Showing new listings for Monday, 10 November 2025

New submissions (showing 6 of 6 entries)

We study chaotic synchronization in a five-dimensional Hindmarsh-Rose neuron model augmented with electromagnetic induction and a switchable memristive autapse. For two diffusively coupled neurons, we derive the linearized error dynamics and prove global asymptotic stability of the synchronization manifold via a quadratic Lyapunov function. Verifiable sufficient conditions follow from Sylvester's criterion on the leading principal minors, and convergence is established using Barbalat's lemma. Leveraging Helmholtz's decomposition, we separate the error field into conservative and dissipative parts and obtain a closed-form expression for the synchronization energy, along with its dissipation law, providing a quantitative measure of the energetic cost of synchrony. Numerical simulations confirm complete synchronization, overall decay of the synchronization energy, and close agreement between Lyapunov and Hamiltonian diagnostics across parameter sweeps. Building on these results, we introduce a port-Hamiltonian physics-informed neural network that embeds the conservative/dissipative structure in training through physically motivated losses and structural priors. The learned Hamiltonian and energy-rate match analytical benchmarks. The framework narrows the gap between dynamical systems theory and data-driven discovery, and provides a template for energy-aware modeling and control of nonlinear neuronal synchronization.

Infinitesimal volumes stretch and contract as they coevolve with classical phase space trajectories according to a linearized dynamics. Unless these tangent space dynamics are modified, the underlying chaotic dynamics will cause the volume to vanish as tangent vectors collapse on the most expanding direction. Here, we propose an alternative linearized dynamics and rectify the generalized Liouville equation to preserve phase space volume, even for non-Hamiltonian systems. Within a classical density matrix theory, we define the time evolution operator from the anti-symmetric part of the stability matrix so that phase space volume is time-invariant. The operator generates orthogonal transformations without distorting volume elements, providing an invariant measure for dissipative dynamics and a evolution equation for the density matrix akin to the quantum mechanical Liouville-von Neumann equation. The compressibility of volume elements is determined by a non-orthogonal operator made from the symmetric part of the stability matrix. We analyze complete sets of basis vectors for the tangent space dynamics of chaotic systems, which may be dissipative, transient or driven, without re-orthogonalization of tangent vectors. The linear harmonic oscillator, the Lorenz-Fetter model, and the Hénon-Heiles system demonstrate the computation of the instantaneous Lyapunov exponent spectrum and the local Gibbs entropy flow rate using these bases and show that it is numerically convenient.

We introduce and systematically develop two classes of discrete integrable operators: those with 2times 2 matrix kernels and those possessing general differential kernels, thereby generalizing the discrete analogue previously studied. A central finding is their inherent connection to higher-order pole solutions of integrable hierarchies, contrasting sharply with standard operators linked to simple poles. This work not only provides explicit resolvent formulas for matrix kernels and differential operator analogues but also offers discrete integrable structures that encode higher-order behaviour.

The linear stability of two exact stationary solutions of the parametrically driven, damped nonlinear Dirac equation is investigated. Stability is ascertained through the resolution of the eigenvalue problem, which stems from the linearization of this equation around the exact solutions. On the one hand, it is proven that one of these solutions is always unstable, which confirms previous analysis based on a variational method. On the other hand, it is shown that sufficiently large dissipation guarantees the stability of the second solution. Specifically, we determine the stability curve that separates stable and unstable regions in the parameter space. The dependence of the stability diagram on the driven frequency is also studied, and it is shown that low-frequency solitons are stable across the entire parameter space. These results have been corroborated with extensive simulations of the parametrically driven and damped nonlinear Dirac equation by employing a novel and recently proposed numerical algorithm that minimizes discretization errors.

We use Clifford's geometric algebra to extend the Stuart-Landau system to dimensions D >2 and give an exact solution of the oscillator equations in the general case. At the supercritical Hopf bifurcation marked by a transition from stable fixed-point dynamics to oscillatory motion, the Jacobian matrix evaluated at the fixed point has N=lfloor{D/2}rfloor pairs of complex conjugate eigenvalues which cross the imaginary axis simultaneously. For odd D there is an additional purely real eigenvalue that does the same. Oscillatory dynamics is asymptotically confined to a hypersphere mathbb{S}^{D-1} and is characterised by extreme multistability, namely the coexistence of an infinite number of limiting orbits each of which has the geometry of a torus mathbb{T}^N on which the motion is either periodic or quasiperiodic. We also comment on similar Clifford extensions of other limit cycle oscillator systems and their generalisations.

We investigate the effects on solitons dynamics of introducing aPT-symmetric complex potential in a specific family of the cubic Dirac equation in (1+1)-dimensions, called the ABS model. The potential is introduced taking advantage of the fact that the nonlinear Dirac equation admits a Lagrangian formalism. As a consequence, the imaginary part of the potential, associated with gains and losses, behaves as a spatially periodic damping (changing from positive to negative, and back) that acts at the same time on the two spinor components. A collective coordinates theory is developed by making an ansatz for a moving soliton where the position, rapidity, momentum, frequency, and phase are all functions of time. We consider the complex potential as a perturbation and verify that numerical solutions of the equation of motions for the collective coordinates are in agreement with simulations of the nonlinear Dirac equation. The main effect of the imaginary part of the potencial is to induce oscillations in the charge and energy (they are conserved for real potentials) with the same frequency and phase as the momentum. We find long-lived solitons even with very large charge and energy oscillations. Additionally, we extend to the nonlinear Dirac equation an empirical stability criterion, previously employed successfully in the nonlinear Schrödinger equation.

Cross submissions (showing 2 of 2 entries)

Coupled lasers offer a promising approach to scaling the power output of photonic devices for applications demanding high frequency precision and beam coherence. However, maintaining coherence among lasers remains a fundamental challenge due to desynchronizing instabilities arising from time delay in the optical coupling. Here, we depart from the conventional notion that disorder is detrimental to synchronization and instead propose an interpretable mechanism through which heterogeneity in the laser parameters can be harnessed to promote synchronization. Our approach allows stabilization of pre-specified synchronous states that, while abundant, are often unstable in systems of identical lasers. The results show that stable synchronization enabling coherence can be frequently achieved by introducing intermediate levels of random mismatches in any of several laser constructive parameters. Our results establish a principled framework for enhancing coherence in large laser networks, offering a robust strategy for power scaling in photonic systems.

We solve the Yang-Baxter-like matrix equation AXA = XAX for a general given matrix A to get all anti-commuting solutions, by using the Jordan canonical form of A and applying some new facts on a general homogeneous Sylvester equation. Our main result provides all the anti-commuting solutions of the nonlinear matrix equation.

Replacement submissions (showing 5 of 5 entries)

Causal emergence (CE) based on effective information (EI) demonstrates that macro-states can exhibit stronger causal effects than micro-states in dynamics. However, the identification of CE and the maximization of EI both rely on coarse-graining strategies, which is a key challenge. A recently proposed CE framework based on approximate dynamical reversibility, utilizing singular value decomposition (SVD), is independent of coarse-graining. Still, it is limited to transition probability matrices (TPM) in discrete states. To address this, this article proposes a novel CE quantification framework for Gaussian iterative systems (GIS), based on approximate dynamical reversibility derived from the SVD of inverse covariance matrices in forward and backward dynamics. The positive correlation between SVD-based and EI-based CE, along with the equivalence condition, is given analytically. After that, we provide precise coarse-graining strategies directly from singular value spectra and orthogonal matrices. This new framework can be applied to any dynamical system with continuous states and Gaussian noise, such as auto-regressive growth models, Markov-Gaussian systems, and even SIR modeling using neural networks (NN). Numerical simulations on typical cases validate our theory and offer a new approach to studying the CE phenomenon, emphasizing noise and covariance over dynamical functions in both known models and machine learning.

We investigate the dynamics of the Ikeda map in the conservative limit, where it is represented as a two-dimensional area-preserving map governed by two control parameters, theta and phi. We demonstrate that the map can be interpreted as a composition of a rotation and a translation of the state vector. In the integrable case (phi = 0), the map reduces to a uniform rotation by angle theta about a fixed point, independent of initial conditions. For phi ne 0, the system becomes nonintegrable, and the rotation angle acquires a coordinate dependence. The resulting rotation number profile exhibits extrema as a function of position, indicating the formation of shearless barriers. We analyze the emergence, persistence, and breakup of these barriers as the control parameters vary.

The agent-based modelling community has a debate on how ``intelligent'' artificial agents should be, and in what ways their local intelligence relates to the emergence of a collective intelligence. I approach this debate by endowing the preys and predators of the Lotka-Volterra model with behavioral algorithms characterized by different levels of sophistication. The main finding is that by endowing both preys and predators with the capability of making predictions based on linear extrapolation a novel sort of dynamic equilibrium appears, where both species co-exist while both populations grow indefinitely. While this broadly confirms that, in general, relatively simple agents favor the emergence of complex collective behavior, it also suggests that one fundamental mechanism is that the capability of individuals to take first-order derivatives of one other's behavior can allow the collective computation of derivatives of any order.

Many mathematical models of interacting agents assume that individual interactions scale down in proportion to the network size, ensuring that the combined input received from the network does not diverge. In theoretical neuroscience, Sompolinsky and Van Vreeswijk proposed in 1996 that, should these scalings be violated (and under appropriate conditions), the system may not diverge but rather approach a balanced state where the inputs to each neuron compensate each other (in neuroscience, where inhibitory currents compensate the excitatory ones). We come back to this observation and formulate here a mathematical conjecture for the occurrence of such behaviors in general stochastic systems of interacting agents. From a mathematical viewpoint, this conjecture can be viewed as a double-limit problem in the space of probability measures, which we discuss in detail, as it provides several possible mathematical avenues for proving this result. We provide some numerical and theoretical explorations of the conjecture in classical models of neuronal networks. Moreover, we provide a complete proof of an asymptotic result consistent with one of the double-limit problems in a one-dimensional model with separable coupling inspired by models of chemically-coupled neurons. This proof relies on asymptotic methods, and particularly desingularization techniques used in some PDEs, that we apply here to the mean-field limit of the network as the coupling is made to diverge. From the applications viewpoint, this theory provides an alternative, minimalistic explanation for the widely observed balance of excitation and inhibition in the cerebral cortex not requiring the assumption of the existence of complex regulatory mechanisms.

Recent high-resolution, high-Reynolds-number simulations have shown that the initial total circulation, quantified by the vorticity packing fraction (VPF), strongly influences the late-time Eulerian statistical equilibria of decaying incom- pressible two-dimensional Navier-Stokes turbulence (Biswas et al., 2022, Physics of Fluids 34, 065101), revealing a transition from point-vortex--dominated to finite-size (patch-vortex) equilibria with increasing vortex packing, and emphasizing the role of of the classical exclusion principle (i.e., incompressibility) and total circulation in determining the final statistical states. The present study examines how the associated Lagrangian tracer transport evolves with VPF across the early (linear-nonlinear turbulence onset), intermediate (turbulence development), and late (coherent dipole evolution) stages, and how it correlates with the corresponding Eulerian states. Turbulence, triggered by the Kelvin-Helmholtz instability and sustained by inverse energy cascades, forms large-scale coherent vortices that govern long-time transport. Tracer dynamics, analyzed via mean-square displacement and position-velocity probability distri- bution functions (PDFs), reveal that increasing VPF accelerates turbulence onset, drives a transition from sub- to super- diffusive transport with decreasing anisotropy in the intermediate stage, and determines late-time behavior dominated by either orbital coherent vortex trapping (sub-diffusive) or linear translational dipole motion (super-diffusive). These dis- tinct long-time transport characteristics, evolving from sub- to super-diffusive behavior with increasing vorticity pack- ing, demonstrate a strong correspondence between the transition from point-vortex- to finite-size-vortex-dominated Eulerian equilibria and the underlying Lagrangian transport in decaying incompressible 2D Navier-Stokes turbulence.