When parameters of a dynamical system change sufficiently fast, critical transitions can take place even in the absence of bifurcations. This phenomenon is known as rate-induced tipping and has been reported in a variety of systems, from simple ordinary differential equations and maps to mathematical models in climate sciences and ecology. In most examples, the transition happens at a critical rate of parameter change, a rate-induced tipping point, and is associated with a simple unstable orbit (edge state). In this work, we show how this simple picture changes when non-attracting fractal sets exist in the autonomous system, a ubiquitous situation in non-linear dynamics. We show that these fractals in phase space induce fractals in parameter space, which control the rates and parameter changes that result in tipping. We explain how such rate-induced fractals appear and how the fractal dimensions of the different sets are related to each other. We illustrate our general theory in three paradigmatic systems: a piecewise linear one-dimensional map, the two-dimensional Hénon map, and a forced pendulum.

We investigate integrable boundary states in the anisotropic Heisenberg chain under periodic or twisted boundary conditions, for both even and odd system lengths. Our work demonstrates that the concept of integrable boundary states can be readily generalized. For the XXZ spin chain, we present a set of factorized integrable boundary states using the KT-relation, and these states are also applicable to the XYZ chain. It is shown that a specific set of eigenstates of the transfer matrix can be selected by each boundary state, resulting in an explicit selection rule for the Bethe roots.

When the phase shift of X-shaped solutions before and after interaction is finite but approaches infinity, the vertices of the two V-shaped structures become separated due to the phase shift and are connected by a localized structure, which is referred to as the stem structure. This special type of elastic collision is known as a quasi-resonant collision. This study investigates quasi-resonant solutions and the associated localized stem structures in the context of the KPII and KPI equations. For the KPII equation, we classify quasi-resonant 2-solitons into weakly and strongly types, depending on whether the phase approaches \(-\infty\) or \(+\infty\). We analyze their asymptotic forms to detail the trajectories, amplitudes, velocities, and lengths of their stem structures. These results of quasi-resonant 2-solitons are used to to provide analytical descriptions of interesting patterns of the water waves observed on shallow water surface. Similarly, for the KPI equation, we construct quasi-resonant breather-soliton solutions and classify them into weakly and strongly types, based on the behavior of their internal parameters. We compare the similarities and differences between the stem structures in the quasi-resonant soliton and the quasi-resonant breather-soliton. Additionally, we provide a comprehensive and rigorous analysis of their asymptotic forms and stem structures. Our results indicate that the resonant solution, i.e. resonant breather-soliton of the KPI and soliton for the KPII, represents the limiting case of the quasi-resonant solution as phase approaches \(\infty\).

Tipping behavior can occur when an equilibrium of a dynamical system loses stability in response to a slowly varying parameter crossing a bifurcation threshold, or where noise drives a system from one attractor to another, or some combination of these effects. Similar behavior can be expected when a multistable system is forced by a chaotic deterministic system rather than by noise. In this context, the chaotic tipping window was recently introduced and investigated for discrete-time dynamics. In this paper, we find tipping windows for continuous-time nonlinear systems forced by chaos. We characterize the tipping window in terms of forcing by unstable periodic orbits of the chaos, and we show how the location and structure of this window depend on the relative timescales between the forcing and the responding system. We illustrate this by finding tipping windows for two examples of coupled bistable ODEs forced with chaos. Additionally, we describe the dynamic tipping window in the setting of a changing system parameter.

The real part of the focusing modified Korteweg-de Vries (MKdV) equation defined over the complex field $\mathbb{C}$ is reduced to the focusing gauged MKdV (FGMKdV) equation. In this paper, we construct the real hyperelliptic solutions of FGMKdV equation in terms of data of the hyperelliptic curves of genus $g$ and demonstrate the closed hyperelliptic plane curves of genus $g=5$ whose curvature obeys the FGMKdV equation by extending the previous results of genus three (Matsutani, {\it{J. Geom. Phys}} {\bf{215}} (2025) 105540). These are a generalization of Euler's elasticae.

Information processing abilities of active matter are studied in the reservoir computing (RC) paradigm to infer the future state of a chaotic signal. We uncover an exceptional regime of agent dynamics that has been overlooked previously. It appears robustly optimal for performance under many conditions, thus providing valuable insights into computation with physical systems more generally. The key to forming effective mechanisms for information processing appears in the system's intrinsic relaxation abilities. These are probed without actually enforcing a specific inference goal. The dynamical regime that achieves optimal computation is located just below a critical damping threshold, involving a relaxation with multiple stages, and is readable at the single-particle level. At the many-body level, it yields substrates robustly optimal for RC across varying physical parameters and inference tasks. A system in this regime exhibits a strong diversity of dynamic mechanisms under highly fluctuating driving forces. Correlations of agent dynamics can express a tight relationship between the responding system and the fluctuating forces driving it. As this model is interpretable in physical terms, it facilitates re-framing inquiries regarding learning and unconventional computing with a fresh rationale for many-body physics out of equilibrium.