Randomized Infinite-Modal Map

• The paper introduces a randomized infinite-modal map that replaces deterministic phase oscillations with uniformly distributed angles to derive explicit burst-height and interevent interval formulas.
• It shows that the logarithmic radial dynamics conform to a Gaussian distribution, leading to a log-normal stationary distribution for burst magnitudes in chaotic regimes.
• The framework provides a simple yet effective parameter estimation method for both stationary and slowly time-varying regimes, enhancing extreme-event prediction in homoclinic systems.

A randomized infinite-modal map is a stochastic surrogate for an infinite-modal return map, introduced to analyze the statistical properties of homoclinic bursting and to support prediction of extreme events. In the formulation developed for the PRV map of Pacifico–Rovella–Viana, the deterministic oscillatory phase is replaced by independent uniformly distributed angles, yielding explicit formulas for burst-height statistics, interevent interval statistics, and parameter estimation in both stationary and slowly time-dependent regimes (Nakagawa, 9 Sep 2025).

1. Definition and dynamical setting

In this context, the central deterministic object is the PRV map, a two-dimensional infinite-modal map arising as a Poincaré return map near a homoclinic orbit to a saddle-focus. The map is “infinite-modal” because it has countably many critical points, which in this context come from the oscillatory logarithmic phase dependence created by spiraling near the saddle-focus (Nakagawa, 9 Sep 2025).

The construction begins from the linearized saddle-focus dynamics near the origin,

$$\begin{pmatrix} \dot{x} \ \dot{y} \ \dot{z} \end{pmatrix} = \begin{pmatrix} -\alpha & \omega & 0 \ -\omega & -\alpha & 0 \ 0 & 0 & \beta \end{pmatrix} \begin{pmatrix} x \ y \ z \end{pmatrix},$$

with $\alpha,\beta,\omega>0$ and the Shilnikov condition

$\alpha<\beta,$

which indicates possible chaotic dynamics near the homoclinic orbit. Two sections are chosen, $P_0$ on $y=0$ and $P_1$ on $z=h>0$. The flow from $P_0\to P_1$ gives

$\tau=\frac{1}{\beta}\log\frac{h}{z_0},$

and therefore the map

$$T_1: \begin{dcases} x_1 = x_0 \left(\frac{z_0}{h}\right)^a \cos\left[b\log\left(\frac{z_0}{h}\right)\right], \ y_1 = x_0 \left(\frac{z_0}{h}\right)^a \sin\left[b\log\left(\frac{z_0}{h}\right)\right], \end{dcases}$$

where

$\alpha,\beta,\omega>0$0

The return from $\alpha,\beta,\omega>0$1 is modeled as a smooth translation plus rotation,

$\alpha,\beta,\omega>0$2

Composing these yields the “unclosed” map $\alpha,\beta,\omega>0$3,

$\alpha,\beta,\omega>0$4

Pacifico et al. “close” it by extending it to all $\alpha,\beta,\omega>0$5, which yields the PRV map

$\alpha,\beta,\omega>0$6

$\alpha,\beta,\omega>0$7

Here $\alpha,\beta,\omega>0$8 and $\alpha,\beta,\omega>0$9 are the main parameters; $\alpha<\beta,$0 are subparameters. In the numerical studies the paper fixes

$\alpha<\beta,$1

and often also takes $\alpha<\beta,$2, because large $\alpha<\beta,$3 does not materially change the statistical properties of interest. The radial variable used throughout is

$\alpha<\beta,$4

2. Randomization theory and the randomized PRV map

The theoretical analysis is built on the randomization theory of infinite-modal maps. Its central assumption is the uniform distribution hypothesis: because the phase

$\alpha<\beta,$5

appears highly oscillatory and empirically looks uniform, one replaces it by i.i.d. random variables $\alpha<\beta,$6 (Nakagawa, 9 Sep 2025).

This produces the randomized PRV map

$\alpha<\beta,$7

$\alpha<\beta,$8

with $\alpha<\beta,$9 independent and identically distributed. The paper verifies the hypothesis by showing that the empirical histogram of $P_0$0 is close to uniform (Nakagawa, 9 Sep 2025).

Under the simplifying regime

$P_0$1

the map reduces to a radial recursion

$P_0$2

where $P_0$3. Defining

$P_0$4

gives the logarithmic radial map

$P_0$5

The paper identifies this as the main analytical reduction. It is the same structural form as the 1D AP-map theory, but with the extra factor $P_0$6 multiplying $P_0$7, which changes the formulas for mean and variance (Nakagawa, 9 Sep 2025).

3. Stationary statistics and asymptotic distribution

From the linear random recursion, the paper derives the stationary mean under the stationarity assumption. Writing

$P_0$8

one has

$P_0$9

hence

$y=0$0

The variance is derived under the assumption

$y=0$1

The paper emphasizes that this variance depends only on $y=0$2, making it especially useful for parameter estimation (Nakagawa, 9 Sep 2025).

For the stationary distribution, the analysis standardizes the recursion and introduces characteristic functions

$y=0$3

They satisfy

$y=0$4

Following earlier randomization theory, when $y=0$5 and $y=0$6, $y=0$7. Thus $y=0$8 becomes asymptotically standard normal for $y=0$9 close to 1, so $P_1$0 is Gaussian. Equivalently, $P_1$1 has a log-normal stationary distribution. This analytic result matches the numerically observed height histograms of $P_1$2 near the strongly intermittent regime $P_1$3 (Nakagawa, 9 Sep 2025).

The paper reports that these results were numerically checked against direct PRV-map simulations and found to agree well, especially near $P_1$4. This suggests that the randomized map is not merely a formal replacement of the deterministic phase, but a quantitatively informative approximation in the intermittent regime.

4. Extreme-event statistics, bifurcation structure, and Lyapunov behavior

The paper studies the PRV map as a surrogate model for extreme-event bursting. On the numerical side, it shows that as $P_1$5, the PRV map exhibits stronger intermittency and smaller burst heights in $P_1$6, $P_1$7, and $P_1$8 (Nakagawa, 9 Sep 2025).

The bifurcation diagrams of $P_1$9 and $z=h>0$0 are almost identical, and the radial variable inherits the same structure. The first Lyapunov exponent $z=h>0$1 depends mainly on $z=h>0$2 and increases as $z=h>0$3 increases, while the second Lyapunov exponent $z=h>0$4 depends mainly on $z=h>0$5 and decreases as $z=h>0$6 increases. A broad region in parameter space still has at least one positive Lyapunov exponent, confirming chaos (Nakagawa, 9 Sep 2025).

The paper also studies interevent interval statistics. An “event” is declared when $z=h>0$7 crosses a threshold $z=h>0$8, and the interevent interval probability distribution is defined as

$z=h>0$9

The threshold is often compared to

$P_0\to P_1$0

which is the typical event scale. Numerically, if $P_0\to P_1$1, the interval distribution shows a short-time power-law regime followed by long-time exponential decay. If $P_0\to P_1$2, a shoulder appears in the long-time region before the eventual exponential tail. The mean and variance of interevent intervals depend approximately linearly on $P_0\to P_1$3 near $P_0\to P_1$4, but become nearly constant when the threshold is much larger than $P_0\to P_1$5 (Nakagawa, 9 Sep 2025).

A common misconception is that the height statistics and timing statistics are controlled in the same way. The paper separates them: the height distribution is captured by the randomized-map reduction through the asymptotic Gaussian law for $P_0\to P_1$6, whereas the interval statistics depend strongly on both $P_0\to P_1$7 and the threshold.

5. Parameter estimation and non-stationary regimes

A practical part of the framework is the parameter estimation method for the PRV map. Since the stationary variance is one-to-one in $P_0\to P_1$8, the paper proposes a direct estimator from the sample variance of the observed $P_0\to P_1$9. The three-step procedure is: compute the unbiased sample variance $\tau=\frac{1}{\beta}\log\frac{h}{z_0},$0 from a window of past data, estimate

$\tau=\frac{1}{\beta}\log\frac{h}{z_0},$1

and then apply an empirical correction

$\tau=\frac{1}{\beta}\log\frac{h}{z_0},$2

The residual linear bias is reported numerically as

$\tau=\frac{1}{\beta}\log\frac{h}{z_0},$3

so the correction is empirical rather than derived (Nakagawa, 9 Sep 2025).

Once $\tau=\frac{1}{\beta}\log\frac{h}{z_0},$4 is estimated, $\tau=\frac{1}{\beta}\log\frac{h}{z_0},$5 can be recovered from the mean formula:

$\tau=\frac{1}{\beta}\log\frac{h}{z_0},$6

where $\tau=\frac{1}{\beta}\log\frac{h}{z_0},$7 is the sample mean of $\tau=\frac{1}{\beta}\log\frac{h}{z_0},$8. The remaining parameters $\tau=\frac{1}{\beta}\log\frac{h}{z_0},$9 and $$T_1: \begin{dcases} x_1 = x_0 \left(\frac{z_0}{h}\right)^a \cos\left[b\log\left(\frac{z_0}{h}\right)\right], \ y_1 = x_0 \left(\frac{z_0}{h}\right)^a \sin\left[b\log\left(\frac{z_0}{h}\right)\right], \end{dcases}$$0 are treated as less important because the statistical properties are insensitive to them when $$T_1: \begin{dcases} x_1 = x_0 \left(\frac{z_0}{h}\right)^a \cos\left[b\log\left(\frac{z_0}{h}\right)\right], \ y_1 = x_0 \left(\frac{z_0}{h}\right)^a \sin\left[b\log\left(\frac{z_0}{h}\right)\right], \end{dcases}$$1 is large; $$T_1: \begin{dcases} x_1 = x_0 \left(\frac{z_0}{h}\right)^a \cos\left[b\log\left(\frac{z_0}{h}\right)\right], \ y_1 = x_0 \left(\frac{z_0}{h}\right)^a \sin\left[b\log\left(\frac{z_0}{h}\right)\right], \end{dcases}$$2 and $$T_1: \begin{dcases} x_1 = x_0 \left(\frac{z_0}{h}\right)^a \cos\left[b\log\left(\frac{z_0}{h}\right)\right], \ y_1 = x_0 \left(\frac{z_0}{h}\right)^a \sin\left[b\log\left(\frac{z_0}{h}\right)\right], \end{dcases}$$3 are assumed known from the orbit center (Nakagawa, 9 Sep 2025).

The same methodology is tested on non-stationary data with time-dependent $$T_1: \begin{dcases} x_1 = x_0 \left(\frac{z_0}{h}\right)^a \cos\left[b\log\left(\frac{z_0}{h}\right)\right], \ y_1 = x_0 \left(\frac{z_0}{h}\right)^a \sin\left[b\log\left(\frac{z_0}{h}\right)\right], \end{dcases}$$4. Two examples are considered: a sinusoidal parameter

$$T_1: \begin{dcases} x_1 = x_0 \left(\frac{z_0}{h}\right)^a \cos\left[b\log\left(\frac{z_0}{h}\right)\right], \ y_1 = x_0 \left(\frac{z_0}{h}\right)^a \sin\left[b\log\left(\frac{z_0}{h}\right)\right], \end{dcases}$$5

with $$T_1: \begin{dcases} x_1 = x_0 \left(\frac{z_0}{h}\right)^a \cos\left[b\log\left(\frac{z_0}{h}\right)\right], \ y_1 = x_0 \left(\frac{z_0}{h}\right)^a \sin\left[b\log\left(\frac{z_0}{h}\right)\right], \end{dcases}$$6, $$T_1: \begin{dcases} x_1 = x_0 \left(\frac{z_0}{h}\right)^a \cos\left[b\log\left(\frac{z_0}{h}\right)\right], \ y_1 = x_0 \left(\frac{z_0}{h}\right)^a \sin\left[b\log\left(\frac{z_0}{h}\right)\right], \end{dcases}$$7, $$T_1: \begin{dcases} x_1 = x_0 \left(\frac{z_0}{h}\right)^a \cos\left[b\log\left(\frac{z_0}{h}\right)\right], \ y_1 = x_0 \left(\frac{z_0}{h}\right)^a \sin\left[b\log\left(\frac{z_0}{h}\right)\right], \end{dcases}$$8, and a piecewise linear parameter

$$T_1: \begin{dcases} x_1 = x_0 \left(\frac{z_0}{h}\right)^a \cos\left[b\log\left(\frac{z_0}{h}\right)\right], \ y_1 = x_0 \left(\frac{z_0}{h}\right)^a \sin\left[b\log\left(\frac{z_0}{h}\right)\right], \end{dcases}$$9

with $\alpha,\beta,\omega>0$00, $\alpha,\beta,\omega>0$01, $\alpha,\beta,\omega>0$02, $\alpha,\beta,\omega>0$03. Using a sliding window variance estimate with $\alpha,\beta,\omega>0$04, the corrected estimator tracks the time variation reasonably well, though the window size must be chosen carefully: too short gives noisy estimates, while too long over-averages the changes (Nakagawa, 9 Sep 2025).

6. Scope, neighboring theories, and terminological boundaries

The randomized infinite-modal map framework should be distinguished from several neighboring uses of “random map.” The closest neighboring dynamical-systems theory in the supplied literature is the study of critically finite random interval maps with finitely many monotonicity intervals. That work considers random multimodal $\alpha,\beta,\omega>0$05 maps with negative Schwarzian derivative, defined on a finite union of closed intervals in $\alpha,\beta,\omega>0$06, and proves existence and uniqueness properties of $\alpha,\beta,\omega>0$07-conformal random measures, monotonicity and Lipschitz continuity of the expected topological pressure, a pressure-zero parameter $\alpha,\beta,\omega>0$08, and the Bowen-type formula

$\alpha,\beta,\omega>0$09

for $\alpha,\beta,\omega>0$10-almost every $\alpha,\beta,\omega>0$11 (Atnip et al., 2018).

That theory is explicitly not a theory of arbitrary random interval maps with infinitely many turning points. The finite family $\alpha,\beta,\omega>0$12 of monotonicity intervals, the finite critical set, the restriction of critical values to $\alpha,\beta,\omega>0$13, and the use of negative Schwarzian distortion control are central to its arguments. In the wording of the supplied description, it “does not treat a fully general ‘randomized infinite-modal map’ in the sense of an arbitrary random interval map with infinitely many turning points” (Atnip et al., 2018). This distinction is substantive: the randomized PRV construction starts from an infinite-modal deterministic map and randomizes the phase variable, whereas the critically finite interval-map framework studies a random skew product with finite multimodality.

A second terminological boundary concerns random maps in probability and combinatorics. In the literature on planar maps, a “map” is a planar graph embedding rather than a self-map of a phase space. For example, one paper constructs a random infinite squaring of a rectangle as a canonical planar embedding of the uniform infinite 3-connected planar map (Addario-Berry et al., 2014), while another develops $\alpha,\beta,\omega>0$14-random feuilletages as candidates for the role of the Brownian map in higher dimensions (Lionni et al., 2019). These uses of “random map” are unrelated to infinite-modal return maps in dynamical systems.

Within these boundaries, the randomized infinite-modal map serves as a mechanism-based statistical theory for extreme events. The supplied description states that it connects the local homoclinic geometry to global burst statistics through a randomized surrogate, yielding explicit formulas for the mean, variance, and asymptotic distribution of burst heights and enabling a simple parameter-estimation scheme, even in slowly time-varying settings (Nakagawa, 9 Sep 2025).