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 .
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 . . .
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]
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[edit] 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.[3] 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[edit] Block diagram representation of the linear state-space equations inputs, outputs and where: ). .
Limit-cycle Stable limit cycle (shown in bold) and two other trajectories spiraling into it In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of a great many real world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854-1912). Definition[edit] We consider a two-dimensional dynamical system of the form where is a smooth function. with values in which satisfies this differential equation. such that for all . Properties[edit] By the Jordan curve theorem, every closed trajectory divides the plane into two regions, the interior and the exterior of the curve. Stable, unstable and semi-stable limit cycles[edit] Finding limit cycles[edit] .
Laplace transform History[edit] 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.[2] 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.[3][4][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[edit] The parameter s is the complex number frequency: with real numbers and ω. instead of F.