3,049 research outputs found
Slow invariant manifolds as curvature of the flow of dynamical systems
Considering trajectory curves, integral of n-dimensional dynamical systems,
within the framework of Differential Geometry as curves in Euclidean n-space,
it will be established in this article that the curvature of the flow, i.e. the
curvature of the trajectory curves of any n-dimensional dynamical system
directly provides its slow manifold analytical equation the invariance of which
will be then proved according to Darboux theory. Thus, it will be stated that
the flow curvature method, which uses neither eigenvectors nor asymptotic
expansions but only involves time derivatives of the velocity vector field,
constitutes a general method simplifying and improving the slow invariant
manifold analytical equation determination of high-dimensional dynamical
systems. Moreover, it will be shown that this method generalizes the Tangent
Linear System Approximation and encompasses the so-called Geometric Singular
Perturbation Theory. Then, slow invariant manifolds analytical equation of
paradigmatic Chua's piecewise linear and cubic models of dimensions three, four
and five will be provided as tutorial examples exemplifying this method as well
as those of high-dimensional dynamical systems
Slow invariant manifold of heartbeat model
A new approach called Flow Curvature Method has been recently developed in a
book entitled Differential Geometry Applied to Dynamical Systems. It consists
in considering the trajectory curve, integral of any n-dimensional dynamical
system as a curve in Euclidean n-space that enables to analytically compute the
curvature of the trajectory - or the flow. Hence, it has been stated on the one
hand that the location of the points where the curvature of the flow vanishes
defines a manifold called flow curvature manifold and on the other hand that
such a manifold associated with any n-dimensional dynamical system directly
provides its slow manifold analytical equation the invariance of which has been
proved according to Darboux theory. The Flow Curvature Method has been already
applied to many types of autonomous dynamical systems either singularly
perturbed such as Van der Pol Model, FitzHugh-Nagumo Model, Chua's Model, ...)
or non-singularly perturbed such as Pikovskii-Rabinovich-Trakhtengerts Model,
Rikitake Model, Lorenz Model,... More- over, it has been also applied to
non-autonomous dynamical systems such as the Forced Van der Pol Model. In this
article it will be used for the first time to analytically compute the slow
invariant manifold analytical equation of the four-dimensional Unforced and
Forced Heartbeat Model. Its slow invariant manifold equation which can be
considered as a "state equation" linking all variables could then be used in
heart prediction and control according to the strong correspondence between the
model and the physiological cardiovascular system behavior.Comment: arXiv admin note: substantial text overlap with arXiv:1408.171
Canards and curvature: nonsmooth approximation by pinching
In multiple time-scale (singularly perturbed) dynamical systems, canards are counterintuitive solutions that evolve along both attracting and repelling invariant manifolds. In two dimensions, canards result in periodic oscillations whose amplitude and period grow in a highly nonlinear way: they are slowly varying with respect to a control parameter, except for an exponentially small range of values where they grow extremely rapidly. This sudden growth, called a canard explosion, has been encountered in many applications ranging from chemistry to neuronal dynamics, aerospace engineering and ecology. Canards were initially studied using nonstandard analysis, and later the same results were proved by standard techniques such as matched asymptotics, invariant manifold theory and parameter blow-up. More recently, canard-like behaviour has been linked to surfaces of discontinuity in piecewise-smooth dynamical systems. This paper provides a new perspective on the canard phenomenon by showing that the nonstandard analysis of canard explosions can be recast into the framework of piecewise-smooth dynamical systems. An exponential coordinate scaling is applied to a singularly perturbed system of ordinary differential equations. The scaling acts as a lens that resolves dynamics across all time-scales. The changes of local curvature that are responsible for canard explosions are then analyzed. Regions where different time-scales dominate are separated by hypersurfaces, and these are pinched together to obtain a piecewise-smooth system, in which curvature changes manifest as discontinuity-induced bifurcations. The method is used to classify canards in arbitrary dimensions, and to derive the parameter values over which canards form either small cycles (canards without head) or large cycles (canards with head)
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