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  2. Lyapunov exponent - Wikipedia

    en.wikipedia.org/wiki/Lyapunov_exponent

    Lyapunov proved that if the system of the first approximation is regular (e.g., all systems with constant and periodic coefficients are regular) and its largest Lyapunov exponent is negative, then the solution of the original system is asymptotically Lyapunov stable. Later, it was stated by O. Perron that the requirement of regularity of the ...

  3. Logistic map - Wikipedia

    en.wikipedia.org/wiki/Logistic_map

    When the parameter r = 4, the behavior becomes chaotic over the entire range [0, 1]. At this time, the Lyapunov exponent λ is maximized, and the state is the most chaotic. The value of λ for the logistic map at r = 4 can be calculated precisely, and its value is λ = log 2.

  4. Floquet theory - Wikipedia

    en.wikipedia.org/wiki/Floquet_theory

    The real parts of the Floquet exponents are called Lyapunov exponents. The zero solution is asymptotically stable if all Lyapunov exponents are negative, Lyapunov stable if the Lyapunov exponents are nonpositive and unstable otherwise. Floquet theory is very important for the study of dynamical systems, such as the Mathieu equation.

  5. Lyapunov stability - Wikipedia

    en.wikipedia.org/wiki/Lyapunov_stability

    The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations.

  6. Exponential stability - Wikipedia

    en.wikipedia.org/wiki/Exponential_stability

    In control theory, a continuous linear time-invariant system (LTI) is exponentially stable if and only if the system has eigenvalues (i.e., the poles of input-to-output systems) with strictly negative real parts (i.e., in the left half of the complex plane). [1]

  7. Lyapunov function - Wikipedia

    en.wikipedia.org/wiki/Lyapunov_function

    A Lyapunov function for an autonomous dynamical system {: ˙ = ()with an equilibrium point at = is a scalar function: that is continuous, has continuous first derivatives, is strictly positive for , and for which the time derivative ˙ = is non positive (these conditions are required on some region containing the origin).

  8. Lyapunov dimension - Wikipedia

    en.wikipedia.org/wiki/Lyapunov_dimension

    The exact limit values of finite-time Lyapunov exponents, if they exist and are the same for all , are called the absolute ones [3] {+ (,)} = {()} {} and used in the Kaplan–Yorke formula. Examples of the rigorous use of the ergodic theory for the computation of the Lyapunov exponents and dimension can be found in. [ 11 ] [ 12 ] [ 13 ]

  9. Chaotic scattering - Wikipedia

    en.wikipedia.org/wiki/Chaotic_scattering

    A fitted straight line of negative slope, = is overlaid. The path-length, s , is equivalent to the decay time, T , provided we scale the (constant) speed appropriately. Note that an exponential decay rate is a property specifically of hyperbolic chaotic scattering.