30 research outputs found
Identification of Stochastically Perturbed Autonomous Systems from Temporal Sequences of Probability Density Functions
The paper introduces a method for reconstructing one-dimensional iterated maps that are driven by an external control input and subjected to an additive stochastic perturbation, from sequences of probability density functions that are generated by the stochastic dynamical systems and observed experimentally
Reconstruction of one-dimensional chaotic maps from sequences of probability density functions
In many practical situations, it is impossible to measure the individual trajectories generated by an unknown chaotic system, but we can observe the evolution of probability density functions generated by such a system. The paper proposes for the first time a matrix-based approach to solve the generalized inverse Frobenius–Perron problem, that is, to reconstruct an unknown one-dimensional chaotic transformation, based on a temporal sequence of probability density functions generated by the transformation. Numerical examples are used to demonstrate the applicability of the proposed approach and evaluate its robustness with respect to constantly applied stochastic perturbations
First-passage times in complex scale-invariant media
How long does it take a random walker to reach a given target point? This
quantity, known as a first passage time (FPT), has led to a growing number of
theoretical investigations over the last decade1. The importance of FPTs
originates from the crucial role played by first encounter properties in
various real situations, including transport in disordered media, neuron firing
dynamics, spreading of diseases or target search processes. Most methods to
determine the FPT properties in confining domains have been limited to
effective 1D geometries, or for space dimensions larger than one only to
homogeneous media1. Here we propose a general theory which allows one to
accurately evaluate the mean FPT (MFPT) in complex media. Remarkably, this
analytical approach provides a universal scaling dependence of the MFPT on both
the volume of the confining domain and the source-target distance. This
analysis is applicable to a broad range of stochastic processes characterized
by length scale invariant properties. Our theoretical predictions are confirmed
by numerical simulations for several emblematic models of disordered media,
fractals, anomalous diffusion and scale free networks.Comment: Submitted version. Supplementary Informations available on Nature
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Tutte polynomial of pseudofractal scale-free web
The Tutte polynomial of a graph is a 2-variable polynomial which is quite
important in both combinatorics and statistical physics. It contains various
numerical invariants and polynomial invariants, such as the number of spanning
trees, the number of spanning forests, the number of acyclic orientations, the
reliability polynomial, chromatic polynomial and flow polynomial. In this
paper, we study and gain recursive formulas for the Tutte polynomial of
pseudofractal scale-free web (PSW) which implies logarithmic complexity
algorithm is obtained to calculate the Tutte polynomial of PSW although it is
NP-hard for general graph. We also obtain the rigorous solution for the the
number of spanning trees of PSW by solving the recurrence relations derived
from Tutte polynomial, which give an alternative approach for explicitly
determining the number of spanning trees of PSW. Further more, we analysis the
all-terminal reliability of PSW and compare the results with that of Sierpinski
gasket which has the same number of nodes and edges with PSW. In contrast with
the well-known conclusion that scale-free networks are more robust against
removal of nodes than homogeneous networks (e.g., exponential networks and
regular networks). Our results show that Sierpinski gasket (which is a regular
network) are more robust against random edge failures than PSW (which is a
scale-free network). Whether it is true for any regular networks and scale-free
networks, is still a unresolved problem.Comment: 19pages,7figures. arXiv admin note: text overlap with arXiv:1006.533
Identifying stochastic basin hopping by partitioning with graph modularity
It has been known that noise in a stochastically perturbed dynamical system can destroy what was the original zero-noise case barriers in the phase space (pseudobarrier). Noise can cause the basin hopping. We use the Frobenius-Perron operator and its finite rank approximation by the Ulam-Galerkin method to study transport mechanism of a noisy map. In order to identify the regions of high transport activity in the phase space and to determine flux across the pseudobarriers, we adapt a new graph theoretical method which was developed to detect active pseudobarriers in the original phase space of the stochastic dynamic. Previous methods to identify basins and basin barriers require a priori knowledge of a mathematical model of the system, and hence cannot be applied to observed time series data of which a mathematical model is not known. Here we describe a novel graph method based on optimization of the modularity measure of a network and introduce its application for determining pseudobarriers in the phase space of a multi-stable system only known through observed data. © 2007 Elsevier Ltd. All rights reserved
Applied and Computational Measurable Dynamics
This book connects many concepts in dynamical systems with mathematical tools from areas such as graph theory and ergodic theory
Applied and Computational Measurable Dynamics
This book connects many concepts in dynamical systems with mathematical tools from areas such as graph theory and ergodic theory
The infinitesimal operator for the semigroup of the Frobenius-Perron operator from image sequence data: vector fields and transport barriers from movies.
In this paper, we present an approach to approximate the Frobenius-Perron transfer operator from a sequence of time-ordered images, that is, a movie dataset. Unlike time-series data, successive images do not provide a direct access to a trajectory of a point in a phase space; more precisely, a pixel in an image plane. Therefore, we reconstruct the velocity field from image sequences based on the infinitesimal generator of the Frobenius-Perron operator. Moreover, we relate this problem to the well-known optical flow problem from the computer vision community and we validate the continuity equation derived from the infinitesimal operator as a constraint equation for the optical flow problem. Once the vector field and then a discrete transfer operator are found, then, in addition, we present a graph modularity method as a tool to discover basin structure in the phase space. Together with a tool to reconstruct a velocity field, this graph-based partition method provides us with a way to study transport behavior and other ergodic properties of measurable dynamical systems captured only through image sequences