## ::Dynamical system

### ::concepts

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In mathematics, a **dynamical system** is a set of relationships among two or more measurable quantities, in which a fixed rule describes how the quantities evolve over time in response to their own values. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

At any given time a dynamical system has a state given by a set of real numbers (a vector) that can be represented by a point in an appropriate state space (a geometrical manifold). The *evolution rule* of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic; in other words, for a given time interval only one future state follows from the current state;<ref>Strogatz, S. H. (2001). Nonlinear dynamics and chaos: with applications to physics, biology and chemistry. Perseus publishing.</ref><ref>Katok, A., & Hasselblatt, B. (1995). Introduction to the modern theory of dynamical systems. Cambridge, Cambridge.</ref> however, some systems are stochastic, in that random events also affect the evolution of the state variables.

**Dynamical system sections**

Intro Overview History Basic definitions Linear dynamical systems Local dynamics Bifurcation theory Ergodic systems Examples of dynamical systems Multidimensional generalization See also References Further reading External links

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Systems::system Space::point Phase::''x'' Theory::systems Title::author ''t''::theorem

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In mathematics, a **dynamical system** is a set of relationships among two or more measurable quantities, in which a fixed rule describes how the quantities evolve over time in response to their own values. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

At any given time a dynamical system has a state given by a set of real numbers (a vector) that can be represented by a point in an appropriate state space (a geometrical manifold). The *evolution rule* of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic; in other words, for a given time interval only one future state follows from the current state;<ref>Strogatz, S. H. (2001). Nonlinear dynamics and chaos: with applications to physics, biology and chemistry. Perseus publishing.</ref><ref>Katok, A., & Hasselblatt, B. (1995). Introduction to the modern theory of dynamical systems. Cambridge, Cambridge.</ref> however, some systems are stochastic, in that random events also affect the evolution of the state variables.

**Dynamical system sections**

Intro Overview History Basic definitions Linear dynamical systems Local dynamics Bifurcation theory Ergodic systems Examples of dynamical systems Multidimensional generalization See also References Further reading External links

PREVIOUS: Intro | NEXT: Overview |

<< | >> |