## 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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### References:

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