Phase Transition Point in Classical Dynamics

This short note sketches a mean-field route to the stability/chaos transition in a simple classical random dynamical system. The model is a continuous-time analogue of a random neural network and the result is the familiar effective-gain threshold at which the maximal Lyapunov exponent changes sign.

Model

We consider

where , is a smooth nonlinearity (e.g. ), and is a gain parameter.

In the large- limit, averages over and over the disorder become self-averaging.

We will use the two-time correlations

assuming stationarity so that the correlations depend on the time difference only. (Indices are dummy under the mean-field average.)

Tangent dynamics and a two-time PDE

Introduce a small perturbation . The perturbed dynamics is

Linearizing Eqs. (1)–(3) yields the tangent (variational) equation

The same equation at a shifted time is

Multiply (4) and (5), average over and disorder, and define the perturbation correlator

Using -independence and gives the closed large- equation

Now switch to “center” and “difference” times

Then (7) becomes

Mean-field closure for the unperturbed correlations

For the unperturbed process (1), the standard mean-field closure gives (under stationarity or after Fourier transforming in )

It is convenient to view (10) as a Newton equation for a “particle” at coordinate in an effective potential :

where dots denote -derivatives. The dependence follows from a Gaussian closure: for large the field is approximately Gaussian and two-time statistics of are functionals of .

A related identity under the same closure is

Schrödinger reduction for the growth mode

Assume stationarity in the background () and look for separable solutions to (9) of the form

Plugging (13) into (9) gives a time-independent “Schrödinger” equation

The ground-state eigenvalue controls the largest growth rate. Near one may approximate by its value at the origin. Using (11)–(12) one finds

With a flat potential approximation around , the ground-state energy is

where the last equality uses stationarity of the one-time marginal of .

Equating with from (14) gives

where we picked the positive branch corresponding to growth.

Maximal Lyapunov exponent and the transition

The maximal Lyapunov exponent is

With and the ansatz (13),

Combining (17)–(19) yields the compact expression

Thus the phase transition occurs at

below which perturbations decay () and above which they grow exponentially (). For a linear system , one has and the formula reduces to the intuitive .

Remarks

• The effective gain automatically incorporates the nonlinearity through the stationary variance of . In practice, must be found self-consistently from (10)–(12).
• The flat- approximation around captures the threshold and leading behavior. Keeping the full -dependence refines the spectrum but does not move the transition at .
• For sigmoids such as , saturation reduces below , pushing upward compared to the linear case.