We prove that the Lorenz equations support a strange attractor, as conjectured by Ed-ward Lorenz in 1963. We also prove that the attractor is robust, i.e., it persists under small perturbations of the coeecients in the underlying diierential equations. The proof is based on a combination of normal form theory and rigorous numerical computations.

A six-dimensional Rössler-Lorenz hybrid has two coexistent attractors. Both, either or neither may be strange. Introduction. The Rössler [1] and Lorenz [2] equations are textbook examples of chaotic dynamical systems. Each is three-dimensional, continuoustime, smooth and autonomous, with a single strange attractor for certain parameter values. This paper draws attention to a six-dimensional hyb...

Since the seminal remark by Pecora and Carroll [Phys. Rev. Lett. 64, 821 (1990)] that one can synchronize chaotic systems, the main example in the related literature has been the Lorenz equations. Yet this literature contains a mixture of true and false, and of justified and unsubstantiated claims about the synchronization properties of the Lorenz equations. In this note we clarify some of the ...

1. Much of the recent literature on chaos in dynamtransitions can be intermittent. ical systems has been concerned with pointing out the All these phenomena, and particularly hysteresis, behavioural similarity of many different types of can be explained on the basis of a suitable non-monomodels. The transition to aperiodic motion via the tone difference equation [11], such as that relating (Fei...

Inspired by numerical solutions of the Lorenz equations, we model the Poincar e map of the ow by a one-parameter map of the unit interval. For a certain region in the parameter space of the Lorenz equations, we show that the corresponding one-dimensional map is chaotic, imposing only minimal conditions on its derivative. Perturbing the map, we get a two-dimensional map corresponding to a ow in ...

In this paper we study coupled nonidentical Lorenz equations with three different boundary conditions. Coupling rules and boundary conditions play essential roles in the qualitative analysis of solutions of coupled systems. By using Lyapunov stability theory, a sufficient condition is obtained for the global stability of trivial equilibrium of coupled system with Dirichlet condition. Then we re...

The classical Lorenz lowest order system of three nonlinear ordinary differential equations, capable of producing chaotic solutions, has been generalized by various authors in two main directions: (i) for number of equations larger than three (Curry1978) and (ii) for the case of complex variables and parameters. Problems of laser physics and geophysical fluid dynamics (baroclinic instability, g...