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On the magnetization chaotic dynamics in the ferromagnetic resonance region
AI41
Journal of Magnetism and Magnetic Materials 110 (1992) 202-208 North-Holland
III
I I
On the magnetization chaotic dynamics in the ferromagnetic resonance region A.B. B o r i s o v a, B.N. F i l i p p o v a, V.V. Z v e r e v b a n d B.Ya. R u b i n s t e i n a a Institute of Metal Physics, GSP-170, 620219 Ekatarinburg, Russia h Department of Higher Mathematics, Ural Politechnical Institute, 1(-2, 620002 Ekatarinburg, Russia
Received 16 April 1991
The regular and chaotic motion of the magnetization vector in the uniaxial ferromagnet is investigated. The various configurations of the external magnetic field applied to the magnet are considered. Numerical results are presented and different types of the dynamics are discussed.
1. Introduction In recent years chaotic motion in deterministic nonlinear systems is getting continuous attention. It is evident today that chaotic behaviour is not an exceptional phenomenon and is widely spread in nature. Chaotic motion in the above-mentioned sense is discovered in biological, chemical, physical and other systems (see refs. [1-3]). Although chaotical behaviour is typical for highly nonlinear dynamical systems, usually it is impossible to obtain the corresponding analytical solutions of the dynamical equations, and so the most effective approach for investigations in this field is the numerical analysis of the simple nonlinear models. Now it is realized that low-dimensional models (with discrete or continuous time scale) are an important laboratory for nonlinear studies. In our opinion phenomenological models of magnetic media are appropriate for study of the chaotic dynamics. Various types of nonlinear behaviour as magnetic resonance, spin wave parametric generation, soliton motion, etc. have been studied intensively. However, as we know, routes to chaos and chaotic dynamics (including Feigenbaum period doubling, intermittency and so on) were not observed experimentally and investigated theoretically.
In this paper we shall study the magnetization dynamics in a ferromagnet using the L a n d a u Lifshitz equation [4] (LL model). We restrict our consideration to the spatially homogeneous case and so neglect the spin wave excitation process and other spatio-temporal phenomena. Of course, it may be incorrect in some real physical situations when the magnetization deviation from the initial equilibrium state is not small and the homogeneous rotation is unstable. However, we think that in any case it is important to study qualitative pecularities of dynamics for the simplest and fundamental physical models such as the homogeneous LL model of a ferromagnet. Moreover, we can hope that our approach also gives the proper description for some experimentally realised situations. For example, we can indicate at least one case when homogeneous rotation (without the magnetization breaking and the domain creation) was observed experimentally in a system being far from the equilibrium state; this is the impulse remagnetization of the thin magnetic films (TMF) [5-10]. This phenomenon and also the local remagnetization of the film are interesting in view of practical applications of the T M F in the magneto-optical memory units (see ref. [11]). In the present paper we focus on the LL model for a ferromagnet with
0304-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
A.B. Borisov et aL / On the magnetization chaoticdynamics uniaxial anisotropy. Note that our results may be valid also for ferrimagnets and weak ferromagnets.
2. Equations of motion We consider a spatially homogeneous evolution of the magnetization in a ferromagnet using the Landau-Lifshitz equation with the Hilbert dissipation term [12]: d M / d t = - 7 [ M X Heft] + a M s - ' [ M X d M / d t ] . (1) In this equation 7 is the gyromagnetic constant and a the relaxation parameter; M = E 3- iMiei is the magnetization vector ( I M I = Ms is conserved); vectors • i, i = 1, 2, 3, compose the orthonormal basis in space. In the uniaxial anisotropy case Herf = H ( t ) - / 3 M 3 e 3, where H ( t ) = H o + H a ( t ) is the external time dependent magnetic field a n d / 3 is the anisotropy parameter (we consider an easy axis case (/3 < 0)). Eq. 1 can be rewritten in the form: dm i
1
(
,
(ii) h a = e3hrf cos tar lies along the anisotropy axis and h 0 = elh o lies in the orthogonal (e 2, e 2) plane; (iii) the magnetization is driven by the pulsing magnetic field hrf = elh p ( r ) having the rectangular pulse form; this field lies in the (e l, • 2) plane and has the amplitude hpa, where hpa >>/3M~ (anisotropy of light plane type). The habitual conclusion getting from the linear F M R theory consists in the statement that both lineary oscillating and rotating magnetic fields cause almost identical resonant phenomena. Note that in our highly nonlinear case this statement is wrong. Taking into account the time dependence of the coefficient in (2) and the conservation law I M I =M~ we can deduce that in general the system under investigation has 3 / 2 degrees of freedom. However, if the driving field rotates in the (el, • 2) plane: h t + i h 2 = h + e i'°~ (h+ is constant), the equations of motion (2) takes the form d m + / d r - itam+= [i(1 + i a m 3 ) ( ( h 3 - m 3 ) m + - m 3 h + ) - o t m +(m +h_ -m_h+)/2]/(1
+ a2),
}
(3a)
E Eijkmjqk , 1 + a 2 qi + Otj,k= l
dr
i--1,2,3,
(2)
where qi =
203
E
j,k=l
eUk(~j3mjm,+mjh,),
~ij, is the unit antisymmetric tensor; ~Sik is the Kronecker delta symbol; m = M / M , , ( I m I = 1), h = h o + h r f = H / / 3 M ~ , r = t a o t and t a o = y / 3 M v In this paper we investigate the magnetization behaviour in the long-time asymptotics (i.e. when the initial conditions are forgotten) supposing that the external magnetic field is of the order /3Ms. We analyse several configurations in space: (i) h o = e3h o lies along the anisotropy axis e 3 and hrf=ethrf cos tar lies in the orthogonal (e i, e 2) plane; here h 0 and hrf are the time-independent amplitudes;
I
O
I
-n/q
I
I
I
0
~./~I
X
m
_
I 1
2
1 ! I-, r f ] Fig. 1. Bifurcationdiagramfor the case (i) for h0= 0. The points on the horizontal show X(~'N) for r N N'rrf, N~ 1, 2, 3.... and for fixed hrf values. =
A.B. Borisor et al. / On the magnetization chaotic dynamics
204
LO
' ' ,]l
;~1
U3
d"
,
x
,
,
;21
,
,
'
:11
'
'
0 ' ~0
....
d
Lt3
11
cl
CD
01.5
,
d! q,
O
11
4
2t
i
t:
LD
Fig. 2. Fragments of phase trajectories for the case (i): X(¢) (in units of 7 ) vs. 6 ( ¢ ) (in units of 2 7 ) for the continuous time. Arrows show the direction of the time flow. Oblique crosses appear periodically and mark the time moments ~'N = N~'rf, N = 1, 2 , . . . ; their projections on the ordinate axis are marked with touches. (a) A regular nutation with periodic oscillations for hrf = 0.4; (b) the transition to the motion in both upper and lower hemispheres for hrf z 1.05; regular (c) and chaotic (d) nutation around the axis that lies orthogonally to the anisotropy axis for hrf ~ 1.4 and 2.2 respectively; (e) complex chaotic motion on the sphere for hrf = 2.4.
A.B. Borisov et aL / On the magnetization chaotic dynamics ..J_ ,TJ dm3/d,r
1
= ~[a(h
205
I
3-m3)m+m_
- ft." t-I
i
I
0
~/'--I
X
+ i r a _ h + ( 1 + iam3) + c.c.] /(1
(3b)
+ O~2 )
in the rotating frame (here m + = ( m ~ + im2)e-i'°% m _ = ( m + ) * , h _ = ( h + ) * ) . Coefficients in (3) are time-independent and so we deal now with the nonlinear system having only one degree of freedom. It is well-known that in this case the dynamical stochastization is impossible. In this paper we will investigate more closely a general situation. For any case (i)-(iii) one can integrate the eqs. (2) using numerical procedures, but in the case (iii) it is more convenient to employ the analytical formulae. Let ml(,r ) = cos X(,r) cos 6(,r) = sin 0(,r), m2(,r ) = cos x ( ' r ) sin ~b(±r) = cos 0(,r) sin tb(t), m3(t ) = sin
X(,r)= cos
0 ( r ) cos cb(,r).
°.
1
.
°
Using angular variables defined by (4) we can integrate the eqs. (2) straightforwardly. It is easy to obtain that (a) during the action of the external field (neglecting the anisotropy effects) the result of integration may be expressed as
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
-) r"
f
......
Fig. 3. Bifurcation diagram for case (i) for h 0 = 0.2.
Combining the expressions (5), (6) and using (4), we obtain the two-dimensional evolution map: ON+ I = F ( ON,
(4)
.
°
~N+I
=
~)N; Og, ~, "r]),
(7)
G ( O N ' ~)N; Ol, X, "l']),
where ON - 0(N(Tp + ,r0)), tbN = ~b(N(% + ,r0)). How we can make the calculations using the formulae (7) instead of the numerical integration. Thus we obtain the system's "phase trajectory" in the discrete time scale (with temporal step ,rp + r 0) by means of the step-by-step procedure.
sin 0 ( T + % ) = (th 77 + sin 0(~')) / ( 1 + th 77 sin 0(~')), 4,(,r + % ) = 4 ' ( O
(5)
where 77--hp'rp/(1 + a 2) and rp is the pulse length; (b) when the external field is switched off the solution takes the form sin X ( ¢ + % ) = A
,'~I
0
......
0.18
sin X(T)
/ ( 1 - (1 - A2)sin2x(r)) '/2, g,(,r + ,r0) = 0 ( T )
-.
- n,
(6)
O. 20
-
where ~ ( T ) = a -I ln{[A sin X ( T ) + ( 1 --(1 ~2) s i n 2 x ( , r ) ) l / 2 ] / ( 1 + sin X(T))}, A = exp(--cr%/(1 + a2)) and To is the interpulse gap.
C~
- -
Fig. 4. Period-doubling bifurcations in case (ii). T h e points on the horizontal s h o w X(rN) for a f'uced a value.
206
A.B. Borisoc et al. / On the magnetization chaotic dynamics
3. Numerical studies In this section the dynamical behaviour of m is investigated by numerical solving the eqs. (2). Bifurcation diagrams (see figs. 1, 3 and 5) are constructed as plots of numerically obtained X(~') (see formulae (4)) versus a bifurcation parameter. The points on the horizontal lines represent the values X(z N) for ~'N = N'rrf, N = 1, 2, 3 . . . . ; here ~'rf = 2'rr/tOrf is the period of the external sinusoidal if-field. Also to illustrate the pecularities of motion we use the plots of X(1") vs. ~(~') for the continuous time.
d"
x
al
S
q;
0
0
o0 3
01.6
i
01.6
?-
?d
We first consider bifurcation diagrams and phase trajectories for case (i). An example of the bifurcation diagram is shown in fig. 1. It is assumed that to = 1, h 0 = 0, a -- 0.11 and hrf is the bifurcation parameter. This diagram indicates that there is a period '/'re motion for hrf_<0.7 and 1.3 _ 2.1. The phase trajectories images (figs. 2 a - e ) help us to interpret this diagram. Notice that in the case under investigation the inversion m 3 -~ - m 3 leaves the eqs. (2) unchanged. If the if-field amplitude is weak this leads to existence of two equivalent sorts of nutational motions: in
x
c)
S X
d/
0
?-
?-
Fig. 5. F r a g m e n t s of the p h a s e trajectories for the case (ii): X(~') vs. ~,(~') for the c o n t i n u o u s time. A regular motion with periods: (a) ~'rf for a = 0.206; (b) 2%f for a = 0.20; (c) 4'rrf for a = 0.176. (d) A chaotic motion for a = 0.17.
207
A.B. Boriso~" et al. / On the magnetization chaotic dynamics
the upper hemisphere (0 < X < r r / 2 ) and in the lower hemisphere ( - r r / 2 < X < 0) (the respective rotations around the e 3 axis have the opposite directions). It follows that using numerical integration one can observe some different types of motion. Note that for simplification of the figures we draw the trajectory in a few rectangles defined by the expressions - ' r r / 2 < X < rr/2, 2-rrn < g, < 2"rr(n + 1), n -- 0, + 1, + 2 . . . . . Every rectangle corresponds to the whole sphere; after every trip around the • 3 axis the phase point passes from the initial rectangle to the neighbour one (n ~ n + 1). As a first example, a regular nutation with periodically oscillating X(z) for a weak incident field is shown in fig. 2a (the situation that the trajectory is placed in the upper or lower hemisphere depends on the initial conditions). The
(a)
increase of the if-field intensity causes abrupt complication of the phase trajectories. Now the phase point often passes an equator and moves chaotically in both the upper and lower hemispheres; this is shown in fig. 2b (note that a qualitatively similar behaviour have been discovered for the first time in the well-known Lorenz model [13]). A further increase of the rf-field amplitude causes the regular (fig. 2c) and chaotic (fig 2d) nutations around the e 2 axis, and then the combined chaotic nutation around e 2 and • 3 axes (fig. 2e). One of the bifurcation diagrams for the case (i) for h 0 * 0 is shown in fig. 3; it contains the pattern similar to that of in fig. 1. The bifurcation diagram in the case (ii) with h 0 -- 0.6, h r f -- 0.2, ~o = 0.7 is shown in fig. 4; here a is the bifurcation parameter. In this case we have seen the route to chaos through the se-
,,,.-
II P "
(b)
°
/
/(
'7.
," .
(c)
,,..'
!
°
°
"!
)/
,r o
.j~ i'/"
te
7 ."
! o°o
:
\.
Fig. 6. Phase-portraits constructed by using the map iteration procedure.
o
208
A.B. Borisor et al. / On the magnetization chaotic dynamics
0
0.65
t
'
"0.1
1 ~ ,o
1 -4.5
!" ~HO'!.o~ "~' . ~
that if the MLE is negative then the nearby trajectories converge onto the attractor and the motion is regular; the positive MLE corresponds to the chaotic motion. From fig. 7 one can see the alternation of the regular and chaotic motion zones as 3, changes. Graphs corresponding to the different components of the attractor are marked by points and rings respectively.
r' 0.65 ". 0;67
:°it ~ . !1
"~5
Fig. 7. The maximal Lyapunov exponent for the case (iii) There is the 10-cycle for 0.552 < ,~ < 0.56 and the 12-cycle for 0.564 < ,~ < 0.569. Supplementary frame contains the enlarged fragment of the plot.
quence of the period-doubling bifurcations; the first three bifurcations can be identified in fig. 4. The examples of the phase-portraits of various regimes from periodical with period zrf to chaotic via period-doubling are shown in figs. 5a-d. Note that fig. 5d resembles the well-known Lorenz attractor. In the last case (iii) we have constructed the system's phase portraits (fig. 6) consisting of the isolated phase points (X(rN), ~(TN)), z N = N'rrf. Calculations have been carried out by applying formulae (7) for a = 0.3, r / = 0.785, h is variable. In this case it is easy to observe various attractor transformations including the attractor components merging. Namely, for A = 0.627 (the chaotic motion) there are two symmetrically placed attractor components; either of them takes the form of a loop and is contained in his own basin of attraction (in fig. 6a only one of these components is shown). With A decreases the attractor components merge and this leads to the formation of the new two-component attractor. This attractor consists of the chaotic component (shown in figs. 6 b - d ) and the stable point. It is convenient to combine the phase-portraits analysis with the calculation of the maximal Lyapunov exponent (MLE) [1]. We have to remind
4. Concluding remarks We showed that it is possible to observe the complex regimes of motion even for a simplest model describing the spatial homogeneous magnetization movement in ferromagnets. This system demonstrates some well-known features of motion (such as the cascades of the subharmonic bifurcations and the appearance of the multicomponent attractors) which are typical for nonlinear systems. Thus the magnetization dynamics under the external field action would be interesting to study more comprehensively.
References [1] A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983). [2] H.G. Schuster, Deterministic Chaos (Physik-Verlag, Weinheim, 1984). [3] F.C. Moon, Chaotic Vibrations (John Wiley, New York, 1987). [4] F.H. de Leeuw, R. van der Doel and U. Enz, Rep. Progr. Phys. 43 (1980) 689. [5] G.M. Rodichkov, N.P. Lyakhovskij, P.D. Kim and V.N. Presnetsov, |zvestija Vuzov (Fizika) N7 (1969) 140 (in Russian). [6] C.D. OIson and A.V. Pohm, J. Appl. Phys. 29 (1958) 274. [7] P.R. Gillette and K. Oshima, J. Appl. Phys. 29 (1958) 1465. [8] W. Dietrich and W. Proebster, Electronische Rundschau 14 (1960) 47. [9] D.O. Smith, Phys. Rev. 104 (1956) 1280. ll0] G. Vogler, Phys. Stat. Sol. 2 (1962) 1241. [ll] U. Rosier, Phys. Stat. Sol. 3 (1963) 3. [12] T.L. Gilbert, Phys. Rev. 100 (1955) 1243. [13] E.N. Lorentz, J. Atmos. Sci. 20 (1963) 130.
Journal of Magnetism and Magnetic Materials 110 (1992) 202-208 North-Holland
III
I I
On the magnetization chaotic dynamics in the ferromagnetic resonance region A.B. B o r i s o v a, B.N. F i l i p p o v a, V.V. Z v e r e v b a n d B.Ya. R u b i n s t e i n a a Institute of Metal Physics, GSP-170, 620219 Ekatarinburg, Russia h Department of Higher Mathematics, Ural Politechnical Institute, 1(-2, 620002 Ekatarinburg, Russia
Received 16 April 1991
The regular and chaotic motion of the magnetization vector in the uniaxial ferromagnet is investigated. The various configurations of the external magnetic field applied to the magnet are considered. Numerical results are presented and different types of the dynamics are discussed.
1. Introduction In recent years chaotic motion in deterministic nonlinear systems is getting continuous attention. It is evident today that chaotic behaviour is not an exceptional phenomenon and is widely spread in nature. Chaotic motion in the above-mentioned sense is discovered in biological, chemical, physical and other systems (see refs. [1-3]). Although chaotical behaviour is typical for highly nonlinear dynamical systems, usually it is impossible to obtain the corresponding analytical solutions of the dynamical equations, and so the most effective approach for investigations in this field is the numerical analysis of the simple nonlinear models. Now it is realized that low-dimensional models (with discrete or continuous time scale) are an important laboratory for nonlinear studies. In our opinion phenomenological models of magnetic media are appropriate for study of the chaotic dynamics. Various types of nonlinear behaviour as magnetic resonance, spin wave parametric generation, soliton motion, etc. have been studied intensively. However, as we know, routes to chaos and chaotic dynamics (including Feigenbaum period doubling, intermittency and so on) were not observed experimentally and investigated theoretically.
In this paper we shall study the magnetization dynamics in a ferromagnet using the L a n d a u Lifshitz equation [4] (LL model). We restrict our consideration to the spatially homogeneous case and so neglect the spin wave excitation process and other spatio-temporal phenomena. Of course, it may be incorrect in some real physical situations when the magnetization deviation from the initial equilibrium state is not small and the homogeneous rotation is unstable. However, we think that in any case it is important to study qualitative pecularities of dynamics for the simplest and fundamental physical models such as the homogeneous LL model of a ferromagnet. Moreover, we can hope that our approach also gives the proper description for some experimentally realised situations. For example, we can indicate at least one case when homogeneous rotation (without the magnetization breaking and the domain creation) was observed experimentally in a system being far from the equilibrium state; this is the impulse remagnetization of the thin magnetic films (TMF) [5-10]. This phenomenon and also the local remagnetization of the film are interesting in view of practical applications of the T M F in the magneto-optical memory units (see ref. [11]). In the present paper we focus on the LL model for a ferromagnet with
0304-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
A.B. Borisov et aL / On the magnetization chaoticdynamics uniaxial anisotropy. Note that our results may be valid also for ferrimagnets and weak ferromagnets.
2. Equations of motion We consider a spatially homogeneous evolution of the magnetization in a ferromagnet using the Landau-Lifshitz equation with the Hilbert dissipation term [12]: d M / d t = - 7 [ M X Heft] + a M s - ' [ M X d M / d t ] . (1) In this equation 7 is the gyromagnetic constant and a the relaxation parameter; M = E 3- iMiei is the magnetization vector ( I M I = Ms is conserved); vectors • i, i = 1, 2, 3, compose the orthonormal basis in space. In the uniaxial anisotropy case Herf = H ( t ) - / 3 M 3 e 3, where H ( t ) = H o + H a ( t ) is the external time dependent magnetic field a n d / 3 is the anisotropy parameter (we consider an easy axis case (/3 < 0)). Eq. 1 can be rewritten in the form: dm i
1
(
,
(ii) h a = e3hrf cos tar lies along the anisotropy axis and h 0 = elh o lies in the orthogonal (e 2, e 2) plane; (iii) the magnetization is driven by the pulsing magnetic field hrf = elh p ( r ) having the rectangular pulse form; this field lies in the (e l, • 2) plane and has the amplitude hpa, where hpa >>/3M~ (anisotropy of light plane type). The habitual conclusion getting from the linear F M R theory consists in the statement that both lineary oscillating and rotating magnetic fields cause almost identical resonant phenomena. Note that in our highly nonlinear case this statement is wrong. Taking into account the time dependence of the coefficient in (2) and the conservation law I M I =M~ we can deduce that in general the system under investigation has 3 / 2 degrees of freedom. However, if the driving field rotates in the (el, • 2) plane: h t + i h 2 = h + e i'°~ (h+ is constant), the equations of motion (2) takes the form d m + / d r - itam+= [i(1 + i a m 3 ) ( ( h 3 - m 3 ) m + - m 3 h + ) - o t m +(m +h_ -m_h+)/2]/(1
+ a2),
}
(3a)
E Eijkmjqk , 1 + a 2 qi + Otj,k= l
dr
i--1,2,3,
(2)
where qi =
203
E
j,k=l
eUk(~j3mjm,+mjh,),
~ij, is the unit antisymmetric tensor; ~Sik is the Kronecker delta symbol; m = M / M , , ( I m I = 1), h = h o + h r f = H / / 3 M ~ , r = t a o t and t a o = y / 3 M v In this paper we investigate the magnetization behaviour in the long-time asymptotics (i.e. when the initial conditions are forgotten) supposing that the external magnetic field is of the order /3Ms. We analyse several configurations in space: (i) h o = e3h o lies along the anisotropy axis e 3 and hrf=ethrf cos tar lies in the orthogonal (e i, e 2) plane; here h 0 and hrf are the time-independent amplitudes;
I
O
I
-n/q
I
I
I
0
~./~I
X
m
_
I 1
2
1 ! I-, r f ] Fig. 1. Bifurcationdiagramfor the case (i) for h0= 0. The points on the horizontal show X(~'N) for r N N'rrf, N~ 1, 2, 3.... and for fixed hrf values. =
A.B. Borisor et al. / On the magnetization chaotic dynamics
204
LO
' ' ,]l
;~1
U3
d"
,
x
,
,
;21
,
,
'
:11
'
'
0 ' ~0
....
d
Lt3
11
cl
CD
01.5
,
d! q,
O
11
4
2t
i
t:
LD
Fig. 2. Fragments of phase trajectories for the case (i): X(¢) (in units of 7 ) vs. 6 ( ¢ ) (in units of 2 7 ) for the continuous time. Arrows show the direction of the time flow. Oblique crosses appear periodically and mark the time moments ~'N = N~'rf, N = 1, 2 , . . . ; their projections on the ordinate axis are marked with touches. (a) A regular nutation with periodic oscillations for hrf = 0.4; (b) the transition to the motion in both upper and lower hemispheres for hrf z 1.05; regular (c) and chaotic (d) nutation around the axis that lies orthogonally to the anisotropy axis for hrf ~ 1.4 and 2.2 respectively; (e) complex chaotic motion on the sphere for hrf = 2.4.
A.B. Borisov et aL / On the magnetization chaotic dynamics ..J_ ,TJ dm3/d,r
1
= ~[a(h
205
I
3-m3)m+m_
- ft." t-I
i
I
0
~/'--I
X
+ i r a _ h + ( 1 + iam3) + c.c.] /(1
(3b)
+ O~2 )
in the rotating frame (here m + = ( m ~ + im2)e-i'°% m _ = ( m + ) * , h _ = ( h + ) * ) . Coefficients in (3) are time-independent and so we deal now with the nonlinear system having only one degree of freedom. It is well-known that in this case the dynamical stochastization is impossible. In this paper we will investigate more closely a general situation. For any case (i)-(iii) one can integrate the eqs. (2) using numerical procedures, but in the case (iii) it is more convenient to employ the analytical formulae. Let ml(,r ) = cos X(,r) cos 6(,r) = sin 0(,r), m2(,r ) = cos x ( ' r ) sin ~b(±r) = cos 0(,r) sin tb(t), m3(t ) = sin
X(,r)= cos
0 ( r ) cos cb(,r).
°.
1
.
°
Using angular variables defined by (4) we can integrate the eqs. (2) straightforwardly. It is easy to obtain that (a) during the action of the external field (neglecting the anisotropy effects) the result of integration may be expressed as
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
-) r"
f
......
Fig. 3. Bifurcation diagram for case (i) for h 0 = 0.2.
Combining the expressions (5), (6) and using (4), we obtain the two-dimensional evolution map: ON+ I = F ( ON,
(4)
.
°
~N+I
=
~)N; Og, ~, "r]),
(7)
G ( O N ' ~)N; Ol, X, "l']),
where ON - 0(N(Tp + ,r0)), tbN = ~b(N(% + ,r0)). How we can make the calculations using the formulae (7) instead of the numerical integration. Thus we obtain the system's "phase trajectory" in the discrete time scale (with temporal step ,rp + r 0) by means of the step-by-step procedure.
sin 0 ( T + % ) = (th 77 + sin 0(~')) / ( 1 + th 77 sin 0(~')), 4,(,r + % ) = 4 ' ( O
(5)
where 77--hp'rp/(1 + a 2) and rp is the pulse length; (b) when the external field is switched off the solution takes the form sin X ( ¢ + % ) = A
,'~I
0
......
0.18
sin X(T)
/ ( 1 - (1 - A2)sin2x(r)) '/2, g,(,r + ,r0) = 0 ( T )
-.
- n,
(6)
O. 20
-
where ~ ( T ) = a -I ln{[A sin X ( T ) + ( 1 --(1 ~2) s i n 2 x ( , r ) ) l / 2 ] / ( 1 + sin X(T))}, A = exp(--cr%/(1 + a2)) and To is the interpulse gap.
C~
- -
Fig. 4. Period-doubling bifurcations in case (ii). T h e points on the horizontal s h o w X(rN) for a f'uced a value.
206
A.B. Borisoc et al. / On the magnetization chaotic dynamics
3. Numerical studies In this section the dynamical behaviour of m is investigated by numerical solving the eqs. (2). Bifurcation diagrams (see figs. 1, 3 and 5) are constructed as plots of numerically obtained X(~') (see formulae (4)) versus a bifurcation parameter. The points on the horizontal lines represent the values X(z N) for ~'N = N'rrf, N = 1, 2, 3 . . . . ; here ~'rf = 2'rr/tOrf is the period of the external sinusoidal if-field. Also to illustrate the pecularities of motion we use the plots of X(1") vs. ~(~') for the continuous time.
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A.B. Boriso~" et al. / On the magnetization chaotic dynamics
the upper hemisphere (0 < X < r r / 2 ) and in the lower hemisphere ( - r r / 2 < X < 0) (the respective rotations around the e 3 axis have the opposite directions). It follows that using numerical integration one can observe some different types of motion. Note that for simplification of the figures we draw the trajectory in a few rectangles defined by the expressions - ' r r / 2 < X < rr/2, 2-rrn < g, < 2"rr(n + 1), n -- 0, + 1, + 2 . . . . . Every rectangle corresponds to the whole sphere; after every trip around the • 3 axis the phase point passes from the initial rectangle to the neighbour one (n ~ n + 1). As a first example, a regular nutation with periodically oscillating X(z) for a weak incident field is shown in fig. 2a (the situation that the trajectory is placed in the upper or lower hemisphere depends on the initial conditions). The
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increase of the if-field intensity causes abrupt complication of the phase trajectories. Now the phase point often passes an equator and moves chaotically in both the upper and lower hemispheres; this is shown in fig. 2b (note that a qualitatively similar behaviour have been discovered for the first time in the well-known Lorenz model [13]). A further increase of the rf-field amplitude causes the regular (fig. 2c) and chaotic (fig 2d) nutations around the e 2 axis, and then the combined chaotic nutation around e 2 and • 3 axes (fig. 2e). One of the bifurcation diagrams for the case (i) for h 0 * 0 is shown in fig. 3; it contains the pattern similar to that of in fig. 1. The bifurcation diagram in the case (ii) with h 0 -- 0.6, h r f -- 0.2, ~o = 0.7 is shown in fig. 4; here a is the bifurcation parameter. In this case we have seen the route to chaos through the se-
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that if the MLE is negative then the nearby trajectories converge onto the attractor and the motion is regular; the positive MLE corresponds to the chaotic motion. From fig. 7 one can see the alternation of the regular and chaotic motion zones as 3, changes. Graphs corresponding to the different components of the attractor are marked by points and rings respectively.
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quence of the period-doubling bifurcations; the first three bifurcations can be identified in fig. 4. The examples of the phase-portraits of various regimes from periodical with period zrf to chaotic via period-doubling are shown in figs. 5a-d. Note that fig. 5d resembles the well-known Lorenz attractor. In the last case (iii) we have constructed the system's phase portraits (fig. 6) consisting of the isolated phase points (X(rN), ~(TN)), z N = N'rrf. Calculations have been carried out by applying formulae (7) for a = 0.3, r / = 0.785, h is variable. In this case it is easy to observe various attractor transformations including the attractor components merging. Namely, for A = 0.627 (the chaotic motion) there are two symmetrically placed attractor components; either of them takes the form of a loop and is contained in his own basin of attraction (in fig. 6a only one of these components is shown). With A decreases the attractor components merge and this leads to the formation of the new two-component attractor. This attractor consists of the chaotic component (shown in figs. 6 b - d ) and the stable point. It is convenient to combine the phase-portraits analysis with the calculation of the maximal Lyapunov exponent (MLE) [1]. We have to remind
4. Concluding remarks We showed that it is possible to observe the complex regimes of motion even for a simplest model describing the spatial homogeneous magnetization movement in ferromagnets. This system demonstrates some well-known features of motion (such as the cascades of the subharmonic bifurcations and the appearance of the multicomponent attractors) which are typical for nonlinear systems. Thus the magnetization dynamics under the external field action would be interesting to study more comprehensively.
References [1] A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983). [2] H.G. Schuster, Deterministic Chaos (Physik-Verlag, Weinheim, 1984). [3] F.C. Moon, Chaotic Vibrations (John Wiley, New York, 1987). [4] F.H. de Leeuw, R. van der Doel and U. Enz, Rep. Progr. Phys. 43 (1980) 689. [5] G.M. Rodichkov, N.P. Lyakhovskij, P.D. Kim and V.N. Presnetsov, |zvestija Vuzov (Fizika) N7 (1969) 140 (in Russian). [6] C.D. OIson and A.V. Pohm, J. Appl. Phys. 29 (1958) 274. [7] P.R. Gillette and K. Oshima, J. Appl. Phys. 29 (1958) 1465. [8] W. Dietrich and W. Proebster, Electronische Rundschau 14 (1960) 47. [9] D.O. Smith, Phys. Rev. 104 (1956) 1280. ll0] G. Vogler, Phys. Stat. Sol. 2 (1962) 1241. [ll] U. Rosier, Phys. Stat. Sol. 3 (1963) 3. [12] T.L. Gilbert, Phys. Rev. 100 (1955) 1243. [13] E.N. Lorentz, J. Atmos. Sci. 20 (1963) 130.