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Geometry of Local Instability in Hamiltonian Dynamics
Published in Christos H. Skiadas, Charilaos Skiadas, Handbook of Applications of Chaos Theory, 2017
M. Lewkowicz, J. Levitan, Y. Ben Zion, L. Horwitz
It was recently shown [10,11] that there is a possibility to characterize chaos in Hamiltonian systems by a geometrical approach which takes its point of origin in the curvature associated with a conformal Riemann metric tensor (different from the Jacobi or Eisenhart metric). The method takes its starting point in the equivalence between motions generated by a standard Hamiltonian with a quadratic, kinetic term, and an additional potential term, and a Hamiltonian described by a metric type function of the coordinates multiplied with the momenta in a bilinear form. The Hamilton equations of the original potential model are in this approach contained in the geodesics equations through an inverse map in the tangent space in terms of a geometric embedding. The curvature is obtained from the second covariant derivative of the geodesic deviation and leads to a local criterion for unstable behavior different from the criterion rendered by the Lyapunov criterion. The method can be adapted to include a large class of potential models. Our approach represents a new direction in the use of Riemannian geometry by associating instabilities with an energy-dependent negative curvature appropriate for the geodesic motion and different from that implied by the Jacobi metric. It appears to be more sensitive than calculating the largest Lyapunov exponents or using the Jacobi metric, and it establishes a natural connection between Hamiltonian flows and Anosov flows [12].
Energy decay for a wave equation of variable coefficients with logarithmic nonlinearity source term
Published in Applicable Analysis, 2023
In this paper, we want to use the Riemannian geometry method, so we first introduce some notations about the Riemannian manifold on . Define Denote the inner product and the norm on the tangent space for every by where Then, we can get that is a Riemannian manifold with the metric g. The gradient of u is denoted by and in the Euclidean metric and Riemannian metric g, respectively. We offer [15] to the readers for more information.
An Information Geometry Approach to Robustness Analysis for the Uncertainty Quantification of Computer Codes
Published in Technometrics, 2022
Clement Gauchy, Jerome Stenger, Roman Sueur, Bertrand Iooss
To try to form an intuitive understanding of the consequences of such perturbations on the distribution of the output variable Y, it is necessary to base our perturbation method on a metric which allows us to compare perturbations coming from different inputs distributions in the UQ model. For instance, if one input is Gaussian and another is log uniform, their respective perturbations (associated with the same level of perturbation) should be able to be interpreted in the same way. In particular, the perturbation method should result in identical perturbed densities when applied to two different parametric representations of the same input distribution. The Fisher distance (Rao 1945) is consistent with these wishes. It is based on the local scalar product induced by the Fisher information matrix in a given parametric space, and defines a Riemannian geometry on the corresponding set of probability measures as on any Riemannian manifold with its associated metric.
A different look at the optimal control of the Brockett integrator
Published in International Journal of Control, 2023
Domenico D'Alessandro, Zhifei Zhu
Symmetry reduction has a long and successful history in control theory (see, e.g. Echeverría-Enríquez et al., 2003; Grizzle & Marcus, 1984, 1985; Martínez, 2004; Ohsawa, 2013). The idea is to use symmetries to reduce the complexity of the optimal control problem. In the context of sub-Riemannian geometry which interests us in this paper, for a sub-Riemannian manifold , and a problem with initial condition , we define a symmetry group G as a Lie transformation group acting on M by , , which satisfies the following conditions For every and at every For every , is an isometry on Δ. This means that, for every and for every two tangent vectors and , in we have