## Refinement of a numerical trajectory

The refinement procedure of Grebogi et al.  ([28], hereinafter referred to as GHYS) is analagous to Newton's method for finding a zero of a function. GHYS presented their algorithm for the two dimensional case. We will also be looking closely at a generalization by Quinlan and Tremaine ([62, 63], with [63] hereinafter referred to as QT) to arbitrary-dimensional Hamiltonian systems.

The basic idea is as follows. Let be a trajectory with S steps that has noise , where is the machine precision, and is some constant significantly greater than 1 that allows room for improvement towards the machine precision. Let be the 1-step error at step i+1, where for all i. The set of all 1-step errors is represented by , and is estimated by a numerical integration technique that has higher accuracy than that used to compute . This describes a function, which we call g, which takes as input the entire orbit and whose output is the set of 1-step errors , i.e.,  . Since the 1-step errors are assumed to be small, is small. That is, is close to a zero of g, if one exists. A zero of the function would represent an orbit in which the 1-step errors are identically zero, i.e.,  a true orbit. This is an ideal situation in which to run Newton's method. If the method converges, then a true orbit lies nearby.

Assume we have a noisy D-dimensional orbit , and it has a shadow . Then

where is an approximation to f with a typical accuracy or noise of . Here, should be the accuracy typically used by practitioners of the problem being studied. Now suppose we compute the 1-step errors

although numerically we use , a better approximation to f than . Then represents a correction term that perturbs point towards the shadow. So,

In the spirit of Newton's method, we ignore the in (7), and so one refinement iteration defines the corrections along the entire orbit:

where is the linearized perturbation map. For a discrete map, is just the Jacobian of the map at step i. For a system of ODEs, is the Jacobian of the integral of the ODE from step i to step i+1. The computation of the linear maps , called resolvents for an ODE system, takes most of the CPU time during a refinement. If we let D be the dimension of the problem, then the resolvent has terms in it, and it takes time to compute. Hayes and Jackson [33, 34] list several optimizations to the procedure that speed the process considerably, and should be considered if one is interested in using the algorithm in higher dimensions. Presumably, if one is interested in studying simpler high-dimensional systems, a chaotic map would be a better choice than an ODE system, because no variational equation integration is needed.

For simplicity of explanation, we assume D=2 for the remainder of this subsection, deferring discussion of the higher D case to the next subsection. If the problem were not chaotic, the correction terms could be computed directly from (8). But since will amplify any errors in not lying in the stable direction, computing the 's by iterating (8) forward will amplify errors and typically produce nothing but noise. Therefore, we split the error and correction terms into components in the stable ( ) and unstable ( ) directions at each timestep:

Since it is not known a priori which direction is unstable at each timestep, the unstable vector at time is initialized to an arbitrary unit vector. The linearized map is then iterated forward with

Since magnifies any component that lies in the unstable direction, and assuming we are not so unlucky to choose a that lies too close to the stable direction, then after a few e-folding times will point roughly in the unstable direction at . Similarly, the stable unit direction vectors are computed by initializing to an arbitrary unit vector and iterating backward,

Substituting (10) into (8) yields

While magnifies errors in the unstable direction, it damps them in the stable direction. Likewise, damps errors in the unstable direction and magnifies errors in the stable direction. Thus the terms should be computed backward, and the terms forward. Taking components of (13) in the unstable direction at step i+1, iterate backward on

and taking components in the stable direction, iterate forward on

The initial choices for and are arbitrary as long as they are small -- smaller than the maximum shadowing distance -- because (15) damps initial conditions and (14) damps final conditions. GHYS and QT choose them both as 0. This choice is probably as good as any, but it can be seen here that if one shadow exists, there are infinitely many of them. Another way of looking at these initial choices for and is that they ``pinch'' the growing components at the end point, and the backward-growing components at the initial point, to be small. That is, boundary conditions are being forced on the problem so that the exponential divergence is forcibly masked, if possible, making the solution of (8) numerically stable.

Note that, counter-intuitively, these boundary conditions demand that the initial condition for the shadow and noisy orbits differ along the unstable direction. In fact, this must occur if the change in initial conditions is to have any effect. That is, when looking for a shadow, if perturbations were only allowed in the stable direction, those perturbations would die out, leading the ``shadow'' to follow the true orbit that passes through our initial conditions -- the one that is already known to diverge exponentially from the computed orbit.