Predicting the bounds of large chaotic systems using low-dimensional manifolds

<div><p>Predicting extrema of chaotic systems in high-dimensional phase space remains a challenge. Methods, which give extrema that are valid in the long term, have thus far been restricted to models of only a few variables. Here, a method is presented which treats extrema of chaotic systems as belonging to discretised manifolds of low dimension (low-D) embedded in high-dimensional (high-D) phase space. As a central feature, the method exploits that strange attractor dimension is generally much smaller than parent system phase space dimension. This is important, since the computational cost associated with discretised manifolds depends exponentially on their dimension. Thus, systems that would otherwise be associated with tremendous computational challenges, can be tackled on a laptop. As a test, bounding manifolds are calculated for high-D modifications of the canonical Duffing system. Parameters can be set such that the bounding manifold displays harmonic behaviour even if the underlying system is chaotic. Thus, solving for one post-transient forcing cycle of the bounding manifold predicts the extrema of the underlying chaotic problem indefinitely.</p></div

Transient phase of extrema calculated for Eq (13) (parameter set IV).

<p>The extrema will not be valid before the manifold is post-transient.</p

Schematic showing the evolution of two key eigenvalues along the entire length of the BM, i.e, for <i>m</i> ∈ [0, 1[.

<p>Note that <i>λ</i>₁ has a positive real part in sub-domain I, which is negative in sub-domain II, III and IV. In sub-domain III <i>λ</i>₁ is complex with the complex conjugate <i>λ</i>₂. Due to the requirement of continuity, the local solution space associated with <i>λ</i>₁ and <i>λ</i>₂ must be spanned by <b>v</b>₁ and <b>v</b>₂ along the entire BM. Thus, the fact that one eigenvalue has a positive real part for only a portion of the BM, means that the motion corresponding to two eigenvalues must be spanned along the entire BM.</p

Schematic showing the contraction of a set of initial conditions along off-manifold directions onto the AM and the post-transient harmonic behaviour of the AM and BM.

<p>The parameter <i>m</i> runs from 0 to 1 along the length of the BM. A number of solutions are shown as dots. Initially they lie as a cloud in phase space, but they rapidly contract along stable directions until they converge to the AM. As shall become apparent, it is sufficient to calculate the tangent space of the AM to orientate the BM <b>c</b>(<i>m</i>, <i>t</i>). This is done with a spectral method.</p

Extrema of Eq (12): Plots of extrema (red) and 20 different time series (black) of variables <i>y</i>₁, <i>y</i>₂, <i>y</i>₂₁ and <i>y</i>₂₂ for parameter set I.

<p>The bounds for odd (0th order) variables are somewhat tighter than those of even numbered (1st order) variables.</p

Model parameters for the simulations.

<p>The parameter <i>τ</i> is the time constant for curvature filtering, <i>β</i> is the proportional impedance on curvature, <i>β</i>₂ is the impedance on axial motion, <i>k</i>_<i>t</i>, <i>d</i>_<i>t</i> and <i>m</i>_<i>t</i> are the stiffness, damping and mass of interior point truss elements, <i>α</i> is the non-linear Duffing stiffness, <i>k</i>₁, <i>d</i>₁ and <i>m</i>₁ are the linear stiffness, damping and mass of the Duffing oscillator, <i>f</i>₀ is the forcing term amplitude, <i>ω</i> is the forcing term frequency, <i>k</i>₂, <i>d</i>₂ and <i>m</i>₂ are the stiffness, damping and mass of the linear oscillators, Δ<i>t</i> is the time step for the time integration scheme (BM and interior point position update), Δ<i>t</i>_<i>λ</i> is the time step for eigenvalue calculation and orientation update, <i>K</i> is the number of nodes (and elements) used to discretise <b>c</b>, and <i>K</i>_<i>G</i> is the number of Gauss points for numerical integration in space along <b>c</b>.</p

Extrema of Eq (12): Plots of extrema (red) and 20 different time series (black) of variables <i>y</i>₁, <i>y</i>₂, <i>y</i>₄₁ and <i>y</i>₄₂ for parameter set II.

<p>Extrema of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0179507#pone.0179507.e018" target="_blank">Eq (12)</a>: Plots of extrema (red) and 20 different time series (black) of variables <i>y</i>₁, <i>y</i>₂, <i>y</i>₄₁ and <i>y</i>₄₂ for parameter set II.</p

Snapshots of projections of the BM c(<i>t</i>) onto coordinate surfaces (<i>y</i>₁, <i>y</i>₂), (<i>y</i>₃, <i>y</i>₄), …, (<i>y</i>₈₁, <i>y</i>₈₂) for various times during a simulation for parameter set III in Table 1.

<p>Note that the last two figures are largely similar, indicating that the BM is becoming post-transient.</p

The (<i>y</i>₁, <i>y</i>₂, <i>y</i>₄₂) projection of the AM and BM at times <i>tω</i>/(2<i>π</i>) = {0.005, 0.1, 0.15, 0.3, 0.6, 1.5, 2.0, 4.66} during a simulation.

<p>Overlaid are twenty different time series with tails tracing their motion shown in red. Each tail represents Δ<i>tω</i>/(2<i>π</i>) = 0.05. The time series rapidly converge onto the manifold. The BM will expand to contain those attractors whose basins of attraction it intersects. The interior points are connected with truss elements of zero initial length which are shown in the figure.</p

Schematic of AM and BM.

<p>An off-manifold point with stable orthogonal motion is shown. The unit vector <b>u</b>₁ points in the inward direction and the unit vector <b>u</b>₂ is tangential to the BM. The vectors <b>w</b>₁(<b>x</b>) and <b>w</b>₂(<b>x</b>) span the tangent space of the AM. On the BM (<b>u</b>₁,<b>u</b>₂) and (<b>w</b>₁,<b>w</b>₂) span the same 2-D tangent space. Here, the dimension of the AM, <i>M</i>, is 2.</p