Abstract
Abstract. In this paper we study the spatiotemporal properties of waves in the Lorenz-96 model and their dependence on the dimension parameter n and the forcing parameter F. For F > 0 the first bifurcation is either a supercritical Hopf or a double-Hopf bifurcation and the periodic attractor born at these bifurcations represents a traveling wave. Its spatial wave number increases linearly with n, but its period tends to a finite limit as n → ∞. For F < 0 and odd n, the first bifurcation is again a supercritical Hopf bifurcation, but in this case the period of the traveling wave also grows linearly with n. For F < 0 and even n, however, a Hopf bifurcation is preceded by either one or two pitchfork bifurcations, where the number of the latter bifurcations depends on whether n has remainder 2 or 0 upon division by 4. This bifurcation sequence leads to stationary waves and their spatiotemporal properties also depend on the remainder after dividing n by 4. Finally, we explain how the double-Hopf bifurcation can generate two or more stable waves with different spatiotemporal properties that coexist for the same parameter values n and F.
Highlights
In this paper we study the Lorenz-96 model which is defined by the equations dxj dt= xj−1(xj+1 − xj−2) − xj + F, j= 0, . . ., n − 1, (1)together with the periodic “boundary condition” implied by taking the indices j modulo n
We will approach this question by studying waves represented by periodic attractors that arise through a Hopf bifurcation of a stable equilibrium
At F = 1.1 the equilibrium xF becomes unstable through a supercritical Hopf bifurcation and a stable periodic orbit with wave number 2 is born
Summary
In this paper we study the Lorenz-96 model which is defined by the equations dxj dt. together with the periodic “boundary condition” implied by taking the indices j modulo n. The dimension of the Lorenz 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. 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.
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