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  2. Dynamical system - Wikipedia

    en.wikipedia.org/wiki/Dynamical_system

    A discrete dynamical system, discrete-time dynamical system is a tuple (T, M, Φ), where M is a manifold locally diffeomorphic to a Banach space, and Φ is a function. When T is taken to be the integers, it is a cascade or a map. If T is restricted to the non-negative integers we call the system a semi-cascade. [14]

  3. Dynamical systems theory - Wikipedia

    en.wikipedia.org/wiki/Dynamical_systems_theory

    From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems.

  4. Symbolic dynamics - Wikipedia

    en.wikipedia.org/wiki/Symbolic_dynamics

    In mathematics, symbolic dynamics is the study of dynamical systems defined on a discrete space consisting of infinite sequences of abstract symbols. The evolution of the dynamical system is defined as a simple shift of the sequence. Because of their explicit, discrete nature, such systems are often relatively easy to characterize and understand.

  5. Logistic map - Wikipedia

    en.wikipedia.org/wiki/Logistic_map

    The concept of fixed points is of primary importance in discrete dynamical systems. Another graphical technique that can be used for one-variable mappings is the spider web projection. After determining an initial value x 0 {\displaystyle x_{0}} on the horizontal axis, draw a vertical line from the initial value x 0 {\displaystyle x_{0}} to the ...

  6. Hénon map - Wikipedia

    en.wikipedia.org/wiki/Hénon_map

    In mathematics, the Hénon map, sometimes called Hénon–Pomeau attractor/map, [1] 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 (x n, y n) in the plane and maps it to a new point

  7. Sharkovskii's theorem - Wikipedia

    en.wikipedia.org/wiki/Sharkovskii's_theorem

    Sharkovskii's theorem does not immediately apply to dynamical systems on other topological spaces. It is easy to find a circle map with periodic points of period 3 only: take a rotation by 120 degrees, for example. But some generalizations are possible, typically involving the mapping class group of the space minus a periodic orbit.

  8. Saddle-node bifurcation - Wikipedia

    en.wikipedia.org/wiki/Saddle-node_bifurcation

    In the mathematical area of bifurcation theory a saddle-node bifurcation, tangential bifurcation or fold bifurcation is a local bifurcation in which two fixed points (or equilibria) of a dynamical system collide and annihilate each other. The term 'saddle-node bifurcation' is most often used in reference to continuous dynamical systems.

  9. Arnold's cat map - Wikipedia

    en.wikipedia.org/wiki/Arnold's_cat_map

    The discrete cat map describes the phase space flow corresponding to the discrete dynamics of a bead hopping from site q t (0 ≤ q t < N) to site q t+1 on a circular ring with circumference N, according to the second order equation: + + =