Many physical systems exhibit transient chaos; that is they behave in an erratic, unpredictable way for a period of time, but eventually settle down to rest or to a simple periodic motion.
Rolling dice, tossing coins and other means of generating random outcomes are familiar examples.
A similar phenomenon is observed in experiments on ﬂuids .
Mathematical models also can exhibit long chaotic transients, eg the H´ enon map [2, 3], coupled Van der Pol oscillators  and the kicked double rotor .
Despite the diversity of these examples, they are thought to share in common a universal dynamical mechanism, which is the existence in phase space of an unstable, fractal invariant set on which the dynamics are chaotic [2, 6].
This set has been called a chaotic saddle (owing to its saddle-type instability) or strange repeller.
The existence of a chaotic saddle generates transient chaotic dynamics by the following mechanism.
Phase space orbits originating exactly on the saddle remain chaotic for all time, but due to the saddle’s instability these orbits are not experimentally observable.
Rather, a typical phase space trajectory enters a neighborhood of the saddle via its stable manifold, and thereafter shadows the saddle for a period of time during which it exhibits the erratic motion associated with the chaotic dynamics on the saddle.
After this chaotic transient period the trajectory exits the the saddle along its unstable manifold, eventually to be captured by an attracting set (typically a ﬁxed point or periodic orbit, but possibly a chaotic ∗ Electronic address: firstname.lastname@example.org; URL: http://cariboo.tru.
Ca/advtech/math-sta/faculty/rtaylor † Electronic address: email@example.com; URL: http:// www.math.uwaterloo.ca/∼sacampbe attractor).
Because the the dynamics on unstable sets cannot be observed directly, it is often asserted that such sets have little relevance to experimental observations.
However, in transient chaos it is precisely the transient behavior that is of interest.
Indeed, Kantz and Grassberger  have argued that, owing to the mechanism described above, a uniﬁed understanding of chaotic transients relies on an analysis of the dynamics on the unstable chaotic set.
A fairly complete analysis is possible for systems that exhibit horseshoe-type dynamics , such as the celebrated homo- or hetero-clinic chaos that occurs due to the transversal intersection of stable and unstable manifolds of an equilibrium point or periodic orbit .
In this situation the saddle is known to be a product of Cantor sets, and the dynamics on it are conjugate to a subshift of ﬁnite type (a generalization of the Smale horseshoe ), yielding a symbolic coding of the dynamics on the saddle.
However, it is often diﬃcult to obtain rigorous results for realistic models, so that there is an active literature on the numerical detection and approximation of chaotic saddles.
Here the essential problem is to construct a numerical trajectory that lies very near the saddle for an arbitrarily long time, the idea being that such a trajectory will shadow a true trajectory on the saddle.
This is accomplished by repeatedly making small (ie at the limit of numerical precision) perturbations of a numerical trajectory, so that it remains indeﬁnitely within a (small) neighborhood of the saddle.
Variations on this theme, diﬀering only in the method of choosing suitable perturbations, include the “straddle-orbit method” , “stagger-step method” , the “PIM triple procedure” , and most recently a gradient search algorithm due to Bollt .
All have fairly severe limitations.
The straddle-orbit and PIM triple methods apply only if the unstable manifold of the saddle is one-dimensional.
The 2 other methods suﬀer from the exponential growth of phase space volume with dimension, which greatly hinders the search for successful perturbations if the system dimension is greater than about four.
The construction of a general-purpose algorithm for approximating chaotic saddles remains an open problem.
In this paper we consider the problem of approximating chaotic saddles for delay diﬀerential equations (DDE’s).
DDE’s arise in models of phenomena in which the rate of change of the system state depends explicitly on the state at some past time, as for example in the case of delayed feedback.
Neural systems , respiration regulation , agricultural commodity markets , nonlinear optics , neutrophil populations in the blood [12, 15], and metal cutting  are just a few systems in which delayed feedback leads naturally to models expressed in terms of delay diﬀerential equations.
DDE’s are also interesting because they serve as prototypical dynamical systems of inﬁnite dimension, for which both numerical and analytical methods are intermediate in complexity between ordinary and partial diﬀerential equations.
DDE’s therefore provide a natural ground for developing numerical methods for the analysis of transient chaos in inﬁnite dimensional systems, much as Farmer  has suggested in the context of chaotic attractors.
For simplicity we consider only autonomous, evolutionary delay equations with a single ﬁxed delay time, τ , modeling a process x(t) ∈ I R satisfying dx(t) = f x(t), x(t − τ ) dt 2 In the present work we investigate fractal basins of attraction and transient chaos for DDE’s, taking a particular “logistic” DDE as an example.
We develop an implementation of the stagger-step method applicable to DDE’s of the form (1), and use it to construct and visualize the chaotic saddle for our example.
Since the the saddle is embedded in an inﬁnite dimensional phase space, it is diﬃcult to visualize.
We explore various approaches to visualizing the saddle by using projections onto I R2 and I R3 , and Poincar´ e section techniques to achieve further reductions in dimension.
While being the ﬁrst such investigation for DDE’s in particular, the present work is also novel for giving the ﬁrst numerical construction of a chaotic saddle for a dynamical system of inﬁnite dimension.
This work paves the way for a similar approach to other inﬁnite-dimensional systems, for instance systems modeled by evolutionary PDE’s. II.
BACKGROUND ON DELAY DIFFERENTIAL EQUATIONS (1) We consider one-dimensional autonomous delay diﬀerential equations of the form (1) with x(t) ∈ I R, t ≥ 0.
Properties of such DDE’s and their solutions can be found eg in [24, 25].
Here we summarize the most essential facts relevant to our work.
Without loss of generality we can take the delay time τ to be 1, achieved by an appropriate re-scaling of the time t in equation (1).
Thus the DDE’s we consider have the form x′ (t) = f x(t), x(t − 1) . (2) for some f : I R →I R.
Most of the ideas presented here have obvious generalizations to more general autonomous DDE’s, eg with higher dimension, multiple delays, time varying or distributed delays, and higher derivatives.
Aside from unpublished work referenced in , to date there has been no account of transient chaos in delay diﬀerential equations.
However, there is substantial evidence that transient chaos occurs in some DDE’s.
The present study was motivated by the observations in [18, 19] of fractal basins of attraction (a hallmark of transient chaos ) in delay equations1 .
Transverse homoclinic orbits (hence horseshoe dynamics) have also been proved to occur in some DDE’s [20–22], and there is numerical evidence  for transverse homoclinic orbits in the Mackey-Glass equation .
These results have been presented in discussions of attracting chaos, whereas transient chaos in DDE’s has not been specifically investigated.
In particular, no attempt has been made to identify and construct a chaotic saddle. For equation (2) to deﬁne a unique solution x(t), say for all t ≥ 0, initial data must be furnished in the form of values x(t) for all −1 ≤ t ≤ 0.
Otherwise, the right-hand side will fail to be deﬁned for some t ∈ [0, 1].
In order that equation (2) prescribes a well-deﬁned evolutionary process, we assume that f is such2 that for any continuous “initial function” φ : [−1, 0] → I R, there is a unique solution x(t) satisfying (2) for all t > 0, and the initial condition x(t) = φ(t), t ∈ [−1, 0]. (3) The DDE (2) can be regarded as a dynamical system on the inﬁnite-dimensional phase space C ≡ C [−1, 0], the space of continuous functions on the interval [−1, 0].
To see how this can be so (see for example [25, 26]), consider that a solution x(t) is uniquely determined for all t > 0 only if initial data are given for all t ∈ [−1, 0], in the manner of equation (3).
More generally the continuation 1
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