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A simple mechanism for generating traveling waves which reverse their direction of propagation in a periodic manner is presented. This mechanism is generic in systems possessing O(2) symmetry, and corresponds to a codimension-one symmetry-breaking Hopf bifurcation from a circle of nontrivial steady states.

A symmetry-breaking Hopf bifurcation in an O(2)-symmetric system has eigenvalue of multiplicity two. When the circular symmetry is broken these eigenvalues split into two pairs. The consequences of this splitting in the nonlinear regime are analysed in detail. It is found that the perturbation selects the phase of the standing wave (SW) solutions and that two SW branches, differing in phase by pi , bifurcate from the trivial solution in succession. Pure travelling waves (TW) are no longer possible. Instead two new solution branches denoted by TW' and MW' bifurcate from the SW branches in secondary steady-state and Hopf bifurcations, respectively. In contrast to the TW', the MW' only exist at small amplitudes, terminating on the TW' branch in either global or tertiary Hopf bifurcations. These solutions show remarkable resemblance to the states observed in recent experiments on binary fluid convection in large but finite containers.

The amplitude of a wavetrain in a dispersive system generally evolves according to the cubic nonlinear Schrodinger (NLS) equation. In a reflection-symmetric system, the interaction of right and left travelling wavetrains with amplitudes ϵA+(ξ+,T2) and ϵA-(ξ-, T2) is described by two nonlinear Schrodinger equations with mean field coupling: ±2 i ∂A ± /∂T 2 +( d 2 ω/ d k 2 )×∂ 2 A ± /∂ξ ±2 =μ ± A ± +αA ± ¦A ± ¦ 2 . The independent variables are defined by ξ±=ϵ(cgt∓x), T2=ϵ2t and the coupling arises through μ ± ≡β[(1/P ∓ )∫ P∓ 0 ¦A ∓ ¦ 2 d ξ ∓ ] . The analysis is applied to electromagnetic wave propagation in an optical fibre.

A complete analysis for | μ | , | ν | ≪ 1 \left | \mu \right |, \left | \nu \right | \ll 1 of the equation \[ x ˙ = y , y ˙ = μ x + ν y + M x 2 + Γ x y \dot x = y, \dot y = \mu x + \nu y + M{x^2} + \Gamma xy \] describing a particular unfolding of the Takens-Bogdanov singularity is presented.

The recognition that fluid-dynamical models can yield solutions with less symmetry than the governing equations is not new. Jacobi's discovery that a rotating fluid mass could have equilibrium configurations lacking rotational symmetry is a famous nineteenth-century example. In modern terminology, Jacobi's asymmetric equilibria appear through a symmetry­ breaking bifurcation from a family of symmetric equilibria as the angular momentum (the "bifurcation parameter") increases above a critical value (the "bifurcation point"). Chandrasekhar (1969) gives a brief historical account of this discovery. In this example, as in many others, the presence of symmetry breaking was discovered by solving specific model equations. In contrast to this specificity, it is widely recognized that symmetry-breaking bifurcations are of frequent occurrence in a variety of nonlinear, nonequilibrium physical settings-fluids, chemical reactions, plasmas, and biological systems, to

Binary mixtures with a negative separation ratio are known to exhibit time-independent spatially localized convection when heated from below. Numerical continuation of such states in a closed two-dimensional container with experimental boundary conditions and parameter values reveals the presence of a pinning region in Rayleigh number with multiple stable localized states but no bistability between the conduction state and an independent container-filling state. An explanation for this unusual behavior is offered.

Homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time-independent spatially localized states in a bistable, spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side. This behavior is simplest to understand within the subcritical Swift-Hohenberg equation, but is also present in the subcritical regime of doubly diffusive convection driven by horizontal gradients. In systems that are unbounded in one spatial direction homoclinic snaking continues indefinitely as the localized structure grows to resemble a spatially periodic state of infinite extent. In finite domains or in periodic domains with finite spatial period the process must terminate. In this paper we show that the snaking branches in general turn over once the length of the localized state becomes comparable to the domain, and examine the factors that determine the location of the termination point or points, and their relation to the Eckhaus instability of the spatially periodic state.

A technique is described that should prove invaluable for the computation of effective diffusion coefficients due to wave motion.

We examine the degree to which the transport of a passive scalar contaminant is enhanced by an oscillatory cellular flow. The enhancement is found to depend on the Peclet number Pe and on the ratio η of the oscillation frequency to the shear rate. Analytical expressions are derived for the diffusivity enhancement in the limits of high and low η. Numerical solutions of the advection-diffusion equation confirm these results and connect these two regimes.

Faraday waves near onset in an elliptical container are described by a third-order system of ordinary differential equations with characteristic slow-fast structure. These equations describe the interaction of standing waves with a weakly damped streaming flow driven by Reynolds stresses in boundary layers at the free surface and the rigid walls, and capture the proliferation with decreasing damping of periodic and nonperiodic relaxation oscillations observed near onset in previous simulations. These structures are the result of slow drift through symmetry-related Hopf bifurcations.

International audience; Two-dimensional Marangoni convection in binary mixtures is studied in periodic domains with large spatial period in the horizontal. For negative Soret coefficients convection may set in via growing oscillations which evolve into standing waves. With increasing amplitude these waves undergo a transition to traveling waves, and then to more complex waveforms. Out of this state emerge stable stationary spatially localized structures embedded in a background of small amplitude standing waves. The relation of these states to the time-independent spatially localized states that characterize the so-called pinning region is investigated by exploring the stability properties of the latter, and the associated instabilities are studied using direct numerical simulation in time.

Spatiotemporal control of excitable media is of paramount importance in the development of new applications, ranging from biology to physics. To this end we identify and describe a qualitative property of excitable media that enables us to generate a sequence of traveling pulses of any desired length, using a one-time initial stimulus. The wave trains are produced by a transient pacemaker generated by a one-time suitably tailored spatially localized finite amplitude stimulus, and belong to a family of fast pulse trains. A second family, of slow pulse trains, is also present. The latter are created through a clumping instability of a traveling wave state (in an excitable regime) and are inaccessible to single localized stimuli of the type we use. The results indicate that the presence of a large multiplicity of stable, accessible, multi-pulse states is a general property of simple models of excitable media., Comment: 6 pages, 6 figures

We consider a close relative of plane Couette flow called Waleffe flow in which the fluid is confined between two free-slip walls and the flow driven by a sinusoidal force. We use a reduced model of such flows constructed elsewhere to compute stationary exact coherent structures of Waleffe flow in periodic domains with a large spanwise period. The computations reveal the emergence of stationary states exhibiting strong amplitude and wavelength modulation in the spanwise direction. These modulated states lie on branches exhibiting complex dependence on the Reynolds number but no homoclinic snaking.

New asymptotically exact amplitude equations are derived for a dissipative system near a Hopf bifurcation. Unlike the usual coupled complex Ginzburg-Landau equations these are valid for O(1) group velocities.

A two-parameter analysis of the interaction between two period-doubling modes with azimuthal wavenumbers k and l (l>k≥1) is carried out for systems with circular symmetry. The problem is formulated in terms of an O(2)-equivariant map on C 2. In the generic case all primary, secondary and tertiary solution branches and their stability properties are classified. The results depend on whether k + l is odd or even. When one mode bifurcates subcritically and the other supercritically the pure mode branches lose stability to a branch of reflection-symmetric mixed modes, which in turn can undergo a tertiary Hopf bifurcation to a quasiperiodic reflection-symmetric pattern. We conjecture that this invariant circle can break down with increasing amplitude to produce reflection-symmetric chaos. Additional “phase” instabilities may occur in which the reflection symmetry is broken, and the resulting pattern drifts either clockwise or counterclockwise. The results explain a number of observations by Ciliberto and Gollub on parametrically excited surface waves in circular geometry, and imply several new predictions for such experiments. In an appendix the theory is compared with previous attempts to model the experiments.

The long wavelength of the first instability in Rayleigh-Benard convection between nearly thermally insulating horizontal plates is typical of a variety of physical systems. The evolution of such an instability is described by the equation α t = αα − μ ▿ 2 α −▿ 4 α + κ ▿·‖▿ α ‖ 2 ▿ α + β ▿·▿ 2 α ▿ α − λ ▿· α ▿ α + σ ▿ 2 ‖▿ α‖ 2 , where α is the planform function, μ is the scaled Rayleigh number and κ = ±1. The quantities α,β,λ represent the effects of finite Biot number, asymmetry in the boundary conditions at top and bottom, and departures from the Boussinesq approximation, respectively. The quantity σ = β when the original problem is self-adjoint, but σ ≠ β otherwise. Planform selection is studied for α κ = +1 using equivariant bifurcation theory. On the square lattice both rolls and squares can be stable, depending on the parameters β, λ and σ. The possible secondary bifurcations located near various codimension-two singularities are analyzed. On the hexagonal lattice the primary bifurcation is always degenerate when β = λ = σ =0. Of the six primary solution branches possible in this case the hexagon branch is stable. When β − σ = λ =0, both rolls and hexagons bifurcate supercritically, but rolls are stable. Finally, when β ≠ σ and/or ψ ≠ 0 a hysteretic transition to H + or H - occurs depending on sgn( β + λ k 2 c −σ ); if ‖ β ‖, ‖ λ ‖, ‖ σ ‖ ⪡ stable H + , H - coexist at larger amplitudes, but if ‖β + λ k 2 c −σ‖ ⪡ ‖β‖, ‖λ‖, ‖σ‖ = 0(1) , a further hysteretic transition takes place with increasing amplitude in which the hexagons are replaced by stable rolls.

Small amplitude two-dimensional Boussinesq convection in a plane layer with stress-free boundaries rotating uniformly about the vertical is studied. A horizontally unbounded layer is modelled by periodic boundary conditions. When the centrifugal force is balanced by an appropriate pressure gradient the resulting equations are translation invariant, and overstable convection can take the form of travelling waves. In the Prandtl number regime 0.53 < [sgrave] < 0.68 such solutions are preferred over the more usual standing waves. For [sgrave] < 0.53, travelling waves are stable provided the Taylor number is sufficiently large.

A method is presented for constructing a stochastic return map from a stochastic differential equation containing a locally stable limit cycle and small-amplitude [O(e)] additive Gaussian colored noise. The construction is valid provided the correlation time isO(e) orO(1). The effective noise in the return map has nonzeroO(e 2) mean and is state dependent. The method is applied to a model dynamical system, illustrating how the effective noise in the return map depends on both the original noise process and the local deterministic dynamics.

The bistable Swift-Hohenberg equation exhibits multiple stable and unstable spatially localized states of arbitrary length in the vicinity of the Maxwell point between spatially homogeneous and periodic states. These states are organized in a characteristic snakes-and-ladders structure. The origin of this structure in one spatial dimension is reviewed, and the stability properties of the resulting states with respect to perturbations in both one and two dimensions are described. The relevance of the results to several different physical systems is discussed.

International audience; The spatial organization of single-fluid Marangoni convection in vertical cylinders with circular or elliptical horizontal cross section is described. The convection is driven by an imposed heat flux from above through Marangoni stresses at a free but undeformed surface due to temperature-dependent surface tension. The solutions and their stability characteristics are obtained using branch-following techniques together with direct numerical simulations. The changes in the observed patterns with increasing ellipticity are emphasized. In some cases, the deformation of the cylinder results in the presence of oscillations.

Numerical simulations of an asymptotically reduced version of the Craik–Leibovich (CL) equations are described. By filtering surface waves, the CL equations facilitate simulations of surface-wave driven or surface-wave modified phenomena in the upper ocean—most notably, Langmuir circulation (LC)—with time scales long relative to the period of the dominant waves. Although numerical simulations of the fully three-dimensional CL equations are now routine, simulations in spatially extended domains, hundreds of times the characteristic width of individual LC vortices, require immense computing resources. The reduced CL (rCL) equations render numerical simulations in very long domains more feasible by exploiting the strongly anisotropic structure of LC that emerges in the strong CL “vortex force” limit. Here, simulations of the rCL equations are performed to explore the dynamics admitted by the reduced system. Spatially quasi-coherent, three-dimensional flow structures that drift downwind are found to be a prominent feature of the reduced dynamics in the parameter regime explored. The spatial form and temporal variability of these structures are associated with a resonance between pairs of LC vortices with cross-wind (transverse) wavenumbers in a 2:1 ratio. Depending upon the strength of the wind and surface-wave forcing, this spatial resonance leads to either periodic or aperiodic temporal oscillations in the number of LC vortices crossing a fixed downwind station, a process that is accompanied by the formation of Y-shaped features in Lagrangian surface patterns. The stems of these “Y-junctions” are preferentially oriented downwind, as is commonly reported in field observations of LC.

The formation of oscillons in a synchronously oscillating background is studied in the context of both damped and self-exciting oscillatory media. Using the forced complex Ginzburg-Landau equation we show that such states bifurcate from finite amplitude homogenous states near the 2:1 resonance boundary. In each case we identify a region in parameter space containing a finite multiplicity of coexisting stable oscillons with different structure. Stable time-periodic monotonic and nonmonotonic frontlike states are present in an overlapping region. Both types of structure are related to the presence of a Maxwell point between the zero and finite amplitude homogeneous states.

Many powerful techniques from Hamiltonian mechanics are available for the study of ideal hydrodynamics. This article explores some of the consequences of including small viscosity in a study of surface gravitycapillary waves excited by the vertical vibration of a container. It is shown that in this system, as in others, the addition of small viscosity provides a singular perturbation of the ideal fluid problem, and that as a result itse ectsarenontrivial.Therelevanceofexistingstudiesofidealfluid problems is discussed from this point of view.

The Swift-Hohenberg equation with quadratic and cubic nonlinearities exhibits a remarkable wealth of stable spatially localized states. The presence of these states is related to a phenomenon called homoclinic snaking. Numerical computations are used to illustrate the changes in the localized solution as it grows in spatial extent and to determine the stability properties of the resulting states. The evolution of the localized states once they lose stability is illustrated using direct simulations in time. I. INTRODUCTION Ever since the observation that the subcritical complex Ginzburg-Landau equation exhibits stable spatially localized states 1 there has been considerable interest in the properties of these states. The presence of these states has important consequences for other systems described by partial differential equations on the line since the Ginzburg-Landau equation describes the behavior of such systems near bifurcation from the trivial state of the system. Specifically, the complex Ginzburg-Landau equation describes the evolution of a long wavelength oscillatory instability, as well as oscillatory instabilities at finite wavelength in systems with broken reflection symmetry. In contrast, near a steady state bifurcation with finite wavelength the evolution of the instability is described by the real Ginzburg-Landau equation, and this equation possesses only unstable spatially localized states. It is of interest therefore to examine what happens to these unstable states at larger amplitude, where the real Ginzburg-Landau equation no longer provides an adequate description of the system. In this paper we show that the localized states can become stable at such amplitudes, and indeed that there is a large multiplicity of coexisting stable localized states under very general conditions. We are able to relate the existence of these states to a phenomenon sometimes called homoclinic snaking that is well known from the theory of reversible systems with 1:1 resonance, and use this theory to construct a large number of such states. The stability properties of these states are also determined, and the evolution of nonstationary localized states is studied by numerical integration in time. It is an interesting fact that closely related phenomena have already been described in several areas involving pattern formation. The theory was originally developed in the context of water waves, where localized states have been studied by moving into a reference frame of the waves and converting the problem into an ordinary differential equation ODE. The resulting localized states are called solitary waves, and in some cases turn out to be solitons. Kirchgassner 2 has pioneered a successful approach to this type of problem that led to a number of advances in this area. Specifically, the ODE is viewed as a dynamical system in space, and localized states are sought as homoclinic orbits connecting the trivial state to itself. Whether such orbits are possible depends in part on the stability properties of the trivial state: eigenvalues with positive real part indicate that a nontrivial state can grow from x=, while eigenvalues with negative real part indicate that such a state may return, under appropriate conditions, back to the trivial state as x →. The spectrum of the linearization about the trivial state is influenced by spatial symmetries of the system. In many cases, and in particular in the case considered here, the ODE is reversible. As a result the bifurcations that are encountered as a parameter is increased are nongeneric. In the present case the spatial dynamics of the system near the trivial state turn out to be described by the reversible 1:1 resonance. The unfolding of this resonance has been worked out in detail by Iooss and Peroueme 3, and can be used to understand the appearance of a variety of homoclinic orbits in this system, and hence of localized states with different spatial structure.

Simulations of convection in 3He-4He mixtures with a negative separation ratio in two-dimensional containers with realistic boundary conditions and moderately large aspect ratio reveal, at supercritical Rayleigh numbers, the existence of "convectons," i.e., localized states of stationary convection, separated by regions of no convection. The origin and properties of these states are described.

Parametrically driven surface gravity-capillary waves in an elliptically distorted circular cylinder are studied. In the nearly inviscid regime, the waves couple to a streaming flow driven in oscillatory viscous boundary layers. In a cylindrical container, the streaming flow couples to the spatial phase of the waves, but in a distorted cylinder, it couples to their amplitudes as well. This coupling may destabilize pure standing oscillations, and lead to complex time-dependent dynamics at onset. Among the new dynamical behavior that results are relaxation oscillations involving abrupt transitions between standing and quasiperiodic oscillations, and exhibiting ‘canards’.

Linear and weakly nonlinear theory of overstable convection in large but bounded containers is reviewed and the results compared with detailed numerical simulations of binary fluid convection in a two-dimensional domain with realistic boundary conditions. For sufficiently negative separation ratios convection sets in as growing oscillations; the corresponding eigenfunctions take the form of ‘chevrons’ of either odd or even parity. These may bifurcate sub- or supercritically. Simulations of 3He—4He and water-ethanol mixtures show that the oscillations may equilibrate in finite amplitude chevron states, or that these states are unstable to blinking or repeated transient states. The results compare favorably with available experiments.

The onset of convection in systems that are heated via current dissipation in the lower boundary or that lose heat from the top boundary via Newton’s law of cooling is formulated as a bifurcation problem. The Rayleigh number as usually defined is shown to be inappropriate as a bifurcation parameter since the temperature across the layer depends on the amplitude of convection and hence changes as convection evolves at fixed external parameter values. Moreover, the final state of the system is also different since it depends on the details of the applied boundary conditions. A modified Rayleigh number is introduced that does remain constant even when the system is evolving, and solutions obtained with the standard formulation are compared with those obtained via the new one.

The dynamics of parametrically driven, slowly varying counterpropagating wave trains in nearly conservative systems are considered. The system is assumed to be invariant under reflection and translations in one direction, and periodic boundary conditions with period L are imposed, with L large but not too large in order that the effect of detuning be significant. The dynamics near the minima of the resulting resonance tongues are described by a system of coupled nonlocal Schrödinger equations with damping and parametric forcing. Elsewhere the long time behavior of the system is described by a damped complex Duffing equation with real coefficients, whose solutions relax to spatially uniform standing waves. Near the bicritical points where two adjacent resonance tongues intersect a pair of coupled damped complex Duffing equations captures the properties of both pure and mixed modes, and of the periodic solutions resulting from a Hopf bifurcation on the branch of mixed modes. As an application, we consider a Faraday system in an annulus in which a pair of counterpropagating surface gravity-capillary waves are excited parametrically by vertical vibration of the container, including the mean flow driven by time-averaged Reynolds stresses due to oscillatory viscous boundary layers along the bottom and the free surface. This mean flow is shown to have a large effect near the bicritical point, where the mean flow changes the dynamics of the system both quantitatively and qualitatively. In particular, inclusion of the mean flow permits Hopf bifurcations from the branches of pure standing waves, and parity-breaking bifurcations from mixed modes.

Direct numerical simulations of chevrons, blinking states, and repeated transients in binary fluid mixtures with a negative separation ratio heated from below are described. The calculations are performed in two-dimensional containers using realistic boundary conditions and the parameter values used in the experiments of Kolodner [Phys. Rev. E 47, 1038 (1993)]. Particular attention is paid to the multiplicity of states, and their dependence on the applied Rayleigh number and the aspect ratio of the container. Quantitative agreement with the experiments is obtained, and a mechanism explaining the origin and properties of the repeated transients observed in the experiments is proposed.

Faraday waves [1] are gravity-capillary waves that are excited on the surface of a fluid when its container is vibrated vertically and the vertical acceleration exceeds a threshold value. These waves have received much attention in the literature both as a basic fluid dynamical problem and as a paradigm of a pattern-forming system [2],[3],[4]. Unfortunately, in the low viscosity limit, there are several basic issues that remain unresolved, particularly in connection with the generation of mean flows in the bulk. The viscous part of these flows (also called streaming flow or acoustic streaming) is driven by the oscillatory boundary layers attached to the solid walls and the free surface by well-known mechanisms first uncovered by Schlichting [5] and Longuet-Higgins [6]. This mean flow has been shown recently to affect the dynamics of the primary waves at leading order in a related, laterally vibrated system [7]. This is somewhat similar to the effect of an internal circulation on surface wave dynamics in drops [8].

Modulation of cyclic magnetic activity associated with grand minima, as well as modulation associated with breaking of dipole or quadrupole symmetry, can be represented in low-order normal form equations. The behaviour found is robust and can be related to similar spatio-temporal patterns in mean field dynamo models.

Nonlinear doubly diffusive convection in two-dimensional enclosures driven by lateral temperature and concentration differences is studied using a combination of analytical and numerical techniques. The study is organized around a special case that allows a static equilibrium. The stationary states that bifurcate from this equilibrium are either symmetric or antisymmetric with respect to diagonal reflection. Local bifurcation analysis around the critical aspect ratio at which both modes appear simultaneously is complemented using numerical continuation. Perturbation of this situation to one in which no static equilibrium exists provides important information about the multiplicity of steady states in this system.

Long-wavelength instabilities of steady patterns, spatially periodic in three dimensions, are studied. All potentially stable patterns with the symmetries of the simple-, face-centered- and body-centered-cubic lattices are considered. The results generalize the well-known Eckhaus, zigzag, and skew-varicose instabilities to three-dimensional patterns and are applied to two-species reaction-diffusion equations modeling the Turing instability.

Different mechanisms believed to be responsible for the generation of bursts in hydrodynamical systems are reviewed and a new mechanism capable of generating regular or irregular bursts of large dynamic range near threshold is described. The new mechanism is present in the interaction between oscillatory modes of odd and even parity in systems of large but finite aspect ratio, and provides an explanation for the bursting behavior observed in binary fluid convection by Sullivan and Ahlers.

The onset of oscillatory convection in binary fluid mixtures in a two-dimensional domain with realistic boundary conditions on all boundaries is determined as a function of the fluid parameters and the aspect ratio Gamma of the container. The first unstable mode has either odd or even parity under left-right reflection. Depending on Gamma and the separation ratio S, this mode has the form of a standing wave, or a "chevron," consisting of a pair of waves propagating outwards from the cell center (or, in some cases, inwards towards it). Codimension--two points at which odd and even parity modes are simultaneously marginally stable are determined, as are various Takens-Bogdanov points. For fixed SS(TB), all mode interactions among modes of like parity, arising as Gamma varies, are of the nonresonant Hopf-Hopf type; however, the details of the modal interchange are organized by resonant Hopf bifurcations with 1:1 resonance. Particular attention is paid to the asymptotic mode structure as Gamma--infinity, and to the gap (in Rayleigh number and oscillation frequency) between successively unstable modes. The results quantify the parameter regime in which the weakly nonlinear dynamics of the system can be described in terms of the interaction of the first odd and even parity oscillatory modes.

The tendency towards two-dimensionality in rapidly rotating flows is described by the Taylor-Proudman theorem. In applications strict two-dimensionality is usually broken by the presence of boundary and/or thermal forcing, or by initial conditions. Both types of forcing are present in thermal convection in a rapidly rotating horizontal layer, described by the equations $$ \frac{{Du}}{{Dt}} + \hat{\Omega } \times u = - \nabla \pi + E{{\nabla }^{2}}u + \:fracRa\sigma {{E}^{2}}T\hat{z},{\text{ }}\nabla \cdot u = 0 $$ (1) $$ \sigma \frac{{DT}}{{Dt}} = E{{\nabla }^{2}}T $$ (2)

We consider the simplest instabilities involving multiple unstable electrostatic plasma waves corresponding to four-dimensional systems of mode amplitude equations. In each case the coupled amplitude equations are derived up to third order terms. The nonlinear coefficients are singular in the limit in which the linear growth rates vanish together. These singularities are analyzed using techniques developed in previous studies of a single unstable wave. In addition to the singularities familiar from the one mode problem, there are new singularities in coefficients coupling the modes. The new singularities are most severe when the two waves have the same linear phase velocity and satisfy the spatial resonance condition $k_2=2k_1$. As a result the short wave mode saturates at a dramatically smaller amplitude than that predicted for the weak growth rate regime on the basis of single mode theory. In contrast the long wave mode retains the single mode scaling. If these resonance conditions are not satisfied both modes retain their single mode scaling and saturate at comparable amplitudes., 34 pages (Latex), no figures

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