This study explores the integration of a diffusion control parameter into the chaotic dynamics of a modified bouncing ball model. By extending beyond simple elastic collisions, the model introduces elements that affect the diffusive behavior of kinetic energy, offering insights into the interplay between deterministic chaos and stochastic diffusion. The research reinterprets the bouncing ball's interactions with the surface as short-distance collisions that mimic random thermal fluctuations of particles. This refined model reveals complex dynamics, highlighting the synergistic effects between chaos and diffusion in shaping the evolution of the system.

The inverse scattering transform for the defocusing-defocusing coupled Hirota equations is strictly discussed with non-zero boundary conditions at infinity including non-parallel boundary conditions, specifically referring to the asymptotic polarization vectors. To address the non-analyticity encountered in some of the Jost eigenfunctions, the "adjoint" Lax pair is employed. The inverse problem is formulated as an appropriate matrix Riemann-Hilbert problem. A key difference between non-parallel and parallel boundary conditions lies in the asymptotic behavior of the scattering coefficients, which significantly impacts the normalization of the eigenfunctions and the properties of sectionally meromorphic matrices within the Riemann-Hilbert problem framework. When the asymptotic polarization vectors are non-orthogonal, two distinct methodologies are introduced to convert the Riemann-Hilbert problem into a series of linear algebraic-integral equations. In contrast, when the asymptotic polarization vectors are orthogonal, only one method is feasible. Ultimately, it is demonstrated that pure soliton solutions do not exist in both orthogonal and non orthogonal polarization vector cases. This study provides a comprehensive framework for analyzing the defocusing-defocusing coupled Hirota equations using the inverse scattering transform, offering new insights into the characteristics and solutions of the equations.

We consider the variational principle for the Lagrangian 1-form structure for long-range models of Calogero-Moser (CM) type. The multiform variational principle involves variations with respect to both the field variables as well as the independent variables corresponding to deformations of the time-curves in a multi-time space. The ensuing generalised Euler-Lagrange (gEL) equations comprise a system of multi-time EL equations, as well as constraints from so-called `alien derivatives' and `corner equations' arising from how variations on different coordinate curves match up. The closure relation, i.e. closedness of the Lagrange 1-form on solutions of the EL system, guarantees the stationarity of the action functional under deformation of the time-curves, and hence the multidimensional consistency of the corresponding gEL system. Using this as an integrability criterion on the Lagrangian level, we apply the system to some ans\"atze on the kinetic form of the Lagrangian components, associated with models of CM type without specifying the potentials. We show that from this integrability criterion the general elliptic form of the three systems, Calogero-Moser, Ruijsenaars-Schneider, and Goldfish systems, can be derived. We extend the analysis to an associated Hamiltonian formalism, via Noether's theorem and by applying Legendre transformations. Thus, the multiform variational principle leads to a system of generalised Hamilton equations describing Hamiltonian commuting flows for the mentioned elliptic models.

In this work, we study the asymptotic behaviors and dynamics of degenerate and mixed solitons for the coupled Hirota system with strong coherent coupling effects in the isotropic nonlinear medium. Using the binary Darboux transformation, we derive the solutions to represent the degenerate solitons with two eigenvalues that are conjugate to each other. We obtain three types of degenerate solitons and provide their asymptotic expressions. Notably, these degenerate solitons exhibit time-dependent velocities, and the relative distance between the two asymptotic solitons increases logarithmically with the higher-order perturbation parameter $|\varepsilon|$ increasing. We also asymptotically reveal four interaction mechanisms between a degenerate soliton and a bell-shaped soliton: (1) elastic interaction with a position shift; (2) inelastic interaction for the degenerate soliton but elastic for the bell-shaped one; (3) elastic interaction for the degenerate soliton but inelastic for the bell-shaped one; and (4) the coherent interaction during a longer interaction region and elastic interaction based on specific parameter conditions. Besides, we analyze a special degenerate vector soliton that exhibits significant coherence effects, and numerically study the relationship between the robustness of such solitons and parameter $\varepsilon$. Our results indicate that $\varepsilon$ significantly affects the coherence of these solitons, and their robustness decreases when $|\varepsilon|$ increases.

This work presents some results regarding three-dimensional billiards having a non-constant potential of Keplerian type inside a regular domain $D\subset \mathcal R^3$. Two models will be analysed: in the first one, only an inner Keplerian potential is present, and every time the particle encounters the boundary of $D$ is reflected back by keeping constant its tangential component to $\partial D$. The second model is a refractive billiard, where the inner Keplerian potential is coupled with a harmonic outer one; in this case, the interaction with $\partial D$ results in a generalised refraction Snell's law. In both cases, the analysis of a particular type of straight equilibrium trajectories, called \emph{homothetic}, is carried on, and their presence is linked to the topological chaoticity of the dynamics for large inner energies.

The currents in the ocean have a serious impact on ocean dynamics, since they affect the transport of mass and thus the distribution of salinity, nutrients and pollutants. In many physically important situations the current depends quadratic-ally on the depth. We consider a single layer of fluid and study the propagation of the surface waves in the presence of depth-dependent current with quadratic profile. We select the scale of parameters and quantities, which are typical for the Boussinesq propagation regime (long wave and small amplitude limit) and we also derive the well known KdV model for the surface waves interacting with current.

Synchronization is a fundamental dynamical state of interacting oscillators, observed in natural biological rhythms and in the brain. Global synchronization which occurs when non-linear or chaotic oscillators placed on the nodes of a network display the same dynamics as received great attention in network theory. Here we propose and investigate Global Topological Dirac Synchronization on higher-order networks such as cell and simplicial complexes. This is a state where oscillators associated to simplices and cells of arbitrary dimension, coupled by the Topological Dirac operator, operate at unison. By combining algebraic topology with non-linear dynamics and machine learning, we derive the topological conditions under which this state exists and the dynamical conditions under which it is stable. We provide evidence of 1-dimensional simplicial complexes (networks) and 2-dimensional simplicial and cell complexes where Global Topological Dirac Synchronization can be observed. Our results point out that Global Topological Dirac Synchronization is a possible dynamical state of simplicial and cell complexes that occur only in some specific network topologies and geometries, the latter ones being determined by the weights of the higher-order networks

Recent numerical results seem to suggest that in certain regimes of typical particle velocities the gravitational $N-$body problem (for $3\leq N\lesssim 10^3$) is intrinsically less chaotic when the post-Newtonian (PN) force terms are included, with respect to its classical counterpart that exhibits a slightly larger maximal Lyapunov exponent $\Lambda_{\rm max}$. In this work we explore the dynamics of wildly chaotic, regular and nearly regular configurations of the 3-body problem with and without the PN corrective terms aiming at shedding some light on the behaviour of the Lyapunov spectra under the effect of said corrections. Because the interaction of the tangent-space dynamics in gravitating systems, needed to evaluate the Lyapunov exponents, becomes rapidly computationally heavy due to the complexity of the higher order force derivatives involving multiple powers of $v/c$, we introduce a technique to compute a proxy of the Lyapunov spectrum based on the time-dependent diagonalization of the inertia tensor of a cluster of trajectories in phase-space. We find that, for a broad range of orbital configurations, the relativistic 3-body problem has a smaller $\Lambda_{\rm max}$ than its classical counterpart starting with the exact same initial condition. However, the rest of the Lyapunov spectrum can be either lower or larger in the classical case, suggesting that the relativistic precession effectively reduces chaos only along one (or few) directions in phase-space. As a general trend, the dynamical entropy of the relativistic simulations as function of the rescaled speed of light always has a regime in which falls below the classical value.} We observe that, the sole analysis of $\Lambda_{\rm max}$ could induce possibly misleading conclusions on the chaoticity of systems with small (and possibly large $N$.

We study the self-organization of two-dimensional turbulence in a fluid with local interactions. Using simulations and theoretical arguments, we show that the out-of-equilibrium flux to small scales, corresponding to the direct cascade, imposes a constraint on the large-scale emergent flow. As a result, instead of the unique state found in other two-dimensional models, a rich phase diagram of large-scale configurations emerges. We explain what sets the boundaries between the different phases, and show that in the infinite box limit when the range of the direct cascade is kept finite, the large-scale flow exhibits spontaneous symmetry breaking.

Discrete Empirical Interpolation Method (DEIM) estimates a function from its pointwise incomplete observations. In particular, this method can be used to estimate the state of a dynamical system from observational data gathered by sensors. However, when the number of observations are limited, DEIM returns large estimation errors. Sparse DEIM (S-DEIM) was recently developed to address this problem by introducing a kernel vector which previous DEIM-based methods had ignored. Unfortunately, estimating the optimal kernel vector in S-DEIM is a difficult task. Here, we introduce a data-driven method to estimate this kernel vector from sparse observational time series using recurrent neural networks. Using numerical examples, we demonstrate that this machine learning approach together with S-DEIM leads to nearly optimal state estimations.

With the oncologist acting as the ``game leader'', we employ a Stackelberg game-theoretic model involving multiple populations to study prostate cancer. We refine the drug dosing schedule using an empirical Bayes feed-forward analysis, based on clinical data that reflects each patient's prostate-specific drug response. Our methodology aims for a quantitative grasp of the parameter landscape of this adaptive multi-population model, focusing on arresting the growth of drug-resistant prostate cancer by promoting competition across drug-sensitive cancer cell populations. Our findings indicate that not only is it is feasible to considerably extend cancer suppression duration through careful optimization, but even transform metastatic prostate cancer into a chronic condition instead of an acute one for most patients, with supporting clinical and analytical evidence.