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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 ...
The values of the Lyapunov exponents are invariant with respect to a wide range of coordinate transformations. Suppose that g : X → X is a one-to-one map such that ∂ g / ∂ x {\displaystyle \partial g/\partial x} and its inverse exist; then the values of the Lyapunov exponents do not change.
In applied mathematics, the Kaplan–Yorke conjecture concerns the dimension of an attractor, using Lyapunov exponents. [ 1 ] [ 2 ] By arranging the Lyapunov exponents in order from largest to smallest λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \dots \geq \lambda _{n}} , let j be the largest index for which
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. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs.
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 ]
Bethany Joy Lenz is looking back at the “rude awakening” she experienced after leaving a small, ultra-Christian cult in 2012.. During an appearance on the Rooted Recovery Stories with Patrick ...
The 36-year-old quarterback, who was added to the Pro Bowl Games as an alternate when Buffalo's Josh Allen pulled out this week, is a pending free agent after spending one year in Pittsburgh.
In particular, the discrete-time Lyapunov equation (also known as Stein equation) for is A X A H − X + Q = 0 {\displaystyle AXA^{H}-X+Q=0} where Q {\displaystyle Q} is a Hermitian matrix and A H {\displaystyle A^{H}} is the conjugate transpose of A {\displaystyle A} , while the continuous-time Lyapunov equation is