--- title: Randomized Infinite-Modal Map url: https://www.emergentmind.com/topics/randomized-infinite-modal-map type: topic --- # Randomized Infinite-Modal Map 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 [2509.07487]. ## 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 [2509.07487]. 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 $$ a=\alpha/\beta,\qquad b=\omega/\beta. $$ The return from \(P_1\to P_0\) is modeled as a smooth translation plus rotation, $$ T_2: \begin{pmatrix} x_0 \\ z_0 \end{pmatrix} = \begin{pmatrix} \tilde{x} & 0 \\ 0 & \tilde{z} \end{pmatrix} + \begin{pmatrix} \cos\phi & -\sin\phi \\ \sin\phi & \cos\phi \end{pmatrix} \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}. $$ Composing these yields the “unclosed” map \(T_3=T_2\circ T_1\), $$ T_3: \begin{dcases} x' = x \left(\frac{z}{h}\right)^a \cos\left[b\log\left(\frac{z}{h}\right)+\phi\right]+\tilde{x}, \\ z' = x \left(\frac{z}{h}\right)^a \sin\left[b\log\left(\frac{z}{h}\right)+\phi\right]+\tilde{z}. \end{dcases} $$ Pacifico et al. “close” it by extending it to all \(z\in\mathbb{R}\), which yields the PRV map $$ T_\text{PRV}:\mathbf{R}^2\to\mathbf{R}^2, $$ $$ \begin{dcases} x' = x \left(\frac{|z|}{h}\right)^a \cos\left[b\log\left(\frac{|z|}{h}\right)+\phi\right]+\tilde{x}, \\ z' = \mathrm{sgn}(z)\,x \left(\frac{|z|}{h}\right)^a \sin\left[b\log\left(\frac{|z|}{h}\right)+\phi\right]+\tilde{z}. \end{dcases} $$ Here \(a\in(0,1)\) and \(b\in(0,\infty)\) are the main parameters; \(h,\phi,\tilde{x},\tilde{z}\) are subparameters. In the numerical studies the paper fixes $$ h=1,\qquad \phi=0,\qquad \tilde{x}=0.5,\qquad \tilde{z}=0, $$ and often also takes \(b=100\), because large \(b\) does not materially change the statistical properties of interest. The radial variable used throughout is $$ r_n:=\sqrt{(x_n-\tilde{x})^2+(z_n-\tilde{z})^2}. $$ ## 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 $$ \tilde{\theta}_n:= b \log\left(\frac{|z_n|}{h}\right)+\phi \mod 2\pi $$ appears highly oscillatory and empirically looks uniform, one replaces it by i.i.d. random variables \(\theta_n\sim \mathrm{Unif}[0,2\pi)\) [2509.07487]. This produces the **randomized PRV map** $$ T_\text{R-PRV}:\mathbf{R}^2\to\mathbf{R}^2, $$ $$ \begin{dcases} x_{n+1} = x_n \left(\frac{|z_n|}{h}\right)^a \cos\left(\theta_n\right)+\tilde{x}, \\ z_{n+1} = \mathrm{sgn}(z_n)\,x_n \left(\frac{|z_n|}{h}\right)^a \sin\left(\theta_n\right)+\tilde{z}, \end{dcases} $$ with \(\{\theta_n\}\) independent and identically distributed. The paper verifies the hypothesis by showing that the empirical histogram of \(\tilde{\theta}_n\) is close to uniform [2509.07487]. Under the simplifying regime $$ \tilde{z}\ll r_n,\qquad r_n\ll \tilde{x}, $$ the map reduces to a radial recursion $$ r_{n+1}=c\,\xi_n^a\,r_n^a, \qquad c:=h^{-a}|\tilde{x}|, $$ where \(\xi_n=|\sin\varphi_n|\). Defining $$ w_n:=-\log r_n,\qquad \eta_n:=-\log \xi_n, $$ gives the logarithmic radial map $$ w_{n+1}=a w_n + a\eta_n - \log c. $$ 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 \(a\) multiplying \(\eta_n\), which changes the formulas for mean and variance [2509.07487]. ## 3. Stationary statistics and asymptotic distribution From the linear random recursion, the paper derives the stationary mean under the stationarity assumption. Writing $$ \mu:=\mathrm{E}[w_n], $$ one has $$ \mu=a\mu+a\mu_0-\log c,\qquad \mu_0=\mathrm{E}[\eta_n]=\log 2, $$ hence $$ \mu=\frac{a\log 2-\log c}{1-a} =\frac{a\log(2h)-\log|\tilde{x}|}{1-a}. $$ The variance is derived under the assumption $$ \mathrm{E}\!\left[(w_n-\mu)(\eta_n-\mu_0)\right]=0. $$ The paper emphasizes that this variance depends only on \(a\), making it especially useful for parameter estimation [2509.07487]. For the stationary distribution, the analysis standardizes the recursion and introduces characteristic functions $$ \psi_n(t):=\mathrm{E}\!\left[\exp\!\left(it\frac{w_n-\mu}{\sigma}\right)\right],\qquad \varphi_n(t):=\mathrm{E}\!\left[\exp\!\left(it\frac{\eta_n-\mu_0}{\sigma_0}\right)\right]. $$ They satisfy $$ \psi_{n+1}(t)=\psi_n(at)\,\varphi_n\!\left(\sqrt{1-a^2}\,t\right). $$ Following earlier randomization theory, when \(n\to\infty\) and \(a\to 1\), \(\psi_n(t)\to e^{-t^2/2}\). Thus \((w_n-\mu)/\sigma\) becomes asymptotically standard normal for \(a\) close to 1, so \(\log r\) is Gaussian. Equivalently, \(r\) has a **log-normal** stationary distribution. This analytic result matches the numerically observed height histograms of \(\log r_n\) near the strongly intermittent regime \(a\lesssim 1\) [2509.07487]. The paper reports that these results were numerically checked against direct PRV-map simulations and found to agree well, especially near \(a\approx 1\). 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 \(a\to 1\), the PRV map exhibits stronger intermittency and smaller burst heights in \(x_n\), \(z_n\), and \(r_n\) [2509.07487]. The bifurcation diagrams of \(x_n\) and \(z_n\) are almost identical, and the radial variable inherits the same structure. The first Lyapunov exponent \(\lambda_1\) depends mainly on \(b\) and increases as \(b\) increases, while the second Lyapunov exponent \(\lambda_2\) depends mainly on \(a\) and decreases as \(a\) increases. A broad region in parameter space still has at least one positive Lyapunov exponent, confirming chaos [2509.07487]. The paper also studies **interevent interval statistics**. An “event” is declared when \(r_n\) crosses a threshold \(r_\mathrm{th}\), and the interevent interval probability distribution is defined as $$ \Lambda_n=\mathrm{Prob.}\left\{r_1\le r_\mathrm{th},\cdots,r_n\le r_\mathrm{th},r_{n+1}>r_\mathrm{th}\mid r_0>r_\mathrm{th}\right\}. $$ The threshold is often compared to $$ r_a:=\exp\left\{\mathrm{E}[\log r_n]\right\}=e^{-\mu}, $$ which is the typical event scale. Numerically, if \(r_\mathrm{th}\approx r_a\), the interval distribution shows a short-time power-law regime followed by long-time exponential decay. If \(r_\mathrm{th}\gg r_a\), a shoulder appears in the long-time region before the eventual exponential tail. The mean and variance of interevent intervals depend approximately linearly on \(a\) near \(a=1\), but become nearly constant when the threshold is much larger than \(r_a\) [2509.07487]. 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 \(\log r\), whereas the interval statistics depend strongly on both \(a\) 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 \(a\), the paper proposes a direct estimator from the sample variance of the observed \(w_n=-\log r_n\). The three-step procedure is: compute the unbiased sample variance \(s^2\) from a window of past data, estimate $$ \tilde{a}_\text{estimated}=\left(1+\frac{\pi}{\sqrt{12}\,s^2}\right)^{-1/2}, $$ and then apply an empirical correction $$ \tilde{a}_\text{corrected}=\frac{\tilde{a}_\text{estimated}+0.08}{1.08}. $$ The residual linear bias is reported numerically as $$ a_\text{estimated}\approx 1.08\,a_\text{true}-0.08, $$ so the correction is empirical rather than derived [2509.07487]. Once \(a\) is estimated, \(h\) can be recovered from the mean formula: $$ h_\text{estimated} = \frac{1}{2}\exp\!\left[\frac{m(1-\tilde{a}_\text{corrected})+\log|\tilde{x}|}{\tilde{a}_\text{corrected}}\right], $$ where \(m\) is the sample mean of \(w_n\). The remaining parameters \(b\) and \(\phi\) are treated as less important because the statistical properties are insensitive to them when \(b\) is large; \(\tilde{x}\) and \(\tilde{z}\) are assumed known from the orbit center [2509.07487]. The same methodology is tested on **non-stationary data** with time-dependent \(a_n\). Two examples are considered: a sinusoidal parameter $$ a_n=A_0+A_1\sin\!\left(2\pi\frac{n}{T}\right), $$ with \(A_0=0.90\), \(A_1=0.02\), \(T=10^5\), and a piecewise linear parameter $$ a_n= \begin{dcases} B_1 & (n