We apply a new method for the determination of periodic orbits of general dynamical systems to the Lorenz equations. The accuracy of the expectation values obtained using this approach is shown to be much larger and have better convergence properties than the more traditional approach of time averaging over a generic orbit. Finally, we discuss the relevance of the present work to the computatio...

We describe an unconventional method for treating initial value problems. The system of differential equations is discretized on a fixed interval, the initial value is left unspecified, and the underdetermined system is solved by a least squares minimization procedure. When applied to the celebrated Lorenz equations, it produces a simple smooth curve that terminates at a stationary point rather...

We present an example of a new strange attractor which, as we show, belongs to class wild pseudohyperbolic spiral attractors. find this in four-dimensional system differential equations which can be represented extension the Lorenz system.

This chapter deals with bifurcation dynamics in control systems, which are described by ordinary differential equations, partial differential equations and delayed differential equations. In particular, bifurcations related to double Hopf, combination of double zero and Hopf, and chaos are studied in detail. Center manifold theory and normal form theory are applied to simplify the analysis. Exp...

Four types of global error for initial value problems are considered in a common framework. They include classical forward error analysis and shadowing error analysis together with extensions of both to rescaling of time. To determine the amplification of the local error that bounds the global error we present a linear analysis similar in spirit to condition number estimation for linear systems...

Computational models based on discrete dynamical equations are a successful way of approaching the problem of predicting or forecasting the future evolution of dynamical systems. For linear and mildly nonlinear models, the solutions of the numerical algorithms in which they are based, converge to the analytic solutions of the underlying differential equations, for small time-steps and grid-size...

Details of a new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described. The technique combines abstract existence results based on the Conley index theory with rigorous computer assisted computations. As an application of these methods it is proven that for some explicit parameter values the Lorenz equations exhibit chaotic dynamics.