Dynamical systems like the one described by the three-variable Lorenz model may serve as metaphors for complexity in nature. When natural systems are perturbed by external forcing factors, they tend to relax back to their equilibrium conditions after the forcing has shut off. Here we investigate the behavior of such transients in the Lorenz model by studying its trajectories initialized far away from the asymptotic attractor. Perhaps somewhat surprisingly, these transient trajectories exhibit complex routes and, among other things, sensitivity to initial conditions akin to that of the asymptotic behavior on the attractor. Thus, similar extreme events may lead to widely different variations before the perturbed system returns back to its statistical equilibrium.

The by now famous Lorenz system

Because of the great interest in the structural details of this – and
other – chaotic attractors, their numerical simulations are usually
initialized near the attractor itself. In this case the transients,
defined as phase-space trajectories connecting the initial condition
and the attractor, are short and uninteresting

Asymptotically, as time

The first characteristic of the Lorenz-system transients we looked at
was their duration (Fig. 1). A typical time scale associated with
a single revolution of the long-term trajectory around either lobe of
the butterfly-shaped asymptotic attractor for our choice of model
parameters is on the order of unity (not shown). This also happens to
be the duration of the longest transients for initial conditions on
the sphere

We will see later that longer-duration transient trajectories are also
the ones that exhibit the most interesting evolution. A useful
diagnostic for a potentially complex behavior is the
trajectory-averaged maximum local Lyapunov exponents. The local
Lyapunov exponents are defined as the eigenvalues of the dynamical
operator

A typical example of transient behavior for initial states chosen near the ribbon of longest transient times (Fig. 1) or, equivalently, that of largest trajectory-averaged Lyapunov exponents (Fig. 2), is shown in Fig. 3. Here a bunch of trajectories that emanate from close-by initial conditions splits in two diverging sets of trajectories, which follow very different routes prior to reuniting near the asymptotic attractor location; the latter attractor region is evident as a small butterfly-shaped cluster of trajectories close to the origin. The immediate consequence of such transient behavior in the Lorenz system is that it can apparently be as unpredictable as the asymptotic behavior in the sense that similar extreme perturbations may result in completely unrelated transients as the system relaxes back to the state of its statistical equilibrium.

Another interesting observation is that transient approach to the asymptotic attractor may be characterized by fairly complex trajectories, which appear, in some cases, to emulate the attractor itself (Fig. 4a and b). In particular, the trajectories here exhibit larger-scale butterfly-shaped excursions prior to ending up on the similarly shaped asymptotic attractor near the origin. We will refer to this phenomenon as to the “ghost” transient attractor.

Qualitatively, this behavior can be understood in the following
way. We saw previously that some transients are able to stay away from
the attractor for longer time periods than others (Fig. 1). For such
longer transient trajectories, the variables in the model
(Eq.

To present a concrete example of the rescaling mentioned above, we
introduce the following change of variables

Substituting the transformation (Eq.

Note that in the rescaling example Eqs. (

We studied transient behavior in numerical simulations of the
three-variable Lorenz model (Eq.

Persistent chaotic transients in the Lorenz system have been studied
before in the particular case when the Rayleigh number was chosen to
be just below the critical value required for chaotic
behavior

Transient behavior of dynamical systems recently drew a lot of
attention in the ecological literature

The properties of the transient behavior in the Lorenz model discussed here – which are likely to be typical for arbitrary equations exhibiting chaos – are not just beautiful, but may also have important implications in understanding the evolution of complex nonlinear systems such as climate, economy, ecosystems, sociological networks and so on, if these systems are somehow taken far from their equilibrium states. In particular, similar extreme perturbations in such systems may exhibit widely different variations before relaxing back to the statistical equilibrium.

This research was supported by the NSF grant 1243158 (SK) and 2013 University of Wisconsin-Milwaukee Research Growth Initiative grant 101X286 (AAT).