Decomposition of Dynamical Systems

 

arXiv:0903.4617

I’m interested in all kinds of decomposition of all kinds of systems.

I’m planning to write a paper on the decomposition of dynamical systems into fundamental states. This project is a bit vague. I want to make some ideas/results more precise and easier to apply before writing things up.

By a “dynamical system” I mean any dynamical model, which may be quite general: differential or difference equation, evolutionary system, sequential dynamical system, etc. In a dynamicla systems, there is a time variable (discrete or continuous), a configuration or phase space which may be finite or infinite-dimensional (spaces of fields. It can be stochastic, have hidden variables, have many initial values (whenn there are many possible initial values -> possibility to compare solutions, make approximations, etc.)

What kinds of decomposition do we have ?

* Spectral: “Pure” states are eigenvectors of an operator. “Mixed” states are linear combinations of pure states. Works well in linear approximations ?

* Phase space decomposition: an example is Conley’s decomposition. Conley’s theory can be extended to include non-compact and incomplete systems

* Time decomposition ?

* Wavelet decomposition ? Localized operators ? Is this technical or conceptual ?

* Stochastic decomposition ?

* Etc ?

Questions:

* What are the main “fundamental” states ? What is a “state” ?

* How do things go from one state to another one ?

* How do things in different states influence each other ?

* Relations with ergodic theory ? Local normal form theory ?

* Relations among different decompositions ?

* Measurements ?

* Multi-agents ?

References:

* Xiaopeng Chen, Jinqiao Duan, State space decomposition for nonautonomous dynamical systems, Proceedings of the Royal Society of Edinburgh: Section A Mathematics October 2011 141 : pp 957-974.

Decomposition of state spaces into dynamically different components is helpful for the understanding of dynamical behaviors of complex systems. A Conley type decomposition theorem is proved for nonautonomous dynamical systems defined on a non-compact but separable state space. Namely, the state space can be decomposed into a chain recurrent part and a gradient-like part. This result applies to both nonautonomous ordinary differential equations on Euclidean space (which is only locally compact), and nonautonomous partial differential equations on infinite dimensional function space (which is not even locally compact). This decomposition result is demonstrated by discussing a few concrete examples, such as the Lorenz system and the Navier-Stokes system, under time-dependent forcing.

* (Semimartingale decomposition: how to include this stuff ?)

Dynamic Markov bridges motivated by models of insider trading

Authors: Luciano Campi, Umut Çetin, Albina Danilova

Abstract: Given a Markovian Brownian martingale $Z$, we build a process $X$ which is a martingale in its own filtration and satisfies $X_1 = Z_1$. We call $X$ a dynamic bridge, because its terminal value $Z_1$ is not known in advance. We compute explicitly its semimartingale decomposition under both its own filtration $\cF^X$ and the filtration $\cF^{X,Z}$ jointly generated by $X$ and $Z$. Our construction is heavily based on parabolic PDE’s and filtering techniques. As an application, we explicitly solve an equilibrium model with insider trading, that can be viewed as a non-Gaussian generalization of Back and Pedersen’s \cite{BP}, where insider’s additional information evolves over time.

Journal reference: Stochastic processes and their applications, 2011, 121 (3). pp. 534-567

Cite as: arXiv:1202.2980v1 [math.PR]

* Rasmussen: Morse decomposition …

* Decomposition and simulation of sequential dyn systems

* Blokh: decomposition of systems on interval

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