Predictability of Chaotic Dynamics (eBook)

A Finite-time Lyapunov Exponents Approach

Juan C. Vallejo, Miguel A. F. Sanjuan (Autoren)

eBook Download: PDF

2017 | 1st ed. 2017
XV, 136 Seiten
Springer International Publishing (Verlag)
978-3-319-51893-0 (ISBN)

Infos

This book is primarily concerned with the computational aspects of predictability of dynamical systems - in particular those where observation, modeling and computation are strongly interdependent. Unlike with physical systems under control in laboratories, for instance in celestial mechanics, one is confronted with the observation and modeling of systems without the possibility of altering the key parameters of the objects studied. Therefore, the numerical simulations offer an essential tool for analyzing these systems.

With the widespread use of computer simulations to solve complex dynamical systems, the reliability of the numerical calculations is of ever-increasing interest and importance. This reliability is directly related to the regularity and instability properties of the modeled flow. In this interdisciplinary scenario, the underlying physics provide the simulated models, nonlinear dynamics provides their chaoticity and instability properties, and the computer sciences provide the actual numerical implementation.

This book introduces and explores precisely this link between the models and their predictability characterization based on concepts derived from the field of nonlinear dynamics, with a focus on the finite-time Lyapunov exponents approach. The method is illustrated using a number of well-known continuous dynamical systems, including the Contopoulos, Hénon-Heiles and Rössler systems. To help students and newcomers quickly learn to apply these techniques, the appendix provides descriptions of the algorithms used throughout the text and details how to implement them in order to solve a given continuous dynamical system.

Miguel Sanjuan is full professor of physics at the Universidad Rey Juan Carlos in Madrid, Spain, where he founded the Physics Department in 2006. He is a corresponding member of the Spanish Royal Academy of Sciences, section physics and chemistry, and a foreign member of the Lithuanian Academy of Sciences in the areas of physics and mechanical engineering. Prof. Sanjuan is presently the head of the Nonlinear Dynamics, Chaos and Complex Systems Research Group at the Universidad Rey Juan Carlos.

He has published the monograph Nonlinear Resonances (Springer,2015). 

Juan C. Vallejo is an astrophysicist working as BepiColombo SGS Software Development Team Leader for the GMV enterprise group at the European Space Astronomy Centre in Madrid. At the same time he collaborates with the Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Fisica, Universidad Rey Juan Carlos.

 

Inhalt

Preface 8
Contents 11
Acronyms 13
1 Forecasting and Chaos 14
1.1 Historical Introduction 14
1.1.1 The Scientific Method 14
1.1.2 Forecasting and Determinism 16
1.2 Chaotic Dynamics 22
1.3 Computer Numerical Explorations 26
1.3.1 Solving ODEs Numerically 26
1.3.2 Numerical Forecast 27
1.3.3 Symplectic Integrators 30
1.4 Shadowing and Predictability 32
1.5 Concluding Remarks 35
References 35
2 Lyapunov Exponents 38
2.1 Lyapunov Exponents 38
2.2 The Lyapunov Spectrum 40
2.3 The Lyapunov Exponents Family 43
2.4 Finite-Time Exponents 46
2.5 Distributions of Finite-Time Exponents 47
2.6 The Harmonic Oscillator 49
2.7 The Rössler System 51
2.8 The Hénon–Heiles System 56
2.9 Concluding Remarks 67
References 69
3 Dynamical Regimes and Time Scales 73
3.1 Temporal Evolution 73
3.2 Regimes Identification 75
3.3 Transient Behaviours, Sticky Orbits and Transient Chaos 76
3.4 The Hénon-Heiles System 77
3.5 The Contopoulos System 83
3.6 The Rössler System 92
3.7 Hyperbolicity Characterisation Through Finite-Time Exponents 94
3.8 Concluding Remarks 98
References 99
4 Predictability 102
4.1 Numerical Predictability 102
4.2 The Predictability Index 104
4.2.1 The Hénon-Heiles System 107
4.2.2 The Contopoulos System 116
4.2.3 The Rössler System 117
4.2.4 A Galactic System 123
4.3 Concluding Remarks 134
References 137
Erratum to: Predictability of Chaotic Dynamics: A Finite-time Lyapunov Exponents Approach 139
A Numerical Calculation of Lyapunov Exponents 140
A.1 The Variational Equation 140
A.2 Selection of Initial Perturbations 143
A.3 Other Methods 145
A.4 Practical Implementation for Building the Finite-Time Distributions 146
References 147

Produkt-Details

• Erscheint lt. Verlag: 27.3.2017
• Reihe/Serie: Springer Series in Synergetics
• Reihe/Serie: Springer Series in Synergetics
• Zusatzinfo: XV, 136 p. 47 illus., 22 illus. in color.
• Verlagsort: Cham
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Astronomie / Astrophysik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
• Themenwelt: Technik
• Schlagworte: Finite Lyapunov exponents • Finite-time exponent distribution • Forecasting in chaotic systems • Henon-Heiles system • Simulation of galactic dynamics
• ISBN-10: 3-319-51893-3 / 3319518933
• ISBN-13: 978-3-319-51893-0 / 9783319518930