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COMMENT Note on a paper by G Rowlands

By A C Fowler


Abstract. This note discusses the possible use of perturbation methods in studying chaotic trajectories of ordinary differential equations, with particular focus on a recent paper on this topic by Rowlands. In a recent paper, Rowlands (1983) has presented an approximate analysis of the Lorenz (1963) equations, which gives a (quantitative) description of the cusp-shaped difference map found by Lorenz, which related successive values in the sequence {M,,} of maxima of one of the variables, Z. Since this non-monotone map provides an explanation of the chaotic (aperiodic) behaviour of numerically computed solutions of the Lorenz equations (Lorenz 1963, Li and Yorke 1975, May 1976, Collet and Eckmann 1980), its prediction would essentially ‘solve ’ the system-at least in principle-and one could then seek to apply Rowlands ’ method to other chaotic systems. The method used is none other than the standard method of multiple scales (as developed by Stuart (1960) to follow disturbances to shear flows into the finite amplitude range) applied to Hopf bifurcation from a steady state solution. This method is well known and widely used, but in the original problem (turbulence in shear flow

Year: 1983
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