Volume 104A, number 1 PHYSICS LETTERS 6 August 1984 LIMIT CYCLES IN A FORCED LORENZ SYSTEM J.K. BHATTACHARJEE, K. BANERJEE, D. CHOWDHURY, R. SARAVANAN Department of Physics, Indian Institute of Technology, Kanpur 208016, India and S. MANNA Saha Institute of Nuclear Physics, Calcutta 700009, India Received 17 May 1984 We consider a Lorenz system where the control parameter is sinusoidally modulated. Limit cycles appear where pre- viously there were strange attractors. Perturbation theory is used to estimate the critical amplitude of the modulation for which limit cycles appear. To facilitate the study of the onset of turbulence in the hydrodynamical example of a fluid heated from below, a three-mode truncation leading to a system of three coupled ordinary differential equa- tions was carried out by Saltzman [ 1 ] and Lorenz [2]. The chaotic behavior of the solutions of the sys- tem of equations was studied by Lorenz [2], who found that the conduction state was followed by a steady convection, which destabilized to a state char- acterized by a strange attractor. The last is the ana- logue of turbulence in hydrodynamics. The route to chaos in the Lorenz system however differs signifi- cantly from the real hydrodynamics, since in the latter the steady convection generally yields a time- periodic state with a single frequency, followed by a quasi-periodic regime and the eventual onset of turbu- lence through period doubling, intermittency or the production of higher tori [3]. Here we show by using perturbation theory [4,5] that if the control param- eter of the Lorenz system is modulated periodically, then it is possible to stabilize the usually unstable limit cycle. A periodic state is thus obtained, restoring dynamic similarity with the real hydrodynamics * 1 .1 Truncation of the hydrodynamic equations by keeping a larger number of Fourier modes yields answers closer to real hydrodynamics, see ref. [6 ]. 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) (unmodulated). The onset of chaos [7] from this limit cycle can now be studied by varying the control parameter. Numerical work [7] on this has been car- ded out. The unmodulated Lorenz system reads X=o(-X + Y), (la) ~'= -XZ + rX - Y, (lb) Z=XY-bZ . (lc) Here r is the control parameter, b a geometric factor and o the Prandtl number for the fluid. For r < 1, the steady conduction state X = Y = Z = 0 is stable. At r = 1 the state loses stability to the steady convec- tion state governed by the fLxed points X* = Y* = +x/b--(r - 1), Z* = r - 1. This pair of stable fixed points undergoes a Hopf bifurcation at r = r y = o(o + b + 3)[(o - b - 1). The bifurcation, however, is inverted and the limit cycle produced for r < r T is unstable. For o = 10, b = 8[3 (the values used by Lorenz), r T = 24.74. For r > r T there is no stable periodic state, numerical integration of the equations by Lorenz for r = 28 showed the existence of a strange attractor. We modu. late the control parameter r as 33