We adapt ideas from geometrical optics and classical billiard dynamics to consider particle trajectories with constant velocity on a cone with specular reflections off an elliptical boundary formed by the intersection with a tilted plane, with tilt angle $\gamma$. We explore the dynamics as a function of $\gamma$ and the cone deficit angle $\chi$ that controls the sharpness of the apex, where a point source of positive Gaussian curvature is concentrated. We find regions of the ($\gamma, \chi$) plane where, depending on the initial conditions, either (A) the trajectories sample the entire cone base and avoid the apex region; (B) sample only a portion of the base region while again avoiding the apex; or (C) sample the entire cone surface much more uniformly, suggestive of ergodicity. The special case of an untilted cone displays only type A trajectories which form a ring caustic at the distance of closest approach to the apex. However, we observe an intricate transition to chaotic dynamics dominated by Type (C) trajectories for sufficiently large $\chi$ and $\gamma$. A Poincaré map that summarizes trajectories decomposed into the geodesic segments interrupted by specular reflections provides a powerful method for visualizing the transition to chaos. We then analyze the similarities and differences of the path to chaos for conical billiards with other area-preserving conservative maps.

Continuous cellular automata (CCAs) have evolved from discrete lookup tables to continuous partial differential equation (PDE) formulations in the search for novel forms of complexity. Despite innovations in qualitative behavior, analytical methods have lagged behind, reinforcing the notion that emergent complexity defies simple explanation. In this paper, we demonstrate that the PDE formulation of Asymptotic Lenia enables rigorous analysis using dynamical systems theory. We apply the concepts of symmetries, attractors, Lyapunov exponents, and fractal dimensions to characterize complex behaviors mathematically. Our contributions include: (1) a mathematical explanation for the four distinct solution classes (solitons, rotators, periodic and chaotic patterns), (2) conditions for the existence of a global attractor with fractal dimension $>4$, (3) identification of Kaplan-Yorke dimension as an effective complexity measure for CCAs, and (4) an efficient open-source implementation for calculating Lyapunov exponents and the covariant Lyapunov vectors for CCAs. We conclude by identifying the minimal set of properties that enable complex behavior in a broader class of CCAs. This framework provides a foundation for understanding and measuring complexity in artificial life systems.

In the present paper, using a modification of the method of vector fields $Z_i$ of the bi-Hamiltonian theory of separation of variables (SoV), we construct symmetric non-Stäckel variable separation for three-dimensional extension of the Clebsch model, which is equivalent (in the bi-Hamiltonian sense) to the system of interacting Manakov (Schottky-Frahm) and Euler tops. For the obtained symmetric SoV (contrary to the previously constructed asymmetric one), all curves of separation are the same and have genus five. It occurred that the difference between the symmetric and asymmetric cases is encoded in the different form of the vector fields $Z$ used to construct separating polynomial. We explicitly construct coordinates and momenta of separation and Abel-type equations in the considered examples of symmetric SoV for the extended Clebsch and Manakov models.

In 1890 W. Hess found the new special case of integrability of the Euler - Poisson equations of motion of a heavy rigid body with a fixed point. In 1963 L.N. Sretensky proved that the special case of integrability, similar to the Hess case, also exists in the problem of the motion of a heavy gyrostat - a heavy rigid body with a fixed point, which contains a rotating homogeneous rotor. Further numerous generalizations of the classical Hess case were proposed, which take place during the motion of a heavy rigid body and a gyrostat with a fixed point in various force fields. The first studies that provided a qualitative description of the motion of a heavy rigid body in the integrable Hess case were published almost immediately after this case was found. In 1892 P.A. Nekrasov proved, that the solution of the problem of motion of a heavy rigid body with a fixed point in the Hess case is reduced to the integration the second order linear homogeneous differential equation with variable coefficients. A similar result regarding the problem of the motion of a heavy gyrostat in the Hess - Sretensky case was presented by Sretensky. In this paper we present the derivation of the corresponding second order linear differential equation and reduce the coefficients of this equation to the form of rational functions. Then, using the Kovacic algorithm we study the problem of the existence of liouvillian solutions of the corresponding second order linear differential equation. We obtain the conditions for the parameters of the problem, under which the liouvillian solutions of the corresponding linear differential equation exist. Under these conditions equations of motion of a heavy gyrostat with a fixed point in the Hess - Sretensky case can be integrated in quadratures.

Many studies on acoustic radiation forces focus on characterizing the behavior of acoustic fields. However, the dynamic response of objects, particularly those larger than the wavelength, remains underexplored. Here we bridge this gap by deriving nonlinear equations of motion for a trapped spherical object under acoustic radiation forces and external excitation, where the Gorkov formulation fails to provide accurate results. Using Sparse Identification of Nonlinear Dynamical Systems (SINDy), we derive the corresponding nonlinear equation of motion from analytical time series data obtained through the Gorkov formulation and external excitation for acoustically small objects, and which recovers the governing equation with less than 0.05% error in coefficient values compared to the analytical solution.. We conduct experiments with the TinyLev levitator with external excitation to generate time series for acoustically large particles. Then, SINDy is applied to reconstruct governing equations from experimental data to see how external excitation amplitude influences the dynamics of acoustically large objects. Our findings demonstrate that the SINDy can effectively be used as a tool for deriving governing equations from complex data to improve and refine theoretical developments; in the present case, for acoustically large objects, where the Gorkov formulation fails to provide an accurate prediction.

A nonlocal version of the massive Thirring model (MTM) and its solutions are presented. We start from a 4-component system that can be reduced to the classical MTM and nonlocal MTM. Bilinear form of the 4-component system and general double Wronskian solutions are derived. By utilizing reduction technique we obtain solutions of the nonlocal MTM. Relations between the nonlocal MTM and the nonlocal Fokas-Lenells equation is discussed. Some solutions of the nonlocal MTM, such as solitons, double-pole solution, algebraic solitons and high order algebraic solitons are analyzed and illustrated.

We study a non-reciprocal version of Model B, as the continuum theory for non-reciprocal particle mixtures. In contrast to non-reciprocal Cahn-Hilliard models, it is important in this context to consider the dependence of mobility coefficients on the local concentrations. We show that a homogeneous state that is linearly stable for one form of the mobility can be unstable for a different form of mobility, an effect that would be impossible in equilibrium and implies a crucial role for mobilities in non-reciprocal mixtures. For unstable homogeneous states we study the spinodal dynamics governing the onset of phase separation. We find, again in contrast to non-reciprocal Cahn-Hilliard models, that exceptional point transitions between static and oscillatory instabilities are generically avoided by first order transitions where the spinodal lengthscale changes discontinuously. At these transitions we find intricate spinodal dynamics with two competing lengthscales, one governing a static instability and the other an oscillatory instability, i.e. one that generates travelling waves. We demonstrate that, depending on interaction strengths, more complex transitions can occur in the spinodal dynamics, including coexistence of three lengthscales and first order transition lines, terminated by critical points, between distinct static instabilities. Finally, we explore the effects of additional non-reciprocity in the interfacial chemical potentials, which would generically be expected when obtaining Model B by coarse graining from a non-reciprocal particle model. We show that interfacial non-reciprocity can increase the region in the spinodal phase diagram where oscillatory instabilities occur, but only up to a certain boundary that we establish analytically and demonstrate numerically.

In this paper, we present a probabilistic analysis of a dynamical particle model for the self-adaptive immune response to cancer, as proposed by the first author in a previous work. The model is motivated by the interplay between immune surveillance and cancer evolution. We rigorously confirm the sharp phase transition in immune system learning predicted in the original work. Additionally, we compute the expected amount of information acquired by the immune system about cancer cells over time. Our analysis relies on time-reversal techniques.

A class of 2x2 random-matrix models is introduced for which the Brody distribution is the exact eigenvalue spacing distribution. The matrix elements consist of constrained finite sums of an exponential random variable raised to various powers that depend on the Brody parameter. The random matrices introduced here differ from those of the Gaussian Orthogonal Ensemble (GOE) in three important ways: the matrix elements are not independent and identically distributed (i.e., not IID) nor Gaussian-distributed, and the matrices are not necessarily real and/or symmetric. The first two features arise from dropping the classical independence assumption, and the third feature stems from dropping the quantum-mechanical conditions imposed in the construction of the GOE. In particular, the hermiticity condition, which in the present class of models, is a sufficient but not necessary condition for the eigenvalues to be real, is not imposed. Consequently, complex non-Hermitian 2x2 random matrices with real or complex eigenvalues can also have spacing distributions that are intermediate between those of the Poisson and Wigner classes. Numerical examples are provided for different types of random matrices, including complex-symmetric matrices with real or complex-conjugate eigenvalues. Various generalizations and extensions are discussed including a simple modification that effectuates cross-over transitions between other classes of eigenvalue spacing statistics. The case of a cross-over transition between semi-Poisson and Ginibre spacing statistics is presented as a novel example.

The general linear group $GL_{n}$, along with its adjoint simple group $PGL_n$, can be described by means of weighted planar networks. In this paper we give a network description for simple Lie groups of types $B$ and $C$. The corresponding networks are axially symmetric modulo a sequence of cluster mutations along the axis of symmetry. We extend to this setting the result of Gekhtman, Shapiro, and Vainshtein on the Poisson property of Postnikov's boundary measurement map. We also show that $B$ and $C$ type networks with positive weights parametrize the totally nonnegative part of the respective group. Finally, we construct network parametrizations of double Bruhat cells in symplectic and odd-dimensional orthogonal groups, and identify the corresponding face weights with Fock-Goncharov cluster coordinates.

We study the spectral stability of smooth, small-amplitude periodic traveling wave solutions of the Novikov equation, which is a Camassa-Holm type equation with cubic nonlinearities. Specifically, we investigate the $L^2(\mathbb{R})$-spectrum of the associated linearized operator, which in this case is an integro-differential operator with periodic coefficients, in a neighborhood of the origin in the spectral plane. Our analysis shows that such small-amplitude periodic solutions are spectrally unstable to long-wavelength perturbations if the wave number if greater than a critical value, bearing out the famous Benmajin-Feir instability for the Novikov equation. On the other hand, such waves with wave number less than the critical value are shown to be spectrally stable. Our methods are based on applying spectral perturbation theory to the associated linearization.

We consider coherent states of weakly interacting bosons under the conditions of external resonant excitation, with a focus on a two-dimensional polariton fluid driven by a plane electromagnetic wave near the ground state. The coherent driving breaks the U(1) symmetry explicitly, which prevents the occurrence of quantum vortices in a uniform scalar condensate. Surprisingly, a spinor (two-component) system of the same kind admits topological excitations, such as domain walls of relative phase or confined half-vortex molecules, typical of a freely evolving spinor Bose system. Opposite-phase domains arise from the spontaneous breakdown of the spin symmetry $(\mathbb{Z}_2)$. Domain walls form with time even when the initial state of the system is uniform or completely disordered; they fall into different topological types distinguished by the total phase variation in the transverse direction.

We revisit a two-temperature Nosé-Hoover wanderer particle embedded in a two-dimensional periodic 2x2 cell with four smooth repulsive corners at $(x,y) = (\pm 1, \pm 1)$ to explore chaos with anisotropic thermostatting. The model employs separate thermostats in the x and y directions, enabling controlled deviations from equilibrium. By integrating the full six-dimensional equations of motion and computing the complete Lyapunov spectrum, we confirm chaos and quantify phase-space contraction with high numerical precision. The total contraction rate, interpreted as entropy production, grows nonlinearly with the thermostat anisotropy and follows a superquadratic power law, $\Lambda\propto -\delta^{2.44}$, deviating from linear-response theory, $\Lambda\propto -\delta^{2}$. We compare this behavior to a quartic correction to the linear-response prediction and discuss both fits in light of their empirical and theoretical implications. Our results demonstrate that nonlinear dissipation scaling emerges naturally even in minimal driven systems. The approximate Kaplan-Yorke dimension reveals a fractal attractor that concentrates as $|T_x - T_y|$ increases. Momentum statistics show significant non-Gaussian behavior under strong driving. Despite its dissipative nature, the model remains strictly time-reversible, offering a pedagogically rich example of microscopic reversibility coexisting with macroscopic entropy production.