scholarly journals Extending the functionality of a symbolic computational dynamic solver by using a novel term-tracking method

2017 ◽  
Vol 232 (19) ◽  
pp. 3439-3452 ◽  
Author(s):  
Niloufar Motazedi ◽  
Matthew P Cartmell ◽  
Jem A Rongong
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Solution Procedure ◽  
Nonlinear Dynamical ◽  
Mathematical Connections ◽  
Approximate Analytical Solutions ◽  
New Feature ◽  
Encoding Method ◽  
The Many ◽  
Robust Process ◽  
First Time

Symbolic computational dynamic solvers are currently under development in order to provide new and powerful tools for modelling nonlinear dynamical systems. Such solvers consist of two parts; the core solver, which comprises an approximate analytical method based on perturbation, averaging, or harmonic balance, and a specialised term-tracker. A term-tracking approach has been introduced to provide a powerful new feature into computational approximate analytical solutions by highlighting the many mathematical connections that exist, but which are invariably lost through processing, between the physical model of the system, the solution procedure itself, and the final result which is usually expressed in equation form. This is achieved by a highly robust process of term-tracking, recording, and identification of all the symbolic mathematical information within the problem. In this paper, the novel source and evolution encoding method is introduced for the first time and an implementation in Mathematica is described through the development of a specialised algorithm.

scholarly journals Symbolic Encoding of Periodic Orbits and Chaos in the Rucklidge System

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Chengwei Dong ◽  
Lian Jia ◽  
Qi Jie ◽  
Hantao Li
Keyword(s):  
Periodic Orbits ◽  
Nonlinear Dynamical Systems ◽  
Unstable Periodic Orbits ◽  
Nonlinear Dynamical ◽  
System A ◽  
Return Maps ◽  
Phase Portrait Analysis ◽  
Encoding Method ◽  
First Return Maps ◽  
Parameter Values

To describe and analyze the unstable periodic orbits of the Rucklidge system, a so-called symbolic encoding method is introduced, which has been proven to be an efficient tool to explore the topological properties concealed in these periodic orbits. In this work, the unstable periodic orbits up to a certain topological length in the Rucklidge system are systematically investigated via a proposed variational method. The dynamics in the Rucklidge system are explored by using phase portrait analysis, Lyapunov exponents, and Poincaré first return maps. Symbolic encodings of the periodic orbits with two and four letters based on the trajectory topology in the phase space are implemented under two sets of parameter values. Meanwhile, the bifurcations of the periodic orbits are explored, significantly improving the understanding of the dynamics of the Rucklidge system. The multiple-letter symbolic encoding method could also be applicable to other nonlinear dynamical systems.

A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE. PART X: PERIOD-1 RULES

2009 ◽  
Vol 19 (05) ◽  
pp. 1425-1654 ◽  
Author(s):  
LEON O. CHUA ◽  
GIOVANNI E. PAZIENZA ◽  
JINWOOK SHIN
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Characteristic Functions ◽  
Theoretical Treatment ◽  
String Length ◽  
Nonlinear Dynamical ◽  
One Dimensional ◽  
First Time ◽  
The Universe ◽  
Science Part ◽  
Group 1

This stage of our journey through the universe of one-dimensional binary Cellular Automata is devoted to period-1 rules, constituting the first of the six groups in which we systematized the 88 globally-independent CA rules. The first part of this article is mainly dedicated to reviewing the terminology and the empirical results found in the previous papers of our quest. We also introduce the concept of the ω-limit orbit with the purpose of linking our work to the classical theory of nonlinear dynamical systems. Moreover, we present the basin tree diagrams of all period-1 rules — except for rule [Formula: see text], which is trivial — along with their Boolean cubes and time-1 characteristic functions. In the second part, we prove a theorem demonstrating that all rules belonging to group 1 have robust period-1 rules for any finite, and infinite, bit-string length L. This is the first time we give analytical results on the behavior of CA local rules for large values of L and, consequently, for bi-infinite bit strings. The theoretical treatment is complemented by two remarkable practical results: an explicit formula for generating isomorphic basin trees, and an algorithm for creating new periodic orbits by concatenation. We also provide several examples of both of them, showing how they help to avoid tedious simulations.

scholarly journals TURBULENCE AND CHAOS IN ANTI-DE-SITTER GRAVITY

2012 ◽  
Vol 21 (11) ◽  
pp. 1242013 ◽  
Author(s):  
H. P. DE OLIVEIRA ◽  
LEOPOLDO A. PANDO ZAYAS ◽  
CÉSAR A. TERRERO-ESCALANTE
Keyword(s):  
Field Theory ◽  
Nonlinear Dynamical Systems ◽  
Interdisciplinary Cooperation ◽  
Harmonic Oscillators ◽  
De Sitter ◽  
Nonlinear Dynamical ◽  
Conformal Field ◽  
Open Questions ◽  
The Many ◽  
Standard Techniques

Due to the Anti-de-Sitter/Conformal Field Theory (AdS/CFT) correspondence, the question of instability of AdS spacetimes sits in the intersection of mathematical and numerical relativities, string theory, field theory and condensed matter physics. In this essay we revisit that important question emphasizing the power of spectral methods and highlighting the effectiveness of standard techniques for studying nonlinear dynamical systems. In particular, we display explicitly how the problem can be modeled as a system on nonlinearly coupled harmonic oscillators. We highlight some of the many open questions that stem from this result and point out that a full understanding will necessarily require the interdisciplinary cooperation of various communities.

Dynamics of a general jerky equation

2018 ◽  
Vol 25 (4) ◽  
pp. 922-932
Author(s):  
Diandian Tang ◽  
Shirui Zhang ◽  
Jingli Ren
Keyword(s):  
Hopf Bifurcation ◽  
Nonlinear Dynamical Systems ◽  
Normal Form Theory ◽  
Hopf Bifurcation Point ◽  
Nonlinear Dynamical ◽  
Center Manifold Theorem ◽  
Fold Bifurcation ◽  
Form Theory ◽  
Jerky Dynamics ◽  
First Time

Some classic nonlinear dynamical systems, such as Rössler's toroidal model, the Genesio model, and 19 Sprott's models, can be classified into seven distinct basic classes of jerky dynamics, labeled by [Formula: see text]. This paper is devoted to the dynamics of a general jerky equation which contains [Formula: see text] as parameters vary. It is shown that the system undergoes fold, Hopf, zero-Hopf, and Bogdanov–Takens bifurcations based on the center manifold theorem and normal form theory. Numerical simulations are also given to make the theoretical results visible and to detect more complicated dynamical behaviors, including degenerate Hopf bifurcation, fold bifurcation of cycle, and limit cycles. Especially, an apple-like attractive portrait is discovered near the zero-Hopf bifurcation point for the first time. Finally, according to the conclusions of the general jerky equation, exact conditions are summarized by two tables on how bifurcations will occur for [Formula: see text], respectively.

scholarly journals Objective Detection of Oceanic Eddies and the Agulhas Leakage

2013 ◽  
Vol 43 (7) ◽  
pp. 1426-1438 ◽  
Author(s):  
Francisco J. Beron-Vera ◽  
Yan Wang ◽  
María J. Olascoaga ◽  
Gustavo J. Goni ◽  
George Haller
Keyword(s):  
Ocean Circulation ◽  
Transport Theory ◽  
Nonlinear Dynamical Systems ◽  
Material Transport ◽  
Nonlinear Dynamical ◽  
Oceanic Eddies ◽  
Objective Detection ◽  
First Time ◽  
Spurious Results ◽  
Satellite Altimetry Data

Abstract Mesoscale oceanic eddies are routinely detected from instantaneous velocities derived from satellite altimetry data. While simple to implement, this approach often gives spurious results and hides true material transport. Here it is shown how geodesic transport theory, a recently developed technique from nonlinear dynamical systems, uncovers eddies objectively. Applying this theory to altimetry-derived velocities in the South Atlantic reveals, for the first time, Agulhas rings that preserve their material coherence for several months, while ring candidates yielded by other approaches tend to disperse or leak within weeks. These findings suggest that available velocity-based estimates for the Agulhas leakage, as well as for its impact on ocean circulation and climate, need revision.

Sequential Estimation of States and Parameters in Noisy Nonlinear Dynamical Systems

1966 ◽  
Vol 88 (2) ◽  
pp. 362-368 ◽  
Author(s):  
D. M. Detchmendy ◽  
R. Sridhar
Keyword(s):  
Differential Equation ◽  
Least Squares ◽  
Measurement Errors ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Sequential Estimation ◽  
Nonlinear Dynamical ◽  
Statistical Assumptions ◽  
Estimation Scheme ◽  
New Feature

The problem considered is the sequential estimation of states and parameters in noisy nonlinear systems. The class of systems considered is those in which the dynamical behavior is described by an ordinary differential equation. No statistical assumptions are required concerning the nature of the unknown inputs to the system or the measurement errors on the output. For estimation purposes, a least-squares criterion is used. The new feature of the approach presented is that a sequential least-squares estimator is obtained for the class of problems considered. This estimator could be implemented in real time. Experimental results from several examples indicate that the proposed estimation scheme is feasible.

Unrivalled Influence

2013 ◽  
Author(s):  
Judith Herrin
Keyword(s):  
Political Life ◽  
General Introduction ◽  
Women And Gender ◽  
Mothers And Daughters ◽  
Diverse Range ◽  
Long Span ◽  
Imperial Court ◽  
And Gender ◽  
The Many ◽  
First Time

This book explores the exceptional roles that women played in the vibrant cultural and political life of medieval Byzantium. This book evokes the complex and exotic world of Byzantium's women, from empresses and saints to uneducated rural widows. Drawing on a diverse range of sources, the book sheds light on the importance of marriage in imperial statecraft, the tense coexistence of empresses in the imperial court, and the critical relationships of mothers and daughters. It looks at women's interactions with eunuchs, the in-between gender in Byzantine society, and shows how women defended their rights to hold land. The book describes how women controlled their inheritances, participated in urban crowds demanding the dismissal of corrupt officials, followed the processions of holy icons and relics, and marked religious feasts with liturgical celebrations, market activity, and holiday pleasures. The vivid portraits that emerge here reveal how women exerted an unrivalled influence on the patriarchal society of Byzantium, and remained active participants in the many changes that occurred throughout the empire's millennial history. The book brings together the author's finest essays on women and gender written throughout the long span of her career. This volume includes three new essays published here for the very first time and a new general introduction. It also provides a concise introduction to each essay that describes how it came to be written and how it fits into her broader views about women and Byzantium.

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