Studies in Applied Mathematics最新文献

This paper investigates the formation and evolution of strange nonchaotic attractors (SNAs) in a discontinuous noninvertible piecewise-smooth map subjected to quasiperiodic forcing. The system is formulated as a two-dimensional skew-product, combining a piecewise-smooth one-dimensional base map with an irrational rotation. The notion of quasiperiodic homoclinic and heteroclinic orbits is introduced, and their potential relevance to the transition from smooth tori to SNAs is explored. Forward images of critical lines are shown to reflect key changes in the attractor and its basin, including disconnected components and the formation of holes. Discontinuities in the attractor are found to result from direct contact with the discontinuity line of the map. Through a set of representative examples, we identify three main routes to SNA formation: (i) contact between the attractor and its basin boundary leading to homoclinic orbits, (ii) contact between the attractor and a repeller, leading to homoclinic orbits, and (iii) contact between the attractor and the discontinuity set. These results clarify how different types of critical curves shape the dynamics, and contribute to the understanding of attractor evolution in discontinuous noninvertible quasiperiodically forced systems.

We formulate a three-dimensional tumor–immune model obtained by adding to an established tumor–host–effector system two phenomenological perturbation terms, whose signs are motivated by the reported tumor-promoting and effector-suppressing influence of regulatory B cells; their coefficients are dynamical control parameters and are not calibrated to biological rates. The perturbed and unperturbed dynamics are compared using phase portraits, Lyapunov exponents, parameter sensitivity, and multifractal indicators. Because trajectory-wise prediction is intrinsically limited in chaotic regimes and initial cell populations are only partially known, we also extend exact response theory, originally developed in nonequilibrium molecular dynamics, to the relevant non-compact positive set of populations values. Admissible initial densities are introduced to control boundary terms and ensure the validity of the exact response identity for finite-time ensemble averages. The resulting formulation quantifies both instantaneous and cumulative changes in tumor, host, and effector-cell observables under finite parameter perturbations, without requiring a small-perturbation approximation. Numerical tests verify the response identity against direct Monte Carlo propagation and assess sampling and time-discretization errors. For the parameter ranges studied, the perturbations can move the model between periodic and chaotic regimes, while ensemble averages remain considerably smoother than individual trajectories, and are capable of significant predictions. Long-time averaged responses are consistent across three admissible initial ensembles, although instantaneous dispersion can remain substantial. The analysis therefore separates attractor geometry from finite-time statistical response and provides a mathematically controlled framework for uncertainty-aware comparison of tumor-growth models, especially relevant in the potentially most interesting transients. The biological interpretation remains qualitative: The perturbation coefficients are not clinically calibrated rates, and any correspondence with regulatory B-cell biology requires independent experimental validation.

We propose the invariant subspace physics-informed neural network (IS-PINN), a novel approach that combines the invariant subspace method (ISM) with deep learning to solve nonlinear diffusion equations. By restricting the solution to a prescribed invariant subspace spanned by exact basis functions, IS-PINN eliminates spatial dependencies and transforms the nonlinear diffusion equation into a dynamical system. The temporal coefficients are learned by a neural network with hard-constrained initial conditions via least-squares projection. We validate IS-PINN on five problems: the semilinear heat equation, the Riabouchinsky–Proudman–Johnson equation (RPJE), the porous medium equation, the quasilinear parabolic equation, and the (1+2)D reaction–diffusion equation. Numerical results demonstrate that IS-PINN outperforms standard PINNs, achieving mean relative $�L_2$ errors of $��1.43�\times �{10}^{�-�5�}�$–$��1.26�\times �{10}^{�-�4�}�$ over five random seeds, with mean training time reduced by approximately 74%–98%.

We are interested in the nonlinear stability of traveling waves for a singular chemotaxis model on the half space with nonzero chemical diffusion. This model describes the dynamics of vascular endothelial cell density attracted by the concentration of vascular endothelial growth factor during the initiation of tumor angiogenesis. We show that under the nonzero flux boundary condition for cells and the Robin boundary condition for the chemical signal, solutions to the system with suitable initial data converge to a shifted traveling wave, whose profile and wave speed are uniquely determined by the parameters involved in the boundary conditions.