Chaotic maps and Lyapunov exponents

For a differentiable one-dimensional map F and orbit x next = F(x), the chain rule gives the derivative of the n-fold iterate as the product of derivatives along the orbit. Taking the logarithm of its absolute value turns that product into a sum. When the average converges, its limit is the Lyapunov exponent for that orbit and tangent direction. A positive value means infinitesimal perturbations grow exponentially on average before nonlinear finite separation invalidates the linear approximation.

The limit need not exist for every map or initial state; a limsup can be reported instead but is a different defined quantity. Zero derivatives can produce minus-infinite logarithms, and nondifferentiable points require special treatment. To replace a time average by a measure average, one needs an invariant probability measure, integrability of the logarithmic derivative, and an applicable ergodic theorem. Ergodicity makes the almost-everywhere limit constant for that measure. A long-looking finite sequence does not supply these hypotheses automatically.

Take the doubling map on the circle, represented by F(x)=2x modulo one on [0,1). The apparent jump at the interval endpoint is an artifact of the circle coordinate; as a circle map it is smooth with derivative two. Every n-step tangent derivative is two to the n, giving exponent log two. Binary digits shift left under iteration, which connects expansion to symbolic information loss. Dyadic endpoints have two binary representations, so a coding argument must choose a convention or handle that countable ambiguity.

A positive exponent does not mean the chosen exact orbit is aperiodic. The rational point one seventh lies on a period-three orbit under doubling, while its infinitesimal exponent is still log two. Nearby points depart from that orbit, but the orbit itself repeats exactly. On the circle, finite separations also wrap around and cannot grow exponentially without bound. The exponent tracks derivatives of arbitrarily small perturbations before saturation, not the distance between two fixed finite-precision trajectories for all time.

The unbounded linear map x next = 2x has the same positive exponent log two, yet expansion alone creates no bounded recurrent chaotic set for its escaping nonzero orbits. Definitions of chaos vary. For a continuous map on an infinite compact metric space without isolated points, topological transitivity together with dense periodic points implies sensitive dependence; these properties give one common topological chaos framework. A positive exponent is related evidence, not a substitute for whichever complete definition or theorem is being used. The next lesson supplies a symbolic invariant-set construction.

In binary floating-point arithmetic, every represented point of [0,1) is dyadic. Repeated exact doubling modulo one eventually shifts all stored fractional bits out and reaches zero, even though the real map supports rich nonperiodic behavior. A scatter plot or estimated exponent can therefore depend strongly on precision and observation length. For smooth higher-dimensional systems, integrate tangent dynamics with periodic normalization to avoid overflow and report convergence checks. Such estimates remain numerical evidence unless accompanied by validated bounds or a separate analytical result.