Chaos theory - Mathematical theory

In mathematics and physics, chaos theory deals with the behavior of certain nonlinear dynamic systems that under certain conditions exhibit a phenomenon known as chaos, which is characterised by a sensitivity to initial conditions (see butterfly effect). As a result of this sensitivity, the behavior of systems that exhibit chaos appears to be random, even though the model of the system is determ ...

Including:

• Chaos theory - Description of the theory
• Chaos theory - Chaotic motion
• Chaos theory - Attractors
• Chaos theory - Strange attractors
• Chaos theory - History
• Chaos theory - Mathematical theory
• Chaos theory - Minimum complexity of a chaotic system
• Chaos theory - Other examples of chaotic systems

Read more here: » Chaos theory: Encyclopedia - Chaos theory

Mathematicians have devised many additional ways to make quantitative statements about chaotic systems. These include: fractal dimension of the attractor Lyapunov exponents recurrence plots Poincaré maps bifurcation diagrams Transfer operator Chaos theory - Minimum complexity of a chaotic system. Many simple systems can also produce chaos without relying on differential equations, such as the logistic map, which is a difference equation (recurrence r ...

See also:

Chaos theory, Chaos theory - Description of the theory, Chaos theory - Chaotic motion, Chaos theory - Attractors, Chaos theory - Strange attractors, Chaos theory - History, Chaos theory - Mathematical theory, Chaos theory - Minimum complexity of a chaotic system, Chaos theory - Other examples of chaotic systems, Chaos theory - Application

Read more here: » Chaos theory: Encyclopedia II - Chaos theory - Mathematical theory

Mathematicians have devised many additional ways to make quantitative statements about chaotic systems. These include: fractal dimension of the attractor Lyapunov exponents recurrence plots Poincaré maps bifurcation diagrams Transfer operator Chaos theory - Minimum complexity of a chaotic system. Many simple systems can also produce chaos without relying on differential equations, such as the logistic map, which is a difference equation (recurrence r ...

See also:

Chaos theory, Chaos theory - Description of the theory, Chaos theory - Chaotic motion, Chaos theory - Attractors, Chaos theory - Strange attractors, Chaos theory - History, Chaos theory - Mathematical theory, Chaos theory - Minimum complexity of a chaotic system, Chaos theory - Other examples of chaotic systems

Read more here: » Chaos theory: Encyclopedia II - Chaos theory - Mathematical theory

A non-linear dynamical system can exhibit one or more of the following types of behavior: forever at rest forever expanding (only for unbounded systems) periodic motion quasi-periodic motion chaotic motion The type of behavior a system may exhibit depends on the initial state of the system and the values of its parameters, if any. The most difficult type of behavior to characterize and predict is chaotic motion, a non-periodic complex motion, for which the theory is named.See also:

Chaos theory, Chaos theory - Description of the theory, Chaos theory - Chaotic motion, Chaos theory - Attractors, Chaos theory - Strange attractors, Chaos theory - History, Chaos theory - Mathematical theory, Chaos theory - Minimum complexity of a chaotic system, Chaos theory - Other examples of chaotic systems

Read more here: » Chaos theory: Encyclopedia II - Chaos theory - Description of the theory

Chaos theory is a collection of results, methods, and visualization techniques used to study dynamical systems. A mathematical model of a natural or human system where numerical quantities are used to represent the state of the system is an example of a dynamical system. Some of these numerical quantities vary with time—they are dynamical—and others remain fixed—they are parameters. In a dynamical system, the numerical quantities are combined in a formula to determine how the dynamical quantities change after a short time period. The c ...

See also:

Chaos theory, Chaos theory - Description of the theory, Chaos theory - Chaotic motion, Chaos theory - Attractors, Chaos theory - Strange attractors, Chaos theory - History, Chaos theory - Mathematical theory, Chaos theory - Minimum complexity of a chaotic system, Chaos theory - Other examples of chaotic systems, Chaos theory - Application

Read more here: » Chaos theory: Encyclopedia II - Chaos theory - Description of the theory

The roots of chaos theory date back to about 1900, in the studies of Henri Poincaré on the problem of the motion of three objects in mutual gravitational attraction, the so-called three-body problem. Poincaré found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point. Later studies, also on the topic of nonlinear differential equations, were carried out by G.D. Birkhoff, A.N. Kolmogorov, M.L. Cartwright, J.E. Littlewood, and Stephen Smale. Except for Smale, who was perhaps the first p ...

See also:

Chaos theory, Chaos theory - Description of the theory, Chaos theory - Chaotic motion, Chaos theory - Attractors, Chaos theory - Strange attractors, Chaos theory - History, Chaos theory - Mathematical theory, Chaos theory - Minimum complexity of a chaotic system, Chaos theory - Other examples of chaotic systems

Read more here: » Chaos theory: Encyclopedia II - Chaos theory - History

The roots of chaos theory date back to about 1900, in the studies of Henri Poincaré on the problem of the motion of three objects in mutual gravitational attraction, the so-called three-body problem. Poincaré found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point. Later studies, also on the topic of nonlinear differential equations, were carried out by G.D. Birkhoff, A.N. Kolmogorov, M.L. Cartwright, J.E. Littlewood, and Stephen Smale. Except for Smale, who was perhaps the first p ...

See also:

Chaos theory, Chaos theory - Description of the theory, Chaos theory - Chaotic motion, Chaos theory - Attractors, Chaos theory - Strange attractors, Chaos theory - History, Chaos theory - Mathematical theory, Chaos theory - Minimum complexity of a chaotic system, Chaos theory - Other examples of chaotic systems, Chaos theory - Application

Read more here: » Chaos theory: Encyclopedia II - Chaos theory - History