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.

Application of a Generalized Frobenius-Perron Operator Scheme to Multivariate Nonlinear Dynamical Systems

1995 ◽  
Vol 50 (12) ◽  
pp. 1123-1127
Author(s):  
R. Stoop ◽  
W.-H. Steeb
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Free Energies ◽  
Nonlinear Dynamical ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Operator Scheme

Abstract The concept of generalized Frobenius-Perron operators is applied to multivariante nonlinear dynamical systems, and the associated generalized free energies are investigated. As important applications, diffusion-related free energies obtained from normally and superlinearly diffusive one-dimensional maps are discussed.

UNIFORMITY AND SPATIAL CHAOS OF SPATIAL PHYSICS KINEMATIC SYSTEM

2010 ◽  
Vol 24 (28) ◽  
pp. 5495-5503
Author(s):  
SHUTANG LIU ◽  
FUYAN SUN ◽  
JIE SUN
Keyword(s):  
Dynamical Systems ◽  
Lattice Model ◽  
Nonlinear Dynamical Systems ◽  
Coupled Map Lattice ◽  
Nonlinear Dynamical ◽  
One Dimensional ◽  
Uniform System ◽  
Spatial Chaos ◽  
Kinematic System ◽  
Bifurcation Behavior

This article summarizes the uniformity law of spatial physics kinematic systems, and studies the chaos and bifurcation behavior of the uniform system in space. In particular, it also fully explains the relation among the uniform system, the coupled map lattice model which has attracted considerable interest currently, and one-dimensional nonlinear dynamical systems.

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.

ON STRONG AND WEAK CHAOTIC PARTIAL SYNCHRONIZATION

2000 ◽  
Vol 10 (01) ◽  
pp. 179-203 ◽  
Author(s):  
Yu. MAISTRENKO ◽  
O. POPOVYCH ◽  
M. HASLER
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Chaotic Behavior ◽  
Chaotic Synchronization ◽  
State Variables ◽  
Large Parameter ◽  
Nonlinear Dynamical ◽  
One Dimensional ◽  
Rich Variety ◽  
Parameter Ranges ◽  
One Dimensional Maps

We study coupled nonlinear dynamical systems with chaotic behavior in the case when two or more (but not all) state variables synchronize, i.e. converge to each other asymptotically in time. It is shown that for symmetrical systems, such partial chaotic synchronization is usually only weak, whereas with nonsymmetrical coupling it can be strong in large parameter ranges. These facts are illustrated with systems of three coupled one-dimensional maps, for which a rich variety of different "partial chaotic synchronizing" phenomena takes place.

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.

Superior Cantor Sets and Superior Devil Staircases

2010 ◽  
Vol 1 (1) ◽  
pp. 78-84 ◽  
Author(s):  
Mamta Rani ◽  
Sanjeev Kumar Prasad
Keyword(s):  
Dynamical Systems ◽  
Real World ◽  
Nonlinear Dynamical Systems ◽  
Cantor Set ◽  
Cantor Sets ◽  
Strange Attractors ◽  
Nonlinear Dynamical ◽  
The Universe ◽  
Real World Problems

Mandelbrot, in 1975, coined the term fractal and included Cantor set as a classical example of fractals. The Cantor set has wide applications in real world problems from strange attractors of nonlinear dynamical systems to the distribution of galaxies in the universe (Schroder, 1990). In this article, we obtain superior Cantor sets and present them graphically by superior devil’s staircases. Further, based on their method of generation, we put them into two categories.

Nonlinear waves, patterns and spatio-temporal chaos in cellular neural networks

1995 ◽  
Vol 353 (1701) ◽  
pp. 101-113 ◽  
Keyword(s):  
Pattern Formation ◽  
Spatial Organization ◽  
Nonlinear Dynamical Systems ◽  
Spiral Wave ◽  
High Dimensional ◽  
Turing Pattern ◽  
Electronic Circuits ◽  
Nonlinear Dynamical ◽  
One Dimensional ◽  
Spatio Temporal

Spatio-temporal pattern formation occurring in discretely coupled nonlinear dynamical systems has been studied numerically. In this paper, we review the possibilities of using arrays of discretely coupled nonlinear electronic circuits to study these systems. Spiral wave initiation and Turing pattern formation are some of the examples. Sidewall forcing of Turing patterns is shown to be capable of driving the system into a perfect spatial organization, namely, a rhombic pattern, where no defects occur. The dynamics of the two layers supporting Turing and Hopf modes, respectively, is analysed as a function of the coupling strength between them. The competition between these two modes is shown to increase with the diffusion between layers. As well, the coexistence of low- and high-dimensional spatio-temporal chaos is shown to occur in one-dimensional arrays.

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.

A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE PART IV: FROM BERNOULLI SHIFT TO 1/f SPECTRUM

2005 ◽  
Vol 15 (04) ◽  
pp. 1045-1183 ◽  
Author(s):  
LEON O. CHUA ◽  
VALERY I. SBITNEV ◽  
SOOK YOON
Keyword(s):  
Bernoulli Shift ◽  
Dynamical Evolution ◽  
Complete Characterization ◽  
Characteristic Functions ◽  
One Dimensional ◽  
New Concepts ◽  
The Right ◽  
Science Part

By exploiting the new concepts of CA characteristic functions and their associated attractor time-τ maps, a complete characterization of the long-term time-asymptotic behaviors of all 256 one-dimensional CA rules are achieved via a single "probing" random input signal. In particular, the graphs of the time-1 maps of the 256 CA rules represent, in some sense, the generalized Green's functions for Cellular Automata. The asymptotic dynamical evolution on any CA attractor, or invariant orbit, of 206 (out of 256) CA rules can be predicted precisely, by inspection. In particular, a total of 112 CA rules are shown to obey a generalized Bernoulli στ-shift rule, which involves the shifting of any binary string on an attractor, or invariant orbit, either to the left, or to the right, by up to 3 pixels, and followed possibly by a complementation of the resulting bit string. The most intriguing result reported in this paper is the discovery that the four Turing-universal rules [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], and only these rules, exhibit a 1/f power spectrum.

scholarly journals Pynamical: Model and visualize discrete nonlinear dynamical systems, chaos, and fractals

2018 ◽  
Author(s):  
Geoff Boeing
Keyword(s):  
Dynamical Systems ◽  
Phase Diagrams ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Models ◽  
Logistic Map ◽  
Three Dimensional ◽  
Basins Of Attraction ◽  
Nonlinear Dynamical ◽  
One Dimensional ◽  
One Dimensional Maps

Pynamical is an educational Python package for introducing the modeling, simulation, and visualization of discrete nonlinear dynamical systems and chaos, focusing on one-dimensional maps (such as the logistic map and the cubic map). Pynamical facilitates defining discrete one-dimensional nonlinear models as Python functions with just-in-time compilation for fast simulation. It comes packaged with the logistic map, the Singer map, and the cubic map predefined. The models may be run with a range of parameter values over a set of time steps, and the resulting numerical output is returned as a pandas DataFrame. Pynamical can then visualize this output in various ways, including with bifurcation diagrams, two-dimensional phase diagrams, three-dimensional phase diagrams, and cobweb plots. These visualizations enable simple qualitative assessments of system behavior including phase transitions, bifurcation points, attractors and limit cycles, basins of attraction, and fractals.

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