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Manifold angles, the concept of self-similarity, and angle-enhanced bifurcation diagrams.

ABSTRACT: Chaos and regularity are routinely discriminated by using Lyapunov exponents distilled from the norm of orthogonalized Lyapunov vectors, propagated during the temporal evolution of the dynamics. Such exponents are mean-field-like averages that, for each degree of freedom, squeeze the whole temporal evolution complexity into just a single number. However, Lyapunov vectors also contain a step-by-step record of what exactly happens with the angles between stable and unstable manifolds during the whole evolution, a big-data information permanently erased by repeated orthogonalizations. Here, we study changes of angles between invariant subspaces as observed during temporal evolution of Hénon's system. Such angles are calculated numerically and analytically and used to characterize self-similarity of a chaotic attractor. In addition, we show how standard tools of dynamical systems may be angle-enhanced by dressing them with informations not difficult to extract. Such angle-enhanced tools reveal unexpected and practical facts that are described in detail. For instance, we present a video showing an angle-enhanced bifurcation diagram that exposes from several perspectives the complex geometrical features underlying the attractors. We believe such findings to be generic for extended classes of systems.

SUBMITTER: Beims MW

PROVIDER: S-EPMC4702165 | biostudies-literature | 2016

REPOSITORIES: biostudies-literature

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Manifold angles, the concept of self-similarity, and angle-enhanced bifurcation diagrams.

Beims Marcus W MW Gallas Jason A C JA

Scientific reports 20160106

Chaos and regularity are routinely discriminated by using Lyapunov exponents distilled from the norm of orthogonalized Lyapunov vectors, propagated during the temporal evolution of the dynamics. Such exponents are mean-field-like averages that, for each degree of freedom, squeeze the whole temporal evolution complexity into just a single number. However, Lyapunov vectors also contain a step-by-step record of what exactly happens with the angles between stable and unstable manifolds during the wh ...[more]

PMID: 26732416

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Project description:This work investigates the effect of arterial bifurcation angulation on atherosclerosis development through in-silico simulations of coupled cell dynamics. The computational model presented here combines cellular pathways, fluid dynamics, and physiologically-realistic vessel geometries as observed in the human vasculature. The coupled cells model includes endothelial cells (ECs) and smooth muscle cells (SMCs) with ion dynamics, hetero and homotypic coupling, as well as electro-diffusive coupling. Three arterial bifurcation surface models were used in the coupled cells simulations. All three simulations showed propagating waves of Ca2+ in both the SMC and EC layers, following the introduction of a luminal agonist, in this case ATP. Immediately following the introduction of ATP concentration Ca2+ waves propagate from the area of high ATP toward the areas of low ATP concentration, forming complex patterns where waves interact with eachother, collide and fade. These dynamic phenomena are repeated with a series of waves of slower velocity. The underlying motivation of this research was to examine the macro-scale phenomena, given that the characteristic length scales of atherosclerotic plaques are much larger than a single cell. The micro-scale dynamics were modeled on macro-scale arterial bifurcation surfaces containing over one million cells. The results of the simulations presented here suggest that susceptibility to atherosclerosis development depends on the bifurcation angulation. In conjunction with findings reported in the literature, the simulation results demonstrate that arterial bifurcations containing wider angles have a more prominent influence on the coupled cells pathways associated with the development of atherosclerosis, by means of disturbed flow and lower SMC Ca2+ concentrations. The discussion of the results considers the findings of this research within the context of the potential link between information transport through frequency encoding of Ca2+ wave dynamics and development of atheroprone conditions.

Dataset Information

Manifold angles, the concept of self-similarity, and angle-enhanced bifurcation diagrams.

Publications

Manifold angles, the concept of self-similarity, and angle-enhanced bifurcation diagrams.

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