Open AccessBook
Chaos: An Introduction to Dynamical Systems
TLDR
One-dimensional maps, two-dimensional map, fractals, and chaotic attraction attractors have been studied in this article for state reconstruction from data, including the state of Washington.Abstract:
One-Dimensional Maps.- Two-Dimensional Maps.- Chaos.- Fractals.- Chaos in Two-Dimensional Maps.- Chaotic Attractors.- Differential Equations.- Periodic Orbits and Limit Sets.- Chaos in Differential Equations.- Stable Manifolds and Crises.- Bifurcations.- Cascades.- State Reconstruction from Data.read more
Citations
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Bifurcations in one-dimensional piecewise smooth maps-theory and applications in switching circuits
TL;DR: In this paper, the authors present a systematic analysis of the bifurcation behavior of power electronic DC-DC converters through a normal form: the piecewise linear approximation in the neighborhood of the border.
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A mathematical framework for critical transitions: Bifurcations, fast–slow systems and stochastic dynamics
TL;DR: In this article, the authors provide an overview of early warning signs for critical transitions in dynamical systems and their application in a wide range of applications from ecosystems and climate change to medicine and finance.
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Nonlinear dynamical system identification from uncertain and indirect measurements
TL;DR: It is shown that the inclusion of information about the time-continuous nature of the underlying trajectories can improve parameter estimation considerably and two approaches, which take into account both the errors-in-variables problem and the problem of complex cost functions, are described in detail.
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Fractal structures in nonlinear dynamics
TL;DR: Fractal basins have become a fundamental ingredient of nonlinear dynamics and chaos theory and have been detected in nature and in most fields of science, with even a certain influence in the arts as mentioned in this paper.
References
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Journal ArticleDOI
Synchronization in chaotic systems
TL;DR: This chapter describes the linking of two chaotic systems with a common signal or signals and highlights that when the signs of the Lyapunov exponents for the subsystems are all negative the systems are synchronized.
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The Fractal Geometry of Nature
TL;DR: A blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) is presented in this article.
Book
Theory of Ordinary Differential Equations
TL;DR: The prerequisite for the study of this book is a knowledge of matrices and the essentials of functions of a complex variable as discussed by the authors, which is a useful text in the application of differential equations as well as for the pure mathematician.