ABOUT UNIVERSAL BASINS OF ATTRACTION IN HIGH-DIMENSIONAL SYSTEMS

2013 ◽  
Vol 23 (12) ◽  
pp. 1350197 ◽  
Author(s):  
ZERAOULIA ELHADJ ◽  
J. C. SPROTT
Keyword(s):  
Dynamical Systems ◽  
Equivalence Class ◽  
Upper Bound ◽  
Vector Fields ◽  
Basins Of Attraction ◽  
Invariant Sets ◽  
High Dimensional ◽  
Time Lags ◽  
Dimensional Vector ◽  
Definition Of

In this letter, we will show the existence of invariant sets called universal basins of attraction for typical nonlinear high-dimensional dynamical systems such as randomly sampled high-dimensional vector fields (ODEs) or maps. The method of analysis is based on the definition of an equivalence class between systems with the same number of neurons, the same number of time lags, and the same upper bound for one family of bifurcation parameters.

INTERTWINED BASINS OF ATTRACTION

2012 ◽  
Vol 22 (06) ◽  
pp. 1250130
Author(s):  
CHANGMING DING
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Basins Of Attraction ◽  
General Definition ◽  
Sufficient Condition ◽  
Topological Equivalence ◽  
Definition Of

This paper deals with intertwined basins of attraction for dynamical systems in a metric space. After giving a general definition of intertwining property, which is preserved by a topological equivalence between dynamical systems, we present a sufficient condition to guarantee the existence of intertwined basins for dynamical systems in ℝn.

scholarly journals Mirror neurons are modulated by grip force and reward expectation in the sensorimotor cortices (S1, M1, PMd, PMv)

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Md Moin Uddin Atique ◽  
Joseph Thachil Francis
Keyword(s):  
Neural Activity ◽  
Visual Information ◽  
Mirror Neurons ◽  
Grip Force ◽  
Neural Representation ◽  
Virtual Object ◽  
Time Lags ◽  
Subjective Value ◽  
Machine Interface ◽  
Definition Of

AbstractMirror Neurons (MNs) respond similarly when primates make or observe grasping movements. Recent work indicates that reward expectation influences rostral M1 (rM1) during manual, observational, and Brain Machine Interface (BMI) reaching movements. Previous work showed MNs are modulated by subjective value. Here we expand on the above work utilizing two non-human primates (NHPs), one male Macaca Radiata (NHP S) and one female Macaca Mulatta (NHP P), that were trained to perform a cued reward level isometric grip-force task, where the NHPs had to apply visually cued grip-force to move and transport a virtual object. We found a population of (S1 area 1–2, rM1, PMd, PMv) units that significantly represented grip-force during manual and observational trials. We found the neural representation of visually cued force was similar during observational trials and manual trials for the same units; however, the representation was weaker during observational trials. Comparing changes in neural time lags between manual and observational tasks indicated that a subpopulation fit the standard MN definition of observational neural activity lagging the visual information. Neural activity in (S1 areas 1–2, rM1, PMd, PMv) significantly represented force and reward expectation. In summary, we present results indicating that sensorimotor cortices have MNs for visually cued force and value.

scholarly journals Tropical Balls and Its Applications to K Nearest Neighbor over the Space of Phylogenetic Trees

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 779
Author(s):  
Ruriko Yoshida
Keyword(s):  
Supervised Learning ◽  
Phylogenetic Trees ◽  
Nearest Neighbor ◽  
Nearest Neighbors ◽  
High Dimensional ◽  
Learning Method ◽  
Dimensional Vector ◽  
K Nearest Neighbor ◽  
K Nearest Neighbors

A tropical ball is a ball defined by the tropical metric over the tropical projective torus. In this paper we show several properties of tropical balls over the tropical projective torus and also over the space of phylogenetic trees with a given set of leaf labels. Then we discuss its application to the K nearest neighbors (KNN) algorithm, a supervised learning method used to classify a high-dimensional vector into given categories by looking at a ball centered at the vector, which contains K vectors in the space.

scholarly journals Off-Shell Noether Currents and Potentials for First-Order General Relativity

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 348
Author(s):  
Merced Montesinos ◽  
Diego Gonzalez ◽  
Rodrigo Romero ◽  
Mariano Celada
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Arbitrary Vector ◽  
Four Dimensions ◽  
First Order ◽  
Direct Form ◽  
Definition Of

We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by n-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local SO(n) or SO(n−1,1) transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do not use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local SO(3,1) or SO(4) off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions.

scholarly journals A Summary of F-Transform Techniques in Data Analysis

Electronics ◽  
2021 ◽  
Vol 10 (15) ◽  
pp. 1771
Author(s):  
Ferdinando Di Martino ◽  
Irina Perfilieva ◽  
Salvatore Sessa
Keyword(s):  
Data Analysis ◽  
Analysis Data ◽  
High Dimensional ◽  
Sufficient Data ◽  
Fuzzy Transform ◽  
Transform Method ◽  
Data Density ◽  
Fuzzy Partitions ◽  
Definition Of ◽  
Performance Results

Fuzzy transform is a technique applied to approximate a function of one or more variables applied by researchers in various image and data analysis. In this work we present a summary of a fuzzy transform method proposed in recent years in different data mining disciplines, such as the detection of relationships between features and the extraction of association rules, time series analysis, data classification. After having given the definition of the concept of Fuzzy Transform in one or more dimensions in which the constraint of sufficient data density with respect to fuzzy partitions is also explored, the data analysis approaches recently proposed in the literature based on the use of the Fuzzy Transform are analyzed. In particular, the strategies adopted in these approaches for managing the constraint of sufficient data density and the performance results obtained, compared with those measured by adopting other methods in the literature, are explored. The last section is dedicated to final considerations and future scenarios for using the Fuzzy Transform for the analysis of massive and high-dimensional data.

scholarly journals Characteristic exponents of dynamical systems in metric spaces

1983 ◽  
Vol 3 (1) ◽  
pp. 119-127 ◽  
Author(s):  
Yuri Kifer
Keyword(s):  
Dynamical Systems ◽  
Metric Spaces ◽  
Invariant Sets ◽  
Characteristic Exponents ◽  
Smooth Case

AbstractWe introduce for dynamical systems in metric spaces some numbers which in the case of smooth dynamical systems turn out to be the maximal and the minimal characteristic exponents. These numbers have some properties similar to the smooth case. Analogous quantities are defined also for invariant sets.

scholarly journals A Twenty-First Century Guidebook for Applied Dynamical Systems

2006 ◽  
Vol 1 (4) ◽  
pp. 279-282 ◽  
Author(s):  
A. R. Champneys
Keyword(s):  
Dynamical Systems ◽  
21St Century ◽  
Vector Fields ◽  
Nonlinear Oscillations ◽  
First Century ◽  
Twenty First Century ◽  
The Impact ◽  
Young Researchers

This paper represents the author’s view on the impact of the book Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields by John Guckenheimer and Philip Holmes, first published in 1983 (Springer-Verlag, Berlin). In particular, the questions addressed are: if one were to write a similar book for the 21st century, which topics should be contained and what form should the book take in order to have a similar impact on the modern generation of young researchers in applied dynamical systems?

scholarly journals Homoclinic orbits and Lie rotated vector fields

2000 ◽  
Vol 24 (3) ◽  
pp. 187-192
Author(s):  
Jie Wang ◽  
Chen Chen
Keyword(s):  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Vector Fields ◽  
Homoclinic Orbits ◽  
Singular Points ◽  
Definition Of

Based on the definition of Lie rotated vector fields in the plane, this paper gives the property of homoclinic orbit as parameter is changed and the singular points are fixed on Lie rotated vector fields. It gives the conditions of yielding limit cycles as well.

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