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Nonlinear Dynamics

posted Dec 6, 2009, 3:57 AM by Olaf Bochmann   [ updated Mar 5, 2010, 8:01 AM ]
The field of nonlinear Dynamics is concerned with analysing Systems that can not be described by linear equations. Nonlinear equations - with the exception of soliton equations - do not follow the superposition principle. In general, they show much more complex behavior. 
The systematic development of the theory of nonlinear dynamics is proposed as follows:
  1. first order differential equation and their bifurcations
  2. phase plane analysis
  3. limit cycles and their bifurcations and Lorenz equations
  4. chaos
  5. iterated maps
  6. period doubling
  7. renormalization
  8. fractals
  9. strange attractors
Dynamics is the subject that deals with systems that change in time, e.g. it settles down to equilibrium, it repeats in cycles, and so on. This behavior can be found in mechanics, chemical kinetics, population biology, and others.
There are two main types of dynamical systems: differential equations and difference equations (iterates maps). Differential equations describe the evolution of a system in continuous time, whereas difference equations describe the evolution in discrete time. For differential equations, one can distinct between ordinary differential equations and partial differential equations. For instance, the equation for a damped harmonic oscillator or the swinging pendulum is an ordinary differential equation, because there is only one independent variable - time. The heat equation is an example of an partial differential equation. It has both, time and space as independent variables. 
In linear systems, as for the damped harmonic oscillator, the right hand side of the general system of first order differential equation is a linear function of the states. These systems can be broken down into parts and the parts can be solveld separately. A linear system is precisely equal to the sum of its parts. This idea enables such methods as normal modes, Laplace transforms, superposition arguments, and Fourier analysis. However, most of everyday live e.g. the swinging pendulum is nonlinear (trigonometric function) and the above mentioned methods fail. 


Maps are discrete time systems - time proceeds in clicks (maps). The modelling tool is difference equation.
In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos. It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied. 
Iterates of the logistic map present periodic cycles. A period tree in a map implies regular cycles of every other length as well as completely chaotic cycles. You can see different periodic cycles on the diagram of bifurcation by observing the number of points for different values of the parameter . The complete diagram of bifurcation can also be seen.


Flows are continuous time systems - time procedes smoothly (flows). The modelling tool is differential equation.
"Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space. For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.
A simple system representing cellular convection is solved numerically. All of the solutions are found to be unstable, and almost all of them are nonperiodic.
The feasibility of very-long-range weather prediction is examined in the light of these results." Edward N. Lorenz [32]

Methods of Solving


Application and Examples

The idea of the Santa Fe time series competition [53] was to put out a bunch of data sets and invites the participants to predict their future. The data sets included laboratory laser, physiological data (sleep apnea), currency rate exchange, RK4 on some chaotic ODE, intensity of some star, and a Bach fugue. Other questions arising from this was what kind of systems produces the data and how much can we learn about the system. Quantitative answers permit direct comparisons.
In his competition entry T. Sauer [45] used a careful implementation of local-linear fitting that had five steps:
Low-pass embed the data to help reduce measurement and quantification noise (smoothing).
Generate more points in embedding space by Fourier interpolation between the points obtained in step 1 (increase the coverage in embedding space).
Find the k-nearest neighbors to the point of prediction (tradeoff between increasing bias and decreasing variance that comes from larger neighborhood).
Use a local SVD to project points onto the local surface. 
Regress a linear model for the neighborhood and use it to generate the forecast.Filtering is another application. Don't use a linear filter for chaotic systems. For nonlinear filter use the stable and the unstable manifold structure on a chaotic attractor. 
If you have a model of the system, you can simulate what happens to each point in forward and backward time. If your system has transverse stable and unstable manifolds, that does useful things to the noise balls. Since all three versions of that data should be identical at the middle time, can average them. Works best if manifolds are perpendicular, butrequires only transversality.
FRACTAL Villages by R. Eglash "I am a mathematician, and I would like to stand on your roof." That is how Ron Eglash greeted many African families he met while researching the fractal patterns hed noticed in villages across the continent. 

Summary of Topics

(CSSS'09 by Liz Bradley)

Introduction; Dynamics of Maps chs 1 & 10 of [50]
a brief tour of nonlinear dynamics [32] (in [17])

an extended example: the logistic map

how to plot its behavior

initial conditions, transients, and fixed points

bifurcations and attractors

chaos: sensitive dependence on initial conditions, $\lambda$, and all that

pitchforks, Feigenbaum, and universality [22] (in [17])

the connection between chaos and fractals [23], ch 11 of [50]

period-3, chaos, and the u-sequence [31, 34] (latter is in [17])

maybe: unstable periodic orbits [2, 25, 49]
Dynamics of Flows [50], sections 2.0-2.3, 2.8, 5, and 6 (except 6.6 and 6.8)
maps vs. flows

time: discrete vs. continuous

axes: state/phase space [9]

an example: the simple harmonic oscillator
some math & physics review [8]

portraying & visualizing the dynamics [9]

trajectories, attractors, basins, and boundaries [9]
dissipation and attractors [42]
how sensitive dependence and the Lyapunov exponent manifest in flows

anatomy of a chaotic attractor: [23]

stretching/folding and the un/stable manifolds

fractal structure and the fractal dimension ch 11 of [50]

unstable periodic orbits [2, 25, 49]

symbol dynamics [26] (in [13]); [28]

ODE solvers and their dynamics [8, 33, 35, 44]

PDE solvers [8, 44]

Poincar´e sections [27]

stability, eigenstuff, un/stable manifolds and a bit of control theory

embedology [29, 30, 39, 46, 47, 45, 52] ([39] is in [37] and [45] is in [53];)

calculating Lyapunov exponents and fractal dimensions [1, 9, 37, 40]

prediction [3, 4, 5, 14, 15, 53]

filtering [20, 21, 24]

control [7, 6, 11, 36, 48] ([36] is in [37])

communication [16, 41]

classical mechanics [10, 43, 51, 54, 55]

music, dance, and image [12, 18, 19]


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