## Abstract

We have undertaken a study of the complex Lorenz equations x ̇ = -σx + σy. y ̇ = (r - z)x - ay. z ̇ = -bz + 1 2(x^{*}y + xy^{*}). where x and y are complex and z is real. The complex parameters r and a are defined by r = r_{1} + ir_{2}; a = 1 - ie and σ and b are real. 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}, namely r1_{c} and r1_{c} . 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 σ <b + 1. If σ > + 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.

Original language | English |
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Pages (from-to) | 139-163 |

Number of pages | 25 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 4 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1982 |

Externally published | Yes |