Bounds on the dissipation in the Lorenz system

In this Letter we find an upper bound on the time-averaged dissipation in the Lorenz system (L. s.) [J. Atmos. Sci. 20 (1963) 244]. Whereas bounding theories were developed and applied to systems described by partial differential equations displaying turbulent behavior, we develop a method similar to the 'background method' [Phys. Rev. E 49 (1994) 4087; Phys. Rev. E 53 (1996) 5957; Phys. Rev. E 51 (1995) 3192; Phys. Plasmas 10 (2003) 4314; Phys. Plasmas 10 (2003) 4324] and apply it to the L. s., which consists of three first-order ordinary differential equations. The bound and the bounding field are explicitly calculated and compared to the numerically computed solutions of the system. For large values of the control parameter, the bound and the time-averaged dissipation differ by less than three percent. We then apply our method to another positive quadratic form defined for the solutions of the L. s