined the stability of homogeneous solutions with respect to spin-wave disturbances of waven u m b e r k. Because of the nonintegrability of the uniform mode (in the general case) we considered special solutions M~(t) established with the help of a computer. We detected two spatially homogeneous Feigenbaum scenarios which consist of subharmonic solutions appearing a f t e r short-lived transients. For solutions of this kind linear stability analysis can be carried out with little numerical effort. It turned out that both scenarios are divided in two parts, one part exhibiting linear stability for all wave-numbers k,
This work was performed within a program of the Sonderforschungsbereich 185 "Nichtlineare D y n a m i k " Frankfurt/Darmstadt. The author wants to express his gratitude to Professor Dr. E. Fick for several helpful discussions.
References [1] H. Suhl, J. Phys. Chem. Solids 1 (1957) 209. ]2] P. Bryant, C. Jeffries and K. Nakamura, Phys. Rev. Lett. 60 (1988) 1185. [3] W. Weitzel and H. Turschner, Phys. Lett. A 140 (1989) 185. [4] L.D. Landau and E.M.-Lifshitz, Phys. Z. Sowjetunion 8 (1935) 153 [see collected papers of L.D. Landau, ed. ter Haar (Pergamon, Oxford, 1965)]. [5] A.I. Akhiezer, V.G. Bar'yakhtar and I.V. Peletminskii, Spin Waves (North-Holland, Amsterdam, 1968). [6] S. Fenyr, T. Frey, Moderne Mathematische Methoden in der Technik, Bd. 1 (Birkh~iuser, Basel, 1967) p. 329. [7] J. Guckenheimer and Ph. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer, Berlin, 1983). ]8] M.J. Feigenbaum, J. Stat. Phys. 19 (1978) 25. I9] T.L. Gill and W.W. Zachary, Phys. Lett. A 128 (1988) 419.