Laser Lorenz equations

Example 13.1 Lorenz equations The strange attractor The Lorenz equations (published in 1963 by Edward N. Lorenz a meteorologist and mathematician) are derived to model some of the unpredictable behavior of weather. The Lorenz equations represent the convective motion of fluid cell that is warmed from below and cooled from above. Later, the Lorenz equations were used in studies of lasers and batteries. For certain settings and initial conditions, Lorenz found that the trajectories of such a system never settle down to a fixed point, never approach a stable limit cycle, yet never diverge to infinity. Attractors in these systems are well-known strange attractors. 

These equations are similar to the Lorenz equations and can exhibit chaotic behavior (Haken 1983, Weiss and Vilaseca 1991). However, many practical lasers do not operate in the chaotic regime. In the simplest case /j, y, K then P and D relax rapidly to steady values, and hence may be adiabatically eliminated, as follows. 

It has been known since 1975 that chaotic behavior is possible in some lasers under specific conditions, due to the similarity between the Lorenz equations (which predict chaos in fluids) and the semiclassical Maxwell-Bloch equations describing single mode lasers including the ef-... 

We present a brief introduction to coupled transport processes described macroscopically by hydrodynamic equations, the Navier-Stokes equations [4]. These are difficult, highly non-linear coupled partial differential equations they are frequently approximated. One such approximation consists of the Lorenz equations [5,6], which are obtained from the Navier-Stokes equations by Fourier transform of the spatial variables in those equations, retention of first order Fourier modes and restriction to small deviations from a bifurcation of an homogeneous motionless stationary state (a conductive state) to an inhomogeneous convective state in Rayleigh-Benard convection (see the next paragraph). The Lorenz equations have been applied successfully in various fields ranging from meteorology to laser physics. 

A major development reported in 1964 was the first numerical solution of the laser equations by Buley and Cummings [15]. They predicted the possibility of undamped chaotic oscillations far above a gain threshold in lasers. Precisely, they numerically found almost random spikes in systems of equations adopted to a model of a single-mode laser with a bad cavity. Thus optical chaos became a subject soon after the appearance Lorenz paper [2]. 

Here 0 are parameters. Ed Lorenz (1963) derived this three-dimensional system from a drastically simpl ified model of convection rolls in the atmosphere. The same equations also arise in models of lasers and dynamos, and as we ll see in Section 9.1, they exactly describe the motion of a certain waterwheel (you might like to build one yourself). 

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