This paper reviews some basic mathematical results on Lyapunov exponents, one of the most fundamental concepts in dynamical systems. The first few sections contain some very general results in nonuniform hyperbolic theory. We consider ( f , μ), where f is an arbitrary dynamical system and μ is an arbitrary invariant measure, and discuss relations between Lyapunov exponents and several dynamical...

The first definition of Lyapunov exponents (depending on a probability measure) for a one-dimensional cellular automaton were introduced by Shereshevsky in 1991. The existence of an almost everywhere constant value for each of the two exponents (left and right), requires particular conditions for the measure. Shereshevsky establishes an inequality involving these two constants and the metric en...

We consider the half-line stochastic heat equation (SHE) with Robin boundary parameter $A = -\frac{1}{2}$. Under narrow wedge initial condition, we compute every positive (including non-integer) Lyapunov exponents of SHE. As a consequence, prove large deviation principle for upper tail KPZ under Neumann -\frac{1}{2}$ rate function $\Phi_+^{\text{hf}} (s) \frac{2}{3} s^{\frac{3}{2}}$. This confi...

Lyapunov exponents measure the exponential rate of divergence of infinitesimally close orbits of a smooth dynamical system. These exponents are intimately related to the global stochastic behavior of the system and are fundamental invariants of a smooth dynamical system. In [EP], Eckmann and Procaccia suggested an analysis of Lyapunov exponents for chaotic dynamical systems. This suggestion was...

The hall mark property of a chaotic attractor, namely sensitive dependence on initial condition, has been associated by the Lyapunov exponents to characterize the degree of exponential divergence/convergence of trajectories arising from nearby initial conditions. At first, we will describe briefly the concept of Lyapunov exponent and the procedure for computing Lyapunov exponents of the flow of...

Lyapunov exponents and direction elds are used to characterize the time-scales and geometry of general linear time-varying (LTV) systems of di erential equations. Lyapunov exponents are already known to correctly characterize the time-scales present in a general LTV system; they reduce to real parts of eigenvalues when computed for linear time-invariant(LTI) systems and real parts of Floquet ex...