Hard transition to chaotic dynamics in Alfve´n wave fronts
J. R. Sanmartin, O. Lopez-Rebollal, and E. del Rio
Escuela Tecnica Superior de Ingenieros Aeronauticos, Universidad Politecnica de Madrid, Madrid, Spain
S. Elaskar
Departamento de Aeronautica, Universidad Nacional de Cordoba, Cordoba, Argentina
共Received 7 August 2003; accepted 4 February 2004; published online 15 April 2004兲
The derivative nonlinear Schro¨dinger 共DNLS兲 equation, describing propagation of circularly polarized Alfve´n waves of finite amplitude in a cold plasma, is truncated to explore the coherent, weakly nonlinear, cubic coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. In a reduced three-wave model共equal dampings of daughter waves, three-dimensional flow for two wave amplitudes and one relative phase兲, no matter how small the growth rate of the unstable wave there exists a parametric domain with the flow exhibiting chaotic relaxation oscillations that are absent for zero growth rate. This hard transition in phase-space behavior occurs for left-hand 共LH兲 polarized waves, paralleling the known fact that only LH time-harmonic solutions of the DNLS equation are modulationally unstable, with damping less than about 共unstable wave frequency兲2/4⫻ion cyclotron frequency. The structural stability of the transition was explored by going into a fully 3-wave model共different dampings of daughter waves, four-dimensional flow兲; both models differ in significant phase-space features but keep common features essential for the transition. © 2004 American Institute of Physics.
关DOI: 10.1063/1.1691453兴
I. INTRODUCTION
Nonlinear Alfve´n-wave interactions may be present in astrophysical, space and laboratory plasmas, with effects that range from heating to driving of current. A recent space ex-ample involves orbiting conductive tethers, which, if in elec-trical contact with the ionosphere, radiate Alfve´n waves.1A steady current flowing along a tether results in continual charge exchange with the ambient plasma, circuit closure being accomplished by charge-carrying Alfve´n waves ex-cited in the ionosphere by the passage of the system. Wave structures 共called Alfve´n wings兲 attached at or near both tether ends present an Airy-functions behavior if linearly described.2 Nonlinear effects, which should appear at the near wave front, might be affected by the magnetic self-field generated by the very current of the tether.3
An Airy-like linear structure is a feature common in fronts of dispersive waves radiated by a moving source, as with ion-acoustic waves excited by a charged body moving mesothermally in a low ion-temperature plasma; the nonlin-ear wave front is then described by the Korteweg–de Vries equation.4In the case of Alfve´n waves, some strong nonlin-ear effects are known to be described by the derivative non-linear Schro¨dinger 共DNLS兲 equation,5 which admits soliton solutions6and has proved amenable to the inverse scattering method for obtaining general solutions.7A variety of behav-iors allowed by the DNLS equation and its modifications have been analyzed.8
Here we show how the DNLS equation may also serve to describe weak nonlinear effects represented by the coher-ent coupling of a few waves, and we explore its complex dynamics. The local, coherent interaction of three waves at or near resonance共3WRI兲is an ubiquitous feature of
nonlin-ear mediums. The 3WRI is especially important in both un-magnetized and un-magnetized plasmas, where dispersive ef-fects can keep nonlinearities weak and electromagnetic waves make coupling to external energy sources easy. The 3WRI has been extensively studied and remains a basic non-linear paradigm.9
The 3WRI evolution for lossless quadratic coupling, with a mode linearly unstable and the two other modes equally damped, is described by a three-dimensional 共3D兲 flow of two wave amplitudes and one relative phase 共a
re-duced 3-wave interaction兲. If damping rates exceed the growth rate⌫of the unstable mode the system is attracted to point sets of vanishing 3D volume, and its long-time behav-ior may be chaotic.10–12For small⌫, a consistent analysis of the flow using a multiple time-scales method, led to an 1D chaotic map.13Actually, the system exhibits a hard transition to complex phase-space dynamics: no matter how small⌫⬎0 there exists a fully developed attractor that is absent at ⌫⭐0 and is chaotic for some parametric domain; this is an ex-ample of a broad scenario for chaos also present in the reso-nant coupling of two oscillators at frequency ratio 2:q, q integer, with the first oscillator unstable.14For quadratic cou-pling共corresponding to q⫽1 in the two-oscillator case兲, the
hard transition and the effects of noise have been
experimen-tally verified using electronic oscillators;15 also, the hard transition was found to persist when the daughter waves had unequal dampings, the flow then being 4D rather than 3D.16,17
Cubic interaction, corresponding to q⫽2 共or 1:1 fre-quency ratio兲in the two-oscillator case, allows a variety of coupling structures. A reduced 3-wave truncation of the non-linear Schro¨dinger equation showed chaotic behavior at finite
2026
⌫;18 a hard transition was encountered in a two-oscillator model of a spherical swing.14,19In the present paper we ex-plore weakly nonlinear dynamics in a truncation of the DNLS equation that shows more complex cubic coupling, in both reduced and full 3-wave models. We want, in particular, to ascertain whether gross features in dynamical behavior, like fully developed phase-space attractors at ⌫⫽0⫹, are structurally stable, as suggested by their appearance for qua-dratic coupling in both 3D and 4D flows, and for the particu-lar cubic coupling of Refs. 14 and 19.
Structural stability would be important because a 3WRI model may fail on a number of conditions it requires. Coher-ence is lost, leading to a random-phase approximation, when the interaction time exceeds the inverse frequency widths of the modes.20,21 The 3WRI model will also, strictly, fail for wave amplitudes so large that the interaction time drops to values comparable with wave periods.22Dynamics just tem-poral, rather than spatiotemtem-poral, may require long uniform wave trains or standing waves;23 multiple waves may be involved.24,25
In Sec. II we present the reduced 3-wave model of the DNLS equation. In Sec. III we analytically determine the
⌫⫽0 attractor of the system. In Sec. IV we derive analytical and numerical results on the small, positive ⌫ attractor共s兲. The full 3-wave model共daughter waves with different damp-ings兲 is considered in Sec. V. A discussion is given in Sec. VI.
II. REDUCED 3-WAVE MODEL OF THE DNLS EQUATION
The derivative nonlinear Schro¨dinger equation describes the evolution of circularly polarized Alfve´n waves of finite amplitude propagating along an unperturbed uniform mag-netic field in a cold, homogeneous and lossless plasma. The description uses a two-fluid, quasineutral approximation
共with electron inertia and current displacement neglected兲. Taking the unperturbed magnetic field B0 in the z direction, the DNLS equation reads5– 8
t ⫹
z
冋
冉
1⫹ 兩兩24
冊册
⫾i
2 2
z2 ⫹␥
ˆ⫽0, 共1兲
where, t, and z are dimensionless perturbed field and vari-ables,
⬅Bx⫾iBy
B0
, cit→t, ci VA
z→z, 共2兲
ci is the ion cyclotron frequency and VA is the Alfve´n
ve-locity. The upper共lower兲sign in Eqs.共1兲and共2兲corresponds to a left-hand 共right-hand兲 circularly polarized wave propa-gating in the z direction; ␥ˆ would be some appropriate
growth/damping linear operator.18 Equation 共1兲 is derived under the following ordering scheme for perturbed quantities
共n andvz are plasma density and velocity along the z axis兲:
Bx
B0⬇ By
B0⬇
冑
n⫺n0n0 ⬇
冑
vz
VA.
To study weakly nonlinear interactions, we consider an approximate solution of Eq. 共1兲consisting of three traveling waves,
⫽2
兺
j⫽1 3
j共t兲eij, j⫽kjz⫺jt, 共3兲
satisfying a resonance condition 2k1⫽k2⫹k3. Wave number and frequency of modes are related by the linear 共lossless兲 dispersion relation for circularly polarized Alfve´n waves at low wave number, j⫽kj⫿kj
2
/2, or in dimensional form (→/ci, k→kVA/ci, k⬅kz⬎0),
j⫽VAkj
冉
1⫿1
2
VAkj
ci
冊
. 共4兲
Both the growth/damping and the nonlinear term in共1兲make the complex amplitudesj vary slowly in time. Introducing
Eq. 共3兲in共1兲and considering only the k1, k2, and k3
com-ponents one arrives at
˙
1⫹␥11⫹ik1关共兩1兩2⫹2兩2兩2⫹2兩3兩2兲1
⫹21*23eit兴⫽0, 共5a兲 ˙
2⫹␥22⫹ik2关共2兩1兩2⫹兩2兩2⫹2兩3兩2兲2
⫹1 2
3
*e⫺it兴⫽0, 共5b兲
˙
3⫹␥33⫹ik3关共2兩1兩2⫹2兩2兩2⫹兩3兩2兲3
⫹1 2
2
*e⫺it兴⫽0, 共5c兲 where ˙j is dj/dt and ⬅21⫺2⫺3 is a frequency
mismatch. We assume that all other components, in particu-lar those involving wave numbers 2k2⫺k1, 2k3⫺k1, 2k2
⫺k3, and 2k3⫺k2, arising from using共3兲in the nonlinear term of Eq. 共1兲, are strongly damped.18
Setting j(t)⫽aj(t)exp关ij(t)兴 in Eqs. 共5a兲–共5c兲 with
aj, j real, and using the resonance condition, the above three complex equations are reduced to four real equations,
a˙1⫽⫺␥1a1⫺共k2⫹k3兲a1a2a3sin, 共6a兲 a˙2⫽⫺␥2a2⫹k2a1
2a
3sin, 共6b兲 a˙3⫽⫺␥3a3⫹k3a12a2sin, 共6c兲
˙⫽⫹
冋
a 1 2冉
k2 a3 a2
⫹k3 a2 a3
冊
⫺2共k2⫹k3兲a2a3
册
⫻cos⫺k2关a12⫺a22兴⫺k3关a12⫺a32兴, 共6d兲 where⬅⫹t⫹2⫹3⫺21.
At this point and throughout Secs. II–IV we restrict the analysis, for simplicity, to the case␥2⫽␥3⬅␥ and return to
the full 3-wave system in Sec. V. Multiplying Eq. 共6b兲 by 2k3a2, Eq. 共6c兲by 2k2a3, and subtracting from each other
there results
d dt共k3a2
2⫺k 2a3
2兲⫽⫺2␥共k 3a2
2⫺k 2a3
2兲; 共7兲
Eq.共7兲shows k3a22⫺k2a32 共but not a22or a32) to decay expo-nentially with time. For a study of the long-time behavior of the system, we may then take k3a2
2⫽ k2a3
2
Note that the frequency mismatch is positive and negative for left-hand共LH兲and right-hand共RH兲polarization, respec-tively, using Eq.共4兲one finds in dimensional form
ci
⬇⫾
冉
1ci
冊
2
冉
k2⫺k3 k2⫹k3冊
2
. 共8兲
This sign difference will later be shown to lead to fundamen-tally different dynamics for the two polarizations. Finally we may both take k3⬍k2and equal signs for a2 and a3with no
loss of generality共for opposite signs, setting→⫹would again leave the system invariant兲; also, we may take all three
a1, a2, a3 positive.
Writing␥1⬅⫺⌫⬍0 and introducing new variables
冑
k2k3a1 2→a1 2, 共k
2⫹k3兲
冑
k3/k2a2 2→a2 2,
system 共6a兲–共6d兲 is reduced to three real nonlinear equa-tions,
a˙1⫽⌫a1⫺a1a2
2sin, 共9a兲
a˙2⫽⫺␥a2⫹a1 2
a2sin, 共9b兲
˙⫽⫺2共a 1 2⫺a
2
2兲共V¯⫺cos兲⫺a 2
2/V¯ , 共9c兲
where
V
¯⬅1⫹k3/k2
2
冑
k3/k2⬎1,
冉
k3k2
⬍1
冊
. 共10兲The limit case V¯⫽1 would exactly recover a truncation of the 1D nonlinear Schro¨dinger equation describing the para-metric excitation of linearly damped waves by the oscillating two-stream instability in plasmas.18We also note that some resonant interactions of two oscillators with frequency ratio 1:1, which have been analyzed by Lo´pez-Rebollal and San-martin, are described by system共9a兲–共9c兲with the last term in共9c兲missing.14,19
III.⌫Ä0 ATTRACTOR FOR THE REDUCED 3-WAVE MODEL
In this section we discuss analytical results that can be readily obtained from system共9a兲–共9c兲and we determine its
⌫⫽0 attractor. A trivial result concerns the flow divergence in共3D兲phase-space a12, a22, and, reading
a12
da1 2
dt ⫹
a22
da2 2 dt ⫹  d
dt⫽2共⌫⫺␥兲.
Nonlinear conservative coupling naturally preserves volume. For ⌫⬍␥, as assumed here, the long-time attractor of the system will be a point-set of vanishing 3D volume.
Again from system共9a兲–共9c兲one obtains equations that would represent conservation laws in the no-dissipation limit,
d dt共a1
2⫹
a22兲⫽2⌫a12⫺2␥a22, 共11兲
d dt
冋
a22
冉
h 0⫺a22
2V¯
冊册
⫽2⌫a22共h
0⫺兲⫺2␥a2 2
冉
h0⫺ a22
V ¯
冊
,共12兲
where
h0共a1,兲⬅˙共a2⫽0兲⫽⫺2a12共V¯⫺cos兲. 共13兲 For ⌫⫽␥⫽0, Eq. 共11兲 rewritten in the original variables would be a Manley–Rowe relation for conservation of action density in wave packets that follow Hamiltonian dynamics. Equation 共7兲 would be a second Manley–Rowe relation: since phasesj only enter Eqs.共6a兲–共6d兲through the
com-bination 21⫺2⫺3, there would be two cyclic angular
coordinates in the full 3-wave case. Equation 共12兲 with
⌫⫽␥⫽0, in turn, can be shown to express energy conserva-tion when the constant-acconserva-tion laws are taken into account.
We note next that Eq.共9b兲shows the plane a2⫽0 to be invariant. When combined with Eq. 共12兲it yields
d dt
冋
h0⫺a22
2V¯
册
⫽⫺2a12sin⫻
冋
h 0⫺a22
2V¯
册
⫹␥a2 2
V
¯ ⫹2⌫共h0⫺兲. 共14兲
In the conservative case, the surface h0(a1,)⫺a2 2
/2V¯⫽0, which only exists for frequency mismatch⬎0, i.e., for LH polarization, would be invariant. In what follows we will only consider LH polarization.
For⌫⬍0, Eq. 共11兲 proves the equilibrium state a1⫽a2
⫽0 to be a global attractor. This equilibrium is unstable for
⌫⬎0. In this section we consider the long-time attractor of system 共9a兲–共9c兲at ⌫⫽0. Note that the entire flow is now asymptotic to the surface a2⫽0, because a1
2⫹ a2
2
will keep diminishing in Eq.共11兲unless a2vanishes. Since that surface
is invariant, trajectories should be asymptotic to its critical elements with transverse stable manifolds.
Consider then the flow on a2⫽0 at⌫⫽0, Eq.共9a兲then yielding a1⫽constant. The intersection of the plane a2⫽0 and the cylindrical surface h0(a1,)⫽0 is a line⌳of fixed points,
a2⫽0, 共15a兲
h0⬅⫺2a12共¯V⫺cos兲⫽0. 共15b兲 Figure 1 shows both ⌳and the surface h0⫽0 for V¯⬎1; ⌳
would reach up to infinity for V¯⫽1. Linearizing the vector field at the fixed points we find eigenvalues 1⫽⫺2a1 2
⫻sin, and 2⫽0, for eigenvectors tangent to the line a1
⫽constant through the corresponding fixed point, and tan-gent to⌳, respectively. From the sign of 1 it follows that, for flow on a2⫽0,⌳points with ⬍ are stable and⬎
points are unstable; two a1⫽constant heteroclinic orbits join
each symmetric pair of ⌳points.
The third eigenvalue is clearly the factor multiplying a2 in Eq.共9b兲,3⫽⫺␥⫹a1
2
sin, with the associated eigenvec-tor parallel to the a2 axis. It follows that for motion off a2
⫽0 the⬎branch is stable, whereas in the branch⬍, under condition
V
¯2⬍1⫹共/2␥兲2, 共16兲
there are points P0and P0*that have3⫽0 and are given by
a12⫽ ␥
sin⫽
V
¯⫿
冑
2⫺4␥2共V¯2⫺1兲for ⫺ and⫹ signs, respectively, with ( P0*)⬍/2 always, but ( P0)⬎/2 for /2␥⬎V¯ . Only ⌳ points in the arc P0P0* are unstable off a2⫽0. For the flow in the entire 3D space the stable fixed points are those in the⬍branch of
⌳ below P0 and above P0*. Note that a1( P0*)→⬁ as V¯
→1.
In the plane a2⫽0 there is another type of critical
ele-ments. There are periodic orbits that move below the bottom
Q of⌳at constant a1⬍a1Q, from ⫽0 to ⫽2, and that
are described by Eq.共9c兲now reading ˙⫽h0(a1,); again,
there are periodic orbits above Q*in Fig. 1.关Their period is /关(V¯ a102 ⫺/2)2⫺a104 兴1/2, which diverges for a10
⫽a1Q(a10⫽a1Q*), when the periodic orbit becomes an
ho-moclinic trajectory at Q (Q*), as seen in the figure.兴Clearly,
a2⫽0 perturbations of any such orbits leave the system moving in another nearby orbit. Also, all these periodic or-bits are stable off a2⫽0: At vanishing a2 we have ˙
⫽O(1) whereas a1 changes at a rate of order a2
2; taking
d(ln a2)/dt from共9b兲, its average over a period is⫺␥⬍0, the
contribution of the sin term vanishing.
Under condition 共16兲 one may say that the stable arc
Q P0 and the periodic orbits below Q make up one attractor
of the flow and the stable arc P0*Q* and the periodic orbits above Q*make up a second attractor. There is a fundamen-tal difference between these two attractors however. Since⌳ points in the arc P0P0* have an 1D unstable manifold
trans-verse to a2⫽0 there exist singular, heteroclinic orbits that
leave this plane at those points, and return to it at a lower a1,
as seen from Eq. 共11兲 with ⌫⫽0. Equation 共14兲 with⌫⫽0 proves that an orbit leaving ⌳between P0 and P0* has the
quantity h0⫺a2 2
/2V¯ , and therefore h0 itself, non-negative
thereafter, corresponding to it moving below the cylindrical surface h0(a1,)⫽0共Fig. 1兲. The singular orbit may reach a
point in the arc Q P0 from the left, keeping⬍throughout,
or may approach the set of periodic orbits. It may also pass just below the surface h0(a1,)⫽0 to reach the range⬎
still off the plane a2⫽0, making a˙1 positive in Eq.共9a兲; the orbit will then emerge at⫽0 with a1⬎a1Qand again reach a point in the arc Q P0 from the left.
IV.⌫\0¿ATTRACTOR
When⌫is made positive under condition共16兲, assumed here, there are just two fixed points, P and P*, given by equations
sin⫽2共␥⫺⌫兲共V¯⫺cos兲⫹⌫/V¯ , 共18a兲
␥/a12⫽sin, 共18b兲
⌫/a22⫽sin, 共18c兲
which recover共17兲for P0 and P0*as⌫→0. The
characteris-tic equation for the stability of those two points is
共⫹2␥⫺2⌫兲共2⫹4␥⌫兲
⫹tan2␥⌫
冋
sin再
1V
¯⫺4共V¯⫺cos兲
冎
⫺2册
⫽0. 共19兲For ⌫⫽0, Eq.共19兲 again recovers the values1⫽⫺2␥, 2
⫽3⫽0 of Sec. III. For ⌫ positive and small, one finds to
order冑⌫,
2,3共P*兲⬇⫾
冑
4 2⫺4␥2共V¯2⫺1兲⫻冑
2⌫⫻a1共P 0*兲; 共20兲
P* at small ⌫ is thus a saddle node with a 1D unstable manifold.
For the stability of P we must go to order⌫,
2,3共P兲⬇⫾i
冑
4 2⫺4␥2共V¯2⫺1兲⫻冑
2⌫⫻a1共P0兲⫹¯⌫, 共21兲
¯⬅ 1
2 tan
冉
␥⫺
1
V ¯ sin
冊
.The sign of¯ is obtained by taking (/␥,V¯ ) from Eq.共17兲 with the upper sign. We find that P, which only exists under condition 共16兲 and is defined for V¯⬎1, is stable above the line (/2␥)2⫽V¯2 关with( P0)⬎/2 as noticed in Sec. III兴
and below the line
冉
2␥
冊
2
⫽ 1
8V¯2⫺4V¯4⫺1 共for V
¯2⬍3
2兲 共22兲
in the parametric plane (/2␥)2, V¯2. Point P goes through a Hopf bifurcation,¯ becoming positive, when crossing either line. Figure 2 summarizes the stability of P for⌫→0⫹. Fig-ure 3 uses Eqs.共8兲and共10兲to represent stability domains in terms of parameters k3/k2 and ci␥/VA
2 k1
2
, where VAk1 ⬇1.
Now consider the long-time behavior of the system for⌫ very small. Away from the surface a2⫽0 the flow will closely follow ⌫⫽0 trajectories. If a trajectory approaches a periodic orbit above Q*, the term⌫a1 in Eq.共9a兲will make
a1ultimately diverge, as the system slowly drifts through the set of periodic orbits; if the ⌫⫽0 trajectory approaches the arc P0*Q*, the system will first have a1 slowly rising along and very close to ⌳, until reaching the set of periodic orbits
FIG. 1. Line⌳of fixed points on invariant plane a2⫽0 at⌫⫽0, and
peri-odic orbits above and below; for⬍only the arcs Q P0and P0*Q* are
at point Q*. The case for⌫⫽0 trajectories approaching ei-ther the arc Q P0 or the periodic orbits below is dramatically
different.
Consider flow in the vicinity of the ⌫⫽0 heteroclinic orbit corresponding to a ⌳-point M on the P0P0*arc, in the
approach back to the surface a2⫽0, below P0. If the orbit
approaches some point m between Q and P0 and because of
the term⌫a1, a1 should eventually start growing at rate ⌫,
keeping close to⌳. In terms of the eigenvalue3 of Sec. III, Eq. 共9b兲can be written as da2/dt⫽3a2; since3 is nega-tive for⌳points below P0and positive from P0 to P0*, and
the a1 rise takes times of order 1/⌫, a2 will become
expo-nentially small (⫺ln a2⬃1/⌫). Once P0is reached, however, a2 will start growing; when values a2⬃冑⌫ are attained, a1
can finally reach a maximum M
⬘
below P0*, and the trajec-tory again start separating from ⌳. If the heteroclinic orbit approaches some periodic orbit below Q, a1will first slowlyincrease while the system drifts among the lower set of pe-riodic orbits to reach⌳.
In the parametric domain of Fig. 2 where P is stable, trajectories starting within some bassin of attraction in phase space have a sequence of points M, m, M
⬘
,..., converging to point P as given, to lowest order in⌫, by Eqs.共17兲 and a2 2⫽⌫⫻a12/␥. The general attractor structure following the loss of stability of P at crossing line B共or C兲at fixed V¯ , giving rise to a limit cycle, depends on the value of V¯ . At V¯ very close to unity the set of periodic orbits is rarely involved in the attractor. Figure 4 shows a limit-cycle attractor for V¯
⫽61/60 (k3/k2⫽25/36), ⌫/␥⫽0.001 and /␥⫽1.3 共with
␥⫽1兲, which is determined by numerically following a single trajectory for long times; Fig. 5 shows a map for the attractor of Fig. 4. Figure 6 shows a chaotic attractor for the same
values of V¯ and⌫/␥, and/␥⫽2.0. Periodic orbits are usu-ally involved at greater V¯ . Figures 7 and 8 show the lower parts of a limit 2-cycle attractor and of a chaotic attractor projected in the a1,  plane, for V¯⫽13/12 (k3/k2⫽4/9), ⌫/␥⫽0.001 and two values of /␥. For V¯2⬎3/2 (k
3/k2⬍2
⫺冑3), and above but close to line A in Fig. 2, the arc P P0is short, leaving a2 still exponentially small when the system reaches P0*; with a2 decreasing and a˙1 remaining positive thereafter, a1 will diverge, as in the case of trajectories ap-proaching the arc P0*Q*.
Consider the limit cycle in Fig. 4. In general, a⌫→0⫹ attractor nested somehow around point P may be described by an exact 1D map representing every maximum of
a1(a1 M⬘) in a trajectory within its bassin of attraction, ver-sus the preceding maximum (a1 M). This map can be deter-mined by a two-step algorithm. In the first step, one numeri-cally follows the heteroclinic orbit from any point M in the
P0P0*arc of⌳to the corresponding point m below P0. The second step is the rise on ⌳ at vanishing rate 共⌫→0, t
⬃1/⌫) up to the next maximum M
⬘
, which can be analyti-cally determined by noting that, no matter how close the solution to a heteroclinic M→m orbit, Eq. 共9a兲 will ulti-mately read da1/dt⫽⌫a1.Using da2/dt⫽3a2, one obtains
a12sin⫺␥
a1
da1⫽⌫d ln a2,
with a1 and  related through the equation h0(a1,)⬅0.
The integral of the left-hand side above, for the entire rise from m to M
⬘
关with ln(1/a2) small compared with 1/⌫ ateither end兴, will vanish in the limit⌫→0⫹. We thus find an equation relating points m and M
⬘
for the slow rise on⌳,G
冉
M⬘,V¯ ,␥
冑
V¯2⫺1冊
⫽G冉
m,V¯ ,␥
冑
V¯2⫺1冊
,⬅共V¯2⫺1兲2 a1
2⫺ V ¯ , FIG. 2. Stability of fixed point P at⌫→0⫹, in parametric plane (/2␥)2,
V
¯2. Lines A, (/2␥)2⫽V¯2⫺1; B, (/2␥)2⫽V¯2; C, given by Eq.共22兲.
FIG. 3. Stability of fixed point P at⌫→0⫹, in parametric planeci␥/VA
2
k1 2
,
k3/k2.
FIG. 4. Limit 1-cycle attractor for V¯⫽61/60 (k3/k2⫽25/36),⌫/␥⫽0.001,
G⬅
冑
1⫺2⫹V¯ sin⫺1⫺冑
V¯2⫺1 sin⫺1冉
1⫹V¯V ¯⫹
冊
⫺2␥
冑
V¯2⫺1
ln共V¯⫹兲.
Figure 5 shows this⌫→0⫹limit map of the sequence of maxima a1 Mfor the values of V¯ ,, and␥of Fig. 4. The map has two fixed points, where it cuts the diagonal. One 共 un-stable兲 fixed point of the map is the fixed point of the flow
P⬇P0, lying at a1 just above unity. The second 共stable兲
fixed point of the map corresponds to the maximum in the limit cycle, lying at a1 just above 1.5. Between both fixed
points the map lies barely above the diagonal. Figure 5 also shows, to the right of the limit-cycle maximum, the ap-proaching sequence of maxima for a single trajectory, at the values of V¯ , , and ␥ indicated, with ⌫/␥⫽0.001, in good agreement with the limit map; this is more clearly seen in the zoom of the inset.
The inset of Fig. 6 shows the lower part of the chaotic attractor projected on the plane a1, . The inset shows clearly a fork in the attractor, with the trajectory sometimes approaching the ⬍ branch of ⌳ from the right. This is possible because the last term in Eq. 共14兲 can make
h0(a1,) negative when a2reaches down to values of order
of ⌫, the trajectory crossing above the cylindrical surface
h0(a1,)⫽0.
Consider now Figs. 7 and 8. If the ⌫⫽0 heteroclinic orbit leaving some point M in the arc P0P0* approaches a
periodic orbit below Q with a1⫽a1m, we will first have a1
slowly increasing while the system drifts among the lower set of periodic orbits to reach ⌳. During this rise Eq. 共9c兲 will read˙⫽h0(a1,)⫽O(1). With the contribution of the
sinterm in共9b兲averaging out during that stage, the overall relation between m and M
⬘
would come out to beG
冉
M⬘,V¯ ,␥
冑
V¯2⫺1冊
⫽⫺ 2共V
¯⫺
冑
V¯2⫺1兲⫺2␥
冑
V¯2⫺1
ln共V¯⫹m兲.
V. FULL 3-WAVE MODEL OF THE DNLS EQUATION
We now briefly consider how changing to a full 3-wave model affects the dynamics of the system. In Eqs.共6a兲–共6d兲 we set
a12→ a1 2
冑
k2k3, a22→ a2 2
k2⫹k3
冑
k2 k3,
FIG. 5. 1D map of a1 Mmaxima for the attractor of Fig. 4, with fixed points
at the fixed point P of the flow, unstable, and at the a1 Mmaximum of the
limit cycle共LC兲, stable. The crosses correspond to the sequence of maxima determined numerically on a single trajectory; the continuous line is the exact⌫→0⫹map. The inset shows another view of the limit map.
FIG. 6. Chaotic attractor for V¯⫽61/60 (k3/k2⫽25/36), ⌫/␥⫽0.001, and
/␥⫽2.0 with␥⫽1. Also shown is the lower part of the attractor projected on the a1– plane.
FIG. 7. Lower part of limit 2-cycle attractor projected on the a1– plane,
a32→ a3 2
k2⫹k3
冑
k3 k2,
and, assuming␥2⬍␥3, introduce a new variable,
r⬅a3/a2,
to replace a3. We then find
a˙1⫽⌫a1⫺ra1a2 2
sin, 共23a兲
a˙2⫽⫺␥2a2⫹ra1 2
a2sin, 共23b兲
r˙⫽⫺共␥3⫺␥2兲r⫹共1⫺r2兲a1 2
sin, 共23c兲
˙⫽⫺2a 1
2
冉
V¯⫺1⫹r 22r cos
冊
⫺2ra22
⫻cos⫹ a2
2
2V¯
冉
k2 k3
⫹r2k3 k2
冊
, 共23d兲
with V¯ (⬍1) and k3/k2still related by Eq.共10兲, where we do
not need k3/k2⬍1. For ␥3⫽␥2 the solution r⫽1 of Eq. 共23c兲 would recover system 共9a兲–共9c兲 in Eqs.共23a兲,共23b兲, and共23d兲. Note also that if a trajectory reaches the surface
r⫽1, r will remain less than unity thereafter in Eq.共23c兲; the 3D space a2⫽0 will then be invariant.
If damping is resistive one has ␥⫽resistivity ⫻⑀0c2k2
or, in dimensionless form 共setting ␥→␥⫻ci, k→k
⫻ci/VA), ␥⬇k2/2cee, where ce and e are electron
cyclotron frequency and Braginskii collision time, respectively.26 We then have ␥3/␥2⫽(k3/k2)2 with k3
⬎k2. For near-parallel propagation at angle Ⰶ冑2冑/ci
and nonzero electron temperature Te, we may have Landau
damping, reading in dimensionless form, ␥⫽k2⫻(me/mi)
⫻(kTe/32meVA
2
)1/2⫻exp(⫺meVA
2
/2kTe).27 Now, ␥3/␥2
⫽k3/k2, with k3⬎k2 again.
As in Sec. III, the long-time attractor of the system will be a point set of vanishing共now 4D兲volume, which can be readily determined for⌫⫽0. First, Eq.共11兲is recovered from
共23a兲 and共23b兲, meaning that共for ⌫⫽0兲 the entire flow is asymptotic to the space a2⫽0, which is an invariant surface,
trajectories being asymptotic to its critical elements with transverse stable manifolds.
Next, we consider the flow on a2⫽0, where a1 is now constant in共23a兲. There exists a line of fixed points⌳given by the equations
共␥3⫺␥2兲r⫽a1 2
共1⫺r2兲sin, 共24a兲
r⫽a12关2V¯ r⫺共1⫹r2兲cos兴. 共24b兲
Eliminating a1in共24a兲and共24b兲yields a relation between
and r that can be written as
cos关⫺*共r兲兴⫽2V¯ r
⌬ , 共25兲
where we defined
*⬅cos⫺1 1⫹r2
⌬ , ⌬⬅
冑
共1⫹r2兲2⫹共1⫺r2兲2¯2,
¯⬅
␥3⫺␥2
.
A solution to Eq.共25兲for (r) only exists if the RHS does not exceed unity, requiring
0⬍r⭐rmax⬍1, or 1/rmax⭐r⬍⬁,
共26兲
rmax2 ⬅␣⫺
冑␣
2⫺1, ␣⬅¯2⫹2V¯2⫺1
¯2⫹1 ⬎1;
only for V¯⫽1, 共i.e., k3⫽k2 making ␣⫽1兲, would r reach
unity. The line of fixed points has therefore two branches,⌳l
for r⬍1 and⌳h for r⬎1.
For branch⌳l, l(r) is given by
l⫽cos⫺1
1⫹r2
⌬ ⫾cos⫺1
2V¯ r
⌬ . 共27兲
The plus sign in Eq.共27兲applies as a1is increased from 0 to a1(rmax), given as
a1共rmax兲/
冑
⫽冑
共1⫹¯2兲V¯ /2¯2共V¯2⫺1兲;l decreases from cos⫺1(1/
冑
1⫹¯2)⫹/2 tocos⫺1(
冑
1⫹¯2/V¯2/冑
1⫹¯2); and r increases from 0 to r max.For greater a1 the minus sign applies in共27兲, and r decreases
with increasing a1; as a1→⬁ one findsl→0 and
r→r⬁⬅V¯⫺
冑
V¯2⫺1. 共28兲 Note that both a1 andl are double-valued functions ofr between r⬁ and rmax. For branch⌳h we have h(r⬎1)
⫽2⫺l(1/r) and a1h(r⬎1)⫽a1l(1/r). Figure 9 shows
the projection of the line of fixed points on the a1– r and
– r planes. It may be shown that throughout the branch⌳l
the right-hand side of Eq. 共24b兲 is positive; ⌳l thus only exists for LH polarization.
Three eigenvalues of the linearized vector field at the fixed points have eigenvectors tangent to the invariant space
a2⫽0, determining the stability of flow on it,
1,2⫽共␥3⫺␥2兲 r2⫹1
r2⫺1⫾i共⫺2a1
2V¯兲r 2⫺1
r2⫹1, 3⫽0,
共29兲
FIG. 8. Lower part of chaotic attractor projected on the a1– plane, for
V
¯⬇13/12 (k
with the null value 3 corresponding to an eigenvector
tan-gent to⌳. The eigenspace associated to1and2is tangent
to the invariant plane a1⫽constant at the respective fixed
point; as seen in共29兲, for flow on the space a2⫽0, points on
the branch ⌳l are stable and points on⌳h are unstable. In
each plane a1⫽a10⬍a1(rmax) within the space a2⫽0 the
flow is determined by Eqs.共23c兲and共23d兲, giving
rd dr⫽
a102 共1⫹r2兲cos⫺共2V¯ a102 ⫺兲r a102 共1⫺r2兲sin⫺共␥3⫺␥2兲r
. 共30兲
One readily verifies that, except for particular separatrices, all trajectories start and end at the fixed points that are inter-sections of branches⌳l and⌳h with the a10plane 共2D foci
with the eigenvalues 1,2). 17
Also, one can verify that once reached the line r⫽1 in that plane all trajectories keep r less than unity. The entire flow in共30兲tends to the stable focus at
r⬍1. In the space a2⫽0, the entire flow moves from the
unstable branch⌳h to the stable branch⌳l.
The eigenvalue for stability of⌳-points off the surface
a2⫽0, which is the factor multiplying a2 in Eq. 共23b兲, 4 ⫽⫺␥2⫹ra1
2
sin, can be rewritten using共24a兲as
4⫽␥3r 2⫺␥
2
1⫺r2 . 共31兲
The associated eigenvector is transverse to the surface a2
⫽0 共parallel to the a2 axis兲. Equation共31兲 shows that, for
motion off that surface, all points on the ⌳h branch are stable, whereas only those points on the⌳l branch with
r⬍
冑
␥2/␥3, 共32兲if any, are stable. Hence, for the flow in the entire 4D space, the stable fixed points of ⌳ are those on the r⬍1 branch satisfying condition共32兲.
Under condition
冑
␥2/␥3⬍rmax, reading as V¯2⬍共␥2⫹␥3兲 2⫹2
4␥2␥3
, 共33兲
there will be a point P0in the arc a1⬍a1(rmax) of⌳lhaving
4⫽0; points on that arc above P0 关i.e., a1⬎a1( P0)] will
be unstable. Note that 共33兲 recovers condition 共16兲 for ␥3
⫽␥2. Under the additional condition r⬁⬍
冑
␥2/␥3there willalso be a point P0* in the arc a1⬎a1(rmax) of⌳l having4 ⫽0; points on that arc above P0* will also be unstable. The full arc would be unstable in the opposite case,
冑
␥2/␥3⬍r⬁, which reads as
V
¯⬍1⫹␥2/␥3
2
冑
␥2/␥3. 共34兲
This condition cannot be satisfied with ␥3⫽␥2, when it
reads V¯⬍1.
We may then conclude that, for⌫⫽0, and under condi-tions 共33兲 and 共34兲, the attractor of the flow is the a1
⬍a1(rmax)⌳l-arc below P0in the space a2⫽0. Note that⌳l
points above P0 have an 1D unstable manifold transverse to a2⫽0, corresponding to the positive sign of the eigenvalue 4. There are thus singular orbits that leave that surface at
those points and end on the⌳l-points below P0, all of which
have stable manifolds transverse to a2⫽0 共and lie in the r
⬍1 domain兲. If the opposite of 共34兲 holds, that is, if P0* exists, the attractor includes the a1⬎a1(rmax),⌳l-arc above
P0*, but singular orbits leaving the unstable arc still end at lower a1 on the stable arc below P0.
The full 3-wave model differs from the reduced 3-wave model, for⌫⫽0, in a number of significant phase-space fea-tures. There are sets of periodic orbits no longer; the line of fixed points ⌳covers the entire 0⬍a1⬍⬁ range; there may
just exist one point on⌳that is neutrally stable off the space
a3⫽a2⫽0. Yet, the fundamental features leading to the hard
transition in phase-space dynamics when⌫is made positive, within some parametric subdomain, are common to both models. For⌫⫽0, the entire flow is asymptotic to the space
a3⫽a2⫽0; there is a line of fixed points ⌳ in that space
with two branches having different stability character; one branch has an stable arc of fixed points at a low a1range and
an arc unstable off a3⫽a2⫽0 at a higher a1 range. These
features are all that is needed for the hard transition to be present.
VI. DISCUSSION OF RESULTS
We have truncated the derivative nonlinear Schro¨dinger
共DNLS兲equation describing the interaction of circularly po-larized Alfve´n waves of finite amplitude, to explore weakly nonlinear dynamics in the coherent cubic coupling of three
FIG. 9. Projection on planes共A兲a1– r and共B兲– r of branches⌳land⌳h of line of fixed points on invariant plane a2⫽0, at⌫⫽0, given by Eqs.共24a兲
waves near resonance 共3WRI兲, wave 1 being linearly un-stable and waves 2 and 3 damped. We have considered a broad scenario for chaos which several 3WRI systems, for both cubic and quadratic coupling, had exhibited: No matter how small the growth rate⌫of the unstable wave there exists certain parametric domain with a fully developed attractor
共chaotic in some subdomain兲that is absent at ⌫⭐0. To ex-plore the structural stability of this hard transition to complex phase-space dynamics we have considered both the reduced 3-wave model 共equal dampings of daughter waves, leading to a 3D flow for wave amplitudes a1, a2 and a relative phase兲 and the fully 3-wave model 共different dampings, 4D flow兲. Structural stability 共suggested by the appearance of that transition elsewhere as mentioned above兲would be im-portant because any 3WRI model has limited validity.
Both reduced and full models showed the hard transition only occurring for left-hand circularly polarized waves, paralelling the known fact that LH time-harmonic solutions of the DNLS equation共for cold plasmas兲are modulationally unstable, a case opposite RH polarized solutions.6In the
re-duced model, transition occurs at damping less than about
0.25⫻共wave-1 frequency兲2/ion cyclotron frequency. A num-ber of features determine the phase-space dynamics of the transition: For ⌫⫽0, the entire flow is asymptotic to the space a3⫽a2⫽0, where a line of fixed points ⌳ covers a
limited a1 range with periodic orbits below and above that
range. A branch of ⌳has an arc of fixed points unstable off the space a3⫽a2⫽0, in between stable arcs; singular,
het-eroclinic orbits off the unstable arc return to that space at lower a1. Chaotic attractors involve repeated slow rises on ⌳, and possibly in the lower set of periodic orbits, followed by fast motion along the heteroclinic orbits. In the full 3-wave model there are no sets of periodic orbits; the line of fixed points ⌳covers the entire 0⬍a1⬍⬁ range; there may
exist just 1 ⌳-arc stable off the space a3⫽a2⫽0.
The easiest feature to detect in the transition would be the associated sudden break in a2 behavior. The strict ⌫ →0⫹ limit has theoretical interest but is impractical; during the slow rises on⌳, time would diverge and a2 would drop below any noise level.15 The basic feature to consider in practical terms is that a2 keeps null for negative⌫whereas,
at ⌫positive and finite, but small compared with dampings and with frequency mismatch, the ratio a2/a1 changes
re-peatedly from almost vanishing values during the long times on⌳to values of order unity in the short times near hetero-clinic orbits. Our scenario for chaos is a hard transition to
relaxation oscillations that can be chaotic.
The DNLS equation has been used in relation to nonlin-ear MHD waves observed in the Earth’s bow shock.28In our model, the required growth rate ⌫ could result, in general, from some plasma instability, due, say, to a beam–plasma interaction or to the plasma carrying a current that makes for negative Landau damping. The plasma does carry a current in the tether case. More to the point, however, tether signal detection would require modulating the current in the tether circuit, the Alfve´n-radiation impedance being weak in the case of a steady tether current.1The magnetic self-field of the tether3would then result in a background magnetic field time modulated 共at some frequency mod).
Such field may excite two growing Alfve´n waves at half the modulation frequency and propagating along the mag-netic field in opposite directions,29each one representing the mother wave 1 of our problem. This would determine a defi-nite wave number k1⬇1/VA⫽21 mod/VA. The ratio k3/k2,
and thus both daughter wave numbers k2 and k3⫽2k1
⫺k2, might then result from minimum-damping or
maximum-amplitude considerations. A proper model may re-quire, however, an analysis that includes more than a pair of daughter waves. It is here important that d(a3/a2)/dt proved
negative at a3/a2⫽1 in Sec. V; that made the surface a2
⫽0 effectively invariant, which was essential for the transi-tion. This suggests the transition could hold under multiple-wave interaction, with the excited and less damped multiple-waves playing the fundamental role.
ACKNOWLEDGMENTS
This work was supported by Ministerio de Ciencia y Tecnologia of Spain, under Grant No. BFM01-3723. Work by S. Elaskar was supported by Ministerio de Ciencia y Tec-nologia of Spain, under Grant No. SB01-0080.
1
S. D. Drell, H. M. Foley, and M. A. Ruderman, J. Geophys. Res. 70, 3131
共1965兲; A. Barnett and S. Olbert, J. Geophys. Res.,关Space Phys.兴 91,
10117共1986兲; J. R. Sanmartı´n and M. Martı´nez-Sanchez, ibid. 100, 1677
共1995兲.
2
J. R. Sanmartin and R. D. Estes, J. Geophys. Res.,关Space Phys.兴 102,
14625共1997兲.
3G. V. Khazanov, N. H. Stone, E. N. Krivorutsky, and E. W. Liemohn, J.
Geophys. Res.,关Space Phys.兴105, 15835共2000兲; G. V. Khazanov, N. H. Stone, E. N. Krivorutsky, K. V. Gamayunov, and E. W. Liemohn, ibid. 106, 10565共2001兲; J. R. Sanmartin and R. D. Estes, ibid. 107, SIA 2-1共2002兲.
4J. R. Sanmartı´n and S. H. Lam, Phys. Fluids 14, 62共1971兲.
5A. Rogister, Phys. Fluids 14, 2733共1971兲; K. Mio, T. Ogino, K. Minami,
and S. Takeda, J. Phys. Soc. Jpn. 41, 265共1976兲.
6E. Mjo”lhus, J. Plasma Phys. 16, 321共1976兲; 19, 437共1978兲. 7
D. J. Kaup and A. C. Newell, J. Math. Phys. 19, 798共1978兲.
8S. R. Spangler and J. P. Sheerin, J. Plasma Phys. 27, 193共1982兲; T. Hada,
C. F. Kennel, and B. Buti, J. Geophys. Res.,关Space Phys.兴94, 65共1989兲; M. V. Medvedev and P. H. Diamond, Phys. Plasmas 3, 863共1996兲; T. Passot and P. L. Sulem, ibid. 10, 3887共2003兲.
9
R. C. Davidson, Methods in Nonlinear Plasma Theory共Academic, New York, 1972兲, Chap. 6; A. Bers, in Plasma Physics—Les Houches 1972, edited by C. De Witt and J. Peyreaud共Gordon and Breach, New York, 1975兲, pp. 117–215; J. Weiland and H. Wilhelmsson, Coherent Nonlinear Interaction of Waves and Plasmas共Pergamon, New York, 1977兲.
10S. Ya. Vyshkind and M. I. Rabinovich, Sov. Phys. JETP 44, 292共1976兲; A.
S. Pikovsky and M. I. Rabinovich, Physica D 2, 8共1981兲.
11J. M. Wersinger, J. M. Finn, and E. Ott, Phys. Rev. Lett. 44, 453共1980兲;
Phys. Fluids 23, 1142共1980兲.
12
M. N. Bussac, Phys. Rev. Lett. 49, 1939共1982兲; C. Meunier, M. N. Bus-sac, and G. Laval, Physica D 4, 236共1982兲.
13D. W. Hughes and M. R. E. Proctor, Physica D 46, 163共1990兲. 14O. Lo´pez-Rebollal and J. R. Sanmartı´n, Physica D 89, 204共1995兲. 15
E. del Rı´o, J. R. Sanmartı´n, and O. Lo´pez-Rebollal, Int. J. Bifurcation Chaos Appl. Sci. Eng. 8, 2255共1998兲.
16D. W. Hughes and M. R. E. Proctor, Wave Motion 20, 201共1994兲. 17O. Lo´pez-Rebollal, J. R. Sanmartı´n, and E. del Rı´o, Phys. Plasmas 5, 2861
共1998兲.
18
M. I. Rabinovich and A. L. Fabrikant, Sov. Phys. JETP 50, 311共1979兲; D. A. Russell and E. Ott, Phys. Fluids 24, 1976共1981兲.
19J. R. Sanmartı´n, O. Lo´pez-Rebollal, and N. de Paola, Physica D 69, 148
共1993兲.
20
P. Terry and W. Horton, Phys. Fluids 25, 491共1982兲; A. M. Martins and J. T. Mendonc¸a, ibid. 31, 3286共1988兲.
21P. A. Robinson and P. M. Drysdale, Phys. Rev. Lett. 77, 2698共1996兲; P. M.
Drysdale and P. A. Robinson, Phys. Plasmas 9, 4896共2002兲.
22G. I. de Oliveira, L. P. L. de Oliveira, and F. B. Rizzato, Physica D 104,
23C. C. Chow, Physica D 81, 237共1995兲. 24
C. S. Kueny and P. J. Morrison, Phys. Plasmas 2, 1926共1995兲; 2, 4149
共1995兲.
25A. C. L. Chian, S. R. Lopes, and J. R. Abaldo, Physica D 99, 269共1996兲;
B. Lefebvre and V. Krasnoselskikh, ibid. 152–153, 742共2001兲.
26
S. Rauf and J. A. Tataronis, Phys. Plasmas 2, 1453共1995兲; J. Plasma Phys.
55, 173共1996兲.
27A. I. Akhiezer, I. A. Akhiezer, R. V. Polovin, A. G. Sitenko, and K. N.
Stepanov, Plasma Electrodynamics共Pergamon, Oxford, 1975兲, Vol. 1, p. 250.
28S. R. Spangler, in Nonlinear Waves and Chaos in Space Plasmas, edited
by T. Hada and H. Matsumoto 共Terrapub, Tokyo, 1997兲, p. 171; E. Mjo”lhus and T. Hada, ibid. p. 121.
29
