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
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 .
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 . . .
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
Random dynamical systemevolving 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
Limit-cycleStable 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 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 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 Finding limit cycles .
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