In multiscale chaotic systems, a basic closure question is how much of the unresolved fine scales is instantaneously determined by the resolved coarse scales on the attractor. In a Fourier description, we formalize this by asking, given a target mode $k$ and a lower-mode cutoff $k_{\rm cut}<k$, how much of mode $k$ is determined by the retained modes $0,\ldots,k_{\rm cut}$. We quantify this relation by the correlation-ratio functional $R(k\mid k_{\rm cut})$, interpreted as conditional-mean explained variance, and use it to build a scale-resolved recoverability map $(k,k_{\rm cut})\mapsto R(k\mid k_{\rm cut})$, whose structure is sharply organized by the nonlinear dynamics. Applying the diagnostic to the Lorenz-96 system for forcings $F=8,16,32,64$, we find that the recoverability maps are strongly nonuniform: low modes remain weakly constrained by still coarser observations, while high modes exhibit finite-band partial slaving once the retained cutoff reaches the energetic intermediate modes. The growth of substantial recoverability is organized around the quadratic triad-access scale $k_{\rm cut}\approx\lceil k/2\rceil$, consistent with the Fourier coupling rule $p+q\equiv k\pmod N$, while remaining shifted by regime-dependent statistics. Increasing $F$ preserves this geometric organization but reduces its amplitude, indicating greater conditional freedom of the unresolved modes in more strongly driven regimes. The maps show that instantaneous deterministic closure varies systematically across scales as a property of the invariant measure: retained modes provide nontrivial deterministic information in some regions, while other regions are dominated by conditional residual variance.

In this note we explore numerically the finite Bernoulli convolutions. We show that with a suitable choice of parameter, it might serve as a toy model for intermittent energy cascade in fully developed turbulence. We then show how the crossings of $\beta$-expansions distribute in $\beta$, and suggest that it might highlight the parameters with enhanced overlap structure that are related to measures that are singular continuous. We later introduce a notion of order to the $\beta$-expansions based on the lexicographical order of the $N$-binary words, and observe that for most sampled adjacent pairs when $\beta=2$, the distance in their order increases exponentially when $\beta$ decreases from 2 to 1. This suggests 'chaotic' behavior, with the 'Lyapunov exponents' bunched into several clusters that depend on $N$. We end the note with some 'order plots' and an interesting connection between the finite $\beta$-compactum with $\beta=g$ (g being the golden ratio) and binary reflected Gray code.

We study the complex eigenvalue statistics of the asymmetric quantum baker map with partial projective openings. The classical asymmetric baker map, with its discontinuity at $q=2/3$, is fully chaotic, has no reflection symmetry, and provides a clean setting with tunable escape rate and fractal repeller dimension. We consider three distinct opening geometries in position space: localized (contiguous channels), random, and uniform (equispaced channels), all controlled by a tunable amplitude reflectivity parameter $\rho$ that interpolates between the fully open ($\rho=0$) and the closed ($\rho=1$) limits. We use the partially truncated circular unitary ensemble (PTCUE) as the random matrix theory benchmark. The main focus is on the joint distribution of the complex spacing ratio $z$, defined as the ratio of the distances from an eigenvalue to its nearest and next-nearest neighbors in the complex plane. We find a smooth crossover from a quasi-1D spectral regime, where eigenvalues cluster near the unit circle and the phase distribution of $z$ is peaked, to a two-dimensional Ginibre-like regime, where the distribution becomes nearly uniform and level repulsion is fully developed. Both the number of open channels $M$ and the reflectivity $\rho$ modulate this crossover, and $\rho$ provides an additional continuous control even at fixed opening size. All three opening models converge to PTCUE statistics at large $M$, while differences are most pronounced for the localized model at small $M$. No evidence of an abrupt transition is found. This crossover which suggests a universal behavior, has deep consequences for open quantum and wave-chaotic experiments.

We propose a phenomenological model of the Global Neuronal Workspace (GNW) in which early sensory processing generates an effective complex-valued landscape governing the dynamics of high-level stimulus representations. This landscape provides a dynamical bridge between sensory encoding and conscious access, enabling both processes to be described within a unified framework. High-level representations are encoded in a cloud function defined on a Hilbert space over a perceptual state space, thereby combining the holistic structure of mental images with a neural implementation. Its dynamics is governed by a nonlinear Schrödinger-type equation in imaginary time with a non-Hermitian, non-normal Hamiltonian and a nonlinear Lotka--Volterra-type term that preserves norm and enables spatially nonlocal interactions. The Hermitian and anti-Hermitian parts of the Hamiltonian generate complementary processes: recognition via dissipative localization at minima of the GNW landscape and information broadcasting via spatial spreading across the state space. The resulting dynamics reproduces the subliminal--preconscious--conscious hierarchy of sensory processing. Conscious access corresponds to the emergence of a bound state, which occurs only when both the GNW landscape depth and the degree of top-down attention exceed threshold values. The resulting framework provides a tractable dynamical description linking sensory processing, attention, and conscious access within a unified dynamical setting.

Chaotic systems often evolve on a low-dimensional attractor whose geometry varies from one region to another. We propose a non-intrusive reduced-order model that reads this local geometry by clustering and uses it to shape a radial basis library whose kernels adapt to each region. Fitting the reduced velocity onto this library by one global regularised least-squares solve gives an explicit, differentiable vector field that reproduces the long-term statistics, that is, the invariant measure, without any use of the governing equations. Since a radial basis field decays away from the data and cannot by itself return an escaped state, the integration is stabilised by a kinematic corrector whose magnitude is reported as a measure of how far each result rests on the learned field rather than on the corrector. On Lorenz-63 the model recovers the attractor, its marginal densities, and the positive and neutral Lyapunov exponents, while under-recovering the strong transverse contraction. On Lorenz-96 its valid prediction time is competitive with tuned neural-network and reservoir-computing forecasters, and the invariant measure is reproduced on both the full state and a reduced observable. On the Kuramoto--Sivashinsky equation and the quasiperiodic Kolmogorov flow the model matches the energy distribution and spectrum of an intrusive quantised-local Galerkin model, and improves on a global Galerkin projection of the same dimension, without ever projecting the governing equations.

The complex sine-Gordon (CSG) model contains an internal phase degree of freedom that strongly modifies the dynamics of its solitary-wave solutions. We present a numerical study of complex kink--kink collisions and determine how the final state depends jointly on the initial velocity and relative phase. In contrast with the elastic collisions of the real sine-Gordon model, the CSG system exhibits scattering, capture, long-lived bion formation, breather-like states, and emission of radiative profiles. The simulations reveal two distinct phase-dependent branches of critical velocity. In one branch, increasing the initial velocity promotes capture, whereas in the other it restores scattering. This dual structure highlights the rich velocity--phase dependence of the collision dynamics. We also compute the energy carried by radiative profiles and examine extreme values of the energy density, kinetic and gradient contributions, and field modulus at the collision center. These quantities show sharp transitions at critical points and provide sensitive diagnostics of phase-controlled dynamics. These results suggest that the relative phase behaves as an effective internal degree of freedom that plays an important role in the collision dynamics of complex solitons.

We investigate the global basin structure of twisted states in nearest-neighbor coupled phase oscillators with a common phase shift $\alpha$. As $\alpha$ increases, basin boundaries become progressively more complex, with their fractal dimension growing toward that of the full ambient phase space. We conjecture that the basins eventually become riddled-like as the system approaches the limit $\alpha\to \frac{\pi}{2}$, where the dynamics becomes volume-preserving. We characterize the transient dynamics via the stabilization time of the winding number and demonstrate that it grows with system size. The scaling accelerates at larger phase shifts, transitioning from logarithmic to power-law behavior. We further analyze the dynamical origin of these long transients. Our results demonstrate how a single phase-shift governs fractal basin complexity and provide new insights into the global geometry and transient dynamics of multistable, yet non-chaotic, coupled phase oscillators.

Collective oscillations in neuronal systems often arise from interactions between excitatory and inhibitory populations rather than from recurrent coupling within a single ensemble. Motivated by the coexistence of strongly and partially synchronized regimes in such systems, we study the Kuramoto Sakaguchi model on a bipartite network. Despite its minimal structure, the model exhibits rich collective dynamics, including both continuous and discontinuous transitions from full synchrony to partial synchrony (PS). In the PS regime, global oscillations fail to entrain one of the two populations, whose oscillators display quasiperiodic dynamics with an average frequency that can significantly deviate from that of the global field, as observed in neuronal networks. We show that this PS state constitutes an example of self-organized quasiperiodicity, arising here in the canonical Kuramoto Sakaguchi model despite its purely linear global coupling.