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The Lyapunov time mirrors the limits of the predictability of the system. By convention, it is defined as the time for the distance between nearby trajectories of the system to increase by a factor of e. However, measures in terms of 2-foldings and 10-foldings are sometimes found, since they correspond to the loss of one bit of information or ...
There are no inherent limitations on the number of variables, parameters etc. Lyap which includes source code written in Fortran, can also calculate the Lyapunov direction vectors and can characterize the singularity of the attractor, which is the main reason for difficulties in calculating the more negative exponents from time series data.
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 ]
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).
Linear Time Invariant (LTI) Systems are those systems in which the parameters , , and are invariant with respect to time. One can observe if the LTI system is or is not controllable simply by looking at the pair ( A , B ) {\displaystyle ({\boldsymbol {A}},{\boldsymbol {B}})} .
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
The ordinary Lyapunov function is used to test whether a dynamical system is (Lyapunov) stable or (more restrictively) asymptotically stable. Lyapunov stability means that if the system starts in a state x ≠ 0 {\displaystyle x\neq 0} in some domain D , then the state will remain in D for all time.
Lyapunov functions are used extensively in control theory to ensure different forms of system stability. The state of a system at a particular time is often described by a multi-dimensional vector. A Lyapunov function is a nonnegative scalar measure of this multi-dimensional state.