Predictability of Chaotic Dynamics (eBook)

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.

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