Journal ArticleDOI
Simple mathematical models with very complicated dynamics
Robert M. May,Robert M. May +1 more
TLDR
This is an interpretive review of first-order difference equations, which can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations.Abstract:
First-order difference equations arise in many contexts in the biological, economic and social sciences. Such equations, even though simple and deterministic, can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations. There are consequently many fascinating problems, some concerned with delicate mathematical aspects of the fine structure of the trajectories, and some concerned with the practical implications and applications. This is an interpretive review of them.read more
Citations
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Journal ArticleDOI
Measuring the Strangeness of Strange Attractors
TL;DR: In this paper, the correlation exponent v is introduced as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise, and algorithms for extracting v from the time series of a single variable are proposed.
Journal ArticleDOI
Oscillation and Chaos in Physiological Control Systems
Michael C. Mackey,Leon Glass +1 more
TL;DR: First-order nonlinear differential-delay equations describing physiological control systems displaying a broad diversity of dynamical behavior including limit cycle oscillations, with a variety of wave forms, and apparently aperiodic or "chaotic" solutions are studied.
Journal ArticleDOI
Quantitative universality for a class of nonlinear transformations
TL;DR: In this article, a large class of recursion relations xn+l = Af(xn) exhibiting infinite bifurcation is shown to possess a rich quantitative structure essentially independent of the recursion function.
Journal ArticleDOI
Deep learning and process understanding for data-driven Earth system science
Markus Reichstein,Gustau Camps-Valls,Bjorn Stevens,Martin Jung,Joachim Denzler,Nuno Carvalhais,Nuno Carvalhais,Prabhat +7 more
TL;DR: It is argued that contextual cues should be used as part of deep learning to gain further process understanding of Earth system science problems, improving the predictive ability of seasonal forecasting and modelling of long-range spatial connections across multiple timescales.
Book
The geometry of fractal sets
TL;DR: In this paper, a rigorous mathematical treatment of the geometrical aspects of sets of both integral and fractional Hausdorff dimension is presented, including questions of local density and the existence of tangents of such sets, and the dimensional properties of their projections in various directions.
References
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Journal ArticleDOI
Deterministic nonperiodic flow
TL;DR: In this paper, it was shown that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states, and systems with bounded solutions are shown to possess bounded numerical solutions.
Book
Foundations of Economic Analysis
TL;DR: Recent statistical techniques, including nonlinear programming, have been added to a basic survey of equilibrium systems, comparative statistics, consumer behavior theory, and cost and production theory as discussed by the authors, and they have been used in a variety of applications.
Book ChapterDOI
Period Three Implies Chaos
TL;DR: In this article, a generalized logistic equation was used to model the distribution of points of impact on a spinning bit for oil well drilling, as mentioned if this distribution is helpful in predicting uneven wear of the bit.
Journal ArticleDOI
Stock and Recruitment
TL;DR: Plotting net reproduction (reproductive potential of the adults obtained) against the density of stock which produced them, for a number of fish and invertebrate populations, gives a domed curve whose apex lies above the line representing replacement reproduction.
Book
Differential Equations, Dynamical Systems, and Linear Algebra
Morris W. Hirsch,Steve Smale +1 more
TL;DR: In this article, the structure theory of linear operators on finite-dimensional vector spaces has been studied and a self-contained treatment of that subject is given, along with a discussion of the relations between dynamical systems and certain fields outside pure mathematics.