Dynamical system

The Lorenz attractor arises in the study of the Lorenz Oscillator, a dynamical system. Overview[edit] 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[edit] Many people regard Henri Poincaré as the founder of dynamical systems.[3] Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). Basic definitions[edit] Flows[edit]

Linear dynamical system Linear dynamical systems are dynamical systems whose evaluation functions are linear. While dynamical systems in general do not have closed-form solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties. Linear systems can also be used to understand the qualitative behavior of general dynamical systems, by calculating the equilibrium points of the system and approximating it as a linear system around each such point. Introduction[edit] In a linear dynamical system, the variation of a state vector (an -dimensional vector denoted ) equals a constant matrix (denoted ) multiplied by varies continuously with time or as a mapping, in which varies in discrete steps These equations are linear in the following sense: if and are two valid solutions, then so is any linear combination of the two solutions, e.g where need not be symmetric. Solution of linear dynamical systems[edit] If the initial vector is aligned with a right eigenvector If ) of the matrix .

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[edit] 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[edit] Examples include: Homomorphisms[edit] 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

Lagrangian The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. The Lagrangian is named after Italian-French mathematician and astronomer Joseph Louis Lagrange. The concept of a Lagrangian was introduced in a reformulation of classical mechanics introduced by Lagrange known as Lagrangian mechanics. Definition[edit] In classical mechanics, the natural form of the Lagrangian is defined as the kinetic energy, T, of the system minus its potential energy, V.[1] In symbols, If the Lagrangian of a system is known, then the equations of motion of the system may be obtained by a direct substitution of the expression for the Lagrangian into the Euler–Lagrange equation. , but solving any equivalent Lagrangians will give the same equations of motion.[2][3] The Lagrangian formulation[edit] Simple example[edit] The trajectory of a thrown ball is characterized by the sum of the Lagrangian values at each time being a (local) minimum. Importance[edit] does not depend on . . .

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[1] 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[edit] 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[edit] 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."[1](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[edit] Types[edit] Positive and negative feedback[edit] Biology[edit]

Dynamical systems theory Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. When difference equations are employed, the theory is called discrete dynamical systems. This theory deals with the long-term qualitative behavior of dynamical systems, and studies the solutions of the equations of motion of systems that are primarily mechanical in nature; although this includes both planetary orbits as well as the behaviour of electronic circuits and the solutions to partial differential equations that arise in biology. This field of study is also called just Dynamical systems, Mathematical Dynamical Systems Theory and Mathematical theory of dynamical systems. Overview[edit] Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems. History[edit]

Random dynamical system evolving according to a succession of maps randomly chosen according to the distribution Q.[1] Motivation: solutions to a stochastic differential equation[edit] 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[edit] 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