The Experts below are selected from a list of 13185 Experts worldwide ranked by ideXlab platform
Euaggelos E Zotos - One of the best experts on this subject based on the ideXlab platform.
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Investigating the Newton???Raphson Basins of Attraction in the restricted three-body problem with modified Newtonian gravity
Journal of Applied Mathematics and Computing, 2018Co-Authors: Euaggelos E ZotosAbstract:The planar circular restricted three-body problem with modified Newtonian gravity is used in order to determine the Newton–Raphson Basins of Attraction associated with the equilibrium points. The evolution of the position of the five Lagrange points is monitored when the value of the power p of the gravitational potential of the second primary varies in predefined intervals. The regions on the configuration (x, y) plane occupied by the Basins of Attraction are revealed using the multivariate version of the Newton–Raphson iterative scheme. The correlations between the Basins of convergence of the equilibrium points and the corresponding number of iterations needed for obtaining the desired accuracy are also illustrated. We conduct a thorough and systematic numerical investigation by demonstrating how the dynamical quantity p influences the shape as well as the geometry of the Basins of Attractions. Our results strongly suggest that the power p is indeed a very influential parameter in both cases of weaker or stronger Newtonian gravity.
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investigating the newton raphson Basins of Attraction in the restricted three body problem with modified newtonian gravity
Journal of Applied Mathematics and Computing, 2018Co-Authors: Euaggelos E ZotosAbstract:The planar circular restricted three-body problem with modified Newtonian gravity is used in order to determine the Newton–Raphson Basins of Attraction associated with the equilibrium points. The evolution of the position of the five Lagrange points is monitored when the value of the power p of the gravitational potential of the second primary varies in predefined intervals. The regions on the configuration (x, y) plane occupied by the Basins of Attraction are revealed using the multivariate version of the Newton–Raphson iterative scheme. The correlations between the Basins of convergence of the equilibrium points and the corresponding number of iterations needed for obtaining the desired accuracy are also illustrated. We conduct a thorough and systematic numerical investigation by demonstrating how the dynamical quantity p influences the shape as well as the geometry of the Basins of Attractions. Our results strongly suggest that the power p is indeed a very influential parameter in both cases of weaker or stronger Newtonian gravity.
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Comparing the fractal Basins of Attraction in the Hill problem with oblateness and radiation
Astrophysics and Space Science, 2017Co-Authors: Euaggelos E ZotosAbstract:The Basins of convergence, associated with the roots (attractors) of a complex equation, are revealed in the Hill problem with oblateness and radiation, using a large variety of numerical methods. Three cases are investigated, regarding the values of the oblateness and radiation. In all cases, a systematic and thorough scan of the complex plane is performed in order to determine the Basins of Attraction of the several iterative schemes. The correlations between the attracting domains and the corresponding required number of iterations are also illustrated and discussed. Our numerical analysis strongly suggests that the Basins of convergence, with the highly fractal basin boundaries, produce extraordinary and beautiful formations on the complex plane.
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Determining the Newton-Raphson Basins of Attraction in the electromagnetic Copenhagen problem
International Journal of Non-linear Mechanics, 2017Co-Authors: Euaggelos E ZotosAbstract:Abstract The Copenhagen problem where the primaries of equal masses are magnetic dipoles is used in order to determine the Newton-Raphson Basins of Attraction associated with the equilibrium points. The parametric variation of the position as well as of the stability of the Lagrange points are monitored when the value of the ratio λ of the magnetic moments varies in predefined intervals. The regions on the configuration (x,y) plane occupied by the Basins of convergence are revealed using the multivariate version of the Newton-Raphson iterative scheme. The correlations between the Basins of Attraction of the libration points and the corresponding number of iterations needed for obtaining the desired accuracy are also illustrated. We perform a thorough and systematic numerical investigation by demonstrating how the dynamical quantity λ influences the shape, the geometry and also the degree of fractality of the attracting domains. Our numerical results strongly indicate that the ratio λ is indeed a very influential parameter in the electromagnetic binary system.
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Fractal Basins of Attraction in the planar circular restricted three-body problem with oblateness and radiation pressure
Astrophysics and Space Science, 2016Co-Authors: Euaggelos E ZotosAbstract:In this paper we use the planar circular restricted three-body problem where one of the primary bodies is an oblate spheroid or an emitter of radiation in order to determine the Basins of Attraction associated with the equilibrium points. The evolution of the position of the five Lagrange points is monitored when the values of the mass ratio \(\mu \), the oblateness coefficient \(A_{1}\), and the radiation pressure factor \(q\) vary in predefined intervals. The regions on the configuration \((x,y)\) plane occupied by the Basins of Attraction are revealed using the multivariate version of the Newton-Raphson method. The correlations between the Basins of convergence of the equilibrium points and the corresponding number of iterations needed in order to obtain the desired accuracy are also illustrated. We conduct a thorough and systematic numerical investigation demonstrating how the dynamical quantities \(\mu \), \(A_{1}\), and \(q\) influence the Basins of Attractions. Our results suggest that the mass ratio and the radiation pressure factor are the most influential parameters, while on the other hand the structure of the Basins of convergence are much less affected by the oblateness coefficient.
Alexander Sprowitz - One of the best experts on this subject based on the ideXlab platform.
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beyond Basins of Attraction quantifying robustness of natural dynamics
IEEE Transactions on Robotics, 2019Co-Authors: Steve Heim, Alexander SprowitzAbstract:Properly designing a system to exhibit favorable natural dynamics can greatly simplify designing or learning the control policy. However, it is still unclear what constitutes favorable natural dynamics and how to quantify its effect. Most studies of simple walking and running models have focused on the Basins of Attraction of passive limit cycles and the notion of self-stability. We instead emphasize the importance of stepping beyond Basins of Attraction. In this paper, we show an approach based on viability theory to quantify robust sets in state-action space. These sets are valid for the family of all robust control policies, which allows us to quantify the robustness inherent to the natural dynamics before designing the control policy or specifying a control objective. We illustrate our formulation using spring-mass models, simple low-dimensional models of running systems. We then show an example application by optimizing robustness of a simulated planar monoped, using a gradient-free optimization scheme. Both case studies result in a nonlinear effective stiffness providing more robustness.
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beyond Basins of Attraction quantifying robustness of natural dynamics
arXiv: Robotics, 2018Co-Authors: Steve Heim, Alexander SprowitzAbstract:Properly designing a system to exhibit favorable natural dynamics can greatly simplify designing or learning the control policy. However, it is still unclear what constitutes favorable natural dynamics and how to quantify its effect. Most studies of simple walking and running models have focused on the Basins of Attraction of passive limit-cycles and the notion of self-stability. We instead emphasize the importance of stepping beyond Basins of Attraction. We show an approach based on viability theory to quantify robust sets in state-action space. These sets are valid for the family of all robust control policies, which allows us to quantify the robustness inherent to the natural dynamics before designing the control policy or specifying a control objective. We illustrate our formulation using spring-mass models, simple low dimensional models of running systems. We then show an example application by optimizing robustness of a simulated planar monoped, using a gradient-free optimization scheme. Both case studies result in a nonlinear effective stiffness providing more robustness.
Yu Mei Jiang - One of the best experts on this subject based on the ideXlab platform.
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Basins of Attraction in piecewise smooth hamiltonian systems
Physical Review E, 2005Co-Authors: Yingcheng Lai, Yu Mei JiangAbstract:Piecewise smooth Hamiltonian systems arise in physical and engineering applications. For such a system typically an infinite number of quasiperiodic ``attractors'' coexist. (Here we use the term ``attractors'' to indicate invariant sets to which typically initial conditions approach, as a result of the piecewise smoothness of the underlying system. These ``attractors'' are therefore characteristically different from the attractors in dissipative dynamical systems.) We find that the Basins of Attraction of different ``attractors'' exhibit a riddledlike feature in that they mix with each other on arbitrarily fine scales. This practically prevents prediction of ``attractors'' from specific initial conditions and parameters. The mechanism leading to the complicated basin structure is found to be characteristically different from those reported previously for similar basin structure in smooth dynamical systems. We demonstrate the phenomenon using a class of electronic relaxation oscillators with voltage protection and provide a theoretical explanation.
Steve Heim - One of the best experts on this subject based on the ideXlab platform.
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beyond Basins of Attraction quantifying robustness of natural dynamics
IEEE Transactions on Robotics, 2019Co-Authors: Steve Heim, Alexander SprowitzAbstract:Properly designing a system to exhibit favorable natural dynamics can greatly simplify designing or learning the control policy. However, it is still unclear what constitutes favorable natural dynamics and how to quantify its effect. Most studies of simple walking and running models have focused on the Basins of Attraction of passive limit cycles and the notion of self-stability. We instead emphasize the importance of stepping beyond Basins of Attraction. In this paper, we show an approach based on viability theory to quantify robust sets in state-action space. These sets are valid for the family of all robust control policies, which allows us to quantify the robustness inherent to the natural dynamics before designing the control policy or specifying a control objective. We illustrate our formulation using spring-mass models, simple low-dimensional models of running systems. We then show an example application by optimizing robustness of a simulated planar monoped, using a gradient-free optimization scheme. Both case studies result in a nonlinear effective stiffness providing more robustness.
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beyond Basins of Attraction quantifying robustness of natural dynamics
arXiv: Robotics, 2018Co-Authors: Steve Heim, Alexander SprowitzAbstract:Properly designing a system to exhibit favorable natural dynamics can greatly simplify designing or learning the control policy. However, it is still unclear what constitutes favorable natural dynamics and how to quantify its effect. Most studies of simple walking and running models have focused on the Basins of Attraction of passive limit-cycles and the notion of self-stability. We instead emphasize the importance of stepping beyond Basins of Attraction. We show an approach based on viability theory to quantify robust sets in state-action space. These sets are valid for the family of all robust control policies, which allows us to quantify the robustness inherent to the natural dynamics before designing the control policy or specifying a control objective. We illustrate our formulation using spring-mass models, simple low dimensional models of running systems. We then show an example application by optimizing robustness of a simulated planar monoped, using a gradient-free optimization scheme. Both case studies result in a nonlinear effective stiffness providing more robustness.
David J. Wales - One of the best experts on this subject based on the ideXlab platform.
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structural analysis of high dimensional Basins of Attraction
Physical Review E, 2016Co-Authors: Stefano Martiniani, Jacob D. Stevenson, David J. Wales, Julian K Schrenk, Daan FrenkelAbstract:We propose an efficient Monte Carlo method for the computation of the volumes of high-dimensional bodies with arbitrary shape. We start with a region of known volume within the interior of the manifold and then use the multistate Bennett acceptance-ratio method to compute the dimensionless free-energy difference between a series of equilibrium simulations performed within this object. The method produces results that are in excellent agreement with thermodynamic integration, as well as a direct estimate of the associated statistical uncertainties. The histogram method also allows us to directly obtain an estimate of the interior radial probability density profile, thus yielding useful insight into the structural properties of such a high-dimensional body. We illustrate the method by analyzing the effect of structural disorder on the Basins of Attraction of mechanically stable packings of soft repulsive spheres.
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Visualizing Basins of Attraction for Different Minimization Algorithms
The journal of physical chemistry. B, 2013Co-Authors: Daniel Asenjo, Jacob D. Stevenson, David J. Wales, Daan FrenkelAbstract:We report a study of the Basins of Attraction for potential energy minima defined by different minimisation algorithms for an atomic system. We find that whereas some minimisation algorithms produce compact Basins, others produce Basins with complex boundaries or Basins consisting of disconnected parts. Such Basins deviate from the "correct" basin of Attraction defined by steepest-descent pathways, and the differences can be controlled to some extent by adjustment of the maximum step size. The choice of the most convenient minimisation algorithm depends on the problem in hand. We show that while L-BFGS is the fastest minimiser, the FIRE algorithm is also quite fast, and can lead to less fragmented Basins of Attraction.
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Basins of Attraction for stationary points on a potential-energy surface
Journal of the Chemical Society Faraday Transactions, 1992Co-Authors: David J. WalesAbstract:Methods to locate stationary points on model potential-energy surfaces, especially minima and transition states, are of great importance. Here we visualise the Basins of Attraction of such points, that is, the regions of the surface for which searches converge to a given structure. The results demonstrate the superiority of the eigenvector-following (EF) method over simple Newton–Raphson (NR) iteration. Furthermore, the boundaries between different Basins of Attraction may apparently be multifractal in character and highly complex. The results also provide some general rules for the efficient characterisation of potential-energy surfaces which are of practical value.
