Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state. A metaphor for this behavior is that a butterfly flapping its wings in Texas can cause a tornado in Brazil.
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it.
The logistic map is a polynomial mapping of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple nonlinear dynamical equations. The map was popularized in a 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation written down by Pierre François Verhulst. Mathematically, the logistic map is written
In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically of a system as a function of a bifurcation parameter in the system. It is usual to represent stable values with a solid line and unstable values with a dotted line, although often the unstable points are omitted. Bifurcation diagrams enable the visualization of bifurcation theory.
In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation vector diverge at a rate given by
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed.
In mathematics, the Hénon map, sometimes called Hénon–Pomeau attractor/map, is a discrete-time dynamical system. It is one of the most studied examples of dynamical systems that exhibit chaotic behavior. The Hénon map takes a point (xn, yn) in the plane and maps it to a new point
James A. Yorke is a Distinguished University Research Professor of Mathematics and Physics and former chair of the Mathematics Department at the University of Maryland, College Park.
The Rössler attractor is the attractor for the Rössler system, a system of three non-linear ordinary differential equations originally studied by Otto Rössler in the 1970s. These differential equations define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor. Rössler interpreted it as a formalization of a taffy-pulling machine.
The Rabinovich–Fabrikant equations are a set of three coupled ordinary differential equations exhibiting chaotic behaviour for certain values of the parameters. They are named after Mikhail Rabinovich and Anatoly Fabrikant, who described them in 1979.
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations occur in both continuous systems and discrete systems.
In dynamical systems, intermittency is the irregular alternation of phases of apparently periodic and chaotic dynamics, or different forms of chaotic dynamics.
In lab experiments that study chaos theory, approaches designed to control chaos are based on certain observed system behaviors. Any chaotic attractor contains an infinite number of unstable, periodic orbits. Chaotic dynamics, then, consists of a motion where the system state moves in the neighborhood of one of these orbits for a while, then falls close to a different unstable, periodic orbit where it remains for a limited time and so forth. This results in a complicated and unpredictable wandering over longer periods of time.
Cybernetical physics is a scientific area on the border of cybernetics and physics which studies physical systems with cybernetical methods. Cybernetical methods are understood as methods developed within control theory, information theory, systems theory and related areas: control design, estimation, identification, optimization, pattern recognition, signal processing, image processing, etc. Physical systems are also understood in a broad sense; they may be either lifeless, living nature or of artificial (engineering) origin, and must have reasonably understood dynamics and models suitable for posing cybernetical problems. Research objectives in cybernetical physics are frequently formulated as analyses of a class of possible system state changes under external (controlling) actions of a certain class. An auxiliary goal is designing the controlling actions required to achieve a prespecified property change. Among typical control action classes are functions which are constant in time, functions which depend only on time, and functions whose value depends on measurement made at the same time or on previous instances. The last class is of special interest since these functions correspond to system analysis by means of external feedback.
Miguel Angel Fernández Sanjuán is a Spanish Theoretical Physicist from Leon, Spain. He is known for his contributions in nonlinear dynamics, chaos theory, and control of chaos, and has published several scientific papers and popular news articles. He has supervised around 20 PhD students in Nonlinear Dynamics, Chaos and Complex Systems.
In the bifurcation theory, a bounded oscillation that is born without loss of stability of stationary set is called a hidden oscillation. In nonlinear control theory, the birth of a hidden oscillation in a time-invariant control system with bounded states means crossing a boundary, in the domain of the parameters, where local stability of the stationary states implies global stability. If a hidden oscillation attracts all nearby oscillations, then it is called a hidden attractor. For a dynamical system with a unique equilibrium point that is globally attractive, the birth of a hidden attractor corresponds to a qualitative change in behaviour from monostability to bi-stability. In the general case, a dynamical system may turn out to be multistable and have coexisting local attractors in the phase space. While trivial attractors, i.e. stable equilibrium points, can be easily found analytically or numerically, the search of periodic and chaotic attractors can turn out to be a challenging problem.
Chaotic scattering is a branch of chaos theory dealing with scattering systems displaying a strong sensitivity to initial conditions. In a classical scattering system there will be one or more impact parameters, b, in which a particle is sent into the scatterer. This gives rise to one or more exit parameters, y, as the particle exits towards infinity. While the particle is traversing the system, there may also be a delay time, T—the time it takes for the particle to exit the system—in addition to the distance travelled, s. In certain systems the two will be equivalent—see below. In a chaotic scattering system, a minute change in the impact parameter, may give rise to a very large change in the exit parameters.
Celso Grebogi is a Brazilian theoretical physicist who works in the area of chaos theory. He is one among the pioneers in the nonlinear and complex systems and chaos theory. Currently he works at the University of Aberdeen as the "Sixth Century Chair in Nonlinear and Complex Systems". He has done extensive research in the field of plasma physics before his work on the theory of dynamical systems. He and his colleagues have shown with a numerical example that one can convert a chaotic attractor to any one of numerous possible attracting time-periodic motions by making only small time-dependent perturbations of an available system parameter. This article is considered as one among the classic works in the control theory of chaos and their control method is known as the OGY method. He was listed in the 2016 Thomson Reuters Citation Laureates.
Edward Ott is an American physicist most noted for his contributions to the development of chaos theory.
In mathematics, a strange nonchaotic attractor (SNA) is a form of attractor which, while converging to a limit, is strange, because it is not piecewise differentiable, and also non-chaotic, in that its Lyapunov exponents are non-positive. SNAs were introduced as a topic of study by Grebogi et al. in 1984. SNAs can be distinguished from periodic, quasiperiodic and chaotic attractors using the 0-1 test for chaos.
