Chapter 1 Introduction
1.1 Examples of deterministic dynamical systems
1.2 Examples of stochastic dynamical systems
1.3 Mathematical modeling with stochastic differential equations
1.4 Outline of this book
1.5 Problems
Chapter 2 Background in Analysis and Probability
2.1 Euclidean space
2.2 Hilbert, Banach and metric spaces
2.3 Taylor expansions
2.4 Improper integrals and Cauchy principal values
2.5 Some useful inequalities
2.5.1 Young''s inequality
2.5.2 Cronwall inequality
2.5.3 Cauchy-Schwaxz inequality
2.5.4 HSlder inequality
2.5.5 Minkowski inequality
2.6 HSlder spaces, Sobolev spaces and related inequalities
2.7 Probability spaces
2.7.1 Scalar random variables
2.7.2 Random vectors
2.7.3 Gaussian random variables
2.7.4 Non-Gaussian random variables
2.8 Stochastic processes
2.9 Coovergence concepts
2.10 Simulation
2.11 Problems
Chapter 3 Noise
3.1 Brownian motion
3.1.1 Brownian motion in R1
3.1.2 Brownian motion in Rn~
3.2 What is Gaussian white noise
3.3* A mathematical model for Gaussian white noise
3.3.1 Generalized derivatives
3.3.2 Gaussian white noise
3.4 Simulation
3.5 Problems
Chapter 4 A Crash Course in Stochastic Differential Equations
4.1 Differential equations with noise
4.2 Riemann-Stieltjes integration
4.3 Stochastic integration and stochastic differential equations
4.3.1 Motivation
4.3.2 Definition of It5 integral
4.3.3 Practical calculations
4.3.4 Stratonovich integral
4.3.5 Examples
4.3.6 Properties of It6 integrals
4.3.7 Stochastic differential equations
4.3.8 SDEs in engineering and science literature
4.3.9 SDEs with two-sided Brownian motions
4.4 It6''s formula
4.4.1 Motivation for stochasticChain rules
4.4.2 ItS''s formula in scalar case
4.4.3 It6''s formula in vector case
4.4.4 Stochastic product rule and integration by parts
4.5 Linear stochastic differential equations
4.6 Nonlinear stochastic differential equations
4.6.1 Existence, uniqueness and smoothness
4.6.2 Probability measure px and expectation Ex associated with an SDE
4.7 Conversion between It5 and Stratonovich stochastic differential
equations
4.7.1 Scalar SDEs
4.7.2 SDE systems
4.8 Impact of noise on dynamics
4.9 Simulation
4.10 Problems
Chapter 5 Deterministic Quantities for Stochastic Dynamics
5.1 Moments
5.2 Probability density functions
5.2.1 Scalar Fokker-Planck equations
5.2.2 Multidimensional Fokker-Planck equations
5.2.3 Existence and uniqueness for Fokker-Planck equations
5.2.4 Likelihood for transitions between different dynamical regimes under
uncertainty
5.3 Most probable phase portraits
5.3.1 Mean phase portraits
5.3.2 Almost sure phase portraits
5.3.3 Most probable phase portraits
5.4 Mean exit time
5.5 Escape probability
5.6 Problems
Chapter 6 Invariant Structures for Stochastic Dynamics
6.1 Deterministic dynamical systems
6.1.1 Concepts for deterministic dynamical systems
6.1.2 The Haxtman-Grobman theorem
6.1.3 Invariant sets
6.1.4 Differentiable manifolds
6.1.5 Deterministic invariant manifolds
6.2 Measurable dynamical systems
6.3 Random dynamical systems
6.3.1 Canonical sample spaces for SDEs
6.3.2 Wiener shift
6.3.3 Cocycles and random dynamical systems
6.3.4 Examples of cocycles
6.3.5 Structural stability and stationary orbits
6.4 Linear stochastic dynamics
6.4.1 Oseledets'' multiplicative ergodic theorem and Lyapunov exponents"
6.4.2 A stochastic Hartman-Grobman theorem
6.5* Random invariant manifolds
6.5.1 Definition of random invariant manifolds
6.5.2 Converting SDEs to RDEs
6.5.3 Local random pseudo-stable and pseudo-unstable manifolds
6.5.4 Local random stable, unstable and center manifolds
6.6 Problems
Chapter 7 Dynamical Systems Driven by Non-Gaussian Levy
Motions
7.1 Modeling via stochastic differential equations with Levy motions
7.2 Levy motions
7.2.1 Functions that have one-side limits
7.2.2 Levy-Ito decomposition
7.2.3 Levy-Khintchine formula
7.2.4 Basic properties of Levy motions
7.3 s-stable Levy motions
7.3.1 Stable random variables
7.3.2 a-stable Levy motions in R1
7.3.3 a-stable Levy motion in Rn
7.4 Stochastic differential equations with Levy motions
7.4.1 Stochastic integration with respect to Levy motions
7.4.2 SDEs with Levy motions
7.4.3 Generators for SDEs with Levy motion
7.5 Mean exit time
7.5.1 Mean exit time for a-stable Levy motion
7.5.2 Mean exit time for SDEs with a-stable Levy motion
7.6 Escape probability and transition phenomena
7.6.1 Balayage-Dirichlet problem for escape probability
7.6.2 Escape probability for a-stable Levy motion
7.6.3 Escape probability for SDEs with a-stable Levy motion
7.7 Fokker-Planck equations
7.7.1 Fokker-Planck equations in R1
7.7.2 Fokker-Planck equations in Rn
7.8 Problems
Hints and Solutions
Further Readings
References
Index
Color Pictures
內容試閱：
Chapter 1
Introduction
Noisy fluctuations are abundant in complex systems。 In some cases， noise is not negligible， whereas in some other situations， noise could even be beneficial。 It is desirable to have a better understanding of the impact of noise on dynamical evo?lution of complex systems。 In other words， it becomes crucial to take randomness into account in mathematical modeling of complex phenomena under uncertainty。
In 1908， Langevin devised a stochastic differential equation for the motion of Brownian particles in a fluid， under random impacts of surrounding fluid molecules。 This stochastic differential equation， although important for understanding Brownian motion， went largely unnoticed in the mathematical community until after stochastic calculus emerged in the late 1940s。 Introductory books on stochastic differential equations SDEs include [8，88，213]。
The goal for this book is to examine and present select dynamical systems concepts， tools， and methods for understanding solutions of SDEs。 To this end， we also need basic information about deterministic dynamical systems modeled by ordinary differential equations ODEs， as presented in the first couple of chapters in one of the references [110，290]。
In this introductory chapter， we present a few examples of deterministic and stochastic dynamical systems， then briefly outline the contents of this book。
1。1 Examples of deterministic dynamical systems
We recall a few examples of deterministic dynamical systems， where short time-scale forcing and nonlinearity can affect dynamics in a profound way。
Example 1。1 A double-well system。
Consider a one-dimensional dynamical system x = x - x3。 It has three equilib?rium states， -1，0 and 1，at which the vector field x - x3 is zero。 Observe that
Note that x = x - x3 = -4，where the potential function Vx = -gx2 + ^x4 has two minimal values sometimes called “wells”），see Figure 1。1。
Figure 1。1 Plot of
A solution curve， or orbit， or trajectory， starting with x0 = xo in -1，0， decreases in time because 士 0 on this interval and approaches the equilibrium state - 1 as t ^ +oo， whereas an orbit starting with x0 = xo in -oo， -1， increases in time because x 0 on this interval and approaches the equilibrium state - 1 as t +oo。 Thus the equilibrium point {-1} is a stable equilibrium state and it is an attractor， i。e。， it attracts nearby orbits。 Likewise {1} is also an attractor。 But the equilibrium state {0} is unstable and is called an repeller。 See Figure 1。2 for a few representative solutions curves。
An orbit starting near one equilibrium state {-1} can not go anywhere near the other equilibrium state {1}， and vice versa。 There is no transition between these two stable states。
If we only look at the solution curves in the state space， E1， wherestate xlives，
we get a state portrait， or as often called， a phase portrait。
Figure 1。3 shows the phase portrait for this double well system。
Figure 1。3 Phase portrait ior x = x - xz
Example 1。2 High-frequency or short time-scale forcing。
Consider a simple one-dimensional nonlinear system with time-periodic forcing with frequency u
Solution curves with frequency a； = 2 and a； = 10 are shown inFigures1。4 and 1。5，
respectively。 The difference between low and high frequency forcing is visible。
Figure 1。4 Solutions of a； = -x -\-x3 e sinct^，x0 = 0。5 with frequency cj = 2: e = 0 no “oscillations” or in blue color and e = 0。35 with “oscillations” or in red color。 See Figure 1。4 in the Color Pictures section at the end of this book
Example 1。3 Small nonlinearity leads to fundamental change in dynamics。
Consider a harmonic oscillator a spring-mass system of mass m and spring constant fc， under damping that is proportional to the cubic of velocity: mx = -kx - ?xs， where 6： is a positive constant。 For simplicity， we take m， k both equal to 1。 This can also be achieved by rescaling the time。 Thus，
Figure 1。5 Solutions of x = -x + x3 + esinojt，x0 = 0。5 with frequency u - 10: e = 0 no “oscillations” or in blue color and e = 0。35 with “oscillations” or in red color。 See Figure 1。5 in the Color Pictures section at the end of this book or equivalently，
where x is the displacement and y is the velocity of the oscillator。 The equilibrium state is 0，0。
Without damping e = 0，the model equations become
Dividing these two equations， we obtain
Integrating this equation， we see that the solution curves xt，yt satisfy the conservation of energy for an arbitrary non-negative constant of integration， c。 Thus， the solution curves are circles， see Figure 1。6。
In the case of damping， i。e。， when e 0， the energy is not conserved:
at all points except the equilibrium point 0，0。 Thus， all orbits approach the equilibrium point 0，0 as t oo， no matter how small the damping coefficient e is， as shown in Figure 1。7。 Comparing Figure 1。6 and Figure 1。7， we see that the dynamics， with or without damping， are drastically different。
Example 1。4 Simple pendulum。
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