Abstract We introduce several new notions of conditional mean dimension for a factor map between two topological dynamical systems. We develop a double variational principle connecting conditional mean topological dimension and conditional rate distortion under the marker property. We also demonstrate that if a dynamical system possesses the marker property, then there exists a metric such that the upper conditional metric mean dimension equals the conditional mean topological dimension.
Abstract This paper is a continuation of our previous study (Segata 2026 Discrete Contin. Dyn. Syst. 51 230–249). on the scattering problem for the Zakharov–Kuznetsov (ZK) equation. When the space dimension is three, we construct a global solution to ZK which scatters to a given free solution without smallness assumption on the asymptotic states.
Abstract In this paper, we prove the existence of monotone heteroclinic solutions for the delayed Nicholson’s blowflies model with harvesting: x ′ ( t ) = − δ x ( t ) − H x ( t − σ ) + ρ x ( t − r ) e − x ( t − r ) . Under the assumption 1 < ρ δ + H ⩽ e , we establish a heteroclinic connection between the equilibria 0 and ln ( ρ / ( δ + H ) ) using the Wu–Zou monotone iteration method adapted to two different delays ( σ < r ). The proof combines the explicit construction of upper and lower solutions with an analysis of the characteristic equation, supported by numerical simulations.
Abstract We use perturbation theory and bifurcation theory to analyse the dynamical behaviour, associated to a model describing a particle moving within a ring around a celestial object. The central body is modelled as a homogeneous triaxial ellipsoid, rotating about its shortest physical axis at a constant angular velocity. It is assumed that the massless ring particle moves within the equatorial plane of the ellipsoid. The dynamics of the particle is studied using epicyclic variables, that lead to a straightforward definition of corotation and Lindblad resonances. These resonances are associated to a Hamiltonian function with two degrees of freedom, for which we compute appropriate expansions for the normal form and the resonant Hamiltonian. Initially, the normal form is verified to be multi–scale non–degenerate, thereby guaranteeing the existence of invariant KAM tori, providing the stability of the resonances, through their confinement in phase space. Subsequently, two test cases are examined: a nearly spherical ellipsoid and a highly aspherical ellipsoid. Furthermore, this study concentrates on three principal resonances: corotation, 1 : 2 , and 1 : 3 , for which we present results concerning their dynamical behaviour obtained analysing the Hamiltonian formulation of the model and the resonant normal form. Specifically, we examine the phase space structure, the amplitude of libration around the resonances, and the occurrence of bifurcations. Remarkably, in none of the two studied test cases, the resonance 1 : 3 presents evidence of bifurcations for relevant values of the eccentricity. Our dynamical study in the current model problem thus supports a higher probability of selecting the resonance 1 : 3 compared to the other resonances.
Abstract Let k ∈ ( d , ∞ ] and consider the k ∗ -distance ‖ μ − ν ‖ k ∗ := sup { | μ ( f ) − ν ( f ) | : f ∈ B b ( R d ) , ‖ f ‖ L ~ k := sup x ∈ R d ‖ 1 B ( x , 1 ) f ‖ L k ⩽ 1 } between probability measures on R d . The exponential ergodicity in 1 -Wasserstein and k ∗ -distances is derived for a class of McKean–Vlasov SDEs with small singular interactions measured by ‖ ⋅ ‖ k ∗ . Moreover, the exponential ergodicity in 2 -Wasserstein distance and relative entropy is derived when the interaction term is given by b ( 0 ) ( x , μ ) := ∫ R d h ( x − y ) μ ( d y ) for some measurable function h : R d → R d with small ‖ h ‖ L ~ k .
Abstract In this article, we further develop the Riemann–Hilbert formalism introduced in our earlier work [Gharakhloo and Its, SIGMA 16 (2020), 100] for the asymptotic analysis of Toeplitz+Hankel determinants with distinct Toeplitz and Hankel symbols. In that work, we showed that the associated Riemann–Hilbert problem can be analyzed by the Deift–Zhou nonlinear steepest descent method under a special restriction on the winding numbers of the symbols. This restriction excludes several natural cases, including the zero–winding configuration for both ϕ and w . The main goal of the present paper is to extend the asymptotic analysis of Toeplitz+Hankel determinants to a broader class of winding–number configurations. As an application, we study the case associated with the Ising model on the zig–zag layered half–plane with critical boundary magnetic field, for which the winding numbers of the Toeplitz and Hankel symbols are 0 and − 1 , respectively. In this setting, we compute the asymptotics of the norms of the corresponding system of orthogonal polynomials.
Wild chaos is a higher-dimensional form of chaotic dynamics that can only arise in vector fields of dimension at least four (or in diffeomorphisms of dimension at least three). It is characterized by the persistent presence of tangencies between stable and unstable manifolds of an invariant set. We study a four-dimensional extension of the classic Lorenz system, which was recently shown (by Gonchenko et al 2021 Nonlinearity 34) to exhibit a so-called wild pseudohyperbolic attractor of spiral type for a specific choice of parameters. Pseudohyperbolicity guarantees that every trajectory in the attractor has a positive maximal Lyapunov exponent, and this property persists under small perturbations of the system. We employ a combination of different advanced numerical techniques to investigate how wild chaotic attractors arise geometrically by analysing the overall bifurcation structure in the (& rhov;,mu)-parameter plane. Here, & rhov; is the standard (Rayleigh) parameter in the classic Lorenz system, and mu is a new parameter that introduces spiralling dynamics near the origin. We begin by identifying the bifurcation structure inherited from the Lorenz system and continuing its homoclinic bifurcations as curves. The computation of kneading diagrams, in combination with Lin's method, allows us to uncover and continue additional global bifurcations intrinsic to the four-dimensional system. To identify different types of attractors, we also compute the Lyapunov spectrum of the attractors on which the one-dimensional unstable manifold of the origin accumulates. Our detailed numerical investigations provide insight into the parameter regions where wild pseudohyperbolic attractors may exist.
We present a new derivation for the asymptotic behaviors for solutions of the compressible Korteweg system with derivatives of arbitrary high order in general L (R)(R-d) ( d >= 2, r is an element of [1,infinity]). It would be shown that solutions decay at the rates of heat kernels for 2 <= r <= infinity while decay slower, asymptotically behaving as so called 'diffusion waves', for 1 <= r < 2. By introducing an initial space with negative regularity B(sic)(1,infinity)(-sigma), we obtain the uniform decay for solutions within L-1 framework, compared to the possible growth in classical results for compressible models. The approach is so-called Gevrey method, based on the parabolic mechanics of the linearized system, which enables us to transform high order derivatives' estimates into decay rates and low-order analytic estimates. In order to control the growth of the radius of analyticity under the particular regularity, especially in case 1 <= r < 2, we shall treat the momentum form of the perturbed system, establish necessary linear analysis based on the semi-group and Gevrey product estimates in sub-analytic families.
In this paper we study the existence of blow-up solutions for the asymmetric sinh-Poisson-type equations with homogeneous Neumann boundary conditions on compact Riemann surfaces, which arise as boundary mean field equations of equilibrium turbulence vortices involving variable intensities and are also of particular interest in the Keller-Segel system for chemotaxis collapse. More precisely, by using the Lyapunov-Schmidt reduction method, we establish sufficient conditions for the existence of solutions that blow up at any prescribed number of distinct points, which may be located both in the interior of the surface and on its boundary.
The finite extensible nonlinear elastic (FENE) dumbbell model couples the Navier-Stokes equations governing fluid velocity with a Fokker-Planck equation describing the evolution of polymer distribution within the fluid. In this work, we study the global-in-time incompressible limit of the compressible FENE dumbbell model on the three-dimensional torus T3, where the incompressible limit is driven by large bulk viscosity. To establish this limit, we develop time-weighted a priori estimates that yield decay rates for strong solutions. A key challenge arises from the fact that increasing the bulk viscosity suppresses the decay of high-frequency components, thereby weakening the dissipation of the density and complicating the derivation of uniform-in-time decay estimates. To overcome this difficulty, we introduce a novel momentum-based estimate and show that the incompressible component of the momentum decays faster in time than the velocity itself. Exploiting this enhanced decay, we successfully close the a priori estimates and establish a time-decreasing convergence rate toward the incompressible limit.
Chaotic systems are characterised by exponential separation between close-by trajectories, which in particular leads to deterministic unpredictability over an infinite time-window. It is now believed, that such butterfly effect is not fully relevant to account for the type of randomness observed in turbulence. For example, tracers in homogeneous isotropic flows are observed to separate algebraically, following an universal growth, independent from the initial separation. This regime, known as Richardon’s regime, suggests that at the level of trajectories, and unlike in chaos theory, randomness may in fact emerge in finite-time. This phenomenon called ‘spontaneous stochasticity’ originates from the singular nature of the underlying dynamics, and provides a candidate framework for turbulent randomness and transport. While spontaneous stochasticity has been mathematically formalised in simplified turbulence models, a precise and systematic tool for quantifying the various facets of this phenomenon is to this day missing. In particular, it is still unclear whether chaos is important for that behaviour to appear. In this paper we introduce a 3d rough flow that can be tuned to present Lagrangian chaos. The flow is inspired by the Weierstrass function and is entitled ‘the WABC model’. After analysing its properties, we define what is spontaneous stochasticity in this context. The provided formal definition is then adapted to better suit for numerical analysis. We present the results from Monte–Carlo simulations of Lagrangian particles in this flow. Within the numerical precision, we quantitatively observe the appearance of spontaneous stochasticity in this model. We investigate the influence of noise type and find that the observed spontaneous stochasticity does not depend on the chosen stochastic regularisations.
In this paper, we first study the complete spectral structure of the relativistic kinetic Fokker-Planck equation near a global Maxwellian in the whole space R-3. It is shown that the equation exhibits a spectral gap in the high-frequency regime, while the asymptotic behaviour of eigenvalues is derived in the low-frequency regime. Based on the spectral analysis, we establish the global existence and algebraic decay rates of the solution in a Sobolev space. These decay rates are proven to be optimal under additional conditions on the initial datum f(0). Moreover, by assuming f(0) has compact support with respect to x, we obtain that the solution vanishes outside a modified light cone < x > = aMt, where a, M > 1. The speed aM can approximate to 1. This result indicates that the relativistic kinetic Fokker-Planck equation has some similarity with wave equation.
This paper investigates periodic orbits in the two-parameter family of tent maps. Building on the known correspondence between equi-kneading curves and equi-entropy curves (Misiurewicz and Visinescu 1991), we derive explicit analytic expressions for these curves. We identify minimal kneading sequences for periods 2n+1 and 2m(2n+1), and obtain closed-form formulas for the corresponding equi-kneading curves and topological entropies. Using these results, we construct a complete hierarchical partition of the parameter space according to the Sharkovsky order, and prove necessary and sufficient conditions for the existence of periodic orbits of any given period. The necessity part, which remained incomplete even in the one-parameter case, is fully established here through combinatorial arguments based on kneading sequence order and renormalisation. Our approach illustrates how explicit symbolic dynamics can yield complete classifications in two-parameter families.
Mathematical models of reaction networks can exhibit very complex dynamics, including multistability, oscillations, and chaotic dynamics. Under certain assumptions on network structure or parameter values, these models may belong to the class of toric dynamical systems, which have remarkably stable dynamics. The concept of a disguised toric dynamical system was introduced to describe the phenomenon where a reaction network generates dynamics that are toric with respect to another (possibly different) network; such systems enjoy all the stability properties of toric dynamical systems but with much fewer restrictions on the networks and parameter values that generate them. The disguised toric locus of a network is the set of parameter values for which the corresponding dynamical system is a disguised toric system. In this work, we develop a general and efficient method for computing the codimension of the disguised toric locus of a reaction network. We further establish simple sufficient conditions for the disguised toric locus of a reaction network to have codimension zero. Additionally, we demonstrate the applicability of our approach by analysing several biological interaction models, including Brusselator-type systems, Thomas-type systems, and circadian clock systems.
We propose the system of kinetic equations as the new analytic tool for achieving a Gaussian approximation to the probability distribution propagated by the four-wave dynamics with strong nonlinearity. The framework forms a comprehensive generalization of the classical kinetic equation developed in the context of nonlinear waves in the weak interaction limit as well as the effective kinetic equation aimed for the wave dynamics in the regime of moderate nonlinearity. Unlike the previous formulations that exclusively determine the variance of the single wave-profile, the system of kinetic equations governs the evolution of the covariance matrix of the duet constructed via augmenting the macrovariable in significant resonance with the target variable. As it was done with the existing kinetic equations for the analysis of the model dynamics with weak and moderate nonlinearity, the system of kinetic equations will prove useful in addressing important issues concerning the statistical features exhibited by the strongly nonlinear four-wave dynamics.
It has been shown in Jeong and Oh (2019 On the Cauchy problem for the Hall and electron magnetohy-drodynamic equations without resistivity I: illposedness near degenerate stationary solutions Ann. PDE 8 15) that the incompressible and irresistive Hall- and electron-magneto-hydrodynamic (MHD) equations are illposed on flat domains M=R-k x T3-k for 0 <= k <= 2. The data and solutions therein were assumed to be independent of one coordinate, which not only significantly simplifies the systems but also allows for a large class of steady states. In this work, we remove the assumption of independence and conclude strong illposedness for compactly supported data in R-3. This is achieved by constructing degenerating wave packets for linearized systems around time-dependent axisymmetric magnetic fields. A few main additional ingredients are: a more systematic application of the generalized energy estimate, use of the Bogovski- operator, and a priori estimates for axisymmetric solutions to the Hall- and electron-MHD systems.
We consider the asymptotic behaviours of the Cucker-Smale model, which includes singular communication rates and bonding forces. We prove that exponential rendezvous occurs regardless of initial data for the power function type singularity and the convergence rate converges to infinity. Due to the double-integrator structure and bonding force term with singularity, we cannot apply standard methodologies to this model. To address this issue and obtain exponential rendezvous results, we employ LaSalle's invariance principle for a singular system and a new Lyapunov functional constructed by adding a new term to the sum of kinetic and configuration energies. We also prove that finite-time rendezvous occurs when the communication rate and bonding force parameters depend on the maximal diameter of agents. To prove this, we define Lyapunov functionals and derive new differential equations that these functionals satisfy, based on the assumptions about the parameters. By analysing this system, we obtain the finite-time rendezvous result. Finally, we verify our results by several numerical simulations.
Many fractals G admit a harmonic immersion into R-n, i.e. an immersion which minimises a natural energy under fixed boundary conditions; we look for harmonic immersions of the Sierpinski gasket into the hyperbolic plane. We show that, given any three points <(A)overtilde>, <(B)overtilde>, <(C)overtilde> in the hyperbolic plane there is a harmonic map bringing the three points A, B, C of the boundary of the gasket to <(A)overtilde>, <(B)overtilde>, <(C)overtilde> respectively. Moreover, if the points <(A)overtilde>, <(B)overtilde>, <(C)overtilde> are sufficiently close in the hyperbolic distance, then the harmonic map is unique and depends differentiably on <(A)overtilde>, <(B)overtilde>, <(C)overtilde>. Lastly, we show that, if the harmonic map phi is injective, then it brings geodesics of the gasket G into geodesics of phi(G).
In this paper, we study the gravitational collapse, a phase of stellar evolution, in a two-dimensional annulus. It is modelled by supersonic solutions to the compressible Euler-Poisson system. We reformulate the Euler-Poisson system into a second-order hyperbolic-elliptic coupled system and a transport equation. The key to the analysis is to obtain higher-order derivative estimates of the associated linearized boundary value problem. The unique existence of supersonic solutions is established by Schauder fixed-point theory and contraction principle.
The zero-electron-mass limit of one-fluid Euler-Poisson systems leads to incompressible Euler equations, a fundamental result that has been extensively studied in previous works. In this paper, we investigate this limit for a two-fluid Euler-Poisson system under general initial conditions. We prove a qualitatively distinct behavior, as the system asymptotically decouples into separate electron and ion dynamics over finite time intervals. Specifically, the ion motion is governed by the compressible Euler-Poisson equations with a semilinear Poisson equation, while the electron dynamics is described by the incompressible type Euler equations. The proof relies on uniform energy estimates with respect to the parameter and compactness arguments applied to the solution sequence. In energy estimates, we first employ the anti-symmetric matrix technique to establish uniform estimates of the solution and its time derivative. Then, we perform an induction on the order of spatial derivatives to complete the energy estimates.
