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  2. Lyapunov stability - Wikipedia

    en.wikipedia.org/wiki/Lyapunov_stability

    Lyapunov was a pioneer in successful endeavors to develop a global approach to the analysis of the stability of nonlinear dynamical systems by comparison with the widely spread local method of linearizing them about points of equilibrium.

  3. 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]

  4. Stability theory - Wikipedia

    en.wikipedia.org/wiki/Stability_theory

    The simplest kind of an orbit is a fixed point, or an equilibrium. If a mechanical system is in a stable equilibrium state then a small push will result in a localized motion, for example, small oscillations as in the case of a pendulum. In a system with damping, a stable equilibrium state is moreover asymptotically stable. On the other hand ...

  5. Linear dynamical system - Wikipedia

    en.wikipedia.org/wiki/Linear_dynamical_system

    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 ...

  6. Conley's fundamental theorem of dynamical systems - Wikipedia

    en.wikipedia.org/wiki/Conley's_fundamental...

    Conley's decomposition is characterized by a function known as complete Lyapunov function. Unlike traditional Lyapunov functions that are used to assert the stability of an equilibrium point (or a fixed point) and can be defined only on the basin of attraction of the corresponding attractor, complete Lyapunov functions must be defined on the whole phase-portrait.

  7. Dynamical systems theory - Wikipedia

    en.wikipedia.org/wiki/Dynamical_systems_theory

    Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems.Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does ...

  8. Saddle-node bifurcation - Wikipedia

    en.wikipedia.org/wiki/Saddle-node_bifurcation

    In discrete dynamical systems, the same bifurcation is often instead called a fold bifurcation. Another name is blue sky bifurcation in reference to the sudden creation of two fixed points. [1] If the phase space is one-dimensional, one of the equilibrium points is unstable (the saddle), while the other is stable (the node).

  9. 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 ...