6,022 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
Fast iteration of cocyles over rotations and Computation of hyperbolic bundles
In this paper, we develop numerical algorithms that use small requirements of
storage and operations for the computation of hyperbolic cocycles over a
rotation. We present fast algorithms for the iteration of the quasi-periodic
cocycles and the computation of the invariant bundles, which is a preliminary
step for the computation of invariant whiskered tori
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Mathematical Structures in Group Decision-Making on Resource Allocation Distributions.
Optimal decisions on the distribution of finite resources are explicitly structured by mathematical models that specify relevant variables, constraints, and objectives. Here we report analysis and evidence that implicit mathematical structures are also involved in group decision-making on resource allocation distributions under conditions of uncertainty that disallow formal optimization. A group's array of initial distribution preferences automatically sets up a geometric decision space of alternative resource distributions. Weighted averaging mechanisms of interpersonal influence reduce the heterogeneity of the group's initial preferences on a suitable distribution. A model of opinion formation based on weighted averaging predicts a distribution that is a feasible point in the group's implicit initial decision space
A geometric mechanism of diffusion: Rigorous verification in a priori unstable Hamiltonian systems
In this paper we consider a representative a priori unstable Hamiltonian
system with 2+1/2 degrees of freedom, to which we apply the geometric mechanism
for diffusion introduced in the paper Delshams et al., Mem. Amer. Math. Soc.
2006, and generalized in Delshams and Huguet, Nonlinearity 2009, and provide
explicit, concrete and easily verifiable conditions for the existence of
diffusing orbits.
The simplification of the hypotheses allows us to perform explicitly the
computations along the proof, which contribute to present in an easily
understandable way the geometric mechanism of diffusion. In particular, we
fully describe the construction of the scattering map and the combination of
two types of dynamics on a normally hyperbolic invariant manifol
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