This paper introduces dual field-theoretic Hamel’s integrator in the form of discrete mechanical systems that utilize dynamic variables measured relative to moving frames. As in the continuous setting, the use of moving frames results in a flexible and overall more efficient formalism. The integrator is derived in a systematic way using the formalism of composition maps. It is then utilized for numerical integration of geometrically exact beams.
We prove the global existence of the solution for one-dimensional cubic fractional ( m⩾ 2 ) nonlinear Schrödinger equations in modulation spaces M_p,p'^s_p for p close to 2 with no smallness condition on initial data, where s_p=(m-2)(1/2-1/p) . The proof adapts a splitting method inspired by the work of Vargas-Vega, Hyakuna-Tsutsumi, Chaichenets et al. to the modulation spaces and exploits polynomial growth of the fractional Schrödinger semigroup on modulation spaces M_p,p' with loss of regularity s_p .
Identifying the optimal stimulation threshold for regulating spatiotemporal bioelectrical activity remains challenging under discontinuous physiological factors. Accordingly, to explore the spatial complexity of neocortical neurons, a non-smooth feedback strategy of memristive autaptic current involving the threshold of membrane potential is proposed, and a delayed reaction–diffusion memristive Wilson neuron model is established under the homogeneous Neumann boundary condition. The existence and local dynamics of the equilibrium point of subsystems are analyzed. In detail, we derive the analytical criteria for saddle-node bifurcation, diffusion-driven Turing bifurcation, and delay-induced Hopf bifurcation. Furthermore, we obtain computable formulas for the stability and direction of spatially periodic solutions caused by Hopf bifurcation. Intriguingly, hidden extreme bistability and fascinating multiple stability switches are triggered under certain delayed self-feedback. Based on differential inclusion theory, we investigate the existence of sliding segments, stable pseudo-equilibrium points, boundary equilibrium points, and tangent points controlled by non-smooth thresholds. Besides, the evolutionary mechanisms of the spatiotemporal self-excited and hidden sliding attractors are revealed by the geometric dynamics approach, and the critical delay corresponding to global sliding-grazing bifurcation and buckling bifurcation is further determined. The obtained results provide substantial theoretical guidance for the diagnosis and rehabilitation of neurological diseases.
The second-order energy- and potential-enstrophy-conserving numerical scheme introduced by Arakawa Lamb (1981) for the shallow-water equations with periodic boundary conditions, and extended by Salmon (2004) in the context of Hamiltonian Poisson-bracket discretisation, is further extended to a fourth-order discretisation for the problem of nonlinear shallow-water sloshing over a corrugated bottom surface in a rectangular rigid basin, with symmetric and non-symmetric porous solid side walls and periodic and non-periodic prescribed inflow-outflow boundary conditions, undergoing a prescribed coupled surge-sway motion. Adaptation to a finite domain with periodic and non-periodic inflow-outflow boundary conditions requires a new approach to the boundary conditions at porous solid boundaries, and the ghost cell grid point approximations in the context of the fourth-order finite-difference discretisation on the Arakawa C-grid. Theoretical Poisson-bracket arguments are used to define the required boundary conditions. In addition, standard biharmonic dissipation is incorporated into the numerical modelling to prevent potential enstrophy from accumulating at the smallest resolved scales, which improves the quality and stability of the approximate solutions, enabling long-term integrations to be carried out. The scheme is implemented, shown to preserve the total mass, energy, and potential enstrophy over long-time integration with bounded fluctuations. The presented higher-order C-bracket sloshing integrator provides a robust, stable, fast and precise building block for long-time computational modelling of floating ocean wave energy devices with flexible components, like a flexible bottom surface or membrane for wave energy extraction.
In this paper we perform a local convergence analysis of smooth nonlinear multidimensional discrete dynamical systems. In previous works we have studied the asymptotic stability at non-hyperbolic fixed points for two dimensional discrete dynamical systems. A similar analysis has been carried out for multidimensional discrete systems whose linearization matrix has all its eigenvalues equal to one. Here we analyze remaining cases according to their linearization matrix. Finally, an application for an aggressive heterogeneous tumor growth model is presented.
In this work, we investigate the global nonlinear normal modes of molecules with an octahedral configuration, with particular focus on sulfur hexafluoride SF_6 . Under the assumption of isotypic nonresonance, we apply the method of equivariant gradient degree to prove the global existence of branches of periodic solutions emanating from the critical orbit of equilibrium. We provide a systematic topological classification of these solutions and identify at least 16 distinct types of symmetries with maximal orbit types.
We characterize the functional structure of deformations that describe the occurrence of possibly irrecoverable sharp folds and cusps in shell-like bodies, while they avoid self-penetration of matter. The combination of special bounded variation maps with one-dimensional jump set and atomic measures provides a functional environment in which it is possible to represent configurations not otherwise described by special bounded Hessian maps, often used for energy minimization of shells. The introduction of atomic measures provides necessary control over the singular component of the generalized gradient measure, ensuring the robust compactness required by the direct method. In the proposed framework, we prove the existence of minimizers under Dirichlet boundary conditions for a class of energies that are appropriate for an ambient setting foreseeing large strains. Although we treat a traditional topic, the analysis goes beyond the classical format of nonlinear elasticity.
Many territorial animals shape their movement decisions using memory of past conflicts, yet the population-level consequences of such nonlocal conflict memory remain poorly understood. We propose and analyze a PDE-ODE hybrid model for a single population moving in response to spatial memory of territorial conflicts. The population density satisfies a diffusion equation with delayed, nonlocal advection generated by a convolution of a conflict variable, while the conflict intensity evolves according to a local ODE encoding a warning mechanism with decay and resetting. Linearization about the positive homogeneous steady state yields a non-self-adjoint operator whose spectrum reduces to a countable family of characteristic equations indexed by Fourier modes. For Gaussian and Laplacian perception kernels, the steady state is linearly asymptotically stable for all perceptual radii in the memoryless case, whereas a top-hat kernel admits diffusion-driven (Turing) instability below a critical radius, producing stationary spatial patterns. For positive memory delay, we identify Hopf bifurcation thresholds at the level of the linearized spectral problem for all three kernels. In the top-hat case, the stationary and oscillatory thresholds may meet in the (R,τ ) -plane, giving a codimension-two Turing-Hopf spectral point. Numerical simulations illustrate these thresholds and the associated spatial and spatiotemporal patterns.
In this paper, we study wave trains (periodic traveling waves) in a damped diatomic Fermi–Pasta–Ulam (FPU) lattice driven by external periodic forces. By applying nonlinear functional analysis, we show the existence and uniqueness of two different periodic waveform functions corresponding to light and heavy particles, respectively. In the case of small forcing and damping, Lyapunov–Schmidt reduction is employed to study the bifurcation of wave trains and the asymptotic expressions of the bifurcating solutions. For monatomic lattices, we adopt a nonstandard assumption of two waveform functions for adjacent particles, which results in two dispersion branches. This differs from traditional models using only one waveform function, where merely a single branch exists. Using this framework, we obtain new conclusions on the bifurcation of small-amplitude traveling waves.
In the paper, localized wave and quasi-periodic wave solutions for two types of generalized nonlocal parity-time (PT)-symmetric Davey–Stewartson-type (DS-type) systems are investigated. These systems have potential applications in many fields such as nonlinear optics, plasma physics, and fluid dynamics. The N-soliton solutions are obtained based on the bilinear theory. For the generalized y-nonlocal DS-type system, three types of breathers and their interaction solutions with periodic line waves are derived. By applying long wave limit and partial long wave limit, three types of lumps and the interaction solutions among lump, breather and periodic line waves are obtained. For the generalized xy-nonlocal DS-type system, two types of periodic wave solutions are obtained. Based on long wave limit, the rogue waves are derived from the N-soliton solutions. The interaction solutions between rogue waves and periodic line waves are obtained by adding constraints to the parameters. Combining the bilinear equation with Riemann-theta function, the N-periodic wave solutions are successfully obtained. The difficulty of solving quasi-periodic wave solutions is transformed into solving the least square problem of the nonlinear systems. A generalized Levenberg–Marquardt (LM) algorithm is proposed to solve this kind of problem. Under the influence of nonlocal symmetry in the y-direction, the quasi-periodic waves exhibit a process of splitting and fusion over time t. A series of new solutions, including the quasi-periodic breathers, the quasi-periodic double-peakon breathers, the quasi-periodic double-chain breathers, are acquired. The quasi-periodic waves of the generalized xy-nonlocal DS-type system exhibit similar dynamical behaviors to the ordinary quasi-periodic waves. Their dynamical behaviors are quantitatively analyzed by using the analysis method of the characteristic line. The asymptotic properties of rogue waves, lump waves and quasi-periodic waves are analyzed and explained. These methods can be further broadened to study other complex wave structures for the nonlocal systems.
Immune effectors and tumor cells often shape the progression of the synthetic stage breast cancer through complex interactions. However, existing models often overlook the spatial and delayed immune dynamics that are critical in capturing immune evasion and tumor resurgence. To address this gap, we developed and analyzed a time-delayed reaction-diffusion model describing tumor-immune interactions during the synthetic stage. The model incorporates key immunological components such as T helper-2 cells, B cells, and cytokine-mediated feedback with spatial diffusion and a biologically motivated time delay. Analysis of the spatially homogeneous system reveals two biologically relevant equilibria: tumor-free and coexistence. The coexistence state is locally stable under baseline conditions, while the tumor-free equilibrium is unstable. Extending the analysis to a spatially distributed system, several cases arise from the dispersion function: stability within a biological parameter regime, diffusion-driven instability in the absence of delay, and delay-driven stability. Our parameter sweep results indicate that diffusion-driven instabilities may occur outside this regime, leading to spatial pattern formation. Numerical simulations using the method of lines reveal that B-cell proliferation occurs predominantly in regions with high tumor cell density, confirming that tumor reduction in the model is primarily driven by B-cell–mediated cytotoxicity, while the delay modulates the system dynamics by influencing transient behavior and, in certain parameter regimes, contributing to enhanced stability. Within biologically relevant regimes, however, the delay does not qualitatively alter the long-term tumor–immune outcome.
This paper investigates the finite-dimensional reduction of the asymptotic dynamics of one-dimensional nonlocal Kirchhoff-type parabolic equations, where the diffusion coefficient depends on the L^2 -norm of the gradient of the solution. Specifically, we establish the existence (without uniqueness) of an ε -regular mild solution and a global attractor for initial data in L^2 . Importantly, by employing a time transformation (one for each solution) that converts the nonlocal term into a nonlinear term, we construct an inertial manifold for initial data in H_0^1 , which implies that the global attractor is the graph of a Lipschitz function over a finite-dimensional subspace of the phase space, thereby revealing that the long-time behavior of this infinite-dimensional system can be completely described by a finite-dimensional system of ordinary differential equations.
Combination immunotherapy, which integrates immune checkpoint inhibition with immunostimulatory approaches, has emerged as a leading strategy for cancer treatment. However, patient responses vary widely due to cancer heterogeneity. In this work, we develop a mathematical model of cancer–immune interactions that incorporates combination therapy, accounting for both the limited efficacy of immune responses and the impact of immune checkpoint inhibitors. Using a hierarchical bifurcation framework, we derive explicit conditions for equilibrium stability and bifurcations, which delineate distinct clinical outcomes: complete tumor eradication, partial control, oscillatory dynamics, tumor escape, and bistability. We further perform global sensitivity analysis to assess the influence of key parameters on tumor progression and immune activity in oscillatory and partially controlled states. Our findings highlight dual parameter effects, where a single factor can suppress tumor growth in partially controlled states yet destabilize tumor–immune dynamics in oscillatory regimes. Numerical simulations not only corroborate the theoretical results but also uncover critical system features, such as immune delay and suppression. Overall, this study demonstrates the sensitivity of tumor–immune dynamics to parameters and initial conditions, and illustrates how dynamical systems analysis can guide the design of personalized immunotherapy strategies.
This research analyzes a predator–prey model with hunting cooperation and an alternative food supply for predators. We demonstrate the presence of a region of invariance first, followed by the boundedness of the system’s trajectories and the permanence of the solutions. Additionally, we show that the equilibrium point (0, 0) possesses the property of being a repeller. The requirements for the occurrence of two positive equilibrium points are explicitly given. The first equilibrium point has the characteristic of being a weak focus, repeller, or attractor, and the second equilibrium point is a saddle. In addition, we identify two crucial scenarios: (i) A separatrix curve Σ̅ divides the solutions into qualitatively different sectors, and (ii) a positive saddle point generates a homoclinic curve through the stable to unstable manifolds. These facts highlight how sensitive the system is to beginning conditions, especially in proximity to the separatrix. Furthermore, the dynamics of the model can be strongly affected by Hopf, Bogdanov–Takens, and transcritical bifurcations that the system may experience. Finally, we validate our analytical results using bifurcation diagrams and numerical simulations.
We propose an energy-based nonlocal p-Laplacian interface problem. Neumann interface conditions are naturally formulated via the energy, while Dirichlet conditions are enforced through a penalty term. A key feature is that the model retains a sharp interface, which facilitates extension to other interface problems; we illustrate this by developing a nonlocal approximation for the p-Laplacian interface problem with membrane conditions. By establishing Γ-convergence and compactness, we prove that as the nonlocal horizon vanishes, minimizers of the nonlocal functionals converge to those of the local counterparts. Numerical experiments using an efficient finite element method confirm the convergence.
Inspired by propagation phenomena occurring on media with hexagonal structures, we propose an idealized system of bistable reaction-diffusion equations posed on a hexagonal lattice, and investigate the existence and speed of traveling wave solutions. We establish a sharp criterion for propagation success (nonzero wave speed) and failure (zero wave speed), where the criterion is determined by an angle-dependent periodic threshold function and the ratio of two parameters in the nonlinearity. Moreover, the sign of the wave speeds is governed by the angle and the parameter ratio. Our results imply that the hexagonal structure and bistability significantly affect the propagation dynamics.
In this paper, we investigate the limit cycles of a class of switching ordinary differential equations consisting of two sub-equations. We propose a general framework for studying the maximum number of limit cycles of the equations. As applications, we analyze three biological models in recent literature. We prove that a general model of single species with seasonal constant-yield harvesting can only possess at most two limit cycles, which improves the work of Xiao (2016) and Han et al. (2018). We also apply our framework to a general model described by the Abel equations with periodic step function coefficients, showing that its maximum number of limit cycles is three. Finally, a population suppression model for mosquitoes considered by Yu (2020) and Zheng et al. (2021) is studied using our approach.
The elementary center–focus problem is studied for a general separable planar analytic Rayleigh–Liénard system dx/dt = y, dy/dt = -g(x) + f(x)ψ (y)y, where g(x), f(x), and ψ (y) are analytic functions. By means of the Melnikov function method, a necessary and sufficient condition is obtained for the origin to be an elementary center: g'(0)>0 and either xf( ϕ ^-1(x)) g( ϕ ^-1(x)) or ψ (y) is an odd function in a neighborhood of the origin, where ϕ (x)= √(2∫ _0^x g(s) ds)·sgn(x).
This paper is mainly devoted to the crossing period annulus and the maximum number of critical periods for planar piecewise linear systems with a straight line of separation. The normal forms of such systems with a crossing period annulus are obtained by an admissible transformation with a linear time scaling, which shows that the systems after transformation are reversible with respect to one of the axes. By further analyzing the monotonicity of the period functions of these normal forms, we prove that the piecewise linear systems with a straight separation line have at most one critical period, which is also sharp.
