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The complex Lorenz equations. (English) Zbl 1194.37039

Summary: We have undertaken a study of the complex Lorenz equations. Behaviour remarkably different from the real Lorenz model occurs. Only the origin is a fixed point except for the special case \(e + r_{2} = 0\). We have been able to determine analytically two critical values of \(r_{1}\). The origin is a stable fixed point for \(0 < r_{1} < r_{1c}\), but for \(r_{1} >r_{1c}\), a Hopf bifurcation to a limit cycle occurs. We have an exact analytic solution for this limit cycle which is always stable if \(\sigma <b + 1\). If \(\sigma > + 1\) then this limit is only stable in the region \(r_{1c} < r_{1} < r_{lc}\). When \(r_{1} >r_{lc}\), a transition to a finite amplitude oscillation about the limit cycle occurs. The nature of this bifurcation is studied in detail by using a multiple time scale analysis to derive the Stuart-Landau amplitude equation from the original equations in a frame rotating with the limit cycle frequency. This latter bifurcation is either a sub- or super-critical Hopf-like bifurcation to a doubly periodic motion, the direction of bifurcation depending on the parameter values. The nature of the bifurcation is complicated by the existence of a zero eigenvalue.

MSC:

37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37C55 Periodic and quasi-periodic flows and diffeomorphisms
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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