In this paper we study the Lorenz-96 model which is defined by the equations dxjdt=xj-1(xj+1-xj-2)-xj+F,j=0,…,n-1, together with the periodic “boundary condition” implied by taking the indices j modulo n. The dimension n∈N and the forcing parameter F∈R are free parameters. interpreted the variables xj as values of some atmospheric quantity in n equispaced sectors of a latitude circle, where the index j plays the role of “longitude”. Hence, a larger value of n can be interpreted as a finer latitude grid. Lorenz also remarked that the vectors (x0,…,xn-1) can be interpreted as wave profiles, and he observed that for F>0 sufficiently large these waves slowly propagate “westward”, i.e., in the direction of decreasing j. Figure  shows a Hovmöller diagram illustrating two traveling waves with wave number 5 for dimension n=24 and the parameter values F=2.75 (in the periodic regime) and F=3.85 (in the chaotic regime).

The Lorenz-96 model was introduced as a tool for numerical experiments in predictability studies, rather than as a physically realistic model. Indeed, wrote that “the physics of the atmosphere is present only to the extent that there are external forcing and internal dissipation, simulated by the constant and linear terms, while the quadratic terms, simulating advection, together conserve the total energy […]”. The value of the Lorenz-96 model primarily lies in the fact that it has a very simple implementation in numerical codes while at the same time it can exhibit very complex dynamics for suitable choices of the parameters n and F. The famous Lorenz-63 model , which does have a clear physical interpretation (namely, as the Galerkin projection of a fluid dynamical model describing Rayleigh–Bénard convection), has two disadvantages. Firstly, it consists of only three ordinary differential equations. Secondly, for the classical parameter values the model has a Lyapunov spectrum (0.91,0,-14.57), which makes the model very dissipative. Such properties are not typical for atmospheric models. In contrast, the dimension of the Lorenz-96 model can be chosen to be arbitrarily large, and for suitable values of the parameters the Lyapunov spectrum is similar to those observed in models obtained from discretizing partial differential equations. For those reasons the Lorenz-96 model has become a test model for a wide range of geophysical applications.

Table  lists some recent papers with applications of the Lorenz-96 model. In most studies the dimension n is chosen ad hoc, but n=36 and n=40 appear to be popular choices. Many applications are related to geophysical problems, but the model has also attracted the attention of mathematicians working in the area of dynamical systems for phenomenological studies in high-dimensional chaos. Note that Eq. () is in fact a family of models parameterized by means of the discrete parameter n. An important question is to what extent both the qualitative and quantitative dynamical properties of Eq. () depend on n. For example, the dimension n has a strong effect on the predictability of large amplitudes of traveling waves in weakly chaotic regimes of the Lorenz-96 model . In general, the statistics of extreme events in dynamical systems strongly depend on topological properties and recurrence properties of the system . Therefore, a coherent overview of the dependence of spatiotemporal properties on the parameters n and F is useful to assess the robustness of results when using the Lorenz-96 model in predictability studies.

In this paper we address the question of how the spatiotemporal properties of waves, such as their period and wave number, in the Lorenz-96 model depend on the dimension n and whether these properties tend to a finite limit as n→∞. We will approach this question by studying waves represented by periodic attractors that arise through a Hopf bifurcation of a stable equilibrium. Along various routes to chaos these periodic attractors can bifurcate into chaotic attractors representing irregular waves which “inherit” their spatiotemporal properties from the periodic attractor. For example, the wave shown in the left panel of Fig.  bifurcates into a three-torus attractor which breaks down and gives rise to the wave in the right panel. Note that both waves have the same wave number. Figure  shows power spectra of these waves, and clearly their dominant peaks are located at roughly the same period. Inheritance of spatiotemporal properties also manifests itself in a shallow water model studied by in which a Hopf bifurcation (related to baroclinic instability) explains the observed timescales of atmospheric low-frequency variability. The Hopf bifurcation plays a key role in explaining the physics of low-frequency variability in many geophysical contexts. Examples are the Atlantic Multidecadal Oscillation , the wind-driven ocean circulation , and laboratory experiments .

In addition to traveling waves, such as illustrated in Fig. , we will also show the existence of stationary waves. In a recent paper by stationary waves have also been discovered in specific regions of the multi-scale Lorenz-96 model. Their paper uses dynamical indicators such as the Lyapunov dimension to identify the parameter regimes with stationary waves. Moreover, we will explain two bifurcation scenarios by which waves with different spatiotemporal properties coexist. This paper complements the results of our previous work , which considers only the case F>0; these two papers together give a comprehensive picture of wave propagation in the Lorenz-96 model.

The remainder of this paper is organized as follows. In Sect.  we explain how to obtain an approximation of the periodic attractor born at a Hopf bifurcation, which enables us to derive spatiotemporal properties of waves in the Lorenz-96 model. In Sect.  we show that, for F>0, periodic attractors indeed represent traveling waves as suggested by Lorenz. Also for F<0 and odd values of n, periodic attractors represent traveling waves, as is demonstrated in Sect. . In Sect. , however, we show analytically that, for n=6 and F<0, stationary waves occur. By means of numerical experiments we show in Sect.  that stationary waves occur in general for even n and F<0. In Sect.  we discuss the bifurcation scenarios by which stable waves with different spatiotemporal properties can coexist for the same values of the parameters n and F.

In this section we consider a general geophysical model in the form of a system of ordinary differential equations: dxdt=f(x,μ),x∈Rn. In this equation, μ∈R is a parameter modeling external circumstances such as forcing. Assume that for the parameter value μ0 the system has an equilibrium x0; this means that f(x0,μ0)=0 and hence x0 is a time-independent solution of Eq. (). In the context of geophysics x0 represents a steady flow, and its linear stability is determined by the eigenvalues of the Jacobian matrix Df(x0,μ0). An equilibrium becomes unstable when eigenvalues of the Jacobian matrix cross the imaginary axis upon variation of the parameter μ. provides an extensive discussion of the physical interpretation of bifurcation behavior.

Assume that Df(x0,μ0) has two eigenvalues ±ωi on the imaginary axis. This indicates the occurrence of a Hopf bifurcation, i.e., the birth of a periodic solution from an equilibrium that changes stability. If the equilibrium x0 is stable for μ<μ0 and unstable for μ>μ0, then under suitable non-degeneracy conditions the Hopf bifurcation is supercritical, which means that a stable periodic orbit exists for μ>μ0 . For small values of ε=μ-μ0 the periodic orbit that is born at the Hopf bifurcation can be approximated by x(t)=x0+εRe[(u+iv)eiωt]+O(ε2), see . Without loss of generality we may assume that the corresponding complex eigenvectors u±iv of the matrix Df(x0,μ0) have unit length. In the context of geophysical applications this first-order approximation of the periodic orbit can be interpreted as a wave-like perturbation imposed on a steady mean flow. The spatiotemporal properties of this wave can now be determined by the vectors x0,u,v, and the frequency ω.

In this section we study waves in the Lorenz-96 model and how their spatiotemporal characteristics depend on the parameters n and F.

The results of Sect.  show that for even n and F<0 either two or four stable periodic orbits coexist for the same parameter values. This phenomenon is referred to as multi-stability in the dynamical systems literature. An overview of the wide range of applications of multi-stability in different disciplines of science is given by .

Multi-stability also occurs when F>0, but for a very different reason. For n=12, Theorem 1 implies that the first bifurcation of the equilibrium xF=(F,…,F) for F>0 is not a Hopf bifurcation, but a double-Hopf bifurcation. Indeed, at F=1 we have two pairs of purely imaginary eigenvalues, namely (λ2,λ10)=(-i3,i3) and (λ3,λ9)=(-i,i). Note that the double-Hopf bifurcation is a codimension-2 bifurcation which means that generically two parameters must be varied in order for the bifurcation to occur . However, symmetries such as those in the Lorenz-96 model can reduce the codimension of a bifurcation.

In previous work we have introduced an embedding of the Lorenz-96 model in a two-parameter family by adding a diffusion-like term multiplied by an additional parameter G: dxjdt=xj-1(xj+1-xj-2)-xj+G(xj-1-2xj+xj+1)+F,j=0,…,n-1. Note that by setting G=0 we retrieve the original Lorenz-96 model in Eq. (). Since the Jacobian matrix of Eq. () is again a circulant matrix we can use Eq. () to determine its eigenvalues: λj=-1-2G(1-cos⁡(2πj/n))+Ff(2πj/n)+Fg(2πj/n)i. Also note that xF=(F,…,F) remains an equilibrium solution of Eq. () for all (F,G). The Hopf bifurcations of xF described in Theorem 1 now occur along the lines G=Ff(2πj/n)-12(1-cos⁡(2πj/n)), and the intersection of two such lines leads to a double-Hopf bifurcation.

Figure  shows a local bifurcation diagram of the two-parameter Lorenz-96 model in the (F,G) plane for n=12 which was numerically computed using MATCONT . A double-Hopf point is located at (F,G)=(1,0), which is indeed implied by Theorem 1. The normal form of a double-Hopf bifurcation depends on the values of two coefficients which determine the unfolding of the bifurcation. In total, there are 11 different bifurcation scenarios to consider. The normal form computation of the double-Hopf bifurcation for n=12 and (F,G)=(1,0) in shows that the unfolding of this particular case is of “type I in the simple case” as described by . This means that from the double-Hopf point only two curves of Neĭmark-Sacker bifurcations emanate. In discrete-time dynamical systems, a Neĭmark-Sacker bifurcation is the birth of a closed invariant curve when a fixed point changes stability through a pair of complex eigenvalues crossing the unit circle in the complex plane. From a continuous-time system, such as Eq. (), we can construct a discrete-time dynamical system by defining a Poincaré return map of a periodic orbit, in which case a Neĭmark-Sacker bifurcation refers to the birth of an invariant two-dimensional torus when the periodic orbit changes stability by a pair of Floquet multipliers crossing the unit circle in the complex plane.

In order to explain the dynamics in a neighborhood around the double-Hopf point, we now use Fig.  to describe the successive bifurcations that occur for G=0.1 fixed and increasing F. At F=1.1 the equilibrium xF becomes unstable through a supercritical Hopf bifurcation (the blue line given by G=F-1) and a stable periodic orbit with wave number 2 is born. At F=1.2 the now unstable equilibrium undergoes a second Hopf bifurcation (the red line given by G=12(F-1)) and an unstable periodic orbit with wave number 3 is born. The latter periodic orbit becomes stable at F≈1.58 through a subcritical Neĭmark-Sacker bifurcation (orange curve) and an unstable two-dimensional invariant torus is born. Hence, for parameter values F>1.58 two stable waves with wave numbers 2 and 3 coexist until one of these waves becomes unstable in a bifurcation. For all fixed values of 0<G<0.15 the same bifurcation scenario occurs, but the values of F are different. For -0.06<G<0 the roles of the two Hopf bifurcations and periodic orbits have to be interchanged.

The scenario described above shows how the presence of two subcritical Neĭmark-Sacker bifurcations emanating from a double-Hopf bifurcation determines a region of the (F,G) plane in which two stable periodic orbits coexist with an unstable two-dimensional invariant torus. We will refer to this region as the “multi-stability lobe”. The scenario described above is not limited to the special case of the Lorenz-96 model, but occurs near a double-Hopf bifurcation of type I in any dynamical system .

Double-Hopf bifurcations are abundant in the two-parameter Lorenz-96 model of Eq. (). The lines described in Eq. () have a different slope for all 0<j<n/2 and j≠n/3, and hence they mutually intersect each other. This implies that the number of double-Hopf points in the (F,G) plane grows quadratically with n, see Appendix . However, not all these points will have an influence on the dynamics: if xF is already unstable, then any dynamical object born through the double-Hopf bifurcation will also be unstable. In what follows, we only consider the double-Hopf bifurcations through which xF can change from stable to unstable. We can find such points as follows. Starting from the line in Eq. () with j=l1+(n) as defined by Eq. (), we first compute double-Hopf points by computing the intersections with all other lines. From these intersections we select those that satisfy the condition max⁡{Reλj:j=0,…,n-1}=0.

Figure  shows the G coordinates of these double-Hopf points as a function of n. Clearly, for large n there exist double-Hopf points which are very close to the F axis, which suggests that the multi-stability lobe that emanates from such points can intersect the F axis and hence influence the dynamics of the original Lorenz-96 model for G=0. Moreover, Fig.  shows that for n>12 there are always two double-Hopf points by which xF can change from stable to unstable. It is then possible that two multi-stability lobes intersect each other, which leads to a region in the (F,G) plane in which at least three stable waves coexist.

Figures – show bifurcation diagrams of three periodic orbits as a function of F for G=0 for n=40,60,80. For each periodic orbit the continuation is started from a Hopf bifurcation of the equilibrium xF. If xF is unstable, then so will be the periodic orbit. However, when the boundary of a multi-stability lobe is crossed, a Neĭmark-Sacker bifurcation occurs, by which a periodic orbit can gain stability. For specific intervals of the parameter F, three stable periodic orbits coexist. Since Fig.  shows that for large values of n the double-Hopf bifurcations are close to the F axis, we expect that the coexistence of three or more stable waves is typical for the Lorenz-96 model.

The double-Hopf bifurcation has been reported in many works on fluid dynamical models. A few examples are baroclinic flows , rotating cylinder flows , Poiseuille flows , rotating annulus flows , and quasi-geostrophic flows . In all of these examples, the coexistence of multiple waves is reported, where the nature of these waves depends on the specific model.

In this paper we have studied spatiotemporal properties of waves in the Lorenz-96 model and their dependence on the dimension n. For F>0 the first bifurcation of the equilibrium xF=(F,…,F) is either a supercritical Hopf or a double-Hopf bifurcation and the periodic attractor born at the Hopf bifurcation represents a traveling wave. The spatial wave number is determined by the index of the eigenpair that crosses the imaginary axis and increases linearly with n, but the period tends to a finite limit as n→∞. For F<0 and n odd, the first bifurcation of xF is always a supercritical Hopf bifurcation and the periodic attractor that appears after the bifurcation is again a traveling wave. In this case the wave number equals (n-1)/2 and the period is O(4n).

For n even and F<0 the first bifurcation of xF is a pitchfork bifurcation which occurs at F=-12 and leads to two stable equilibria. If n=4k+2 for some k∈N, then each of these equilibria undergoes a Hopf bifurcation which leads to the coexistence of two stationary waves. The role of the pitchfork bifurcation is to change the mean flow which in turn changes the propagation of the wave. If n=4k for some k∈N, then two pitchfork bifurcations take place at F=-12 and F=-3 before a Hopf bifurcation occurs, which leads to the coexistence of four stationary waves.

The occurrence of pitchfork bifurcations before the Hopf bifurcation leads to multi-stability, i.e., the coexistence of different waves for the same parameter settings. A second scenario that leads to multi-stability is via the double-Hopf bifurcation. For n=12 the equilibrium xF loses stability through a double-Hopf bifurcation. By adding a second parameter G to the Lorenz-96 model we have studied the unfolding of this codimension-2 bifurcation. Two Neĭmark-Sacker bifurcation curves emanating from the double-Hopf point bound a lobe-shaped region in the (F,G) plane in which two stable traveling waves with different wave numbers coexist. For dimensions n>12 we find double-Hopf bifurcations near the F axis, which can create two multi-stability lobes intersecting each other, and in turn this can lead to the coexistence of three stable waves coexisting for G=0 and a range of F values. Hence, adding a parameter G to the Lorenz-96 model helps to explain the dynamics, which is observed in the original model for G=0.

Our results provide a coherent overview of the spatiotemporal properties of the Lorenz-96 model for n≥4 and F∈R. Since the Lorenz-96 model is often used as a model for testing purposes, our results can be used to select the most appropriate values of n and F for a particular application. The periodic attractors representing traveling or stationary waves can bifurcate into chaotic attractors representing irregular versions of these waves, and their spatiotemporal properties are inherited from the periodic attractor (see for example Figs. ) and . This means that our results on the spatiotemporal properties of waves apply to broader parameter ranges of the parameter F than just in a small neighborhood of the Hopf bifurcation.

The results presented in this paper also illustrate another important point: both qualitative and quantitative aspects of the dynamics of the Lorenz-96 model depend on the parity of n. This phenomenon also manifests itself in discretized partial differential equations. For example, for discretizations of Burgers' equation, observed that for odd degrees of freedom the dynamics were confined to an invariant subspace, whereas for even degrees of freedom this was not the case. For the Lorenz-96 model the parity of n also determines the possible symmetries of the model. We will investigate these symmetries and their consequences on bifurcation sequences using techniques from equivariant bifurcation theory in forthcoming work .