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
Related: System Theory
Linear dynamical systemLinear 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 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 If the initial vector is aligned with a right eigenvector If ) of the matrix .
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
Measure-preserving dynamical systemIn 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
LagrangianThe 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 In classical mechanics, the natural form of the Lagrangian is defined as the kinetic energy, T, of the system minus its potential energy, V. 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. The Lagrangian formulation Simple example The trajectory of a thrown ball is characterized by the sum of the Lagrangian values at each time being a (local) minimum. Importance does not depend on . . .
CalculusHistory Modern calculus was developed in 17th century Europe by Isaac Newton and Gottfried Wilhelm Leibniz (see the Leibniz–Newton calculus controversy), but elements of it have appeared in ancient Greece, China, medieval Europe, India, and the Middle East. Ancient The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Medieval Modern In Europe, the foundational work was a treatise due to Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton. He is now regarded as an independent inventor of and contributor to calculus. Leibniz and Newton are usually both credited with the invention of calculus. Foundations
Interval exchange transformationGraph 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
Dynamical systems theoryDynamical 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 Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems. History
State-space representationIn 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: ). .