# What does the Lyapunov exponent tell us?

## What does the Lyapunov exponent tell us?

The exponent λ measured for a long period of time (ideally t→∞) is the Lyapunov exponent. If λ>0, small distances grow indeﬁnitely over time, which means the stretching mechanism is in effect. Or if λ<0, small distances don’t grow indeﬁnitely, i.e., the system settles down into a periodic trajectory eventually.

### What does a negative Lyapunov exponent mean?

Basic properties If the system is conservative (i.e., there is no dissipation), a volume element of the phase space will stay the same along a trajectory. If the system is dissipative, the sum of Lyapunov exponents is negative.

What is maximum Lyapunov exponent?

The maximum Lyapunov exponent (MLE) has often been suggested as the prominent measure for evaluation of dynamic stability of locomotion in pathological and healthy population.

Can Lyapunov exponent be negative?

Negative Lyapunov exponents are characteristic of dissipative or non-conservative systems (the damped harmonic oscillator for instance). Such systems exhibit asymptotic stability; the more negative the exponent, the greater the stability.

## What are Lyapunov exponents and why are they interesting?

Lyapunov exponents play a key role in three areas of Avila’s research: smooth ergodic theory, billiards and translation surfaces, and the spectral theory of 1-dimensional Schrödinger operators.

### How do you get the Lyapunov exponent?

The finite-time Lyapunov exponents are computed by solving the variational equation, that reflects the growth rate of the orthogonal semiaxes (equivalent to the initial deviation vectors) of one ellipse centred at the initial position as the system evolves [2].

How do you calculate Lyapunov exponent?

How do you find the exponent of Lyapunov?

Starts here9:41Dynamical Systems And Chaos: Lyapunov Exponents (Optional)YouTube

## What is the maximum Lyapunov exponent of Chaos?

It is common to refer to the largest one as the maximal Lyapunov exponent (MLE), because it determines a notion of predictability for a dynamical system. A positive MLE is usually taken as an indication that the system is chaotic (provided some other conditions are met, e.g., phase space compactness).

### What are the basic properties of Lyapunov?

Basic properties. If the system is conservative (i.e., there is no dissipation), a volume element of the phase space will stay the same along a trajectory. Thus the sum of all Lyapunov exponents must be zero. If the system is dissipative, the sum of Lyapunov exponents is negative.

How do you calculate Lyapunov exponents?

Generally the calculation of Lyapunov exponents, as defined above, cannot be carried out analytically, and in most cases one must resort to numerical techniques. An early example, which also constituted the first demonstration of the exponential divergence of chaotic trajectories, was carried out by R. H. Miller in 1964.

What is the difference between conditional exponents and synchronization?

The conditional exponents are those of the response system with the drive system treated as simply the source of a (chaotic) drive signal. Synchronization occurs when all of the conditional exponents are negative.