Strange Attractors and Chaos in Nonlinear Mechanics

1983 ◽  
Vol 50 (4b) ◽  
pp. 1021-1032 ◽  
Author(s):  
P. J. Holmes ◽  
F. C. Moon
Keyword(s):  
Mathematical Models ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Nonlinear Differential Equations ◽  
Homoclinic Orbits ◽  
Strange Attractors ◽  
Electrical Systems ◽  
Nonlinear Mechanical ◽  
Sensitive Dependence

We review several examples of nonlinear mechanical and electrical systems and related mathematical models that display chaotic dynamics or strange attractors. Some simple mathematical models — iterated piecewise linear mappings — are introduced to explain and illustrate the concepts of sensitive dependence on initial conditions and chaos. In particular, we describe the role of homoclinic orbits and the horseshoe map in the generation of chaos, and indicate how the existence of such features can be detected in specific nonlinear differential equations.

PREDICTING REGIMES OF INDETERMINATE JUMPS TO RESONANCE BY ASSESSING FRACTAL BOUNDARIES IN CONTROL SPACE

1994 ◽  
Vol 04 (06) ◽  
pp. 1645-1653 ◽  
Author(s):  
MOHAMED S. SOLIMAN
Keyword(s):  
Initial Conditions ◽  
Initial Study ◽  
Control Space ◽  
Electrical Systems ◽  
Wide Range ◽  
Nonlinear Mechanical ◽  
Sensitive Dependence ◽  
Simplified Method ◽  
Local And Global Bifurcations ◽  
Parameter Values

For nonlinear mechanical, structural and electrical systems, it is important to know, over a wide range of parameter values, whether jumps to resonance exist. Since the outcome may often be indeterminate, we have to identify parameters which place bounds on the determinancy of such jumps. Such an analysis is often quite complex since it involves determining both the main local and global bifurcations (the homoclinic and heteroclinic connections that induce a fractal basin structure at the bifurcation) that can lead to an indeterminate jump to resonance. Utilizing the fact that a sensitive dependence on the initial conditions implies a sensitive dependence on parameters, in this paper we propose a simplified method, based on assessing fractal boundaries in control space, that may be used to reveal aspects of indeterminate bifurcational behaviour of the system. Such an analysis which allows us to identify approximate regimes of indeterminate jumps to resonance, may either be used as an initial study for further detailed analysis, or as a simplified engineering approach in the analysis of systems liable to this type of resonant behaviour.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

ORDER AND CHAOS FOR A CLASS OF PIECEWISE LINEAR MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1379-1394 ◽  
Author(s):  
VÍCTOR JIMÉNEZ LÓPEZ
Keyword(s):  
Positive Measure ◽  
Invariant Measures ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Absolutely Continuous ◽  
Linear Maps ◽  
Sensitive Dependence ◽  
Piecewise Linear Maps ◽  
Almost All ◽  
Absolutely Continuous Invariant

For a class of piecewise linear maps f: I → I from a compact interval I into itself, we describe the asymptotic behavior of the sequence [Formula: see text] for almost all x ∈ I. We also study in this setting the relations among sensitive dependence on initial conditions, existence of scrambled sets of positive measure and existence of absolutely continuous invariant measures.

Chairman’s introduction

1987 ◽  
Vol 413 (1844) ◽  
pp. 3-4
Keyword(s):  
Asymptotic Behaviour ◽  
Frequency Spectrum ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Strange Attractors ◽  
Qualitative Properties ◽  
Differential Topology ◽  
Deterministic Systems ◽  
Sensitive Dependence ◽  
Quantitative Properties

The understanding of chaos and strange attractors is one of the most exciting areas of mathematics today. It is the question of how the asymptotic behaviour of deterministic systems can exhibit unpredictability and apparent chaos, due to sensitive dependence upon initial conditions, and yet at the same time preserve a coherent global structure. The field represents a remarkable confluence of several different strands of thought. 1. Firstly came the influence of differential topology, giving global geometric insight and emphasis on qualitative properties. By qualitative properties I mean invariants under differentiable changes of coordinates, as opposed to quantitative properties which are invariant only under linear changes of coordinates. To give an example of this influence, I recall a year-long symposium at Warwick in 1979/80, which involved sustained interaction between pure mathematicians and experimentalists, and one of the most striking consequences of that interaction was a transformation in the way that experimentalists now present their data. It is generally in a much more translucent form: instead of merely plotting a frequency spectrum and calling the incomprehensible part ‘noise’, they began to draw computer pictures of underlying three-dimensional strange attractors.

Pieces of Mind

2018 ◽  
Author(s):  
Carrie Figdor
Keyword(s):  
Mathematical Models ◽  
Moral Status ◽  
Entire Range ◽  
Fruit Flies ◽  
Literal Interpretation ◽  
Psychological Capacities

Many people accept that chimpanzees, dolphins, and some other animals can think and feel. But these cases are just the tip of a growing iceberg. If biologists are right, fruit flies and plants make decisions, worms and honeybees can be trained, bacteria communicate linguistically, and neurons have preferences. Just how far does cognition go? This book is the first to critically consider this question from the perspective of the entire range of new ascriptions of psychological capacities throughout biology. It is also the first to consider the role of mathematical models and other quantitative forms of evidence in prompting and supporting the new ascriptions. It defends a default literal interpretation of psychological terms across biological domains. It also considers the implications of the literal view for efforts to explain the mind’s place in nature and for traditional ways of distinguishing the superior moral status of humans relative to other living beings.

scholarly journals On some kinds of dense orbits in general nonautonomous dynamical systems

2020 ◽  
Vol 7 (1) ◽  
pp. 163-175
Author(s):  
Mehdi Pourbarat
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Point Of View ◽  
Topological Conjugacy ◽  
Topological Transitivity ◽  
Nonautonomous Systems ◽  
Nonautonomous Dynamical Systems ◽  
Sensitive Dependence ◽  
Dense Orbits

AbstractWe study the theory of universality for the nonautonomous dynamical systems from topological point of view related to hypercyclicity. The conditions are provided in a way that Birkhoff transitivity theorem can be extended. In the context of generalized linear nonautonomous systems, we show that either one of the topological transitivity or hypercyclicity give sensitive dependence on initial conditions. Meanwhile, some examples are presented for topological transitivity, hypercyclicity and topological conjugacy.

IS SENSITIVE DEPENDENCE ON INITIAL CONDITIONS NATURE’S SENSORY DEVICE?

1992 ◽  
Vol 02 (01) ◽  
pp. 193-199 ◽  
Author(s):  
RAY BROWN ◽  
LEON CHUA ◽  
BECKY POPP
Keyword(s):  
Initial Conditions ◽  
Sensory Perception ◽  
Sensory Features ◽  
Sensitive Dependence ◽  
Nonlinear Devices

In this letter we illustrate three methods of using nonlinear devices as sensors. We show that the sensory features of these devices is a result of sensitive dependence on parameters which we show is equivalent to sensitive dependence on initial conditions. As a result, we conjecture that sensitive dependence on initial conditions is nature’s sensory device in cases where remarkable feats of sensory perception are seen.

Reassessment of transient permeability measurement for tight rocks: The role of boundary and initial conditions

2021 ◽  
pp. 104173
Author(s):  
Yue Wang ◽  
Zhiguo Tian ◽  
Steffen Nolte ◽  
Alexandra Amann-Hildenbrand ◽  
Bernhard M. Krooss ◽  
...  
Keyword(s):  
Initial Conditions ◽  
Permeability Measurement ◽  
Tight Rocks
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