 # Dynamical system The Lorenz attractor arises in the study of the Lorenz Oscillator, a dynamical system. Overview Before the advent of computers, finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system. For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. History Many people regard Henri Poincaré as the founder of dynamical systems. Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). Basic definitions Flows

Measure-preserving dynamical system In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Definition A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation on it. In more detail, it is a system with the following structure: is a set, is a σ-algebra over , is a probability measure, so that μ(X) = 1, and μ(∅) = 0, is a measurable transformation which preserves the measure , i.e., . , the identity function on X;, whenever all the terms are well-defined;, whenever all the terms are well-defined. The earlier, simpler case fits into this framework by definingTs = Ts for s ∈ N. The existence of invariant measures for certain maps and Markov processes is established by the Krylov–Bogolyubov theorem. Examples Examples include: Homomorphisms The concept of a homomorphism and an isomorphism may be defined. Consider two dynamical systems and . The system is defined as

THE GENERAL SYSTEM? REVIEWED by Charles Francois, Editor, International Encyclopedia of Cybernetics and Systemics By Thomas Mandel "A human being is part of the Whole...He experiences himself, his thoughts and feelings, as something separated from the rest...a kind of optical delusion of his consciousness. "My friend, all theory is gray, and the Golden tree of life is green." It is time we, especially we in the systems movement, stop fighting amongst ourselves. "In contrast to the mechanistic Cartesian view of the world, the world-view emerging from modern physics can be characterized by words like organic, holistic, and ecological. The subject of a General Principle, a.k.a. But some others go on to create an entire new general system of their own, which they attain by particularizing the general definition somewhat, in effect creating a "sister" GST. The former derivative process results in the same thing being said but in different ways. The idea of a General System is not necessarily new. By Tom Mandel

Interval exchange transformation Graph of interval exchange transformation (in black) with and . In blue, the orbit generated starting from In mathematics, an interval exchange transformation is a kind of dynamical system that generalises circle rotation. The phase space consists of the unit interval, and the transformation acts by cutting the interval into several subintervals, and then permuting these subintervals. Formal definition Let and let be a permutation on of positive real numbers (the widths of the subintervals), satisfying Define a map called the interval exchange transformation associated to the pair as follows. let Then for , define if lies in the subinterval . acts on each subinterval of the form is moved to position Properties Any interval exchange transformation is a bijection of to itself preserves the Lebesgue measure. The inverse of the interval exchange transformation is again an interval exchange transformation. where for all If is just a circle rotation. is irrational, then is uniquely ergodic. such that

Feedback "...'feedback' exists between two parts when each affects the other."(p53, §4/11) A feedback loop where all outputs of a process are available as causal inputs to that process "Simple causal reasoning about a feedback system is difficult because the first system influences the second and second system influences the first, leading to a circular argument. In this context, the term "feedback" has also been used as an abbreviation for: Feedback signal – the conveyance of information fed back from an output, or measurement, to an input, or effector, that affects the system.Feedback loop – the closed path made up of the system itself and the path that transmits the feedback about the system from its origin (for example, a sensor) to its destination (for example, an actuator).Negative feedback – the case where the fed-back information acts to control or regulate a system by opposing changes in the output or measurement. History Types Positive and negative feedback Biology

Random dynamical system evolving according to a succession of maps randomly chosen according to the distribution Q. Motivation: solutions to a stochastic differential equation Let be a -dimensional vector field, and let . to the stochastic differential equation exists for all positive time and some (small) interval of negative time dependent upon , where denotes a -dimensional Wiener process (Brownian motion). In this context, the Wiener process is the coordinate process. Now define a flow map or (solution operator) by (whenever the right hand side is well-defined). (or, more precisely, the pair ) is a (local, left-sided) random dynamical system. Formal definition Formally, a random dynamical system consists of a base flow, the "noise", and a cocycle dynamical system on the "physical" phase space. be a probability space, the noise space. as follows: for each "time" , let be a measure-preserving measurable function: for all and Suppose also that That is, . ; in these cases, the maps is ergodic. Now let , the base flow

State-space representation In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a vector within that space. To abstract from the number of inputs, outputs and states, these variables are expressed as vectors. inputs and outputs, we would otherwise have to write down Laplace transforms to encode all the information about a system. State variables The internal state variables are the smallest possible subset of system variables that can represent the entire state of the system at any given time. The minimum number of state variables required to represent a given system, , is usually equal to the order of the system's defining differential equation. Linear systems Block diagram representation of the linear state-space equations inputs, outputs and where: ). .

Base flow (random dynamical systems) In mathematics, the base flow of a random dynamical system is the dynamical system defined on the "noise" probability space that describes how to "fast forward" or "rewind" the noise when one wishes to change the time at which one "starts" the random dynamical system. In the definition of a random dynamical system, one is given a family of maps on a probability space . The measure-preserving dynamical system is known as the base flow of the random dynamical system. are often known as shift maps since they "shift" time. The parameter may be chosen to run over (a two-sided continuous-time dynamical system); (a one-sided continuous-time dynamical system); (a two-sided discrete-time dynamical system); (a one-sided discrete-time dynamical system). Each map is required to be a -measurable function: for all , to preserve the measure : for all , . Furthermore, as a family, the maps satisfy the relations In other words, the maps form a commutative monoid (in the cases and , where would be given by

Laplace transform History The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform (now called z transform) in his work on probability theory. The current widespread use of the transform came about soon after World War II although it had been used in the 19th century by Abel, Lerch, Heaviside, and Bromwich. From 1744, Leonhard Euler investigated integrals of the form as solutions of differential equations but did not pursue the matter very far. Joseph Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form which some modern historians have interpreted within modern Laplace transform theory.[clarification needed] akin to a Mellin transform, to transform the whole of a difference equation, in order to look for solutions of the transformed equation. Formal definition The parameter s is the complex number frequency: with real numbers and ω. instead of F.

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