A GALLERY OF CHUA ATTRACTORS: PART III

2007 ◽  
Vol 17 (03) ◽  
pp. 657-734 ◽  
Author(s):  
ELEONORA BILOTTA ◽  
FAUSTO STRANGES ◽  
PIETRO PANTANO
Keyword(s):  
Parameter Space ◽  
Information Seeking ◽  
Ad Hoc ◽  
Three Dimensional ◽  
High Dimensional ◽  
Special Focus ◽  
Processes Of Change ◽  
Parameter Spaces ◽  
Chua’S Oscillator ◽  
Components Analysis

The visualization of patterns related to chaos is a challenge for those who are part of today's dynamical systems community, especially when we consider the aim of providing users with the ability to visually analyze and explore large, complex datasets related to chaos. Thus visualization could be considered a useful element in the discovery of unexpected relationships and dependencies that may exist inside the domain of chaos, both in the phase and the parameter spaces. In the second part of "A Gallery of Chua attractors", we presented an overview of forms which can only be produced by the physical circuit. In Part III, we illustrate the variety and beauty of the strange attractors produced by the dimensionless version of the system. As in our earlier work, we have used ad hoc methods, such as bifurcation maps and software tools, allowing rapid exploration of parameter space. Applying these techniques, we show how it is possible, starting from attractors described in the literature, to find new families of patterns, with a special focus on the cognitive side of information seeking and on qualitative processes of change in chaos, thus demonstrating that traditional categories of chaos exploration need to be renewed. After a brief introduction to dimensionless equations for Chua's oscillator, we show 150 attractors, which we represent using three-dimensional images, time series and FFT diagrams. For the most important patterns, we also report Lyapunov exponents. To show the position of dimensionless attractors in parameter space, we use parallel coordinate techniques that facilitate the visualization of high dimensional spaces. We use Principal Components Analysis (PCA) and Mahalanobis Distance to provide additional tools for the exploration and visualization of the structure of the parameter space.

scholarly journals Navigator function for the conformal bootstrap

2021 ◽  
Vol 11 (3) ◽  
Author(s):  
Marten Reehorst ◽  
Slava Rychkov ◽  
David Simmons-Duffin ◽  
Benoit Sirois ◽  
Ning Su ◽  
...  
Keyword(s):  
Parameter Space ◽  
Optimization Methods ◽  
High Dimensional ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
Standard Methods ◽  
Conformal Bootstrap ◽  
Bootstrap Techniques ◽  
Binary Information ◽  
Quasi Newton

Current numerical conformal bootstrap techniques carve out islands in theory space by repeatedly checking whether points are allowed or excluded. We propose a new method for searching theory space that replaces the binary information "allowed"/"excluded" with a continuous "navigator" function that is negative in the allowed region and positive in the excluded region. Such a navigator function allows one to efficiently explore high-dimensional parameter spaces and smoothly sail towards any islands they may contain. The specific functions we introduce have several attractive features: they are well-defined in large regions of parameter space, can be computed with standard methods, and evaluation of their gradient is immediate due to an SDP gradient formula that we provide. The latter property allows for the use of efficient quasi-Newton optimization methods, which we illustrate by navigating towards the 3d Ising island.

scholarly journals Bootstrapping boundary-localized interactions

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Connor Behan ◽  
Lorenzo Di Pietro ◽  
Edoardo Lauria ◽  
Balt C. van Rees
Keyword(s):  
Boundary Condition ◽  
Scalar Field ◽  
Boundary Conditions ◽  
Parameter Space ◽  
Three Dimensional ◽  
Conformal Boundary ◽  
Dimensional Boundary ◽  
Dirichlet And Neumann Conditions ◽  
Boundary Fields ◽  
Boundary Dynamics

Abstract We study conformal boundary conditions for the theory of a single real scalar to investigate whether the known Dirichlet and Neumann conditions are the only possibilities. For this free bulk theory there are strong restrictions on the possible boundary dynamics. In particular, we find that the bulk-to-boundary operator expansion of the bulk field involves at most a ‘shadow pair’ of boundary fields, irrespective of the conformal boundary condition. We numerically analyze the four-point crossing equations for this shadow pair in the case of a three-dimensional boundary (so a four-dimensional scalar field) and find that large ranges of parameter space are excluded. However a ‘kink’ in the numerical bounds obeys all our consistency checks and might be an indication of a new conformal boundary condition.

scholarly journals Experimental & FEM Analysis of Orthodontic Mini-Implant Design on Primary Stability

2021 ◽  
Vol 11 (12) ◽  
pp. 5461
Author(s):  
Elmedin Mešić ◽  
Enis Muratović ◽  
Lejla Redžepagić-Vražalica ◽  
Nedim Pervan ◽  
Adis J. Muminović ◽  
...  
Keyword(s):  
Finite Element ◽  
Three Dimensional ◽  
Primary Stability ◽  
Fem Analysis ◽  
Implant Design ◽  
Special Focus ◽  
Engineering System ◽  
Mini Implants ◽  
Pull Out ◽  
Computer Aided

The main objective of this research is to establish a connection between orthodontic mini-implant design, pull-out force and primary stability by comparing two commercial mini-implants or temporary anchorage devices, Tomas®-pin and Perfect Anchor. Mini-implant geometric analysis and quantification of bone characteristics are performed, whereupon experimental in vitro pull-out test is conducted. With the use of the CATIA (Computer Aided Three-dimensional Interactive Application) CAD (Computer Aided Design)/CAM (Computer Aided Manufacturing)/CAE (Computer Aided Engineering) system, 3D (Three-dimensional) geometric models of mini-implants and bone segments are created. Afterwards, those same models are imported into Abaqus software, where finite element models are generated with a special focus on material properties, boundary conditions and interactions. FEM (Finite Element Method) analysis is used to simulate the pull-out test. Then, the results of the structural analysis are compared with the experimental results. The FEM analysis results contain information about maximum stresses on implant–bone system caused due to the pull-out force. It is determined that the core diameter of a screw thread and conicity are the main factors of the mini-implant design that have a direct impact on primary stability. Additionally, stresses generated on the Tomas®-pin model are lower than stresses on Perfect Anchor, even though Tomas®-pin endures greater pull-out forces, the implant system with implemented Tomas®-pin still represents a more stressed system due to the uniform distribution of stresses with bigger values.

scholarly journals High-order compact LOD methods for solving high-dimensional advection equations

2021 ◽  
Vol 40 (3) ◽  
Author(s):  
Bo Hou ◽  
Yongbin Ge
Keyword(s):  
Truncation Error ◽  
Three Dimensional ◽  
Taylor Series Expansion ◽  
High Order ◽  
High Dimensional ◽  
Analysis Method ◽  
Order Accuracy ◽  
Von Neumann ◽  
One Dimensional ◽  
Von Neumann Analysis

AbstractIn this paper, by using the local one-dimensional (LOD) method, Taylor series expansion and correction for the third derivatives in the truncation error remainder, two high-order compact LOD schemes are established for solving the two- and three- dimensional advection equations, respectively. They have the fourth-order accuracy in both time and space. By the von Neumann analysis method, it shows that the two schemes are unconditionally stable. Besides, the consistency and convergence of them are also proved. Finally, numerical experiments are given to confirm the accuracy and efficiency of the present schemes.

scholarly journals The Formation of a Cavity in a 3D Flux Rope

2013 ◽  
Vol 8 (S300) ◽  
pp. 147-150 ◽  
Author(s):  
Donald Schmit ◽  
Sarah Gibson
Keyword(s):  
Ad Hoc ◽  
Numerical Models ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Flux Rope ◽  
Multiple Structures ◽  
Field Lines ◽  
Hydrostatic Balance ◽  
Distinct Field ◽  
Three Dimensional Space

AbstractThere are currently no three dimensional numerical models which describe the magnetic and energetic formation of prominences self-consistently. Consequently, there has not been significant progress made in understanding the connection between the dense prominence plasma and the coronal cavity. We have taken an ad-hoc approach to understanding the energetic implications of the magnetic models of prominence structure. We extract one dimensional magnetic field lines from a 3D MHD model of a flux rope and solve for hydrostatic balance along these field lines incorporating field-aligned thermal conduction, uniform heating, and radiative losses. The 1D hydrostatic solutions for density and temperature are then mapped back into three dimensional space, which allows us to consider the projection of multiple structures. We find that the 3D flux rope is composed of several distinct field line types. A majority of the flux rope interior field lines are twisted but not dipped. These field lines are density-reduced relative to unsheared arcade field lines. We suggest the cavity may form along these short interior field lines which are surrounded by a sheath of dipped field lines. This geometric arrangement would create a cavity on top of a prominence, but the two structures would not share field lines or plasma.

scholarly journals Robust simulation of mineral precipitation–dissolution problems with variable mineral surface area

2021 ◽  
Vol 129 (1) ◽  
Author(s):  
Serge Kräutle ◽  
Jan Hodai ◽  
Peter Knabner
Keyword(s):  
Surface Area ◽  
Ad Hoc ◽  
Chemical Species ◽  
Special Focus ◽  
Proof Of Concept ◽  
Mineral Surface ◽  
Mineral Precipitation ◽  
Numerical Tests ◽  
Ill Posed ◽  
Mineral Surface Area

AbstractWe consider a macroscale model of transport and reaction of chemical species in a porous medium with a special focus on mineral precipitation–dissolution processes. In the literature, it is frequently proposed that the reaction rate should depend on the reactive mineral surface area, and so on the amount of mineral. We point out that a frequently used model is ill posed in the sense that it admits non-unique solutions. We investigate what consequences this non-uniqueness has on the numerical solution of the model. The main novelty in this article is our proposal of a certain substitution which removes the ill-posedness from the system and which leads to better numerical results than some “ad hoc methods.” We think that the proposed substitution is a rather elegant way to get rid of the non-uniqueness and the numerical difficulties and is much less technical than other ideas. As a proof of concept, we present some numerical tests and simulations for the new model.

Reducing Undesired Wave Reflection at Domain Boundaries in 3D Finite Volume-Based Flow Simulations via Forcing Zones

2020 ◽  
Vol 64 (01) ◽  
pp. 23-47
Author(s):  
Robinson Peric ◽  
Moustafa Abdel-Maksoud
Keyword(s):  
Finite Volume ◽  
Wave Reflection ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Special Focus ◽  
Computational Domain ◽  
Wave Reflections ◽  
Flow Simulations ◽  
Large Eddy ◽  
Domain Boundaries

This article reviews different types of forcing zones (sponge layers, damping zones, relaxation zones, etc.) as used in finite volume-based flow simulations to reduce undesired wave reflections at domain boundaries, with special focus on the case of strongly reflecting bodies subjected to long-crested incidence waves. Limitations and possible sources of errors are discussed. A novel forcing-zone arrangement is presented and validated via three-dimensional (3D) flow simulations. Furthermore, a recently published theory for predicting the forcing-zone behavior was investigated with regard to its relevance for practical 3D hydrodynamics problems. It was found that the theory can be used to optimally tune the case-dependent parameters of the forcing zones before running the simulations. 1. Introduction Wave reflections at the boundaries of the computational domain can cause significant errors in flow simulations, and must therefore be reduced. In contrast to boundary element codes, where much progress in this respect has been made decades ago (see e.g., Clement 1996; Grilli &Horillo 1997), for finite volume-based flow solvers, there are many unresolved questions, especially:How to reliably reduce reflections and disturbances from the domain boundaries?How to predict the amount of undesired wave reflection before running the simulation? This work aims to provide further insight to these questions for flow simulations based on Navier-Stokes-type equations (Reynolds-averaged Navier-Stokes, Euler equations, Large Eddy Simulations, etc.), when using forcing zones to reduce undesired reflections. The term "forcing zones" is used here to describe approaches that gradually force the solution in the vicinity of the boundary towards some reference solution, as described in Section 2; some examples are absorbing layers, sponge layers, damping zones, relaxation zones, or the Euler overlay method (Mayer et al. 1998; Park et al. 1999; Chen et al. 2006; Choi &Yoon 2009; Jacobsen et al. 2012; Kimet al. 2012; Schmitt & Elsaesser 2015; Perić & Abdel-Maksoud 2016a; Vukčević et al. 2016).

Sign in / Sign up

Export Citation Format

Share Document